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N O N L I N Nonlinear Regression Analysis Program Phillip H. Sherrod Member, Association of Shareware Professionals (ASP) Nonlin allows you to perform statistical regression analyses to estimate the values of parameters for linear, multivariate, polynomial, logistic, exponential, and general nonlinear functions. The regression analysis determines the values of the parameters which cause the function to best fit the observed data that you provide. This process is also called "curve fitting." Nonlin allows you to specify the function whose parameters are being estimated using ordinary algebraic notation. In addition to determining the parameter estimates, Nonlin can be directed to generate an output file with predicted values and residuals. It can also plot the data observations and the computed function. Although designed for regression analysis, Nonlin can also be used to find the root (zero point) or minimum absolute value of a nonlinear expression. Nonlin is in use at many engineering and research centers around the world. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction to Regression Analysis . . . . . . . . . . . . 1 1.2 Introduction to Nonlin . . . . . . . . . . . . . . . . . . 2 1.3 Installing Nonlin . . . . . . . . . . . . . . . . . . . . . 4 2. Using Nonlin . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Statement Syntax . . . . . . . . . . . . . . . . . . . . . 6 2.2 Variables and Parameters . . . . . . . . . . . . . . . . . 6 2.3 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Overview of Computation Process . . . . . . . . . . . . . . 8 2.5 Function Specification . . . . . . . . . . . . . . . . . . 9 2.5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . 9 2.5.2 Numeric Constants . . . . . . . . . . . . . . . . . . 10 2.5.3 Symbolic Constants . . . . . . . . . . . . . . . . . 11 2.5.4 Built-in Constant . . . . . . . . . . . . . . . . . . 11 2.5.5 Built-in Functions . . . . . . . . . . . . . . . . . 11 2.6 Nonlin Command Files . . . . . . . . . . . . . . . . . . 16 2.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 Include Files . . . . . . . . . . . . . . . . . . . . . . 16 2.9 Required Statements . . . . . . . . . . . . . . . . . . . 17 3. Nonlin Statements . . . . . . . . . . . . . . . . . . . . . 18 3.1 TITLE . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 DOUBLE . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5 CONSTANT . . . . . . . . . . . . . . . . . . . . . . . . 20 3.6 CONSTRAIN . . . . . . . . . . . . . . . . . . . . . . . . 20 3.7 SWEEP . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.8 FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . 21 3.9 CORRELATE . . . . . . . . . . . . . . . . . . . . . . . . 22 3.10 COVARIANCE . . . . . . . . . . . . . . . . . . . . . . . 22 3.11 CONFIDENCE . . . . . . . . . . . . . . . . . . . . . . . 22 3.12 TOLERANCE . . . . . . . . . . . . . . . . . . . . . . . 23 3.13 ITERATIONS . . . . . . . . . . . . . . . . . . . . . . . 23 3.14 ANGLETYPE . . . . . . . . . . . . . . . . . . . . . . . 23 3.15 OUTPUT . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.16 POUTPUT . . . . . . . . . . . . . . . . . . . . . . . . 24 3.17 PLOT . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.18 SPLOT . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.19 RPLOT . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.20 NPLOT . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.21 PRESOLUTION . . . . . . . . . . . . . . . . . . . . . . 31 3.22 WIDTH . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.23 NOECHO . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.24 Assignment Statement . . . . . . . . . . . . . . . . . . 32 3.25 IF Statement . . . . . . . . . . . . . . . . . . . . . . 32 i Contents ii 3.26 WHILE Statement . . . . . . . . . . . . . . . . . . . . 33 3.27 DO Statement . . . . . . . . . . . . . . . . . . . . . . 33 3.28 FOR Statement . . . . . . . . . . . . . . . . . . . . . 34 3.29 BREAK Statement . . . . . . . . . . . . . . . . . . . . 34 3.30 CONTINUE Statement . . . . . . . . . . . . . . . . . . . 35 3.31 STOP Statement . . . . . . . . . . . . . . . . . . . . . 35 3.32 DATA . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4. Understanding The Results . . . . . . . . . . . . . . . . . 37 4.1 Descriptive Statistics for Variables . . . . . . . . . . 37 4.2 Parameter Estimates . . . . . . . . . . . . . . . . . . . 37 4.3 t Statistic . . . . . . . . . . . . . . . . . . . . . . . 37 4.4 Prob(t) . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.5 Final Sum of Squared Deviations . . . . . . . . . . . . . 38 4.6 Average and Maximum Deviation . . . . . . . . . . . . . . 38 4.7 Proportion of Variance Explained . . . . . . . . . . . . 39 4.8 Adjusted Coefficient of Multiple Determination . . . . . 39 4.9 Durbin-Watson Statistic . . . . . . . . . . . . . . . . . 39 4.10 Analysis of Variance Table . . . . . . . . . . . . . . . 41 4.11 Correlation Matrix . . . . . . . . . . . . . . . . . . . 41 5. Theory of Operation . . . . . . . . . . . . . . . . . . . . 43 5.1 Minimization Algorithm . . . . . . . . . . . . . . . . . 43 5.2 Convergence Criterion . . . . . . . . . . . . . . . . . . 43 6. Hints for Nonlin Use . . . . . . . . . . . . . . . . . . . . 45 6.1 Convergence Failures . . . . . . . . . . . . . . . . . . 45 6.2 Singular Matrix Problems . . . . . . . . . . . . . . . . 46 6.3 Performance Issues . . . . . . . . . . . . . . . . . . . 46 6.4 Program Limits . . . . . . . . . . . . . . . . . . . . . 47 7. Example Analyses . . . . . . . . . . . . . . . . . . . . . . 48 8. Special Applications . . . . . . . . . . . . . . . . . . . . 52 8.1 Omitted Dependent Variable . . . . . . . . . . . . . . . 52 8.2 Root Finding and Expression Minimization . . . . . . . . 53 8.2.1 Function Minimization Examples . . . . . . . . . . . 55 9. Acknowledgement and Use of Nonlin . . . . . . . . . . . . . 56 9.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . 56 9.2 Use and Distribution of Nonlin . . . . . . . . . . . . . 56 9.3 Association of Shareware Professionals . . . . . . . . . 57 9.4 Copyright Notice . . . . . . . . . . . . . . . . . . . . 57 9.5 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . 57 10. Other Software . . . . . . . . . . . . . . . . . . . . . . 59 10.1 Mathplot -- Mathematical Function Plotting Program . . . 59 10.2 TSX-32 -- Multi-User Operating System . . . . . . . . . 59 10.3 SIMSTAT -- Interactive Statistics Program . . . . . . . 60 11. Software Order Form . . . . . . . . . . . . . . . . . . . . 61 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 1 Introduction 1.1 Introduction to Regression Analysis The goal of regression analysis is to determine the values of parameters for a function that cause the function to best fit a set of data observations that you provide. In linear regression, the function is a linear (straight line) equation. For example, if we assume the value of an automobile decreases by a constant amount each year after its purchase, and for each mile driven, the following linear function would predict its value (the dependent variable on the left side of the equal sign) as a function of the two independent variables which are age and miles: value = price + depage*age + depmiles*miles where 'value', the dependent variable, is the value of the car, 'age' is the age of the car, and 'miles' is the number of miles that the car has been driven. The regression analysis performed by Nonlin will determine the best values of the three parameters, 'price', the estimated value when age is 0 (i.e., when the car was new), 'depage', the depreciation that takes place each year, and 'depmiles', the depreciation for each mile driven. The values of 'depage' and 'depmiles' will be negative because the car loses value as age and miles increase. In a problem such as this car depreciation example, you must provide a data file containing the values of the dependent and independent variables for a set of observations. In this example each observation record would contain three numbers: value, age, and miles, collected from used car ads for the same model car. The more observations you provide, the more accurate will be the estimate of the parameters. The Nonlin statements to perform this regression are shown below: Variables value,age,miles; Parameters price,depage,depmiles; Function value = price + depage*age + depmiles*miles; Data; (data values go here) Once the values of the parameters are determined by Nonlin, you can use the formula to predict the value of a car based on its age 1 Chapter 1. Introduction 2 and miles driven. For example, if Nonlin computed a value of 16000 for price, -1000 for depage, and -0.15 for depmiles, then the function value = 16000 - 1000*age - 0.15*miles could be used to estimate the value of a car with a known age and number of miles. If a perfect fit existed between the function and the actual data, the actual value of each car in your data file would exactly equal the predicted value. Typically, however, this is not the case, and the difference between the actual value of the dependent variable and its predicted value for a particular observation is the error of the estimate which is known as the "deviation" or "residual". The goal of regression analysis is to determine the values of the parameters which minimize the sum of the squared residual values for the set of observations. This is known as a "least squares" regression fit. 1.2 Introduction to Nonlin Nonlin is a very powerful regression analysis program. Using it you can perform multivariate, linear, polynomial, exponential, logistic, and general nonlinear regression. What this means is that you specify the form of the function to be fitted to the data, and the function may include nonlinear terms such as variables raised to powers and library functions such as log, exponential, sine, etc. For complex analyses Nonlin allows you to specify function models using conditional statements (IF, ELSE), looping (FOR, DO, WHILE), work variables, and arrays. Nonlin uses a state-of-the-art regression algorithm that works as well, or better, than any you are likely to find in commercial statistical packages. As an example of nonlinear regression, consider another depreciation problem. The value of a used airplane decreases for each year of its age. Assuming the value of a plane falls by the same amount each year, a linear function relating value to age is: value = p0 + p1*Age Where 'p0' and 'p1' are the parameters whose values are to be determined. However, it is a well known fact that planes (and automobiles) lose more value the first year than the second, and more the second than the third, etc. This means that a linear (straight line) function cannot accurately model this situation. A better, nonlinear, function is: value = p0 + p1*exp(-p2*Age) Where the 'exp' function is the value of e (2.7182818...) raised to a power. This type of function is known as "negative exponential" and is appropriate for modeling a value whose rate of decrease is proportional to the difference between the value and Chapter 1. Introduction 3 some base value. The F33YEAR.NLR example command file fits a linear function to the value of used airplanes. The F33EXP.NLR example fits a negative exponential function to the same data. Run both examples and compare the fitted functions. See F33.NLR for an example of a multiple regression using three independent variables. Much of the convenience of Nonlin comes from the fact that you can enter complicated functions using ordinary algebraic notation. Examples of functions that can be handled with Nonlin include: Linear: Y = p0 + p1*X Quadratic: Y = p0 + p1*X + p2*X^2 Multivariate: Y = p0 + p1*X + p2*Z + p3*X*Z Exponential: Y = p0 + p1*exp(X) Periodic: Y = p0 + p1*sin(p2*X) Misc: Y = p0 + p1*Y + p2*exp(Y) + p3*sin(Z) In other words, the function is a general expression involving one dependent variable (on the left of the equal sign), one or more independent variables, and one or more parameters whose values are to be estimated. Because of its generality, Nonlin can perform all of the regressions handled by ordinary linear or multivariate regression programs as well as nonlinear regression. However, in order to handle nonlinear functions, Nonlin uses an iterative function optimization algorithm which is slower than the simple linear regression algorithm and has the potential for not converging to a solution. Note: Some other regression programs claim to perform nonlinear regression but actually do it by transforming the values of the variables such that the function is converted to linear form. They then perform a linear regression on the transformed function. This technique has a major flaw: it determines the values of the parameters that minimize the squared residuals for the transformed, linearized function rather than the original function. This is different than minimizing the squared residuals for the actual function and the estimated values of the parameters may not produce the best fit of the original function to the data. Nonlin uses a true nonlinear regression technique that minimizes the squared residuals for the actual function. Nonlin can also handle functions that cannot be transformed to a linear form. Chapter 1. Introduction 4 1.3 Installing Nonlin The Nonlin system consists of the following files: NONLIN.EXE -- The executable program. NONLIN.DOC -- Documentation file. NONLIN.FON -- Font file used if you request a plot. NONLIN.LJF -- LaserJet font file (registered version only). *.NLR -- Example command files. REGISTER.DOC -- Form used to register your use of Nonlin. To install Nonlin, copy the files into the directory of your choice. The registered version of Nonlin includes a file named NONLIN.LJF with the fonts needed for printing plots on HP LaserJet printers. If you do not plan to generated hard copy output for a LaserJet printer, you may delete the NONLIN.LJF file. If the NONLIN.FON and NONLIN.LJF files are not in your current directory, you must place a command of the following form in your AUTOEXEC.BAT file to tell Nonlin where to look for its font files: SET NONLIN=directory Where "directory" is the name of the device and directory where the files are located. For example, if the files are located in a directory named NONLIN on the C disk, the following command could be used: SET NONLIN=C:\NONLIN Chapter 2 Using Nonlin Once Nonlin has been installed, it can be started using a DOS command of the form: NONLIN command_file [listing_file] where "command_file" is the name of a file containing Nonlin commands that control the analysis. The sections that follow describe these commands. If you specify a command file name without an extension, ".NLR" is used as the default extension. A "listing_file" parameter may be specified on the command line. If you specify a file name, the output (results) of the regression analysis are written to this file. If no file name is specified, the output is written to a file with the same name as the command_file but with the extension ".LST". If you specify a listing file name without an extension, ".LST" is provided as the default extension. Specify NUL for the listing_file if you do not want to generate an output file. For example, to process a command file named LINEAR.NLR, directing output to a file named LINEAR.LST, use the following command: NONLIN LINEAR To do the same analysis, directing the output to a file named MODEL1.LST, use the following command: NONLIN LINEAR MODEL1 Normally Nonlin commands and computed results are displayed on the screen and written to the listing file; however, if you place a NOECHO command in your command file the screen display is suppressed but the output is still written to the listing file. At this point, I suggest you pause in your reading and try running a Nonlin example to get a feel for how it works. Several example files with the extension ".NLR" are provided with the distribution. LINEAR.NLR is a good one to start with followed by AIDS.NLR. If you do not have a graphics monitor, edit the LINEAR.NLR command file (and other example files) and remove the PLOT command. 5 Chapter 2. Using Nonlin 6 2.1 Statement Syntax The syntax for Nonlin statements follows the style of the C programming language. Each statement must end with a semicolon character, you may place more than one statement on a line, and text strings and file names are enclosed in quote marks. Nonlin has the same arithmetic and logical operators as C, and brace characters ('{' and '}') are used to group statements. The following keywords are reserved by Nonlin and may not be used as the names of variables or parameters: angletype, break, confidence, connect, connect2, constant, constrain, continue, correlate, covariance, data, do, domain, double, else, for, function, grid, if, iteration, iterations, model, noecho, nogrid, nopause, notitle, noxlabel, noylabel, nplot, output, parameter, parameters, plot, poutput, presolution, print, register, residual, rplot, splot, stop, sweep, title, to, tolerance, values, var, variable, variables, while, width, xlabel, xvar, xvar2, ylabel, yvar, yvar2. 2.2 Variables and Parameters Nonlin allows you to use four types of variables: input, computed, system, and constant. Input variables are variables whose values are set by observations read from your data file. Input variables are declared using the VARIABLES statement. For each step of the analysis Nonlin cycles through each data observation and executes the statements in your Nonlin command file. Each execution begins by setting the input variables to the values for a specific observation. If you want to transform input variables you should assign the transformed values to computed variables rather than modifying the values of input variables. Computed variables are variables whose values are computed and assigned in the Nonlin command file. They are declared using the DOUBLE statement. You can use computed variables to hold intermediate results of calculations or to hold transformed values of input variables. Computed variables may hold single values (scalars), lists of items (vectors), or arrays of values. You may assign initial values to computed variables when they are declared with the DOUBLE statement and you may modify the values during the course of execution. System variables hold values calculated by Nonlin during the execution of your commands. There are three system variables: PREDICTED --- the value of the dependent variable computed by executing your function using the current parameter and input variable values. RESIDUAL --- the difference between predicted and actual value of the dependent variable of the function. Chapter 2. Using Nonlin 7 OBS --- the number of the observation record from your data file that is currently being executed, the first record is number 1. The PREDICTED and RESIDUAL variables only have defined values after the FUNCTION statement has been executed. They are primarily useful in SPLOT and OUTPUT statements. Symbolic constants are declared using the CONSTANT statement and assigned values with their declaration. The values may not be altered after the declaration. Symbolic constants may not appear on the left side of an equal sign, otherwise they may be used wherever variables or constants may be used. There are two other designations for variables that should be mentioned. An "independent" variable is a variable that appears in the FUNCTION statement on the right side of the equal sign. You may have many independent variables. The "dependent" variable appears in the FUNCTION statement on the left side of the equal sign. There will be only one dependent variable. If you use multiple FUNCTION statements, the same dependent variable must be used in each one. Input and computed variables may be used as independent and dependent variables. In addition to variables you will use the PARAMETERS statement to declare parameters whose values are to be calculated by Nonlin. Parameters are used like variables but there values are neither read from the input file nor computed by statements in your program, rather their value is determined by Nonlin so as to cause the function to best fit your data observations. 2.3 Plots Nonlin allows you to generate four types of plots as part of the analysis. Any combination of these plot types may be requested and you can generate multiple scatter plots by including more than one SPLOT statement. With the registered version of Nonlin you can cause hard copy images of the plots to be generated on HP LaserJet printers. The plots are requested using the following statements: PLOT --- Generate a plot showing the computed value of the function superimposed on a scatter plot of the input data values. Nonlin evaluates the function using the computed values of the parameters at many data points over the domain and plots it as a smooth curve. Because Nonlin must evaluate the function at points between the observations the PLOT statement can only be used if your function has a single independent variable which is an input variable (not a computed variable). You may use symbolic constants in the function. SPLOT --- Generate a scatter plot with marks at (X,Y) coordinates of data points. You may plot two sets of points on the same graph for comparison purposes. Unlike the PLOT command, you Chapter 2. Using Nonlin 8 may use any type of variable with the SPLOT command: input, computed, system, and constant. Options are available to cause the plots to be connected by straight line segments but unlike the PLOT statement Nonlin will not compute curved segments between the points. You can use multiple SPLOT statements in your program to cause multiple scatter plots to be generated. RPLOT --- Generate a plot showing the residual values of the function on the vertical (Y) axis. A "residual" value (or error deviation) is the difference between an actual value of the dependent variable of the function for an observation and the predicted value based on the function fitted by the regression analysis. A residual plot is useful for determining where, and by how much, the fitted function fails to predict the actual observations. NPLOT --- Display a normal probability plot of the residual values. In this plot, the actual value of each residual is plotted on the vertical (Y) axis and the expected value of the residual, assuming the residuals are normally distributed, is plotted on the horizontal (X) axis. If the residuals are normally distributed, the resulting plot will be a straight line passing through the origin with a slope of 1. A normal probability plot is useful for determining whether the residuals are normally distributed. If they are not normally distributed then the form of the function being fitted may be inappropriate for the data. 2.4 Overview of Computation Process Before getting into the details of how you present an analysis to Nonlin it is a good idea to review the basic computation process. You prepare a command file containing Nonlin statements to declare variables, perform computations, and compute the value of a function that you wish to have fitted to your data observations. The function will have a dependent variable on the left side of the equal sign and one or more independent variables and parameters on the right side of the equal sign. Either input or computed variables may be used as dependent and independent variables. For each observation record in your data file Nonlin executes your statements and computes the value of the function. The computed value of the function is assigned to the PREDICTED system variable. The predicted value is then subtracted from the actual value of the dependent variable and this difference is assigned to the RESIDUAL system variable. This process is repeated for each observation in the data file and the squared residual values are added together. After all of the observations have been processed Nonlin adjusts the values of the parameters whose values are to be determined and repeats the process attempting to minimize the sum of the squared residuals. Chapter 2. Using Nonlin 9 It is important to understand what happens when Nonlin executes a FUNCTION statement such as Function y = a + b*x; Unlike an assignment statement, this FUNCTION statement does not assign a new value to the y variable. Rather it computes the value of the expression on the right side of the equal sign using the current values of the 'a' and 'b' parameters and assigns this value to the PREDICTED system variable. It then subtracts this from the current value of the y variable to determine the residual. 2.5 Function Specification Much of the power of Nonlin comes from its ability to estimate the value of parameters that are part of complicated functions that you enter in ordinary algebraic form. Nonlin provides you with great power in defining the function. Not only can you use a complicated expression to specify the function, you can also use multiple statements complete with intermediate work variables, conditional control (IF, ELSE), and looping. The only requirement is that a FUNCTION statement must be executed to define the estimated value of the dependent variable. The following section explains the arithmetic operators and built-in functions that are used to specify a function. 2.5.1 Arithmetic Operators The following arithmetic operators may be used in expressions: ++ add 1 to a variable -- subtract 1 from a variable + addition - subtraction or unary minus * multiplication / division % modulo ** or ^ exponentiation The "++" and "--" operators may be used either immediately before or after a variable name. If they are used before the name, the increment or decrement is performed before the value of the variable is used in the expression. If they are used after the name, the value of the variable before being modified is used in the expression and then the increment or decrement takes place. For example, the sequence: a = 3; b = 3; x = ++a; y = b++; Chapter 2. Using Nonlin 10 assigns the value 4 to x and 3 to y. At the end of the sequence, both a and b have the value 4. The following assignment operators can be used in expressions: variable = expression; // Assign expression to variable variable += expression; // Add expression to variable variable -= expression; // Subtract expression from variable variable *= expression; // Multiply variable by expression variable /= expression; // Divide variable by expression The following operators compare two values and produce a value of 1 if the comparison is true, or 0 if the comparison is false: == Equal != Not equal <= Less than or equal >= Greater than or equal < Less than > Greater than The following logical operators may be used: ! Logical NOT (negates true and false) && AND || OR The conditional operator has the form: operand1 ? operand2 : operand3 The value of operand1 is evaluated. If it is true (not zero) then the value of operand2 is the result of the expression. If the value of operand1 is false (zero) then operand3 is the result of the expression. There are two other special operators: "[...]" (square brackets) which enclose subscripts on arrays (see the description of the DOUBLE statement for information about arrays), and "," (comma) which is used to specify left-to-right, sequential evaluation of a list of expressions. Operator precedence, in decreasing order, is as follows: subscript, unary minus, logical NOT, ++ and --, exponentiation, multiplication, division and modulo, addition and subtraction, relational (comparison), logical AND, logical OR, conditional, assignment, comma. Parentheses may be used to group terms. 2.5.2 Numeric Constants Numeric constants may be written in their natural form (1, 0, 1.5, .0003, etc.) or in exponential form, n.nnnEppp, where n.nnn is the base value and ppp is the power of ten by which the base is multiplied. For example, the number 1.5E4 is equivalent to 15000. All numbers are treated as "floating point" values (actually, Chapter 2. Using Nonlin 11 double precision), regardless of whether a decimal point is specified or not. 2.5.3 Symbolic Constants You can use the CONSTANT statement to associate symbolic names with constant numeric values. When you use the symbolic name in the function the numeric value is substituted for the symbolic name. See Section 3.5, page 20. The PIECE.NLR example contains a symbolic constant. 2.5.4 Built-in Constant The symbolic name "PI" is equivalent to the value of pi, 3.14159... You may write PI using either upper or lower case. 2.5.5 Built-in Functions The following functions are built into Nonlin and may be used in expressions. The ANGLETYPE statement controls whether trigonometric functions operate on angles in units of degrees or radians (see Section 3.14, page 23). ABS(x) -- Absolute value of x. ACOS(x) -- Arc cosine of x. ASIN(x) -- Arc sine of x. ATAN(x) -- Arc tangent of x. BETAI(x,a,b) -- Incomplete beta function: Ix(a,b). The incomplete beta function can be used to compute a variety of statistical functions. For example, the probability of Student's t with 'df' degrees of freedom can be computed with BETAI(df/(df+t^2),.5*df,.5). The probability of the F statistic with df1 and df2 degrees of freedom can be computed with 2*BETAI(df2/(df2+df1*f),.5*df2,.5*df1). CEIL(x) -- Ceiling of x (an equivalent name for this function is INT). Returns the smallest integer that is at least as large as x. For example, CEIL(1.5)=2; CEIL(4)=4; CEIL(-2.6)=-2. COS(x) -- Cosine of x. COSH(x) -- Hyperbolic cosine of x. COT(x) -- Cotangent of x. (COT(x) = 1/TAN(x)). CSC(X) -- Cosecant of x. (CSC(x) = 1/SIN(x)). CTOP(angle) -- Convert an angle in the compass coordinate system to a polar coordinate angle. The polar coordinate system has the origin of an angle along the positive X axis and the angle increases in a counter-clockwise direction. The compass Chapter 2. Using Nonlin 12 coordinate system has the positive Y axis as the origin (i.e., north) and the angle increases in a clockwise direction. The PTOC function performs the reverse transformation. DEG(x) -- Converts an angle, x, measured in radians to the equivalent number of degrees. See also the description of the ANGLETYPE statement. EI1(alpha,phi) -- Elliptic integral of the first kind. Computes the integral from 0 to phi (degrees or radians) of the function d.phi/sqrt(1-k**2*sin(phi)**2), where k = sin(alpha). alpha and phi must be in the range 0 to 90 degrees or pi/2 radians. The ANGLETYPE statement determines whether the angles are in units of degrees or radians. EI2(alpha,phi) -- Elliptic integral of the second kind. Computes the integral from 0 to phi degrees or radians of the function sqrt(1-k**2*sin(phi)**2)*d.phi, where k = sin(alpha). alpha and phi must be in the range 0 to 90 degrees or pi/2 radians depending on the setting of ANGLETYPE. EIC1(alpha) -- Complete elliptic integral of the first kind. Computes the integral from 0 to 90 degrees (or pi/2 radians) of the function d.phi/sqrt(1-k**2*sin(phi)**2), where k = sin(alpha). alpha must be in the range 0 to (less than) 90 degrees (or pi/2 radians) depending on the setting of ANGLETYPE. EIC2(alpha) -- Complete elliptic integral of the second kind. Computes the integral from 0 to 90 degrees (or pi/2 radians) of the function sqrt(1-k**2*sin(phi)**2)*d.phi, where k = sin(alpha). alpha must be in the range 0 to 90 degrees or pi/2 radians depending on the setting of ANGLETYPE. ERF(x) -- Standard error function of x. EXP(x) -- e (base of natural logarithms) raised to the x power. FAC(x) -- x factorial (x!). Note, the FAC function is computed using the GAMMA function (FAC(x)=GAMMA(x+1)) so non-integer argument values may be computed. FLOOR(x) -- Floor of x. Returns the largest integer that is less than or equal to x. For example, FLOOR(2.5)=2; FLOOR(4)=4; FLOOR(-3.6)=-4. GAMMA(x) -- Gamma function. Note, GAMMA(x+1) = x! (x factorial). GAMMAI(x) -- Reciprocal of GAMMA function (GAMMAI(x) = 1/GAMMA(x)). GAMMALN(x) -- Log (base e) of the GAMMA function. HAV(x) -- Haversine of x. (HAV(x) = (1-COS(x))/2). Chapter 2. Using Nonlin 13 INT(x) -- Ceiling of x (an equivalent name for this function is CEIL). Returns the smallest integer that is at least as large as x. For example, INT(1.5)=2; INT(4)=4; INT(-2.6)=-2. J0(x) -- Bessel function of the first kind, order zero. J1(x) -- Bessel function of the first kind, order one. JN(n,x) -- Bessel function of the first kind, order n. LOG(x) -- Natural logarithm of x. LOG10(x) -- Base 10 logarithm of x. LOG2(x) -- Base 2 logarithm of x. MAX(x1,x2) -- Maximum value of x1 or x2. MIN(x1,x2) -- Minimum value of x1 or x2. NORMAL(x) -- Normal probability distribution of x. X is in units of standard deviations from the mean. See also the NPD function. NORMAL(x) = NPD(x,0,1); NPD(x,mean,std) -- Normal probability distribution of x with specified mean and standard deviation. X is in units of standard deviations from the mean. PAREA(x) -- Area under the normal probability distribution curve from -infinity to x. (i.e., integral from -infinity to x of NORMAL(x)). PRINTF("format",value1,value2,...) -- Format and print a series of values. The Nonlin printf function has the same syntax and function as the printf function in the C language. It causes a string to be written to your terminal and also the listing file for the analysis. Printf is primarily useful as a diagnostic tool to give you a way to observe what is happening during an analysis. Note: since your commands are executed for each data observation and each iteration, the printf may generate a great deal of output. The first argument to printf is a quoted string that contains characters to be printed, control codes, and (if values are to be printed) formatting specifications. If you are familiar with the C programming language, the Nonlin formatting string has the same form and control codes. Ordinary characters and numbers in the format string are printed just as they appear. Use the control code '\n' to cause a carriage-return, line-feed sequence to be printed to terminate a line. For example, the following command prints a line of text: Chapter 2. Using Nonlin 14 printf("Beginning of analysis\n"); If you wish to insert formatted values in the string, specify one or more expressions after the format string. Place in the format string at the location where you want to insert the formatted value the sequence '%lf' (percent sign, lower case L, f) if you want the number formatted in the style nnnn.nnnn or '%lE' if you want exponential notation (nnn.nnnEnnn). Optionally, you may specify the width of the formatted value and the number of decimal places between '%' and 'l'. For example, the following sequence produces a formatted value with 8 total characters and 4 decimal places: %8.4lf. Here are several examples: printf("Processing observation %lf\n",obs); printf("X = %lf, Y = %lf\n",x,y); printf("Predicted = %14.6lE\n",predicted); PTOC(angle) -- Convert an angle in the polar coordinate system to a compass coordinate angle. The polar coordinate system has the origin of an angle along the positive X axis and the angle increases in a counter-clockwise direction. The compass coordinate system has the positive Y axis as the origin (i.e., north) and the angle increases in a clockwise direction. The CTOP function performs the reverse transformation. PTORX(angle,distance) -- Convert a position in polar coordinates to the corresponding rectangular coordinate. This function returns the X coordinate of the position; use PTORY to obtain the Y coordinate. Note: polar coordinates are specified with the positive X axis being the origin for the angle and with the angle increasing in the counter-clockwise direction. PTORY(angle,distance) -- Convert a position in polar coordinates to the corresponding rectangular coordinate. This function returns the Y coordinate of the position; use PTORX to obtain the X coordinate. Note: polar coordinates are specified with the positive X axis being the origin for the angle and with the angle increasing in the counter-clockwise direction. PULSE(a,x,b) -- Pulse function. If the value of x is less than a or greater than b, the value of the function is 0. If x is greater than or equal to a and less than or equal to b, the value of the function is 1. In other words, it is 1 for the domain (a,b) and zero elsewhere. If you need a function that is zero in the domain (a,b) and 1 elsewhere, use the expression (1-PULSE(a,x,b)). RAD(x) -- Converts an angle measured in degrees to the equivalent number of radians. See also the description of the ANGLETYPE statement. RANDOM() -- Returns a random value uniformly distributed in the range 0 to 1. Chapter 2. Using Nonlin 15 ROUND(x) -- Rounds x to the nearest integer. For example, ROUND(1.1)=1; ROUND(1.8)=2; ROUND(-2.8)=-3; RTOPA(x,y) -- Convert a rectangular coordinate (x,y) to the corresponding polar coordinate (angle,distance). This function returns the angle, use RTOPD to get the distance coordinate. Note: polar coordinates are specified with the positive X axis being the origin for the angle and with the angle increasing in the counter-clockwise direction. RTOPD(x,y) -- Convert a rectangular coordinate to the corresponding polar coordinate. This function returns the distance from the origin, use RTOPA to get the angle. Note: polar coordinates are specified with the positive X axis being the origin for the angle and with the angle increasing in the counter-clockwise direction. SEC(x) -- Secant of x. (SEC(x) = 1/COS(x)). SEL(a1,a2,v1,v2) -- If a1 is less than a2 then the value of the function is v1. If a1 is greater than or equal to a2, then the value of the function is v2. SIN(x) -- Sine of x. See TREND.NLR for an example of a function with a sin term. See also Section 4.9 for additional information about using sin terms in functions. SINH(x) -- Hyperbolic sine of x. SQRT(x) -- Square root of x. STEP(a,x) -- Step function. If x is less than a, the value of the function is 0. If x is greater than or equal to a, the value of the function is 1. If you need a function which is 1 up to a certain value and then 0 beyond that value, use the expression STEP(x,a). See PIECE.NLR for an example of this function. T(n,x) -- Chebyshev polynomial of order n. TAN(x) -- Tangent of x. TANH(x) -- Hyperbolic tangent of x. Y0(x) -- Bessel function of the second kind, order zero. Y1(x) -- Bessel function of the second kind, order one. YN(n,x) -- Bessel function of the second kind, order n. Chapter 2. Using Nonlin 16 2.6 Nonlin Command Files The commands described in this section are placed in a command file. When you start Nonlin, you specify the name of the command file as a parameter on the command line. For example, if the command file name is CAR.NLR, the following command would cause Nonlin to execute the commands in the command file: NONLIN CAR.NLR If you do not specify a file name extension for the command file, ".NLR" is used by default. The output of the regression for this example would be written to a file named CAR.LST. Command files can be created using a text editor such as EDIT-32, EDLIN, the DOS EDIT program, or any other editor or word processor that is capable of creating an ascii text file without formatting codes. 2.7 Comments The beginning of a comment is denoted with "//" (two consecutive slash characters). Everything from the "//" sequence to the end of the line is treated as a comment. Comments may be on lines by themselves or on the ends of other statements. You can also specify a comment by beginning the comment with the "/*" character sequence. All characters following this are treated as comments up to the matching "*/" sequence. The following lines illustrate both types of comments: // Function to be fitted y = a + b*x; // Simple linear equation /* * This is a comment. */ z = y / 5; /* This is a comment too */ 2.8 Include Files Nonlin provides a #INCLUDE statement that you may place in your command file to cause another file to be inserted in the command file at the point where the #INCLUDE statement occurs. The included file may contain any valid Nonlin statements or data that would be appropriate at the specified point in the command file. Processing of the statements in the original command file resume when the end of the included file (and any nested files) is reached. The form of the statement is: #include "file" where 'file' is the name of the file whose contents are to be inserted. If no extension is specified, ".NLR" is used by default. Include files may be nested up to 10 levels deep. The following is an example of a Nonlin command file that includes the function specification: Chapter 2. Using Nonlin 17 Title "Example of file inclusion"; Variables X,Y; Parameters a,b; #include "fun1"; // Function statement is in "fun1.nlr" data; (data records follow) 2.9 Required Statements Every command file must contain the following statements: VARIABLES, PARAMETERS, FUNCTION, and DATA. The DATA statement introduces the data for the analysis and must be the last statement in the file (data records may follow it). Other, optional, statements may be interspersed in the command file. The following is an example of a complete command file: title "Depreciation Example"; variables value,age,miles; parameters base,depage,depmiles; function value = base + depage*age + depmiles*miles; data; (data records follow) Chapter 3 Nonlin Statements The following is a list of the valid Nonlin statements that can be placed in a Nonlin command file. Nonlin statements are not case sensitive. Remember to end each statement with a semicolon. 3.1 TITLE TITLE "string"; (optional) -- Specifies a title line that is printed with the results of the analysis. Note: the title string must be enclosed in quote signs. 3.2 VARIABLES VARIABLES var1,var2,...; (required) -- Specifies the names of the input variables whose values will be read from your data file. The order of the variable names must match the order of the data values on each observation record. You may define more variables than you actually use in the function specification. A maximum of 25 variables may be specified. The length of a variable name is limited to 10 characters. Capitalize the variable names as you want them displayed in the results. The keyword "VARIABLE" may be used instead of "VARIABLES". You may specify all of the variables on a single statement or you may use multiple VARIABLES statements. If you use multiple statements, the order in which they appear in the command file must match the order of the variable values on each observation record. The VARIABLES statement must precede the FUNCTION statement. See F33.NLR for an example of a multiple regression using three independent variables. You can also use the DOUBLE statement to declare variables (see below). The difference is that the VARIABLES statement declares variables that are read from the input file whereas the DOUBLE statement declares variables whose values will be computed by statements in your command file. 3.3 PARAMETERS PARAMETERS param1[=initial1],param2[=initial2],...; (required) -- Specifies the names of the parameters whose values are to be determined by Nonlin. Nonlin is capable of handling up to 25 parameters. The parameter names may not exceed 10 characters in length. Do not specify any parameters that are not used in the 18 Chapter 3. Nonlin Statements 19 analysis. The PARAMETERS statement must precede the FUNCTION statement. The keyword "PARAMETER" may be used instead of "PARAMETERS". Optionally, an initial estimate of the parameter value may be specified by following the parameter name with an equal sign and the value. If no value is specified, 1 is used by default. Specifying an initial value that is near the actual value usually speeds up the operation of Nonlin and may enable it to successfully converge to a solution. If Nonlin is unable to converge to a solution, try specifying different starting values for the parameters. Try to specify a value that at least has the correct sign as the expected final value. The CONSTRAIN statement (see page 20) can be used to limit the range of values for parameters. The SWEEP statement (see page 21) can be used to perform the regression analysis with a range of parameter initial values. 3.4 DOUBLE DOUBLE var1[=value],var2[=value],...; (optional) -- Specifies the names of computed variables that you may use subsequently to hold calculated values. Nonlin allows you to define up to 30 computed variables. All variables hold double precision (64 bit) floating point values. Optionally, the name of a variable may be followed by an equal sign and a value to which the variable is initialized. If you do not specify an initial value, the variable is initialized to 0. The following are examples of DOUBLE statements: double t1,t2; double roomtemp=73; It is convenient to use computed variables for intermediate calculations such as transformed values of input variables. Nonlin allows you to declare arrays with one or two dimensions. To do this follow the name of the variable with number of array elements enclosed in square brackets. If the array has two dimensions specify the number of rows, then the number of columns, separated by a comma. (Note: this is different than the C language syntax for declaring a two-dimensional array). The following statements declare a one dimensional array (i.e., a vector) with 20 elements and a two dimensional array with 5 rows and 10 columns: double xvec[20]; double ya[5,10]; You may assign initial values to arrays by following the variable declaration with an equal sign and a list of values enclosed in curly braces. In the case of a two dimensional array, the values should be specified by rows (i.e., the last subscript varies most Chapter 3. Nonlin Statements 20 rapidly). The following are examples of array declarations with initializations: double xvec[5] = {2,5,7,1,0}; double xa[2,3] = {2.3,7.5,1.2,4.4,2.6,7.3}; When used in expressions the subscript values are 0 based. That is, the first element of the array is referenced using a subscript value of 0 and the last element is referenced using a subscript value equal to one less than the number of elements in the array. For example, the following statements would declare an array with 100 elements and initialize it: double xsq[100],i; for (i=0; i<100; i++) { xsq[i] = i; } 3.5 CONSTANT CONSTANT variable=value; (optional) -- Specifies the name of a symbolic constant and associates a numeric value. You can then use the symbolic name where you would use a number and the corresponding constant numeric value will be substituted. This is useful when you are trying out different models and want to easily be able to change a constant value for each run. The following is an example of a symbolic constant named "Roomtemp" that causes the value 73 to be substituted in the function: Variable Time; // Cooling time in seconds Variable Temp; // Temperature of object Constant Roomtemp = 73; // Ambient temperature Parameter InitTemp; // Initial temperature Parameter Coolrate; // Cooling rate factor Function Temp = Roomtemp + InitTemp * exp(-Coolrate * Time); 3.6 CONSTRAIN CONSTRAIN parameter=lowvalue,highvalue; (optional) -- Specifies a lower and upper limit on the range of a parameter value. During the solution process, Nonlin may allow a parameter's value to temporarily move in a direction away from its final value. With some functions it may be necessary to constrain the parameter's value so that it does not go negative (e.g., if the function takes the square root of the parameter), or zero (if the parameter is in a denominator). If a parameter is tightly constrained, Nonlin may report "singular convergence" because it is unable to converge to an optimum value of the parameter; however, the estimated values of other parameters may be useful. Only a single parameter and its associated limits may be specified on each CONSTRAIN statement, but you may use multiple CONSTRAIN statements. The PARAMETERS statement must precede the CONSTRAIN statement. Use the CONSTANT statement if you wish to define a parameter with a fixed value. Chapter 3. Nonlin Statements 21 The parameter value is allowed to range from 'lowvalue' to 'highvalue'. If you want to prevent a parameter value from going to zero, you must specify a value greater than zero for the low value (specifying zero would allow it to reach, but not go below, zero). For example, the following statement constrains the value of 'age' to be greater than zero and less than or equal to 100: constrain age = .0001,100; See the COOLING.NLR, F33EXP.NLR, and POWER.NLR files for examples of the CONSTRAIN statement. 3.7 SWEEP SWEEP parameter=lowvalue,highvalue,stepsize; (optional) -- Specifies that the regression analysis is to be performed repeatedly with a set of starting values for the parameter. The first analysis is performed with the parameter having the 'lowvalue'; the value of 'stepsize' is then added to the parameter's initial value and the analysis is performed again. The process is repeated until the value of the parameter reaches 'highvalue'. Each time the analysis is performed the value of the residual sum of squares is compared with the best previous result. The estimated values of the parameters for the best starting value are saved and used for the final analysis and report. Only one parameter may be specified on each SWEEP statement, but you may have as many SWEEP statements as there are parameters. The number of regression analyses performed will be equal to the product of the number of parameter values for each SWEEP statement. The SWEEP statement is useful when you are trying to fit a complicated function that may have "local minimum" values other than the "global minimum". Periodic functions (sin, cos, etc.) are especially troublesome. See the SINE.NLR command file for an example of the SWEEP statement. 3.8 FUNCTION FUNCTION depvar = function; (required) -- Specifies the form of the function whose parameters are to be determined. The dependent variable must be the only thing to the left of the equal sign. The expression to the right of the equal sign may contain variables, parameters, constants, operators, and library functions such as sqrt, sin, exp, etc. The VARIABLES and PARAMETERS statements must appear in the command file before the FUNCTION statement. The function may be specified using parameters, input variables, computed variables (declared using the DOUBLE statement), constants, and library functions. You may use more than one FUNCTION statement if you use IF or other conditional Chapter 3. Nonlin Statements 22 statements to select which one will be executed. However, during each execution of your command file one, and only one, FUNCTION statement must be executed. Some example FUNCTION statements are show below: Function y = p0 + p1*x; Function distance = .5 * accel * time^2; Function value = price + yrdep*age + miledep*miles; Function populatn = base * growrate * exp(time); 3.9 CORRELATE CORRELATE [var1,var2,...]; (optional) -- Causes Nonlin to compute and print a correlation matrix. If you do not specify a list of variables the correlation matrix includes all input variables. If you wish to control exactly which variables are included in the matrix or if you wish to include computed variables (declared with a DOUBLE statement) you may specify a list of variables. See Section 4.11 on page 41 for more information about correlation. The F33.NLR example includes a CORRELATE statement. The following are examples of the CORRELATE statement: correlate; correlate x1,x2,x3,y; 3.10 COVARIANCE COVARIANCE; (optional) -- Causes the variance-covariance matrix for the parameters to be printed. 3.11 CONFIDENCE CONFIDENCE [percent]; (optional) -- Specifies that a confidence interval is to be printed for each estimated parameter. The purpose of regression analysis is to determine the best estimate of parameter values. However, as with most statistical calculations, the values determined are estimates of the true values. The CONFIDENCE statement causes Nonlin to print a table showing the range of possible values for each parameter given a specified confidence value. The "percent" parameter specifies the probability that that the actual value of the parameter is within the confidence interval to be computed. For example, the statement Confidence 95; specifies that the confidence interval(s) are to be computed such that there is a 95 percent probability that the actual values of the parameters are within the intervals (or that there is a 5 percent chance that the parameters are outside the intervals). The "percent" parameter may range from 50 to 99.999. If the Chapter 3. Nonlin Statements 23 CONFIDENCE statement is used without specifying a percent value, 90 is used by default. 3.12 TOLERANCE TOLERANCE value; (optional, default=1E-10) -- Specifies the tolerance factor that is used to determine when the algorithm has converged to a solution. Reducing the tolerance value may produce a slightly more accurate result but will increase the number of iterations and the running time. The tolerance value must be in the range 1E-15 to 1E-1. See Section 5.2 for additional information about how the tolerance value is used to determine when the function has converged. 3.13 ITERATIONS ITERATIONS value; (optional, default=50) -- Specifies the maximum number of iterations that should be attempted by the algorithm. If the solution does not converge to the limit specified by the TOLERANCE statement (or to the default tolerance) before the maximum number of iterations is reached, the process is stopped and the results are printed. Failure to converge before the specified number of iterations could be caused by one of three things: 1. The maximum allowed number of iterations may be too small. Try using an ITERATIONS statement with a larger value. 2. The tolerance factor may be too small. Even a properly converging solution will eventually "level off" or oscillate around a good, but non-zero, sum of squares value. Try using the TOLERANCE statement to increase the tolerance value. 3. The function may not be converging. Try specifying better (or at least different) starting values for the parameters on the PARAMETERS statement. Consider using the SWEEP statement to specify a range of parameter starting values. 3.14 ANGLETYPE ANGLETYPE DEGREES or RADIANS; (optional) -- Specifies whether trigonometric library functions such as SIN, COS, TAN, etc. are to operate in units of degrees or radians. The default setting is degrees. You may only declare the angle type once in your program and the declaration must come before any statements that use trig functions. You can also use the DEG() and RAD() functions to convert between degrees and radians. The following are example ANGLETYPE statements: angletype degrees; angletype radians; Chapter 3. Nonlin Statements 24 3.15 OUTPUT OUTPUT [TO "file"] var1,var2,...; (optional) -- Specifies that after the analysis is completed, data values are to be written to a file. One record is written for each data observation in the input file. If the TO "file" portion of the statement is specified, the output is written to the specified file. If this portion of the statement is omitted, the output values are written to the listing file along with the results of the analysis. If a file name is specified without an extension, ".OUT" is used by default. The list of variable names determines which variables are written to the file and the order in which the values appear in each output record. Any variable previously declared with a VARIABLES or DOUBLE statement may be specified. In addition, the following system variable names may appear in the output list: OBS -- The observation record number, starting at 1 and increasing by 1. PREDICTED -- The predicted value for the dependent variable for the observation, given the independent variable values and the parameters as calculated by the analysis. RESIDUAL -- The difference between the actual value of the dependent variable and its predicted value. Examples of OUTPUT statements are shown below: output age,miles,value,predicted,residual; output to "growth.dat" obs,time,populatn,predicted; 3.16 POUTPUT POUTPUT "file"; (optional) -- The POUTPUT statement specifies that Nonlin is to write the final estimated values of the parameters to a file. Each parameter value is written to a separate line of the file. This statement is useful to create a file of estimated parameter values to be fed into another analysis program. This statement can also be used to determine the parameter estimates to more significant digits than displayed in the printed listing because the format used by the POUTPUT statement writes the values with 18 significant digits. The following is an example of a POUTPUT statement: poutput "params.dat"; 3.17 PLOT PLOT [options]; (optional) -- Display a plot of the calculated function and the data observations. Each data point is displayed with a blue 'X'; the function that Nonlin fits to the data is superimposed as a yellow curve. Chapter 3. Nonlin Statements 25 The PLOT statement can only be used if the FUNCTION declaration meets the following requirements: (1) there is must only a single independent variable; (2) the independent variable must be an input variable (i.e., declared with a VARIABLES statement not a DECLARE statement). You may use symbolic constants declared with the CONSTANT statement. If the function does not meet these requirements you may produce different types of plots using the SPLOT, RPLOT and NPLOT statements. You must have a CGA, EGA, or VGA monitor to use the PLOT statement, and the NONLIN.FON font file must be in the current directory or in a directory specified by the NONLIN environment variable. Press Return to proceed with the analysis after you finish looking at the plot. The following options may be specified on the PLOT statement: NOGRID -- suppress the grid lines that are normally displayed with the plot. TITLE="string" -- specify a title to be displayed with the plot. If no title is specified the title defined by the TITLE statement is used. NOTITLE -- suppresses the title for the plot which, by default, is the title specified with the TITLE statement. XLABEL="string" -- specify a label to be printed along the X axis. If you do not use this qualifier, the name of variable whose values determine the X coordinates is used as the default label. NOXLABEL -- suppress printing any label along the X axis. YLABEL="string" -- specify a label to be printed along the Y axis. If you do not use this qualifier, the name of variable whose values determine the Y coordinates is used as the default label. NOYLABEL -- suppress printing any label along the Y axis. DOMAIN=lowvalue,hivalue -- specifies the domain over which the plot is to be generated. If no domain is specified, Nonlin uses the range of the independent variable for the domain. RESIDUAL -- draw vertical lines from each observed data point to the corresponding point on the calculated function line. These lines represent the "residual" value that Nonlin is attempting to minimize. See also the descriptions of the RPLOT and NPLOT statements on pages 28 and 30. ITERATION -- draw a plot for each iteration of the regression analysis. Normally, the plot is drawn after the analysis has converged to a solution; you may use the ITERATION option to Chapter 3. Nonlin Statements 26 observe the function during each iteration of the analysis as it converges to fit the data. VALUES -- use in conjunction with the ITERATION option to cause the current parameter values to be displayed before the plot for the current iteration. PRINT -- print a copy of the plot on an HP LaserJet printer. This option is only available in the registered version of Nonlin. Nonlin writes the plot to the PRN device which much be attached to an HP Series II or Series III printer. The NONLIN.LJF font file must be in the current directory or in a directory specified by the NONLIN environment variable. NOPAUSE -- do not pause after the plot is displayed. Normally, Nonlin pauses after displaying a plot to allow you time to examine it; you press Enter to continue execution once you have finished looking at the plot. The NOPAUSE option causes Nonlin to continue with execution without pausing after the plot is displayed. This is useful in conjunction with the PRINT option when Nonlin is run in a batch file and you want to generate a hardcopy plot but not pause after the screen display. If more than one option is specified, separate them with commas. For example, to produce a plot with X and Y axis labels use a statement with the following form: PLOT XLABEL="Time",YLABEL="Blood concentration"; 3.18 SPLOT SPLOT [options]; (optional) -- Display a scatter plot of (X,Y) data points. Using the XVAR and YVAR options (see below) you can specify which variable is used for the vertical (Y) dimension and which is used for the horizontal (X) dimension. Any type of variable may be specified including input variables, computed variables (declared with the DOUBLE statement), the dependent variable of the function, and the system variables PREDICTED, RESIDUAL, and OBS (see Section 2.2 on page 6). You may display two scatter plots on the same image. This is useful for comparing computed values with input values. To do this use the XVAR2 and YVAR2 options to specify the variables for the X and Y dimensions for the second plot. Each data point for the primary plot (specified by XVAR and YVAR) is marked with a blue 'X'. The data points for the second plot (specified by XVAR2 and YVAR2) are marked with yellow triangles. You can use the CONNECT and CONNECT2 options to draw straight line segments through the points. The NOMARK and NOMARK2 options may be used to suppress the data point markers. The following options may be specified on the SPLOT statement: Chapter 3. Nonlin Statements 27 XVAR=variable -- specify the variable to be used for the horizontal (X) dimension of the first set of plotted points. This can be any type of variable, input or computed. If you do not specify this option and there is only a single independent variable in the function, it is used by default. YVAR=variable -- specify the variable to be used for the vertical (Y) dimension. This can be any type of variable, input or computed. If you do not specify this option then the dependent variable of the function (i.e., the one on the left of the equal sign) is used by default. XVAR2=variable -- specify the variable to be used for the horizontal (X) dimension of the second set of plotted points. This can be any type of variable. If you specify YVAR2 but not XVAR2, the default is the same variable as specified by XVAR. CONNECT -- Connect the first set of points by straight line segments. The points are displayed and connected in the same order that they appear in the data file. CONNECT2 -- Connect the second set of points by straight line segments. NOMARK -- Suppress the display of the 'X' symbols that normally mark the first set of data points. This can be used with CONNECT to cause only the line to be drawn. NOMARK2 -- Suppress the display of the triangle symbols that normally mark the second set of data points. NOGRID -- suppress the grid lines that are normally displayed with the plot. TITLE="string" -- specify a title to be displayed with the plot. If no title is specified the title defined by the TITLE statement is used. NOTITLE -- suppresses the title for the plot which, by default, is the title specified with the TITLE statement. XLABEL="string" -- specify a label to be printed along the X axis. If you do not use this qualifier, the name of variable whose values determine the X coordinates is used as the default label. NOXLABEL -- suppress printing any label along the X axis. YLABEL="string" -- specify a label to be printed along the Y axis. If you do not use this qualifier, the name of variable whose values determine the Y coordinates is used as the default label. Chapter 3. Nonlin Statements 28 NOYLABEL -- suppress printing any label along the Y axis. DOMAIN=lowvalue,hivalue -- specifies the domain over which the plot is to be generated. If no domain is specified, Nonlin uses the range of the horizontal variable(s) for the domain. PRINT -- print a copy of the plot on an HP LaserJet printer. This option is only available in the registered version of Nonlin. Nonlin writes the plot to the PRN device which much be attached to an HP Series II or Series III printer. The NONLIN.LJF font file must be in the current directory or in a directory specified by the NONLIN environment variable. NOPAUSE -- do not pause after the plot is displayed. Normally, Nonlin pauses after displaying a plot to allow you time to examine it; you press Enter to continue execution once you have finished looking at the plot. The NOPAUSE option causes Nonlin to continue with execution without pausing after the plot is displayed. If there is more than one option, separate them with commas. The following is an example SPLOT statement: splot xvar=time,yvar=sodium,yvar2=potassium,connect,connect2, title="Blood concentration over time", xlabel="Time (hours)",ylabel="Sodium & Potassium"; 3.19 RPLOT RPLOT [options]; (optional) -- Display a plot of the residual values. A "residual" value (or error deviation) is the difference between an actual value of the dependent variable for an observation and the predicted value based on the function fitted by the regression analysis. If the calculated function exactly predicted the actual observation values, all of the residual values would be zero. However, this is usually not the case and the residual values show where, and by how much, the fitted function fails to predict the actual observations. The RPLOT statement causes Nonlin to display a plot showing the residual values on the vertical (Y) axis. The variable plotted along the horizontal (X) axis may be specified using the XVAR option (see below). You may specify any variable including the dependent variable and computed variables declared with the DOUBLE statement. If you do not specify a variable and there is a single independent variable in the function it is used. The X axis label indicates which variable was used. A residual plot is very useful for determining if the form of the function being fitted is appropriate for the data values. If the residual values are randomly distributed in positive and negative directions then the form (shape) of the fitted function is probably appropriate for the data and the deviations are due to random measurement errors. If, however, the residuals show a systematic pattern such as a periodic cycle, then the function may Chapter 3. Nonlin Statements 29 not be appropriate for the data values. See the discussion of the Durbin-Watson statistic in Section 4.9, page 39, for additional information about autocorrelated residual values. The PLOT, RPLOT, SPLOT, and NPLOT statements may be used in the same command file. Press Return to proceed with the analysis after you have finished looking at the plot. The following options may be specified on the RPLOT statement: XVAR=variable -- specify which variable is to be used for the horizontal (X) dimension of the plot. You may specify any variable including independent input variables, the dependent variable of the function (i.e., the one on the left of the equal sign), and computed or transformed variables declared with the DOUBLE statement. If there is only a single independent variable Nonlin will use it by default. The label along the X axis indicates which variable was used. NOGRID -- suppress the grid lines that are normally displayed with the plot. TITLE="string" -- specify a title to be displayed with the plot. If this option is not specified, the default title is "Plot of residuals". NOTITLE -- suppresses the title for the plot which, by default, is "Plot of residuals". XLABEL="string" -- specify a label to be printed along the X axis. If you do not use this qualifier, the name of variable whose values determine the X coordinates is used as the default label. NOXLABEL -- suppress printing any label along the X axis. YLABEL="string" -- specify a label to be printed along the Y axis. If you do not use this qualifier, the default label is "Residual". NOYLABEL -- suppress printing any label along the Y axis. DOMAIN=lowvalue,hivalue -- specifies the domain over which the plot is to be generated. If no domain is specified, Nonlin uses the range of the X dimension variable. ITERATION -- draw a plot for each iteration of the regression analysis. Normally, the plot is drawn after the analysis has converged to a solution; you may use the ITERATION option to observe the function during each iteration of the analysis as it converges to fit the data. VALUES -- use in conjunction with the ITERATION option to cause the current parameter values to be displayed before the plot for the current iteration. Chapter 3. Nonlin Statements 30 PRINT -- print a copy of the plot on an HP LaserJet printer. This option is only available in the registered version of Nonlin. Nonlin writes the plot to the PRN device which much be attached to an HP Series II or Series III printer. The NONLIN.LJF font file must be in the current directory or in a directory specified by the NONLIN environment variable. NOPAUSE -- do not pause after the plot is displayed. Normally, Nonlin pauses after displaying a plot to allow you time to examine it; you press Enter to continue execution once you have finished looking at the plot. The NOPAUSE option causes Nonlin to continue with execution without pausing after the plot is displayed. If more than one option is specified, separate them with commas. 3.20 NPLOT NPLOT [options] (optional) -- Display a normal probability plot of the residual values. In this plot, the actual value of each residual is plotted on the vertical (Y) axis and the expected value of the residual, assuming the residuals are normally distributed, is plotted on the horizontal (X) axis. If the residuals are normally distributed, the resulting plot will be a straight line passing through the origin with a slope of 1 (i.e., the actual value of each residual should equal the expected value from the normal distribution). If the residuals are not normally distributed, the plot will deviate from a straight line. Nonlin displays a red line along which the X marks should be displayed if the residuals are normally distributed. This plot also computes the correlation between the actual residual values and their expected values and displays the correlation coefficient in the title line "(r=n.nnn)". If the residual values are normally distributed, the correlation should be close to 1.000. A correlation value less than 0.940 suggests that the residuals are not normally distributed. The PLOT, RPLOT, SPLOT, and NPLOT statements may be used in the same command file. Press Return to proceed with the analysis after you have finished looking at the plot. The following options may be specified on the NPLOT statement: GRID -- display grid lines to make it easier to estimate values. TITLE="string" -- specify a title to be displayed with the plot. If no title is specified the default title is "Normal probability plot". NOTITLE -- suppresses the title for the plot. XLABEL="string" -- specify a label to be printed along the X axis. If you do not use this qualifier, default label is "Expected residuals". Chapter 3. Nonlin Statements 31 NOXLABEL -- suppress printing any label along the X axis. YLABEL="string" -- specify a label to be printed along the Y axis. If you do not use this qualifier, the default label is "Actual residuals". NOYLABEL -- suppress printing any label along the Y axis. ITERATION -- draw a plot for each iteration of the regression analysis. Normally, the plot is drawn after the analysis has converged to a solution; you may use the ITERATION option to observe the function during each iteration of the analysis as it converges to fit the data. VALUES -- use in conjunction with the ITERATION option to cause the current parameter values to be displayed before the plot for the current iteration. PRINT -- print a copy of the plot on an HP LaserJet printer. This option is only available in the registered version of Nonlin. Nonlin writes the plot to the PRN device which much be attached to an HP Series II or Series III printer. NOPAUSE -- do not pause after the plot is displayed. Normally, Nonlin pauses after displaying a plot to allow you time to examine it; you press Enter to continue execution once you have finished looking at the plot. The NOPAUSE option causes Nonlin to continue with execution without pausing after the plot is displayed. If more than one option is specified, separate them with commas. 3.21 PRESOLUTION PRESOLUTION value; (optional) -- Specifies whether plots sent to HP LaserJet printers should use 150 or 300 dot-per-inch resolution. This option is only available in the registered version of Nonlin. The value parameter must be 150 or 300. The default value is 150 causes the plots to use most of the horizontal width of an 8.5x11 inch page. These plots are suitable for direct transfer to overhead transparencies. Specifying 300 for the resolution produces smaller plots that are suitable for inclusion in printed documents. 3.22 WIDTH WIDTH value; (optional) -- Specify the width, in inches, of printed plots. This option is only available in the registered version of Nonlin. Due to memory space considerations, the maximum width is limited to about 7.9 inches for 150 DPI resolution and 4.5 inches for 300 DPI resolution. If you have limited memory space, you may have to reduce the width to be able to produce printed plots. This statement is ignored unless you request that a plot be printed. Chapter 3. Nonlin Statements 32 3.23 NOECHO NOECHO; (optional) -- Specifies that the statements and computed results are not to be listed on the screen. The output is still written to the listing file and any requested plots are displayed on the screen. 3.24 Assignment Statement The assignment statement is an executable statement that evaluates an expression and assigns its value to a variable. The syntax for an assignment statement is: variable = expression; // Assign expression to variable variable += expression; // Add expression to variable variable -= expression; // Subtract expression from variable variable *= expression; // Multiply variable by expression variable /= expression; // Divide variable by expression where "variable" is a variable that was previously declared using a DOUBLE statement. The variable may be subscripted if it is an array. "expression" is a valid arithmetic or logical expression following the rules explained earlier. If the expression involves a relational comparison operator (e.g., <, >, >=, etc.) or a logical operation (&&, ||, !), the value 1 is used for true and 0 for false. The expression may contain any type of variable (input, computed, or constant) along with parameters and library functions. 3.25 IF Statement The form of the IF statement is: IF (expression) statement1 [ELSE statement2] If the expression is true (not zero) statement1 is executed, if the expression is false (0) and the ELSE clause is specified, statement2 is executed. The ELSE clause and the second set of controlled statements are optional. You may control groups of statements by enclosing them in braces. The following are examples of valid IF statements: if (x > bigx) bigx = x; if (x < Pivot) { Function Y = B0+B1*(X-Pivot); } else { Function Y = B0+B2*(X-Pivot); } The PIECE.NLR command file contains an example of an IF statement. Chapter 3. Nonlin Statements 33 3.26 WHILE Statement The WHILE statement loops until the controlling expression becomes false (0) or a BREAK statement is executed within the loop. The form of the WHILE statement is: WHILE (expression) { << controlled statements >> } Each time around the loop the expression is evaluated. If it is true (non zero) the controlled statements are executed and then the process repeats until the expression becomes false. If a BREAK statement is executed within the loop, execution of the loop terminates and control is transferred to the first statement beyond the end of the loop. If a CONTINUE statement is executed in the loop, control is transferred to the conditional test at the top of the loop. The following is an example of a WHILE statement: while (x < 5) { x = x + xmove; y = y + ymove; } 3.27 DO Statement The DO statement is very similar to the WHILE statement except the control expression is evaluated at the end of the loop rather than the beginning. This causes the loop always to be executed at least once. The form of the DO statement is: DO { << controlled statements >> WHILE (expression); For each iteration of the loop the controlled statements are executed and then the conditional expression is evaluated. If it is true (non-zero) control transfers to the first controlled statement at the top of the loop. A BREAK statement may be used to terminate the loop before the conditional expression is evaluated. A CONTINUE statement can be used to cause control to be transferred from within the loop to the point where the conditional expression is evaluated. The following is an example of a DO statement: do { x += xstep; y += ystep; } while (x < limit); Chapter 3. Nonlin Statements 34 3.28 FOR Statement The FOR statement is a looping control statement similar to the WHILE statement; however, the FOR statement also allows you to specify initialization expressions that are executed once at the beginning of the loop, and loop-end expressions that are executed at the end of each loop cycle. The form of the FOR statement is: FOR (expression1; expression2; expression3) statement; Execution of a FOR statement proceeds as follows: 1. Evaluate expression1. Typically this expression will include assignment operators ("=") to set initial values for loop variables. If you need more than one initial expression, specify them as a list separated by commas. 2. Evaluate expression2. If its value is false (0) terminate the FOR statement and transfer control to the statement that follows the controlled statement. If expression2 is true, proceed to the next step. 3. Execute the controlled statement. If more than one statement is to be controlled, enclose them with brace characters ("{" "}"). 4. Evaluate expression3. This expression will typically contain operators such as "++", "+=", "--", or "-=" to modify the value of a loop variable. 5. Transfer control to step 2, where expression2 is once again evaluated. The following is an example of a FOR statement: for (time=starttime; time<endtime; time+=timestep) { << controlled statements >> } 3.29 BREAK Statement The BREAK statement can be used in FOR, WHILE, and DO loops to terminate the loop and cause control to transfer to the statement beyond the end of the loop. The following is an example of a BREAK statement: time = 0; x = 0; while (time < endtime) { x += delta * xspeed; if (x > 10) break; } Chapter 3. Nonlin Statements 35 3.30 CONTINUE Statement The CONTINUE statement can be used in FOR, WHILE, and DO loops to terminate the current iteration and begin the next one. When CONTINUE is executed in a WHILE or DO statement, control is transferred to the point in the loop where the loop control expression is evaluated. When CONTINUE is executed in a FOR statement, control is transferred to the bottom of the loop where expression3 is evaluated (which normally augments the values of the loop variables for the next iteration). The form of the CONTINUE statement is: continue; 3.31 STOP Statement The STOP statement terminates the calculations for the current iteration. The last value of the independent variable (as specified with a FUNCTION statement) is used as the calculated value of the function. An implicit stop occurs if you "fall through" the last executable statement. The form of the STOP statement is: stop; 3.32 DATA DATA ["file"]; (required) -- Specifies the name of the file containing the data records, or introduces the data records which follow the statement. If a file name is specified on the DATA statement, the file is opened, its data records are read, and the regression analysis is performed. If a file name is specified without an extension, ".DAT" is used by default. Note that if you specify a file name it must be enclosed in quote marks. If no file name is specified on the DATA statement, the data records must immediately follow the DATA statement in the command file. Each data record must contain at least as many data values as the number of variables specified on the VARIABLES statement(s). The order of the variables as specified on the VARIABLES statement must match the order of the values in each observation. Any data values beyond those required for the specified variables are ignored. Each observation must begin on a new line. The data values must be separated by one or more spaces and/or a comma. You may place a comment on the end of a data record by beginning the comment with "//". Data values may contain decimal points and may be expressed in exponential notation (i.e., n.nnnnEppp). The DATA statement must be the last statement in the command file. If no file name is specified on the DATA statement, the data records must immediately follow the DATA statement in the command Chapter 3. Nonlin Statements 36 file. The following is an example of a complete command file including data records: Variables age,miles,value; Parameters base,depage,depmiles; Function value = base + depage*age + depmiles*miles; Data; 2 10000 13000 4 42000 9000 1 7000 17000 6 52000 6000 5 48000 8000 If the data records had been placed in a separate file named CAR.DAT, the statements would read as follows: Variables age,miles,value; Parameters base,depage,depmiles; Function value = base + depage*age + depmiles*miles; Data "car.dat"; Chapter 4 Understanding The Results 4.1 Descriptive Statistics for Variables Nonlin prints a variety of statistics at the end of each analysis. For each variable, Nonlin lists the minimum value, the maximum value, the mean value, and the standard deviation. You should confirm that these values are within the ranges you expect. 4.2 Parameter Estimates For each parameter, Nonlin displays the initial parameter estimate (which you specified on the PARAMETER statement, or 1 by default), the final (maximum likelihood) estimate, the standard error of the estimated parameter value, the "t" statistic comparing the estimated parameter value with zero, and the significance of the t statistic. Nine significant digits are displayed for the parameter estimates. If you need to determine the parameters to greater precision, use the POUTPUT statement. The final estimate parameter values are the results of the analysis. By substituting these values in the equation you specified to be fitted to the data, you will have a function that can be used to predict the value of the dependent variable based on a set of values for the independent variables. For example, if the equation being fitted is y = p0 + p1*x and the final estimates are 1.5 for p0 and 3 for p1, then the equation y = 1.5 + 3*x is the best equation of this form that will predict the value of y based on the value of x. 4.3 t Statistic The "t" statistic is computed by dividing the estimated value of the parameter by its standard error. This statistic is a measure of the likelihood that the actual value of the parameter is not zero. The larger the absolute value of t, the less likely that the actual value of the parameter could be zero. 37 Chapter 4. Understanding The Results 38 4.4 Prob(t) The "Prob(t)" value is the probability of obtaining the estimated value of the parameter if the actual parameter value is zero. The smaller the value of Prob(t), the more significant the parameter and the less likely that the actual parameter value is zero. For example, assume the estimated value of a parameter is 1.0 and its standard error is 0.7. Then the t value would be 1.43 (1.0/0.7). If the computed Prob(t) value was 0.05 then this indicates that there is only a 0.05 (5%) chance that the actual value of the parameter could be zero. If Prob(t) was 0.001 this indicates there is only 1 chance in 1000 that the parameter could be zero. If Prob(t) was 0.92 this indicates that there is a 92% probability that the actual value of the parameter could be zero; this implies that the term of the regression equation containing the parameter can be eliminated without significantly affecting the accuracy of the regression. One thing that can cause Prob(t) to be 1.00 (or near 1.00) is having redundant parameters. If at the end of an analysis several parameters have Prob(t) values of 1.00, check the function carefully to see if one or more of the parameters can be removed. Also try using a DOUBLE statement to set one or more of the parameters to a reasonable fixed value; if the other parameters suddenly become significant (i.e., Prob(t) much less than 1.00) then the parameters are mutually dependent and one or more should be removed. See Section 6.2 for more information about mutually dependent parameters. The t statistic probability is computed using a two-sided test. The CONFIDENCE statement can be used to cause Nonlin to print confidence intervals for parameter values. The SQUARE.NLR example regression includes an extraneous parameter (p0) whose estimated value is much smaller than its standard error; the Prob(t) value is 0.99982 indicating that there is a high probability that the value is zero. 4.5 Final Sum of Squared Deviations In addition to the variable and parameter values, Nonlin displays several statistics that indicate how well the equation fits the data. The "Final sum of squared deviations" is the sum of the squared differences between the actual value of the dependent variable for each observation and the value predicted by the function, using the final parameter estimates. 4.6 Average and Maximum Deviation The "Average deviation" is the average over all observations of the absolute value of the difference between the actual value of the dependent variable and its predicted value. The "Maximum deviation for any observation" is the maximum difference (ignoring sign) between the actual and predicted value of the dependent variable for any observation. Chapter 4. Understanding The Results 39 4.7 Proportion of Variance Explained The "Proportion of variance explained (R^2)" indicates how much better the function predicts the dependent variable than just using the mean value of the dependent variable. This is also known as the "coefficient of multiple determination." It is computed as follows: Suppose that we did not fit an equation to the data and ignored all information about the independent variables in each observation. Then, the best prediction for the dependent variable value for any observation would be the mean value of the dependent variable over all observations. The "variance" is the sum of the squared differences between the mean value and the value of the dependent variable for each observation. Now, if we use our fitted function to predict the value of the dependent variable, rather than using the mean value, a second kind of variance can be computed by taking the sum of the squared difference between the value of the dependent variable predicted by the function and the actual value. Hopefully, the variance computed by using the values predicted by the function is better (i.e., a smaller value) than the variance computed using the mean value. The "Proportion of variance explained" is computed as 1 - (variance using predicted value / variance using mean). If the function perfectly predicts the observed data, the value of this statistic will be 1.00 (100%). If the function does no better a job of predicting the dependent variable than using the mean, the value will be 0.00. 4.8 Adjusted Coefficient of Multiple Determination The "adjusted coefficient of multiple determination (Ra^2)" is an R^2 statistic adjusted for the number of parameters in the equation and the number of data observations. It is a more conservative estimate of the percent of variance explained, especially when the sample size is small compared to the number of parameters. It is computed using the formula: Ra^2 = 1 - (n-1)/(n-p) * (1-R^2) where 'n' is the number of observations, 'p' is the number of parameters, and 'R^2' is the unadjusted coefficient of multiple determination. 4.9 Durbin-Watson Statistic The "Durbin-Watson test for autocorrelation" is a statistic that indicates the likelihood that the deviation (error) values for the regression have a first-order autoregression component. The regression models assume that the error deviations are uncorrelated. In business and economics, many regression applications involve time series data. If a non-periodic function, such as a straight line, is fitted to periodic data the deviations have a periodic form and are positively correlated over time; these deviations are said to be "autocorrelated" or "serially correlated." Chapter 4. Understanding The Results 40 Autocorrelated deviations may also indicate that the form (shape) of the function being fitted is inappropriate for the data values (e.g., a linear equation fitted to quadratic data). If the deviations are autocorrelated, there may be a number of consequences for the computed results: 1) The estimated regression coefficients no longer have the minimum variance property; 2) the mean square error (MSE) may seriously underestimate the variance of the error terms; 3) the computed standard error of the estimated parameter values may underestimate the true standard error, in which case the t values and confidence intervals may be incorrect. Note that if an appropriate periodic function is fitted to periodic data, the deviations from the regression will be uncorrelated because the cycle of the data values is accounted for by the fitted function. Small values of the Durbin-Watson statistic indicate the presence of autocorrelation. Consult significance tables in a good statistics book for exact interpretations; however, a value less than 0.80 usually indicates that autocorrelation is likely. If the Durbin-Watson statistic indicates that the residual values are autocorrelated, it is recommended that you use the RPLOT and/or NPLOT statements to display a plot of the residual values. If the data has a regular, periodic component you can try including a sin term in your function. The TREND.NLR example fits a function with a sin term to data that has a linear growth with a superimposed sin component. With the sin term the function has a residual value of 29.39 and a Durbin-Watson value of 2.001; without the sin term (i.e., fitting only a linear function) the residual value is 119.16 and the Durbin-Watson value is 0.624 indicating strong autocorrelation. The general form of a sin term is amplitude*sin(2*pi*(x-phase)/period) where 'amplitude' is a parameter that determines the magnitude of the sin component, 'period' determines the period of the oscillation, and 'phase' determines the phase relative to the starting value. If you know the period (e.g., 12 for monthly data with an annual cycle) you should specify it rather than having Nonlin attempt to determine it. If an NPLOT statement is used to produce a normal probability plot of the residuals, the correlation between the residuals and their expected values (assuming they are normally distributed) is printed in the listing. If the residuals are normally distributed, the correlation should be close to 1.00. A correlation less than 0.94 suggests that the residuals are not normally distributed. Chapter 4. Understanding The Results 41 4.10 Analysis of Variance Table An "Analysis of Variance" table provides statistics about the overall significance of the model being fitted. 4.11 Correlation Matrix The CORRELATE statement can be used to cause Nonlin to print a correlation matrix. A "correlation coefficient" is a value that indicates whether there is a linear relationship between two variables. The absolute value of the correlation coeffecient will be in the range 0 to 1. A value of 0 indicates that there is no relationship whereas a value of 1 indicates that there is a perfect correlation and the two variables vary together. The sign of the correlation coefficient will be negative if there is an inverse relationship between the variables (i.e., as one increases the other decreases). For example, consider a study measuring the height and weight of a group of individuals. The correlation coefficient between height and weight will likely have a positive value somewhat less than one because tall people tend to weight more than short people. A study comparing number of cigarettes smoked with age at death will probably have a negative correlation value. A correlation matrix shows the correlation between each pair of variables. The diagonal of the matrix has values of 1.00 because a variable always has a perfect correlation with itself. The matrix is symmetric about the diagonal because X correlated with Y is the same as Y correlated with X. Problems occur in regression analysis when a function is specified that has multiple independent variables that are highly correlated. The common interpretation of the computed regression parameters as measuring the change in the expected value of the dependent variable when the corresponding independent variable is varied while all other independent variables are held constant is not fully applicable when a high degree of correlation exists. This is due to the fact that with highly correlated independent variables it is difficult to attribute changes in the dependent variable to one of the independent variables rather than another. The following are effects of fitting a function with high correlated independent variables: 1. Large changes in the estimated regression parameters may occur when a variable is added or deleted, or when an observation is added or deleted. 2. Individual tests on the regression parameters may show the parameters to be nonsignificant. 3. Regression parameters may have the oppsite algebraic sign than expected from theoretical or practical considerations. Chapter 4. Understanding The Results 42 4. The conficence intervals for important regression parameters may be be much wider than would otherwise be the case. The solution to these problems may be to select the most significent of the correlated variables and use only it in the function. Note: the correlation coefficients indicate the degree of linear association between variables. Variables may be highly related in a nonlinear fashion and still have a correlation coefficent near 0. Chapter 5 Theory of Operation 5.1 Minimization Algorithm Nonlin uses a model/trust-region technique along with an adaptive choice of the model Hessian. The algorithm is essentially a combination of Gauss-Newton and Levenberg-Marquardt methods; however, the adaptive algorithm often works much better than either of these methods alone. The basis for the minimization technique used by Nonlin is to compute the sum of the squared residuals for one set of parameter values and then slightly alter each parameter value and recompute the sum of squared residuals to see how the parameter value change affects the sum of the squared residuals. By dividing the difference between the original and new sum of squared residual values by the amount the parameter was altered, Nonlin is able to determine the approximate partial derivative with respect to the parameter. This partial derivative is used by Nonlin to decide how to alter the value of the parameter for the next iteration. If the function being modeled is well behaved, and the starting value for the parameter is not too far from the optimum value, the procedure will eventually converge to the best estimate for the parameter. This procedure is carried out simultaneously for all parameters and is, in fact, a minimization problem in n-dimensional space, where 'n' is the number of parameters. For a much more detailed explanation of the regression algorithm used by Nonlin see ACM Transactions on Mathematical Software 7,3 (Sept. 1981) "Dennis, J.E., Gay, D.M., and Welsch, R.E. -- An adaptive nonlinear least-squares algorithm." 5.2 Convergence Criterion Nonlin has several convergence criteria that stop the iterative minimization procedure. The TOLERANCE statement can be used to alter the convergence tolerance value. Two internal variables are used to determine when convergence has occurred. RFCTOL has a default value of 1E-10 and can be altered by use of the TOLERANCE statement. AFCTOL has a default value of 1E-20 and is only altered by the TOLERANCE statement if the value specified is less than the default value. In the discussion which 43 Chapter 5. Theory of Operation 44 follows the "function value" is half the sum of the squared residuals computed using the current parameter estimates. "Relative function convergence" is reported if the predicted maximum possible function reduction is at most RFCTOL*ABS(F0) where F0 is the function value at the start of the current iteration, and if the last step attempted achieved no more than twice the predicted function decrease. "Absolute function convergence" is reported if the function value is less than AFCTOL. Chapter 6 Hints for Nonlin Use 6.1 Convergence Failures One of the potential problems that confronts any nonlinear minimization procedure is non-convergence. Non-convergence is usually not a problem for regressions using a linear model, but becomes a more serious consideration when using complicated nonlinear functions; increasing the number of parameters aggravates the problem. Non-convergence can occur in two ways: the solution may diverge or it may converge to the wrong solution -- a local minimum rather than the global minimum. Periodic functions, such as sin, and cos, are particularly prone to convergence problems. For example, consider a nonlinear regression performed with the function: y = offset + amplitude * sin(frequency * x) where x and y are variables, and offset, amplitude, and frequency are the parameters whose values are to be determined. If the starting value for frequency is not reasonably close to the correct value, the solution may converge to a harmonic (multiple) or subharmonic (fundamental) value of the frequency. A command file named SINE.NLR is supplied with the statements and data to perform this analysis. The SWEEP statement can be very useful in cases like the sine example. In the SINE.NLR example analysis, the actual value of the frequency is 3; the function converges to the correct solution if the starting value is in the range 2.6 to 3.3. However, this example is quite insensitive to the starting value of the amplitude parameter. With an actual value of 2, the correct solution is found with starting values from 1 through 10000. Similarly, the offset parameter, which had an actual value of 10, was successfully determined with starting values ranging from 1 to over 50000. Another example which is sensitive to a parameter starting value is POWER.NLR which attempts to determine the values of the parameters p0, p1, and p2 for the function y = p0 + p1*x^p2 45 Chapter 6. Hints for Nonlin Use 46 (where "x^p2" means x raised to the p2 power). The actual value of p2 in the example data is 2; the solution converges correctly if the starting value of p2 is in the range 1.8 to 3.8. As with the other example, the solution is relatively insensitive to the starting values of p0 and p1. 6.2 Singular Matrix Problems Another possible problem is that the analysis may stop with the message "Singular convergence. Mutually dependent parameters?". This is usually due to one of two things: (1) a redundant parameter that is co-dependent with another parameter, or (2) a situation where the value of one parameter "blocks" the effect of other parameters. As an example of a redundant parameter, consider the function y = p0 + p1*p2*x This is a simple linear equation except there are two parameters, p1, and p2, which are both factors to the variable x. It should be clear that there is no unique solution to this problem since any value of p1 is possible if the right value of p2 is chosen. Similarly, the function y = p0 + p1 + p2*x has no unique solution since either p0 or p1 is redundant. Similarly, in the equation y = p0 + p1*exp(x+p2) either p1 or p2 is redundant. The second type of singular matrix problem can be illustrated by the function y = p0 + p1*x^p2 If, during the solution process, p1 takes on the value 0, then varying the value of p2 has no effect on the equation and Nonlin cannot figure out which way to change the value of p2 to move toward convergence. The solution to this problem is to assign a starting value that is not zero to p1, and use the CONSTRAIN statement to force p1 to remain non-zero. 6.3 Performance Issues Nonlin is carefully programmed and compiled with an optimizing compiler for maximum performance. However, Nonlin is a real "number cruncher," and the nonlinear regression algorithm is mathematically very elaborate. During each iteration, Nonlin computes gradients, Jacobians, Hessians, and eigenvalues, and performs QR and Cholesky matrix decompositions. All calculations are carried out using double precision (64 bit) floating point. Chapter 6. Hints for Nonlin Use 47 Nonlin does not require an 80x87 numeric coprocessor, but its performance is greatly enhanced if one is present. In fact, an 8088 CPU with an 8087 numeric coprocessor can perform regression analyses faster than a 20 MHz 80386 that does not have a coprocessor. If you have an 8088 without a coprocessor, be patient -- Nonlin is probably giving it the workout of its life. Very long running times can result if you use the SWEEP statement with many starting values. The problem is compounded if you have multiple SWEEP statements. If you use the SWEEP statement to try a large number of starting parameter values, you can save time by using the ITERATIONS statement to specify a small number of iterations (such as 5) during the initial attempt to find a solution. Once a feasible set of starting parameter values has been determined, remove the SWEEP statement, specify the starting values on the PARAMETERS statement, increase the number of iterations, and rerun the analysis to get the final result. 6.4 Program Limits The following is a summary of the Nonlin program limitations: Maximum number of variables = 25 Maximum number of parameters = 25 Maximum length of variable or parameter names = 10 The maximum number of data observations that Nonlin can handle depends on the number of parameters as shown by the table that follows: # Parameters Max Observations 1 2019 2 1611 3 1339 4 1144 5 997 6 883 7 791 8 715 9 652 10 599 Chapter 7 Example Analyses A number of example regression analysis files are provided with your Nonlin distribution. All of the example command files have the extension ".NLR". Some of the important ones are described below, others contain comment lines that explain what they do. LINEAR.NLR -- Simple linear regression with plotted function and data. QUAD.NLR -- Fit a quadratic equation. Plot the function and the data. ASYMPTOT.NLR -- Fit an asymptotic function Y = 12 - 10/X. AIDS.NLR -- A logistic curve is a growth curve used to model functions which increase gradually at first, more rapidly in the middle growth period, and slowly at the end, leveling off at a maximum value after some period of time. This type of curve is frequently used to model biological growth patterns where there is an initial exponential growth period followed by a leveling off as more of the population is infected or as the food supply or some other factor limits further growth. The form of the symmetric logistic growth function is: y = k / (1 + exp(a + b*x)) where 'k', 'a', and 'b' are parameters that shape and scale the function. The value of 'b' is negative. The AIDS.NLR example fits a logistic curve to the number of new cases of AIDS reported in the United States during the period 1981 through 1992. The computed function fits the data remarkably well showing that the AIDS infection rate is following a classic logistic curve and should level off at about 47,500 new cases per year (in the United States). The DOMAIN option on the PLOT statement causes Nonlin to extrapolate the plot of the function through 1995. F33.NLR -- Multivariate linear regression (multiple regression). Calculate the value of a used Beech F33 Bonanza airplane using a linear model based on its age, the number of hours on its airframe, and the number of hours on its engine. The t value and Prob(t) indicate that the number of hours on the engine ('Engdep' parameter) is not significant to the regression 48 Chapter 7. Example Analyses 49 model; the other parameters are significant but airframe hours is less significant than the base price and age of the plane. F33YEAR.NLR -- Similar to F33.NLR except the price of the Bonanza is calculated based on a linear function of only the age. F33EXP.NLR -- Similar to F33YEAR.NLR except a negative exponential function is used rather than a linear function. Compare the fit of this model with that of the F33YEAR.NLR example. SINE.NLR -- Fit an equation involving a sin function. The SWEEP statement is used to find a starting point that will converge. TREND.NLR -- Fit a function that has a linear growth term and a periodic component involving a sin term. See Section 4.9 for additional information about this example. SQUARE.NLR -- Fit a sine series to a square wave. Note in this example that the 'p0' parameter, which represents the constant term of the equation, has an estimated value of 9.22715E-006 (very nearly zero) and a standard error of 0.0398754. This yields a t value of nearly zero and Prob(t) of 0.99982 which means that there is a 99.982% chance that the actual value of p0 may be zero (it is in fact zero). This illustrates how you can use the t value and Prob(t) to identify extraneous parameters. COOLING.NLR -- Fit an equation involving an exponential function. If a heated object is allowed to cool, the rate of cooling at any instant is proportional to the difference between the object's temperature and the ambient (room) temperature. In other words, an object cools faster at first, while it is hot, and the rate of cooling slows down as the temperature of the object approaches the ambient temperature. The function that relates the object's temperature to time is: Temperature = Roomtemp+InitTemp*exp(-Coolrate*Time) Where InitTemp is the number of degrees above room temperature at time 0, and Coolrate is a factor that depends on the mass of the object, how well it is insulated, etc. The exp function is the value of e (2.7182818...) raised to a power. The COOLING.NLR example determines the parameters InitTemp and Coolrate to fit an equation of this form to some data the author collected. BOIL.NLR -- The boiling point of water decreases as the pressure in the vessel containing the water decreases. "Clapeyron's equation" shows that the boiling point is related to pressure according to the following function: Temperature = b / log(Pressure/a) - 459.7 Where 'Temperature' is in degrees Fahrenheit (the 459.7 constant converts degrees Fahrenheit to degrees Rankine -- Chapter 7. Example Analyses 50 relative to absolute zero), 'Pressure' is the pressure in the vessel in pounds per square inch, and 'a' and 'b' are parameters whose values are to be determined. The data for this example was collected by the author's son for a science project. MAGNET.NLR -- Fit a function involving an arc tangent and a variable to the third power. This is an interesting physics problem. If a magnet is placed due east of a compass, the deflection of the compass needle from north is equal to the arc tangent of the ratio of the strength of the magnet's field relative to the earth's magnetic field. The strength of the magnet's field at the compass is inversely proportional to the cube of the distance from the magnet to the compass. Thus, the function relating these terms is Deflection = deg(atan(Strength / Distance ^ 3)) The deg function converts an angle in radians to degrees. In the example, Deflection and Distance are the variables, and the value of the Strength parameter is determined. DIODE.NLR -- The current through a diode increases sharply as the voltage across the diode is increased. An equation that approximates the current flow as a function of the voltage is: I = exp(b*(V-c)) where 'I' is the current, 'V' is the voltage, and 'b', and 'c' are parameters that are to be estimated by the nonlinear regression. AVLTIME.NLR -- An AVL tree is a balanced binary tree used to store information in a computer's memory. Because the entries in an AVL tree are kept in sorted order, and the tree is kept in a balanced form, it is possible to rapidly find any entry in the tree. The time required to create an AVL tree with N entries is approximately equal to: Time = a + b*N*log2(N) where 'a' is a constant term equal to the overhead involved in starting and completing a tree creation, and 'b' is a growth coefficient that depends on the speed of the computer. The log2(N) function is the log base 2 of N (the number of entries). The AVLTIME.NLR example fits an equation to a data set that relates the time in seconds required to create an AVL tree with the number of entries in the tree. PIECE.NLR -- Piecewise linear function. Fit a function consisting of two linear pieces that bend at X=5. When X is less than 5, the slope of the function is B1. When X is greater than or equal to 5, the slope is B2. B0 is the Y value of the function at X=5 (i.e., at the pivot point). The IF statement is used to control which function model is used depending on Chapter 7. Example Analyses 51 whether the value of the dependent variable is greater than or less than the pivot point. Chapter 8 Special Applications 8.1 Omitted Dependent Variable There is a class of nonlinear regression problems that can be best expressed by omitting the dependent variable (i.e., the variable on the left of the equal sign). To understand what this means first consider the normal regression case with a dependent variable. For each observation the function is evaluated and the computed value is subtracted from the corresponding value of the dependent variable for that observation. This residual value is then squared and added to the other squared residual values. The goal is to minimize the total sum of squared residuals. In the case where the dependent variable is omitted, the function is computed for each observation and the value of the function is squared (i.e., it is treated as the residual) and added to the other squared values. The goal is to minimize the sum of the squared values of the function. Thus, for a perfect fit the computed value of the function for every observation would be zero. To perform this type of analysis omit the dependent variable and equal sign from the left side of the function specification. As an example of this type of analysis consider the problem of fitting a circle to a set of points that form a roughly circular pattern (i.e., a "circular regression"). Our goal is to determine the center point of the circle (Xc,Yc) and the radius (R) which will make the circle best fit the points so that the sum of the squared distances between the points and the perimeter of the circle is minimized (the points are as close to the perimeter of the circle as possible). For this problem we have three parameters whose values are to be determined: Xc, Yc, and R. There will be one data observation for each point to which the circle is being fitted. For each point there are two variables, Xp and Yp, the X and Y coordinates of the point's position. Since our goal is to minimize the sum of the squared distances from the points to the perimeter of the circle, we need a function that will compute this distance for each point. If the center of the circle is at (Xc,Yc) and the position of a point is (Xp,Yp) then, from the theorem of Pythagoras, we know the distance from the center to the point is 52 Chapter 8. Special Applications 53 sqrt((Xp-Xc)^2 + (Yp-Yc)^2) But we are interested in the distance from the perimeter to the point. Since the radius of the circle is R, the distance from the perimeter to the point (along a straight line from the center to the point) is sqrt((Xp-Xc)^2 + (Yp-Yc)^2) - R That is, the distance from the perimeter to the point is equal to the distance from the center to the point less the distance from the center to the perimeter (the radius). The distance will be positive or negative depending on whether the point is outside or inside the circle but this does not matter since the value is squared as part of the minimization process. The Nonlin statements for this analysis are as follows: Variables Xp,Yp; Parameters Xc,Yc,R; Function sqrt((Xp-Xc)^2 + (Yp-Yc)^2) - R; Note that there is no dependent variable or equal sign to the left of the function. Nonlin will determine the values of the parameters Yp, Yc, and R such that the sum of the squared values of the function (i.e., the sum of the squared distances) is minimized. The CIRCLE.NLR file contains a full example of this analysis. As a second example similar to the first one, consider a town that is trying to decide where to place a fire station. The location should be central such that the sum of the squared distances from the station to each house is minimized. Nonlin can be used to determine the coordinates of the station (Xc,Yc) given a set of coordinates for each house location (Xh,Yh) by using a slightly simpler function than the first example: Function sqrt((Xh-Xc)^2 + (Yh-Yc)^2); 8.2 Root Finding and Expression Minimization Although it is designed for nonlinear regression analysis, Nonlin can also be used to find the root (zero point) or minimum absolute value of a nonlinear expression. To use Nonlin in this fashion follow these steps: . Do not use any VARIABLE statements. . Use PARAMETER statements to specify the names and optional starting values for the parameters whose values are to be determined as the roots or minimum value of the expression. . Use the FUNCTION statement to specify the expression whose roots or minimum value is to be found; do NOT specify a Chapter 8. Special Applications 54 dependent variable and equal sign -- specify only the expression that is to be minimized. . Do not include any data records after the DATA statement; it simply signals the end of the command file and causes the analysis to begin. The following is an example command file to find the root of the expression sin(x)-log(x): Parameter x; Function sin(x) - log(x); Data; Notice that the "variable" in the expression, X, is not declared to be a variable but rather a parameter. This example is included in the file MINSL.NLR which you can run. For this type of analysis, Nonlin determines the values of the parameters that minimize the absolute value of the expression. If the expression has a zero value (i.e., a root), that value is found since that is the smallest possible absolute value. If the expression does not have a zero point, Nonlin determines the values of the parameters that produce the smallest absolute value of the expression. For example, the expression 2*X^2-3*X+10 does not have a root but reaches a minimum value of 8.875 when X is 0.75. The MINPAROB.NLR command file contains this example. There are a number of cautions that you should keep in mind when using Nonlin to find roots or minimum values: . Nonlin will find only one root or minimum value per analysis. For example, the expression 9-X^2 has two roots: -3 and +3. Nonlin will find one of the roots; which one it finds depends on the starting value specified for X. . Nonlin will find only real roots, not complex. . If the expression contains a local minimum, Nonlin may find it rather than the global minimum or root. Of course, if you are looking for a local minimum in a certain region this could be considered a feature. For example, the expression 0.5*X^3+5*(X-2)^2+15 has a local minimum at X=1.61 and a root at X=-13.38. If the starting value of X is less than -8.3 the root is found; if the starting value is greater than -8.3, the local minimum is found. If the expression contains only a single variable, use the Mathplot program to graphically display the expression and determine a good starting value for the variable (see page 59 for additional information about Mathplot). The SWEEP statement can also be used to try multiple starting values when searching for a global minimum. Chapter 8. Special Applications 55 8.2.1 Function Minimization Examples MINFALL.NLR -- The time taken for an object to slide down a frictionless guide from position (0,h) to another position (d,0) (i.e., falling through a distance 'h' while moving horizontally a distance 'd') depends on the path that the object takes as it follows the guide. It turns out that the path that minimizes the descent time is not a straight line from (0,h) to (d,0) but rather a curve called a brachistochrone with a steeper slope near the beginning, that gives the object a chance to accelerate quickly, and then a shallower slope further on. Finding the shape of this curve is a classic problem in the branch of mathematics called the Calculus of Variations. The MINFALL example solves a simpler case of this problem: the object slides along a straight guide from (0,1000) to an intermediate position (px,py), and then along another straight guide from (px,py) to (1000,0). What point, (px,py), minimizes the descent time? Note concerning the answer: The fall time for the object if it follows a straight guide from (0,1000) to (1000,0) is 2.0203 seconds; the fall time if it follows the two straight segments found by MINFALL is 1.8748; the fall time if it follows the ideal curved brachistochrone is 1.8590. The speed of the object at the end of the fall is the same regardless of the path taken due to conservation of energy. MINFUEL.NLR -- A lunar lander is hovering above the surface of the moon looking for a suitable landing site. Available fuel is critical and the desired site is 200 meters away. How long should the horizontal thruster be fired to start and stop the motion over the ground? The vertical thruster must be used continuously to keep the lander from being pulled to the surface. If too little horizontal thrust is used the spacecraft will move slowly and much fuel will be consumed by the vertical thruster counterbalancing the downward gravitational pull while hovering over the surface. On the other hand, if the horizontal thruster is fired for a long time, the spacecraft will move quickly (minimizing the hovering time) but excessive fuel will be used during the horizontal acceleration and deceleration. MINFUEL.NLR determines how long the thruster should be fired during the start and stop accelerations such that the total fuel consumption (start thrust + stop thrust + hover) is minimized. Chapter 9 Acknowledgement and Use of Nonlin 9.1 Acknowledgement The nonlinear regression algorithm used by Nonlin was published in ACM Transactions on Mathematical Software 7,3 (Sept. 1981) "Dennis, J.E., Gay, D.M., and Welsch, R.E. -- An adaptive nonlinear least-squares algorithm." 9.2 Use and Distribution of Nonlin There are two versions of the Nonlin program: shareware and registered. You are welcome to make copies of the shareware version of Nonlin and pass them on to friends or post this program on bulletin boards or distribute it via disk catalog services, CD ROMS, or other means provided the entire Nonlin distribution is included in its original, unmodified form. A distribution fee may be charged for the cost of the diskette, shipping and handling. Vendors are encouraged to contact the author to get the most recent version of Nonlin. As a shareware product, you are granted a no-cost, trial period of 30 days during which you may evaluate Nonlin. If you find Nonlin to be useful, educational, and/or entertaining, and continue to use it beyond the 30 day trial period, you are required to compensate the author by sending the registration form printed at the end of this document (and in REGISTER.DOC) with the appropriate registration fee to help cover the development and support of Nonlin. In return for registering, you will be authorized to continue using Nonlin beyond the trial period and you will receive a registered version of the program, a bound typeset manual, and three months of support via telephone, mail, or CompuServe. Your registration fee will be refunded if you encounter a serious bug that cannot be corrected. The registered version of Nonlin omits the shareware notification screen at the start of the run and does not require you to press a key to proceed with the analysis. The registered version also includes the ability to print plots on HP LaserJet printers. The registered version of Nonlin is NOT shareware and may not be redistributed or used on more than one computer system. 56 Chapter 9. Acknowledgement and Use of Nonlin 57 The author frequently improves Nonlin and it is likely that the version you have is not the most recent version. Note, the cost of registering Nonlin is insignificant compared with what you would have to pay to purchase a commercial statistical package with an equivalent regression capability. 9.3 Association of Shareware Professionals This program is produced by a member of the Association of Shareware Professionals (ASP). ASP wants to make sure that the shareware principle works for you. If you are unable to resolve a shareware-related problem with an ASP member by contacting the member directly, ASP may be able to help. The ASP Ombudsman can help you resolve a dispute or problem with an ASP member, but does not provide technical support for members' products. Please write to the ASP Ombudsman at 545 Grover Road, Muskegon, MI 49442 or send a CompuServe message via CompuServe Mail to ASP Ombudsman 7007,3536. You are welcome to contact the author: Phillip H. Sherrod 4410 Gerald Place Nashville, TN 37205-3806 USA 615-292-2881 (evenings) CompuServe: 76166,2640 Internet: 76166.2640@compuserve.com 9.4 Copyright Notice Both the Nonlin program and documentation are copyright (c) 1992-1994 by Phillip H. Sherrod. You are not authorized to modify the program. "Nonlin" is a trademark. 9.5 Disclaimer This software and documentation are provided on an "as is" basis. This program may contain "bugs" and inaccuracies, and its results should not be assumed to be correct unless they are verified by independent means. Phillip H. Sherrod disclaims all warranties relating to this software, whether expressed or implied, including but not limited to any implied warranties of merchantability or fitness for a particular purpose. Neither Phillip H. Sherrod nor anyone else who has been involved in the creation, production, or delivery of this software shall be liable for any indirect, consequential, or incidental damages arising out of the use or inability to use such software, even if Phillip H. Sherrod has been advised of the possibility of such damages or claims. The person using the software bears all risk as to the quality and performance of the software. This agreement shall be governed by the laws of the State of Tennessee and shall inure to the benefit of Phillip H. Sherrod and any successors, administrators, heirs and assigns. Any action or proceeding brought by either party against the other arising Chapter 9. Acknowledgement and Use of Nonlin 58 out of or related to this agreement shall be brought only in a state or federal court of competent jurisdiction located in Davidson County, Tennessee. The parties hereby consent to in personam jurisdiction of said courts. Chapter 10 Other Software 10.1 Mathplot -- Mathematical Function Plotting Program If you like Nonlin, you should check out the Mathplot program by the same author. Mathplot allows you to specify complicated mathematical functions using ordinary algebraic expressions and immediately plot them. Four types of functions may be specified: cartesian (y=f(x)); parametric cartesian (y=f(t) and x=f(t)); polar (radius=f(angle)); and parametric polar (radius=f(t) and angle=f(t)). Up to four functions may be plotted simultaneously. Scaling is automatic. Options are available to control axis display and labeling as well as grid lines. Hard copy output may be generated as well as screen display. Mathplot is an ideal tool for engineers, scientists, math and science teachers, and anyone else who needs to quickly visualize mathematical functions. 10.2 TSX-32 -- Multi-User Operating System If you have a need for a multi-user, multi-tasking operating system, you should look into TSX-32. TSX-32 is a full-featured, high performance, multi-user operating system for the 386 and 486 that provides both 32-bit and 16-bit program support. With facilities such as multitasking and multisessions, networking, virtual memory, X-Windows, background batch queues, data caching, file access control, real-time, and dial-in support, TSX-32 provides a solid environment for a wide range of applications. A two user, shareware version of TSX-32 called TSX-Lite is also available. TSX-32 is not a limited, 16-bit, multi-DOS add-on. Rather, it is a complete 32-bit operating system which makes full use of the hardware's potential, including protected mode execution, virtual memory, and demand paging. TSX-32 sites range from small systems with 2-3 terminals to large installations with more than 150 terminals on a single 486. In addition to supporting most popular 16-bit DOS programs, TSX-32 also provides a 32-bit "flat" address space with both Phar Lap and DPMI compatible modes of execution. 59 Chapter 10. Other Software 60 Since the DOS file structure is standard for TSX-32, you can directly read and write DOS disks. And, you can run DOS part of the time and TSX-32 the rest of the time on the same computer. TSX-32 allows each user to control up to 10 sessions. Programs can also "fork" subtasks for multi-threaded applications. The patented Adaptive Scheduling Algorithm provides consistently good response time under varying conditions. The TSX-32 network option provides industry standard TCP/IP networking through Ethernet and serial lines. Programs can access files on remote machines as easily as on their own machine. The SET HOST command allows a user on one machine to log onto another computer in the network. FTP, Telnet, and NFS are available for interoperability with other systems. System requirements: 386 or 486 system, 4MB memory, 12MB of free disk space (Stacker and DoubleSpace are not supported). TSX-32 is, quite simply, the best and most powerful operating system available for the 386 and 486. For additional information contact: S&H Computer Systems, Inc. 1027 17th Avenue South Nashville, TN 37212 USA 615-327-3670 (voice) 615-321-5929 (fax) CompuServe: 71333,27 Internet: 71333.27@compuserve.com 10.3 SIMSTAT -- Interactive Statistics Program If you need a general-purpose statistical package, or would like a menu-oriented, mouse-aware interface to Nonlin, I suggest you try the SIMSTAT program written by Normand Peladeau. SIMSTAT is a menu driven statistical program that provides many basic descriptive and comparative statistics and includes a "bridge" module to allow it to function as a "front end" and data editor for Nonlin. The shareware version of SIMSTAT is available from the IBMAPP forum of CompuServe and from many BBS, or you can contact the author at the address below. SIMSTAT version 3.0 or later is required for use with Nonlin. You must also have the "bridge" module called SIM2NL. SIM2NL.ZIP is included on the distribution disk with the registered version of Nonlin. For information about SIMSTAT, contact: Normand Peladeau, Provalis Research, 5000 Adam Street, Montreal, QC H1V 1W5, Canada, Compuserve: [71760,2103], Internet: 71760.2103@compuserve.com. =============================================================== Software Order Form =============================================================== Name ______________________________________________________ Address ___________________________________________________ City _______________________ State _______ Zip ___________ Country ____________________ Telephone ___________________ Internet address (optional) _______________________________ Nonlin version ____________________________________________ Bulletin board where you found Nonlin _____________________ Comments __________________________________________________ Check the box below which indicates your order type: ___ I wish to register Nonlin ($45). ___ I wish to order Mathplot ($20). ___ I wish to register Nonlin and order Mathplot ($60). Add $5 to any amount shown above if the software is being shipped out of the United States. I cannot accept checks from non-US banks. Visa, MasterCard and American Express credit card charges are accepted but a check, money order, or cash is preferred. If you wish to use a credit card specify the billing name, address, card number, and expiration date. In return for registering, you will receive the registered version of the program, a laser-printed, bound copy of the manual, and three months of telephone or CompuServe support. Your registration fee will be refunded if you find a serious bug that cannot be corrected. Distribution disk choice (check one): 3.50" HD (1.4 MB) ______ 5.25" HD (1.2 MB) ______ 5.25" DD (360 KB) ______ Send this form with the amount indicated to the author: Phillip H. Sherrod 4410 Gerald Place Nashville, TN 37205-3806 USA 615-292-2881 (evenings) CompuServe: 76166,2640 Internet: 76166.2640@compuserve.com Index 62 80x87 coprocessor, 47 CONTINUE statement, 33, 35 ABS function, 11 Convergence criterion, 43 Absolute converge, 44 Convergence failures, 45 Acknowledgement, 56 Cooling example, 49 ACOS function, 11 Copyright notice, 57 Adaptive algorithm, 43 CORRELATE statement, 22, 41 AFCTOL value, 43 Correlation matrix, 41 AIDS growth curve, 48 COS function, 11 AND operator, 10 Cosecant function, 11 ANGLETYPE statement, 23 COSH function, 11 Arc cosine function, 11 COT function, 11 Arc sine function, 11 COVARIANCE statement, 22 Arc tangent function, 11 CSC function, 11 Arithmetic operators, 9 CTOP function, 11 Arrays DATA statement, 35 declaration, 19 DEG function, 12 initialization, 19 Degrees for functions, 23 storage order, 19 Degrees to radians, 14 subscripts, 20 Dependent variable, 7 ASIN function, 11 Deviation ASP, 57 average, 38 Assignment operators, 10, 32 definition, 2 Assn. of Shareware Prof., 57 maximum, 38 Asymptotic function example, Diode current example, 50 48 Disclaimer, 57 ATAN function, 11 DO statement, 33 Author address, 57 DOMAIN option, 25, 28, 29 Autoregression test, 30, 39 DOUBLE statement, 19 Average deviation, 38 DPMI support, 59 AVL tree example, 50 Durbin-Watson statistic, 39 Bessel function, 13, 15 EI2 function, 12 Beta function, 11 EIC1 function, 12 BETAI function, 11 EIC2 function, 12 Brachistochrone, 55 EL1 function, 12 BREAK statement, 33, 34 Elliptic integral function, Build-in functions, 11 12 Built-in constant, 11 ERF function, 12 Calculus of variations, 55 Examples, 48 CEIL function, 11 AIDS growth curve, 48 Chebyshev function, 15 asymptotic function, 48 Circular regression, 52 AVL tree, 50 Clapeyron's equation, 49 boiling water, 49 Comma operator, 10 circular regression, 52 Command files, 16 cooling, 49 Commands, 18 diode current, 50 Comments, 16 fire station location, 53 Comparison operators, 10 function minimization, 55 Compass to polar, 11 linear regression, 48 CompuServe, 57, 60 lunar lander, 55 Conditional operator, 10 magnet force, 50 Confidence intervals, 22 minimum time path, 55 CONFIDENCE statement, 22, 38 multivariate, 48 CONSTANT command, 20 negative exponential, 49 Constant variables, 7 piecewise function, 50 CONSTRAIN statement, 20, 46 quadratic equation, 48 Index 63 square wave, 49 Magnet example, 50 SWEEP statement, 49 Marquardt algorithm, 43 EXP function, 12 Mathplot, 59 Exponentiation operator, 9 MAX function, 13 Expression minimization, 53 Maximum values, 37 FAC function, 12 MIN function, 13 Factorial function, 12 Minimization algorithm, 43 Fire station example, 53 Minimization problem, 53 FLOOR function, 12 Model/trust region, 43 FOR statement, 34 Modulo operator, 9 Function minimization, 53 Multi-user operating system, FUNCTION statement, 21 59 Functions, 11 Multiple determination, 39 GAMMA function, 12 Multivariate regression, 48 GAMMAI function, 12 Mutually dependent, 46 GAMMALN function, 12 Natural log function, 13 Gauss-Newton algorithm, 43 Negative exponential, 2, 49 Growth curve, 48 Networking, 60 HAV function, 12 NOECHO statement, 32 Hessian, 43 NONLIN environment variable, Hyperbolic cosine function, 4 11 NONLIN.DOC file, 4 Hyperbolic sine function, 15 NONLIN.EXE file, 4 Hyperbolic tangent function, NONLIN.FON file, 4, 25 15 NONLIN.LJF file, 4 IF statement, 32 NORMAL function, 13 Include files, 16 Normal probability plot, 30 #INCLUDE statement, 16 NOT operator, 10 Incomplete beta function, 11 NPD function, 13 Independent variables, 7 NPLOT statement, 8, 30 Input variables, 6 autocorrelation test, 40 Installing Nonlin, 4 options, 30 INT function, 13 Numeric constants, 10 Internet, 57, 60 Numeric coprocessor, 47 Inverse gamma function, 12 OBS system variable, 7, 24 ITERATIONS statement, 23 Operators J0 function, 13 assignment, 10 J1 function, 13 comma, 10 JN function, 13 comparison, 10 Keywords, 6 conditional, 10 LaserJet printer, 4, 56 logical, 10 resolution, 31 precedence of, 10 Least squares regression, 2 subscript, 10 Levenberg-Marquardt, 43 OR operator, 10 Limits, 47 Order form, 61 Linear regression, 1, 3 OUTPUT statement, 24 example, 48 PARAMETER statement Listing file, 5 function minimization, 53 LOG function, 13 PARAMETERS statement, 18 Log gamma function, 12 PAREA function, 13 LOG10 function, 13 Performance issues, 46 LOG2 function, 13 Periodic data, 40 Logical operators, 10 PI constant, 11 Logistic curve, 48 Piecewise function example, Lunar lander example, 55 50 Index 64 PLOT statement, 7, 24 RPLOT statement, 8, 28 options, 25 autocorrelation test, 40 Plots options, 29 data and function, 25 RTOPA function, 15 normal probability, 30 RTOPD function, 15 residual values, 28 R^2 statistic, 39 resolution, 31 S&H Computer Systems, Inc., scatter, 26 60 width of, 31 Scatter plots, 26 Polar to compass, 14 SEC function, 15 Polar to rectangular, 14 Secant function, 15 Polynomial equation example, SEL function, 15 48 SET command, 4 POUTPUT statement, 24 Sherrod, Phillip H., 57 extra precision output, 37 SIM2NL, 60 Precedence of operators, 10 SIMSTAT, 60 PREDICTED system variable, 6, SIN function, 15, 40 8, 9, 24 Singular matrix problems, 46 PRESOLUTION statement, 31 SINH function, 15 PRINTF function, 13 SPLOT statement, 7, 26 Prob(t) value, 38 options, 26 Program limits, 47 SQRT function, 15 PTOC function, 14 Square wave example, 49 PTORX function, 14 Standard deviation, 37 PTORY function, 14 Standard error function, 12 PULSE function, 14 Statements, 18 Quadratic equation example, STEP function, 15 48 STOP statement, 35 RAD function, 14 Subscript operator, 10 Radian to degree conversion, Subscripts, 20 12 Support of Nonlin, 56 Radians for functions, 23 SWEEP statement, 21 RANDOM function, 14 convergence failure, 45 Ra^2 statistic, 39 example, 49 Real-time, 59 function minimization, 54 Rectangular to polar, 15 performance issues, 47 REGISTER.DOC file, 4 Symbolic constants, 7, 11, 20 Registering Nonlin, 56 System variables, 6 Registration form, 61 T function, 15 Relational operators, 10 t statistic, 37, 38 Relative convergence, 44 TAN function, 15 Remainder operator (modulo), TANH function, 15 9 TCP/IP, 60 Reserved words, 6 Theory of operation, 43 Residual Time series data, 39 RESIDUAL system variable, 6, TITLE statement, 18 8, 24 TOLERANCE statement, 23 Residual values converge criterion, 43 plotting, 28 Transformed function, 3 average, 38 TREND example, 40 definition, 2 TREND.NLR, 49 maximum, 38 Trigonometric functions RFCTOL value, 43 degrees or radians, 23 Root finding, 53 TSX-32, 59 ROUND function, 15 TSX-Lite, 59 Index 65 Use and distribution, 56 VARIABLES statement, 18 Variance-covariance matrix, 22 Vectors see Arrays, 19 Warranty, 57 WHILE statement, 33 WIDTH statement, 31 X windows, 59 Y0 function, 15 Y1 function, 15 YN function, 15