Total and Effect Deviation Matrices for Linear Regression and Canonical Correlation Analysis
## Default S3 method: lmHmat(x,y,...) ## S3 method for class 'data.frame' lmHmat(x,y,...) ## S3 method for class 'formula' lmHmat(formula,data=NULL,...) ## S3 method for class 'lm' lmHmat(fitdlmmodel,...)
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 |
data |
Data frame from which variables specified in 'formula' are preferentially to be taken. |
fitdlmmodel |
An object of class |
... |
further arguments for the method. |
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.
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 |
call |
The function call which generated the output. |
##------------------------------------------------------------------ ## 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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