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┌─────────────────────────────────────────────────────────────┐
│ ┌─────────────────────────────────────────────────────────┐ │
│ │ ASCII Documentation File for ... PathProj.EXE │ │
│ │ │ │
│ │ Version 9201 │ │
│ ├─────────────────────────────────────────────────────────┤ │
│ │ │ │
│ │ a softRX SHAREWARE module from │ │
│ │ │ │
│ │ Bob Obenchain, CompuServe User [72007,467] │ │
│ └─────────────────────────────────────────────────────────┘ │
└─────────────────────────────────────────────────────────────┘
softRX is a trademark of The Software Prescriptions Company.
"Personal computing treatments for data analysis infirmities."
Copyright(c) 1989-1993 by Bob Obenchain, Ph.D.
ALL RIGHTS RESERVED
*******************
The author of PathProj, Bob Obenchain, is a member of
the Association of Shareware Professionals <ASP>.
┌───────────────────┐
─────────────────────────┤ Table-of-Contents ├─────────────────────────
└───────────────────┘
softRX User Registration . . . . . . . . . . . . . . . . . . . . . . 1
Introduction to PathProj . . . . . . . . . . . . . . . . . . . . . . 2
Shrinkage Regression Background Information. . . . . . . . . . . . . 3
Computing System Requirements for PathProj . . . . . . . . . . . . . 4
softRX Graphical Design Philosophy . . . . . . . . . . . . . . . . . 4
PathProj File Naming Conventions . . . . . . . . . . . . . . . . . . 5
PathProj Walk-Through. . . . . . . . . . . . . . . . . . . . . . . . 6
TECHNICAL APPENDIX: "Extent" and "Shape" of Shrinkage. . . . . . . .12
PathProj Update History. . . . . . . . . . . . . . . . . . . . . . .14
ASP Hub Network. . . . . . . . . . . . . . . . . . . . . . . . . . .15
┌─────────────────────┐
────────────────────────┤ USER REGISTRATION ├────────────────────────
└─────────────────────┘
PathProj.EXE, its ASCII documentation file [PathProj.DOC], and the
included batch input/output (numerical example) files are user-
supported electronic information that we encourage you to examine and
to share with others, provided you have not modified them in any way.
All softRX SHAREWARE modules are distributed in fully functional form
and will remain so without any usage time limits. softRX systems do
not display annoying registration "reminder" screens or messages. And
no (non-commercial) user who is unable to correspond with softRX in
English is expected to register.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 2 of 15
**************************************
**************************************
Only YOU know whether softRX systems are useful to you.
And, if you do use them, you must register!
**************************************
**************************************
The registration fee for PathProj is $25 for use either by a single
individual or for installation on one single-user-at-a-time system.
Three softRX modules (say, RXridge, RXtraces, and PathProj) may be
registered together for $50. softRX offers MAIL-IN SUPPORT SERVICES
(e.g. answers to questions on the theory, application and/or
computation of statistical "shrinkage" regression techniques) only to
its REGISTERED users.
However, registration fees are mandatory for use of softRX modules in
any commercial or government environment. These users must register
and pay for their copies within 30 days of first actual use or else
their license is withdrawn. Site-License arrangements (of at most $15
per module) may also be made by contacting softRX.
Make checks payable to... Bob Obenchain
and mail them to... 5261 Woodfield Drive North
Carmel, IN 46033-8795, USA
!!! Thank You for Supporting the <<<softRX shareware>>> Concept !!!
┌─────────────────────────────┐
──────────────────┤ INTRODUCTION to PathProj ├──────────────────────
└─────────────────────────────┘
PathProj does NOT perform generalized ridge regression computations.
Rather, PathProj generates interactive graphical displays of the results
from other softRX shareware systems...like RXridge.EXE, RelaxR.EXE, or
the RXridge procedures distributed as source code for the GAUSS,
SAS/IML, and S matrix-language interpreters.
PathProj generates INTERACTIVE graphical displays on IBM-compatible
personal computers in CGA (medium resolution, 320x200 pixel) mode.
PathProj helps users interpret the results of a generalized ridge
regression analysis by displaying orthogonal
PROJECTIONS of either regression coefficient estimates
or predictor (regressor) coordinates
from P-dimensional space down onto a user specified 2-dimensional
subspace, where P = [ number of non-constant predictor variables
in the regression model ] can range from 2 to 20.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 3 of 15
You do not need to read this documentation before trying PathProj out.
There is very little damage you can do by giving "wrong" choices for
menu items and responses to prompts. However, first of all, please
> > > MAKE BACK-UP COPIES < < <
of the distribution files and/or of the original softRX shareware
archives. PathProj does not modify its input files, which are output
files created by RXridge or RelaxR. And PathProj does not write
output files (or make screen prints) unless you first press the "S"
key and then press a second, "special" key.
Each PathProj prompt has a [default] value that you may accept simply
by pressing the ENTER key; PathProj menus require you to press a
single key, where any differences between upper and lower case are
ignored. Unless you are already being prompted for input, you can
view a "help" screen simply by pressing function key F1 or the
SpaceBar. And you can always return to the "Main Menu" by repeated
pressing of the ENTER key.
┌─────────────────────────────────────────────┐
────────────┤ Shrinkage Regression Background Information ├────────────
└─────────────────────────────────────────────┘
RXridge and RXtraces are softRX shareware products that interactively
display results of ridge computations using traditional TRACE plots.
In a trace, P quantities (several estimated coefficients, risks,
shrinkage factors, etc.) are plotted vertically against a horizontal
indicator of the extent of shrinkage (from "no" shrinkage at the least
squares estimates all the way to "total" shrinkage, where all
coefficient estimates for non-constant regressor variables are zero.)
PathProj graphics are quite different from traditional trace displays.
Projection plots focus on two-dimensional views that depict the
shrinkage path as a curve from an initial point (representing least
squares) to the origin, [0,0]. There would be no loss in information
in depicting a trace with only two non-constant regressors as a
projection plot. But projection plots usually convey a user-specified
focussing of attention onto some 2-dimensional linear subspace of a
(much) larger space.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 4 of 15
┌───────────────────────────────┐
────────────────────┤ Computing System Requirements ├──────────────────
└───────────────────────────────┘
PathProj.EXE requires your personal computer to be fully IBM-compatible
and to be running IBM PC-DOS or Microsoft MS-DOS version 2.0 or
higher. PathProj uses the screen handling capabilities of a CGA, EGA,
or VGA graphics card/monitor. If you system has only a Monochrome
Graphics Adapter (MGA) card or Hercules Graphics Card (HGC), you could
"simulate" CGA graphics capabilities in order to view trace displays.
For example, if you have SIMCGA.COM (version 4.0, July 1987) by C. P.
Guzis, you could make the SIMCGA system "memory resident" before
invoking PathProj.
DOS Prompt> simcga 3
WARNING: Your CONFIG.SYS file should specify at least...
files=20
┌─────────────────────────────┐
────────────────────┤ Graphical Design Philosophy ├────────────────────
└─────────────────────────────┘
PathProj uses CGA (medium resolution, 320x200 pixel) graphics because
all true graphics hardware for IBM-compatible personal computers has
at least this (minimum) capability. Besides, the (8x8) text font for
this resolution resides in ROM within your system. But using CGA also
means that there isn't enough resolution to draw "nice" labels, tick
marks, and numerical values along axes without greatly reducing the
size of the TRACE plot window. I certainly did not want to use any
sort of reduced-size plot; rather, I decided to emphasize INTERACTIVE
(dynamic, on-screen annotation) methods for high-lighting / reading
numerical values directly off of shrinkage TRACE screen displays.
[You interact with the display using the I, V, H, N, and arrow keys as
explained below.]
The PathProj plot window is square, rather than oblong, to emphasize
that the horizontal and vertical units of measurement are identical
for coefficient projections, i.e. they have an "aspect ratio" of one.
You can view a "help" screen by pressing either the SpaceBar or
function key F1. The resulting HELP screen tells you which keys to
press to activate and move the on-screen cursor and/or how to
dynamically annotate, save, or print the screen display. And you can
always return to the Menus simply by repeated pressing of the ENTER or
Escape keys.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 5 of 15
┌─────────────────────────┐
───────────────────────┤ File Naming Conventions ├─────────────────────
└─────────────────────────┘
PathProj assumes that you have (or someone else has) already performed
ridge calculations using one of our softRX systems, each of which
writes its output to a series of eight (or more) ASCII files. These
files have a common FILENAME (of at most 8 characters) and a
3-character DOS extension, ending with the two letters "RX".
By convention, the filename should not only indicate which set of
regressor and response data are being analyzed but also which
numerical value of the shrinkage path "shape" parameter, Q, was used
in the computations.
For example, the filename "mpg0" labels files associated with
analysis of "gas mileage data" [ Hocking(1976), "The Analysis and
Selection of Variables in Regression," Biometrics 32, 1-51 ] using
Hoerl-Kennard "ordinary" ridge regression, Q = 0.
The 3-character DOS extensions used by softRX software for generalized
ridge regression analysis are:
.PRX ...Parameters and variable names,
.CRX ...generalized ridge Coefficients,
.RRX ...Risk (mean-sq-error) estimates,
.ERX ...Excess Eigenvalues (OLS-Ridge),
.IRX ...Inferior direction cosines, and
.DRX ...shrinkage DELTA factors.
.XRX ...regressor matrix coordinates.
.YRX ...response vector coordinates.
NOTE: .RRX, .ERX, .DRX & .YRX files are not presently used by PathProj.
PathProj cannot make plots for a data set unless it can find the
corresponding .PRX file; it uses the information from the first six
lines of this file in projection plots of both types. For example,
both "mpg0.PRX" and "mpg.PRX" contain...
Line 1: 4 = P, number of regressors (non-constant predictors)
Line 2: 32 = N, number of observations
Line 3: 8 = STEPSPU, steps per unit change in MCAL
Line 4: 0 = QSHAPE of the shrinkage path
Line 5: XNAM= CYLNDS CUBINS HPOWER WEIGHT
Line 6: YNAM= MPG
Line 7: 0.8182 0.0000 0.0000
Line 8: 0.7383 0.1618 0.1250 Note: Lines 7 through 39
*** contain a 33 by 3
Line 38: 2.4907 900.7049 3.8750 matrix with columns
Line 39: 2.7772 +INF 4.0000 TMSE, K, and MCAL.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 6 of 15
NOTE: Lines 3 and 4 are present in "mpg.PRX" simply for format
uniformity; STEPSPU and QSHAPE are meaningless for .XRX files.
The other 3 types of files used by PathProj ( .CRX, .IRX, and .XRX )
will each contain a matrix with P columns and ( 1 + P * STEPSPU )
rows...for MCAL = 0 to MCAL = P. (See the APPENDIX for technical
information about MCAL.)
┌───────────────────────────┐
│ ┌───────────────────────┐ │
═════════════════════════╡ PathProj Walk-Through ╞══════════════════════
│ └───────────────────────┘ │
└───────────────────────────┘
To use PathProj, several RXridge or RelaxR output files (*.PRX, etc.)
MUST be located in the current working directory. Furthermore, the
PathProj.EXE file must also either be located in that same directory
or else in a some location named in your DOS PATH= environment
variable. The first step in use of PathProj is thus usually to
"Change Directories" to the place where your RXridge or RelaxR output
files are located. For example, you might need to enter the following
three sorts of commands at your DOS> prompt...
c:
cd \srx\mydata
pathproj
The above three commands: (1) make the "C" partition of your hard-disk
active, (2) change directories to the "MYDATA" sub-directory of the
C:\SRX directory, and (3) launch PathProj.
Next, you must select the filename of a softRX ".PRX" file using the
bounce-bar menu on the FILENAME Specification Screen:
┌─────────────────────────────────────────────────────────────┐
│ F I L E N A M E Specification Screen │
│ │
│ │
│ ╔FileNames╗ Arrow Keys: Move Highlight Bar Up / Down. │
│ ║MPG ║ │
│ ║MPG0█████║ Return Key: Selects the Highlighted File. │
│ ╚═════════╝ │
│ Escape Key: Quit......Terminate PathProj. │
│ │
│ Example FILENAMES... │
│ mpg0 ...Hocking Q = 0 path. │
│ mpg ...Hocking data coordinates. │
│ cement-5 ...Hald Q = -5 path. │
│ cement ...Hald data coordinates. │
└─────────────────────────────────────────────────────────────┘
PathProj, version 9201 . . . . . . . . . . . . . . . Page 7 of 15
Let us suppose you highlighted "mpg0" and then selected that filename by
pressing the enter key. The next screen will then be...
┌─────────────────────────────────────────────────┐
│ │
│ M A I N * M E N U │
│ │
│ FILENAME : mpg0 │
│ │
│ Using RX Parameters... mpg0.prx │
│ │
│ 1 ...Select Horizontal Axis │
│ 2 ...Select Vertical Axis │
│ 3 ...Display Path Projection Plot │
│ 4 ...Display Predictor Projections │
│ Q ...quit - Exit PathProj. │
│ │
│ Choose both 1 and 2 before either 3 or 4. │
│ │
│ Choice --> ___ Default = 1 │
│ │
│ Press a Single Key : 1 2 3 4 Q or Enter │
│ Use ENTER key for Default Choice. │
│ Press Q key to Exit PathProj now. │
└─────────────────────────────────────────────────┘
Accept the default (choice=1) by simply pressing the ENTER key.
------------------------------------------------------------------------
Choice = 1 : Select Horizontal Axis
------------------------------------------------------------------------
Modify Horizontal Axis ? <Y|N>
-1 -.5 0 +.5 +1
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
1 CYLNDS = 1.0000 |▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
2 CUBINS = 0.0000 |
3 HPOWER = 0.0000 |
4 WEIGHT = 0.0000 |
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
-1 -.5 0 +.5 +1
Press the N key (or the ENTER key) to accept this default choice for the
horizontal axis direction on the projection plot - i.e. parallel to
variable number 1.
But, if you press the Y key instead, you will be able to move DOWN the
predictor variable listing making potential changes to the direction
cosines. As soon as you press the ENTER key with the cursor on the last
direction cosine in the list, PathProj will re-normalize the direction
cosine vector to total length one, redisplay it, and return the cursor
to the <Y|N> prompt line. Pressing the N or ENTER key now signals that
the revised horizontal direction cosine vector is acceptable to you. So
PathProj, version 9201 . . . . . . . . . . . . . . . Page 8 of 15
keep pressing the Y key and modifying the direction cosines until you
have exactly the direction you want for the horizontal axis.
EXAMPLE 1: You might respond y(es) and modify the screen as follows:
Modify Horizontal Axis ? <Y|N> y
-1 -.5 0 +.5 +1
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
1 CYLNDS > 1 |▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
2 CUBINS > 1 |
3 HPOWER > 0 |
4 WEIGHT > 0 |
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
-1 -.5 0 +.5 +1
As soon as you press the ENTER key with the cursor on position 4, above,
PathProj will re-normalize the direction cosine vector to...
Modify Horizontal Axis ? <Y|N>
-1 -.5 0 +.5 +1
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
1 CYLNDS = 0.7071 |▀▀▀▀▀▀▀▀▀▀▀▀▀▀
2 CUBINS = 0.7071 |▀▀▀▀▀▀▀▀▀▀▀▀▀▀
3 HPOWER = 0.0000 |
4 WEIGHT = 0.0000 |
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
-1 -.5 0 +.5 +1
------------------------------------------------------------------------
Choice = 2 : Select Vertical Axis
------------------------------------------------------------------------
Choice number 2 will be the default when you return to the MainMenu
after making choice=1. A screen similar to that for choice = 1 will
appear, except you will now be specifying a vertical projection axis.
Modify Vertical Axis ? <Y|N>
-1 -.5 0 +.5 +1
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
1 CYLNDS = 0.0000 |
2 CUBINS = 1.0000 |▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
3 HPOWER = 0.0000 |
4 WEIGHT = 0.0000 |
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
-1 -.5 0 +.5 +1
PathProj, version 9201 . . . . . . . . . . . . . . . Page 9 of 15
Press the N key (or the ENTER key) to accept the default value for the
vertical axis on projection plots. The default shown above is to use
variable number 2 as the vertical axis on projection plots.
If you press the Y key, you will again be able to move DOWN the
predictor variable listing making potential changes to direction
cosines. When you press the ENTER key with the cursor on the last
variable in the list, PathProj will not only re-normalize the direction
cosine vector to total length one but will also make it...
ORTHOGONAL to the vector of horizontal axis direction cosines.
Pressing the N or ENTER key now signals that the revised vertical
direction cosine vector is acceptable to you. So, again, keep pressing
the Y key and modifying direction cosines until you have exactly what
you want for the vertical axis.
EXAMPLE 2: If you modified the default for axis 1 as described in
example 1, above, the the default for axis 2 automatically becomes:
-1 -.5 0 +.5 +1
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
1 CYLNDS = -0.7071 ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀|
2 CUBINS = 0.7071 |▀▀▀▀▀▀▀▀▀▀▀▀▀▀
3 HPOWER = 0.0000 |
4 WEIGHT = 0.0000 |
No. Name = Direction Cosine +----+----+----+----+----+----+----+----+
EXAMPLE 3: If you want to see somewhat "distorted" projections onto a
pair of oblique (rather than mutually orthogonal) axes, proceed as
follows. First use choice=1 to specify ANY direction orthogonal to your
desired vertical axis. Next, use choice=2 to specify that desired
vertical axis. Then use choice=1 to change the horizontal axis as
desired. Finally, go directly to choice=3 or choice=4 without viewing
or redoing choice=2.
------------------------------------------------------------------------
Choice = 3 : Ridge Coefficient Path-Projection
with Inferior Direction Depiction
------------------------------------------------------------------------
This projection plot has horizontal and vertical magenta axes that
intersect at the ORIGIN, [0,0], which is the end-point of the ridge
path. The projection of the least-squares regression coefficient vector
onto the 2-dimensional subspace you selected is marked with a white BOX.
And the projection of the ridge path is drawn as a cyan curve that
connects the white, least-squares BOX with the shrinkage ORIGIN.
We now list and discuss the "interactive" graphical options you have
each time that PathProj displays a coefficient projection plot...
ENTER or ESCape key...will return you to the menus so you can switch
data sets, axes, and/or to a "coordinate" (rather than
coefficient) projection plot.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 10 of 15
I key....IDENTIFY path points using a cyan on-screen BOX that you can
move along the ridge shrinkage path. The horizontal and
vertical coordinates of the BOX (as well as the MCAL value) are
listed on your screen and updated as you move along the path.
Pressing the right arrow increases MCAL, as if you were moving
to the right on a ridge trace plot...i.e. toward the shrinkage
terminus at [0,0]. This increase in MCAL may move the BOX
left, up or down rather than right on your screen, but you will
always be moving toward the intersection of the magenta axes at
[0,0]. (Pressing the TAB key increases MCAL by 4 steps rather
than 1.)
Pressing the left arrow decreases MCAL, as if you were moving
to the left on a ridge trace plot...i.e. toward the least
squares starting point of the ridge path. This decrease in
MCAL may move the cyan BOX right, up or down rather than left
on your screen, but you will always be moving toward the white
least squares box. (Pressing BackSpace decreases MCAL by 4
steps rather than the 1 step taken when you press left arrow.)
An estimated "Inferior Direction" usually appears as ridge
shrinkage progresses, and the projection of a vector of "unit"
length onto your 2-dimensional subspace is shown as a magenta
ARROW emerging from the BOX that marks your present position
along the ridge path.
It really doesn't make much difference how long this ARROW
"appears" to be, so you may change its apparent length on
your screen with the up and down arrow keys. On the other
hand, you do need to monitor the size of the component that
lies "in" (parallel to rather than orthogonal to) your
2-dimensional subspace. This relative size (projected
length per unit of total length) is listed on your screen
as "INFERIOR LENGTH = ". Note, in particular, that this
relative size does not change when you change the apparent
length of the ARROW.
LENGTH = 0.0000 indicates either that no inferior direction
exists for that MCAL value or that it is orthogonal to your
chosen subspace. LENGTH = 1.0000 would indicate not only
that an inferior direction exists but also that it lies
entirely "within" (parallel to) the 2-dimensional subspace
you are viewing.
It also doesn't really matter whether the projected arrow
lies in a positive or in a negative "orientation" along the
inferior direction. So you can Reverse this orientation
by toggling the R key.
Finally, press ENTER (to eliminate the on-screen BOX around a
path point and the inferior direction projection ARROW) or the
S(pecial) key to save or print the annotated screen.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 11 of 15
H key....to review the Horizontal Axis direction cosine screen.
V key....to review the Vertical Axis direction cosine screen.
N key....Replaces the PathProj plot with a "NAMES and NUMERICS" screen.
Use the right arrow, left arrow, Tab, and BackSpace keys to
change MCAL (by 1 or 4 steps)...same as with the I key, above.
Finally, press ENTER just once to return to graphics mode.
S key....SPECIAL KEY signal. A message will appear to indicate which
keys have been activate for screen saves, dumps, etc. Press
any other key to deactivate the special keys...
L or F10 dumps screen to HP Laserjet
I or F9 dumps to IBM Graphics Matrix
E or F8 dumps to Epson FX, JX, or LQ
O or F7 dumps to Okidata Microline
S = SAVE screen...as filename.PIX
Q = QUIT...Immediate Termination!!!
NOTE: You may also invoke the DOS "graphics" command before
entering PathProj and use the Shift-PrtSc key combination
for screen dumps. With some types of dot-matrix printer,
this method will produce a better reproduction of your
screen than any of the built-in (I, E, or O) keys.
F1 key or SpaceBar...displays a "help" screen that summarizes the above.
------------------------------------------------------------------------
Choice = 4 : Predictor (Coordinate) Projections
------------------------------------------------------------------------
This projection plot also has horizontal and vertical magenta axes that
intersect at the ORIGIN, [0,0]. (This would be the mean of predictor
coordinates IF they have been "centered.") The projections of all N
predictor combinations onto the 2-dimensional subspace you selected are
marked with cyan BOXes.
NOTE: Predictor coordinates and regression coefficient estimates should
be visualized as existing in two different Euclidean spaces, each of
dimension P. Under normal distribution theory, the two sets of contours
of constant likelihood are both hyper-ellipsoids with the same principal
axes (eigenvectors.) But X'X characterizes the dispersion ellipsoids
for predictor coordinates, while the inverse of X'X characterizes the
dispersion ellipsoids of regression coefficients. Thus, coordinate and
coefficient eigenvalues (principal axis variances) are the reciprocals
of each other. Limited "spread" in the observed predictor coordinates
along a certain direction translates directly into high variability in
estimates along that same direction in coefficient space.
We now list and discuss the "interactive" graphical options you have
when PathProj displays a coordinate projection plot...
PathProj, version 9201 . . . . . . . . . . . . . . . Page 12 of 15
ENTER, ESC, or letter M key...will return you to the above menus so
you can switch data sets, axes, and/or to a "coefficient"
(rather than coordinate) projection plot.
I key....Displays a magenta on-screen BOX that you can move from point
to point with the left and right arrow keys. Horizontal and
vertical coordinates (as well as observation number) for the
high-lighted point are also listed on your screen. (The
observation number is simply the row index of a predictor
combination in the X matrix.) Right arrow high-lights the
point with the next higher row index; left arrow high-lights
the point with the next lower row index.
H key....to review the Horizontal Axis direction cosine screen.
V key....to review the Vertical Axis direction cosine screen.
N key....Replaces the PathProj plot with a "NAMES and NUMERICS" screen.
Use the right arrow, left arrow, Tab, and BackSpace keys to
change row indices, 1 to N, of predictor combinations...in
increments of either 1 or 4, as with the I key. Finally,
press ENTER just once to return to graphics mode.
S key....SPECIAL KEYS are the same as for coefficient plots, above.
F1 key or SpaceBar...displays a "help" screen that summarizes the above.
┌────────────────────────────────────────────────────────────────────┐
│ TECHNICAL APPENDIX......"Shape" and "Extent" of Shrinkage in the │
│ Two-Parameter Generalized Ridge Family. │
└────────────────────────────────────────────────────────────────────┘
Q = the ridge parameter that controls the "shape" (or "curvature") of
the ridge path through regression coefficient likelihood space.
Q = +1 ...yields uniform shrinkage (all Shrinkage Factors equal.)
Q = 0 ...yields Hoerl-Kennard "ordinary" ridge regression.
Q = -5 ...is usually very close, numerically, to "Principal
Components Regression," with exact agreement in the
limit as Q approaches minus infinity.
MCAL = the "Multicollinearity Allowance" parameter that indexes
the "extent" of ridge shrinkage along any ridge path.
= P - trace( P x P diagonal matrix of Delta Shrinkage Factors ).
MCAL = 0 ...yields zero shrinkage. This is the "starting point" of the
ridge path, where the ridge estimator coincides with the
Ordinary Least Squares estimator at the left-hand extreme.
MCAL = P ...yields "total" shrinkage. This is the right-hand "end
point" of the path, where the ridge estimator is all ZEROS.
PathProj, version 9201 . . . . . . . . . . . . . . . Page 13 of 15
where...
P = Number of Predictor Variables (non-constant Regressors),
N = Number of Observations (or Regressor Combinations), and
generalized ridge regression "Shrinkage Factors" are of the form...
EigenValue
Delta = ----------------------------------
EigenValue + Konstant*EigenValue^Q
or, equivalently,... 1
= -----------------------------
1 + Konstant*EigenValue^(Q-1)
Empirical evidence that choice of "shape" as well as "extent" of
shrinkage can be rewarding...
Min.MeanSqErr Min.MeanSqErr
Data Set Number of Number of Amount of Shrinkage
Name Observations Predictors Shrinkage Shape
============= ============ ========== ============= =============
FACE data N = 21, P = 10, MCAL = 2.3, Qshape = +.77
Air Pollution N = 60, P = 15, MCAL = 5.4, Qshape = +.07
and Mortality
Acetylene N = 16, P = 9, MCAL = 5.2, Qshape = -.35
Ten-Factor N = 36, P = 10, MCAL = 3.6, Qshape = -.78
Stack Loss N = 15, P = 3, MCAL = 0.24, Qshape = -.95
Mantell, Bell N = 25, P = 3, MCAL = 0.95, Qshape = -1.1
Productivity
Wood Beam N = 10, P = 2, MCAL = 0.26, Qshape = -1.4
Longley N = 16, P = 6, MCAL = 4.0, Qshape = -1.4
Hocking MPG N = 32, P = 10, MCAL = 8.8, Qshape = -7.6
Diesel data N = 44, P = 9, MCAL = 4.9, Qshape = -20
Portland N = 13, P = 4, MCAL = 3.0, Qshape = -INF.
Cement, Hald.
============= ============ ========== ============ =============
Data Set Number of Number of Amount of Shrinkage
Name Observations Predictors Shrinkage Shape
Min.MeanSqErr Min.MeanSqErr
PathProj, version 9201 . . . . . . . . . . . . . . . Page 14 of 15
┌────────────────────────────┐
┌────────────────────┤ PathProj UpDate History... ├────────────────────┐
│ └────────────────────────────┘ │
│ 8904-06 ...developmental versions │
│ 8907 ...beta test version │
│ 8910 ...cosmetic changes to menus, variable name defaults, etc. │
│ 9002 ...add ability to save\print annotated path projections │
│ 9004 ...LZEXE ver0.90 compression (c) 1989 Fabrice Bellard │
│ 9008 ...bounce-bar filename menu; change-of-address; LZEXE v0.91 │
│ 9010 ...change screen save file to .PCX format │
│ 9012 ...fix bounce-bar selection menu for monochrome screens │
│ 9012 ...fix bounce-bar selection menu for monochrome screens │
│ 9103 ...change permanent address; update documentation files │
│ 9201 ...change default responses; shorten screen pause delays │
│ 9303 ...major documentation revision/update. │
└──────────────────────────────────────────────────────────────────────┘
┌─────────┐
┌─────┴───┐ │ (R)
──│ │o │──────────────────
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PathProj, version 9201 . . . . . . . . . . . . . . . Page 15 of 15
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