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subselect (version 0.14)

lmHmat: Total and Effect Deviation Matrices for Linear Regression and Canonical Correlation Analysis

Description

Computes total an effect matrices of Sums of Squares and Cross-Product (SSCP) deviations, divided by a normalizing constant, in linear regression or canonical correlation analysis. These matrices may be used as input to the variable selection search routines anneal, genetic improve or eleaps.

Usage

# S3 method for default
lmHmat(x,y,...)
# S3 method for data.frame
lmHmat(x,y,...)
# S3 method for formula
lmHmat(formula,data=NULL,...)
# S3 method for lm
lmHmat(fitdlmmodel,...)

Arguments

x

A matrix or data frame containing the variables for which the SSCP matrix is to be computed.

y

A matrix or data frame containing the set of fixed variables, the association of x is to be measured with.

formula

A formula of the form 'y ~ x1 + x2 + ...'. That is, the response is the set of fixed variables and the right hand side specifies the variables whose subsets are to be compared.

data

Data frame from which variables specified in 'formula' are preferentially to be taken.

fitdlmmodel

An object of class lm, as produced by R's lm function.

...

further arguments for the method.

Value

A list with four items:

mat

The total SSCP matrix divided by nrow(x)-1

H

The effect SSCP matrix divided by nrow(x)-1

r

The expected rank of the H matrix which, under the assumption of linear independence, equals the minimum between the number of variables in the x and y sets. The true rank of H can be different from r if the linear independence condition fails.

call

The function call which generated the output.

Details

Let x and y be two different groups of linearly independent variables observed on the same set of data units. It is well known that the association between x and y can be measured by their squared canonical correlations which may be found as the positive eigenvalues of certain matrix products. In particular, if \(T_x\) and \(H_{x/y}\) denote SSCP matrices of deviations from the mean, respectively for the original x variables (\(T_x\)) and for their orthogonal projections onto the space spanned by the y's (\(H_{x/y}\)), then the positive eigenvalues of \(T_x^{-1}H_{x/y}\) equal the squared correlations between x and y. Alternatively these correlations could also be found from \(T_y^{-1} H_{y/x}\) but here, assuming a goal of comparing x's subsets for a given fixed set of y's, we will focus on the former product. lmHmat computes a scaled version of \(T_x\) and \(H_{x/y}\) such that \(T_x\) is converted into a covariance matrix. These matrices can be used as input to the search routines anneal, genetic improve and eleaps that try to select x subsets based on several functions of their squared correlations with y. We note that when there is only one variable in the y set, this is equivalent to selecting predictors for linear regression based on the traditional coefficient of determination.

See Also

anneal, genetic, improve, eleaps, lm.

Examples

Run this code
# NOT RUN {
##------------------------------------------------------------------

## 1)  An example of subset selection in the context of Multiple
## Linear Regression. Variable 5 (average price) in the Cars93 MASS
## library is to be regressed on 13 other  variables.  The goal is to
## compare subsets of these 13 variables according to their  ability
## to predict car prices. 

library(MASS)
data(Cars93)
CarsHmat1 <- lmHmat(Cars93[c(7:8,12:15,17:22,25)],Cars93[5])
CarsHmat1

##$mat
##                       MPG.city  MPG.highway   EngineSize    Horsepower
##MPG.city              31.582281    28.283427   -4.1391655 -1.979799e+02
##MPG.highway           28.283427    28.427302   -3.4667602 -1.728655e+02
##EngineSize            -4.139165    -3.466760    1.0761220  3.977700e+01
##Horsepower          -197.979897  -172.865475   39.7769986  2.743079e+03
##RPM                 1217.478962   997.335203 -339.1637447  1.146634e+03
##Rev.per.mile        1941.631019  1555.243104 -424.4118163 -1.561070e+04
##Fuel.tank.capacity   -14.985799   -13.743654    2.5830820  1.222536e+02
##Passengers            -2.433964    -2.583567    0.4017181  5.040907e-01
##Length               -54.673329   -42.267765   11.8197055  4.212964e+02
##Wheelbase            -25.567087   -22.375760    5.1819425  1.738928e+02
##Width                -15.302127   -12.902291    3.3992286  1.275437e+02
##Turn.circle          -12.071061   -10.202782    2.6029453  9.474252e+01
##Weight             -2795.094670 -2549.654628  517.1327139  2.282550e+04
##                            RPM Rev.per.mile Fuel.tank.capacity   Passengers
##MPG.city              1217.4790    1941.6310         -14.985799   -2.4339645
##MPG.highway            997.3352    1555.2431         -13.743654   -2.5835671
##EngineSize            -339.1637    -424.4118           2.583082    0.4017181
##Horsepower            1146.6339  -15610.7036         122.253612    0.5040907
##RPM                 356088.7097  146589.3233        -652.324684 -289.6213184
##Rev.per.mile        146589.3233  246518.7295        -992.747020 -172.8003740
##Fuel.tank.capacity    -652.3247    -992.7470          10.754271    1.6085203
##Passengers            -289.6213    -172.8004           1.608520    1.0794764
##Length               -3844.9158   -5004.3139          33.063850    7.3626695
##Wheelbase            -1903.7693   -2156.2932          16.944811    4.9177186
##Width                -1217.0933   -1464.3712           9.898282    1.9237962
##Turn.circle           -972.5806   -1173.3281           7.096283    1.5037401
##Weight             -150636.1325 -215349.6757        1729.468268  339.0953717
##                        Length    Wheelbase        Width  Turn.circle
##MPG.city             -54.67333   -25.567087   -15.302127   -12.071061
##MPG.highway          -42.26777   -22.375760   -12.902291   -10.202782
##EngineSize            11.81971     5.181942     3.399229     2.602945
##Horsepower           421.29640   173.892824   127.543712    94.742520
##RPM                -3844.91585 -1903.769285 -1217.093268  -972.580645
##Rev.per.mile       -5004.31393 -2156.293245 -1464.371201 -1173.328074
##Fuel.tank.capacity    33.06385    16.944811     9.898282     7.096283
##Passengers             7.36267     4.917719     1.923796     1.503740
##Length               213.22955    82.021973    45.367929    34.780622
##Wheelbase             82.02197    46.507948    20.803062    15.899836
##Width                 45.36793    20.803062    14.280739     9.962015
##Turn.circle           34.78062    15.899836     9.962015    10.389434
##Weight              6945.16129  3507.549088  1950.471599  1479.365358
##                         Weight
##MPG.city             -2795.0947
##MPG.highway          -2549.6546
##EngineSize             517.1327
##Horsepower           22825.5049
##RPM                -150636.1325
##Rev.per.mile       -215349.6757
##Fuel.tank.capacity    1729.4683
##Passengers             339.0954
##Length                6945.1613
##Wheelbase             3507.5491
##Width                 1950.4716
##Turn.circle           1479.3654
##Weight              347977.8927

##$H
##                        MPG.city   MPG.highway    EngineSize   Horsepower
##MPG.city              11.1644681     9.9885440   -2.07077758  -137.938111
##MPG.highway            9.9885440     8.9364770   -1.85266802  -123.409453
##EngineSize            -2.0707776    -1.8526680    0.38408635    25.584662
##Horsepower          -137.9381108  -123.4094525   25.58466246  1704.239046
##RPM                    9.8795182     8.8389345   -1.83244599  -122.062428
##Rev.per.mile         707.3855707   632.8785101 -131.20537141 -8739.818920
##Fuel.tank.capacity    -6.7879209    -6.0729671    1.25901874    83.865437
##Passengers            -0.2008651    -0.1797085    0.03725632     2.481709
##Length               -24.5727044   -21.9845261    4.55772770   303.598201
##Wheelbase            -11.4130722   -10.2109633    2.11688849   141.009639
##Width                 -5.7581866    -5.1516920    1.06802435    71.142967
##Turn.circle           -4.2281864    -3.7828426    0.78424099    52.239662
##Weight             -1275.6139645 -1141.2569026  236.59996884 15760.337110
##                            RPM Rev.per.mile Fuel.tank.capacity    Passengers
##MPG.city               9.879518    707.38557         -6.7879209  -0.200865141
##MPG.highway            8.838935    632.87851         -6.0729671  -0.179708544
##EngineSize            -1.832446   -131.20537          1.2590187   0.037256323
##Horsepower          -122.062428  -8739.81892         83.8654369   2.481708752
##RPM                    8.742457    625.97059         -6.0066801  -0.177747010
##Rev.per.mile         625.970586  44820.25860       -430.0856347 -12.726903044
##Fuel.tank.capacity    -6.006680   -430.08563          4.1270099   0.122124645
##Passengers            -0.177747    -12.72690          0.1221246   0.003613858
##Length               -21.744563  -1556.93728         14.9400378   0.442098962
##Wheelbase            -10.099510   -723.13724          6.9390706   0.205337894
##Width                 -5.095461   -364.84122          3.5009384   0.103598215
##Turn.circle           -3.741553   -267.89973          2.5707087   0.076071269
##Weight             -1128.799984 -80823.45772        775.5646486  22.950164550
##                         Length    Wheelbase        Width   Turn.circle
##MPG.city             -24.572704  -11.4130722   -5.7581866   -4.22818636
##MPG.highway          -21.984526  -10.2109633   -5.1516920   -3.78284262
##EngineSize             4.557728    2.1168885    1.0680243    0.78424099
##Horsepower           303.598201  141.0096393   71.1429669   52.23966202
##RPM                  -21.744563  -10.0995098   -5.0954608   -3.74155256
##Rev.per.mile       -1556.937281 -723.1372362 -364.8412174 -267.89973369
##Fuel.tank.capacity    14.940038    6.9390706    3.5009384    2.57070866
##Passengers             0.442099    0.2053379    0.1035982    0.07607127
##Length                54.083885   25.1198756   12.6736193    9.30612843
##Wheelbase             25.119876   11.6672121    5.8864067    4.32233724
##Width                 12.673619    5.8864067    2.9698426    2.18072961
##Turn.circle            9.306128    4.3223372    2.1807296    1.60129079
##Weight              2807.593227 1304.0186214  657.9107222  483.09812289
##                         Weight
##MPG.city            -1275.61396
##MPG.highway         -1141.25690
##EngineSize            236.59997
##Horsepower          15760.33711
##RPM                 -1128.79998
##Rev.per.mile       -80823.45772
##Fuel.tank.capacity    775.56465
##Passengers             22.95016
##Length               2807.59323
##Wheelbase            1304.01862
##Width                 657.91072
##Turn.circle           483.09812
##Weight             145747.29199

##$r
##[1] 1

##$call
##lmHmat.data.frame(x = Cars93[c(7:8, 12:15, 17:22, 25)], y = Cars93[5])


## 2)  An example of subset selection in the context of Canonical
## Correlation Analysis. Two groups of variables within the Cars93
## MASS library data set are compared. The first group (variables 4th,
## 5th and 6th) relates to price, while the second group is formed by 13
## variables that describe several technical car specifications. The
## goal is to select subsets of the second group that are optimal in
## terms of preserving the canonical correlations with the variables in
## the first group (Warning: the 3-variable "response" group is kept
## intact; subset selection is to be performed only in the 13-variable
## group).  

library(MASS)
data(Cars93)
CarsHmat2 <- lmHmat(Cars93[c(7:8,12:15,17:22,25)],Cars93[4:6])

names(Cars93[4:6])
## [1] "Min.Price" "Price"     "Max.Price"

CarsHmat2

##$mat
##                       MPG.city  MPG.highway   EngineSize    Horsepower
##MPG.city              31.582281    28.283427   -4.1391655 -1.979799e+02
##MPG.highway           28.283427    28.427302   -3.4667602 -1.728655e+02
##EngineSize            -4.139165    -3.466760    1.0761220  3.977700e+01
##Horsepower          -197.979897  -172.865475   39.7769986  2.743079e+03
##RPM                 1217.478962   997.335203 -339.1637447  1.146634e+03
##Rev.per.mile        1941.631019  1555.243104 -424.4118163 -1.561070e+04
##Fuel.tank.capacity   -14.985799   -13.743654    2.5830820  1.222536e+02
##Passengers            -2.433964    -2.583567    0.4017181  5.040907e-01
##Length               -54.673329   -42.267765   11.8197055  4.212964e+02
##Wheelbase            -25.567087   -22.375760    5.1819425  1.738928e+02
##Width                -15.302127   -12.902291    3.3992286  1.275437e+02
##Turn.circle          -12.071061   -10.202782    2.6029453  9.474252e+01
##Weight             -2795.094670 -2549.654628  517.1327139  2.282550e+04
##                            RPM Rev.per.mile Fuel.tank.capacity   Passengers
##MPG.city              1217.4790    1941.6310         -14.985799   -2.4339645
##MPG.highway            997.3352    1555.2431         -13.743654   -2.5835671
##EngineSize            -339.1637    -424.4118           2.583082    0.4017181
##Horsepower            1146.6339  -15610.7036         122.253612    0.5040907
##RPM                 356088.7097  146589.3233        -652.324684 -289.6213184
##Rev.per.mile        146589.3233  246518.7295        -992.747020 -172.8003740
##Fuel.tank.capacity    -652.3247    -992.7470          10.754271    1.6085203
##Passengers            -289.6213    -172.8004           1.608520    1.0794764
##Length               -3844.9158   -5004.3139          33.063850    7.3626695
##Wheelbase            -1903.7693   -2156.2932          16.944811    4.9177186
##Width                -1217.0933   -1464.3712           9.898282    1.9237962
##Turn.circle           -972.5806   -1173.3281           7.096283    1.5037401
##Weight             -150636.1325 -215349.6757        1729.468268  339.0953717
##                        Length    Wheelbase        Width  Turn.circle
##MPG.city             -54.67333   -25.567087   -15.302127   -12.071061
##MPG.highway          -42.26777   -22.375760   -12.902291   -10.202782
##EngineSize            11.81971     5.181942     3.399229     2.602945
##Horsepower           421.29640   173.892824   127.543712    94.742520
##RPM                -3844.91585 -1903.769285 -1217.093268  -972.580645
##Rev.per.mile       -5004.31393 -2156.293245 -1464.371201 -1173.328074
##Fuel.tank.capacity    33.06385    16.944811     9.898282     7.096283
##Passengers             7.36267     4.917719     1.923796     1.503740
##Length               213.22955    82.021973    45.367929    34.780622
##Wheelbase             82.02197    46.507948    20.803062    15.899836
##Width                 45.36793    20.803062    14.280739     9.962015
##Turn.circle           34.78062    15.899836     9.962015    10.389434
##Weight              6945.16129  3507.549088  1950.471599  1479.365358
##                         Weight
##MPG.city             -2795.0947
##MPG.highway          -2549.6546
##EngineSize             517.1327
##Horsepower           22825.5049
##RPM                -150636.1325
##Rev.per.mile       -215349.6757
##Fuel.tank.capacity    1729.4683
##Passengers             339.0954
##Length                6945.1613
##Wheelbase             3507.5491
##Width                 1950.4716
##Turn.circle           1479.3654
##Weight              347977.8927

##$H
##                        MPG.city   MPG.highway    EngineSize   Horsepower
##MPG.city              12.6374638    11.1802504   -2.44856549  -149.055525
##MPG.highway           11.1802504     9.9241995   -2.15551417  -132.381671
##EngineSize            -2.4485655    -2.1555142    0.48131168    28.438641
##Horsepower          -149.0555255  -132.3816709   28.43864077  1788.168412
##RPM                  116.9463468    90.2758380  -29.90735790  -935.019669
##Rev.per.mile         850.6791690   744.7148717 -168.44221351 -9825.172173
##Fuel.tank.capacity    -7.3863845    -6.5473387    1.41367337    88.391549
##Passengers            -0.2756475    -0.2507147    0.05519028     3.036255
##Length               -29.0878749   -25.4205633    5.74148535   337.880225
##Wheelbase            -12.4579187   -11.0208656    2.38906697   148.928887
##Width                 -6.8768553    -6.0641799    1.35405290    79.579106
##Turn.circle           -4.9652258    -4.3460777    0.97719452    57.833523
##Weight             -1399.0819460 -1239.6883974  268.43952022 16693.580681
##                             RPM Rev.per.mile Fuel.tank.capacity   Passengers
##MPG.city              116.946347    850.67917         -7.3863845  -0.27564745
##MPG.highway            90.275838    744.71487         -6.5473387  -0.25071469
##EngineSize            -29.907358   -168.44221          1.4136734   0.05519028
##Horsepower           -935.019669  -9825.17217         88.3915487   3.03625516
##RPM                  8930.289631  11941.01945        -51.6620352  -3.30491485
##Rev.per.mile        11941.019450  59470.19917       -490.0061258 -18.17896445
##Fuel.tank.capacity    -51.662035   -490.00613          4.3742368   0.14814085
##Passengers             -3.304915    -18.17896          0.1481409   0.01208827
##Length               -397.601848  -2033.81167         16.8646785   0.57474210
##Wheelbase             -93.828737   -830.92582          7.3783050   0.24261242
##Width                 -84.771418   -472.37388          3.9523474   0.16370704
##Turn.circle           -64.578815   -345.33527          2.8839031   0.09876958
##Weight             -10423.776629 -93087.56026        826.3348263  28.56899347
##                          Length    Wheelbase        Width   Turn.circle
##MPG.city             -29.0878749  -12.4579187   -6.8768553   -4.96522585
##MPG.highway          -25.4205633  -11.0208656   -6.0641799   -4.34607767
##EngineSize             5.7414854    2.3890670    1.3540529    0.97719452
##Horsepower           337.8802249  148.9288871   79.5791065   57.83352310
##RPM                 -397.6018484  -93.8287370  -84.7714184  -64.57881537
##Rev.per.mile       -2033.8116669 -830.9258201 -472.3738765 -345.33527111
##Fuel.tank.capacity    16.8646785    7.3783050    3.9523474    2.88390313
##Passengers             0.5747421    0.2426124    0.1637070    0.09876958
##Length                69.9185456   28.6482825   16.0342179   11.86931842
##Wheelbase             28.6482825   12.4615297    6.6687394    4.89477408
##Width                 16.0342179    6.6687394    3.8217667    2.73004255
##Turn.circle           11.8693184    4.8947741    2.7300425    2.01640426
##Weight              3199.4701647 1393.7884808  751.2183342  546.92139008
##                         Weight
##MPG.city            -1399.08195
##MPG.highway         -1239.68840
##EngineSize            268.43952
##Horsepower          16693.58068
##RPM                -10423.77663
##Rev.per.mile       -93087.56026
##Fuel.tank.capacity    826.33483
##Passengers             28.56899
##Length               3199.47016
##Wheelbase            1393.78848
##Width                 751.21833
##Turn.circle           546.92139
##Weight             156186.68328

##$r
##[1] 3

##$call
##lmHmat.data.frame(x = Cars93[c(7:8, 12:15, 17:22, 25)], y = Cars93[4:6])


# }

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