The Statistician’s Apprentice: An Introduction to the SWP Operator

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The sweep operator as defined in (Dempster, 1969), commonly referred to as the SWP operator, is a useful tool for a computational statistician working with covariance matrices. In particular, the SWP operator allows a statistician to quickly regress all variables against one specified variable, obtaining OLS estimates for regression coefficients and variances in a single application. Subsequent applications of the SWP operator allows for regressing against more variables.

In this blog post, I will define the sweep operator, provide an application of the sweep data to simulated data, and provide some references for further study. Examples will be provided in R through the ISR3 package (Lisic, 2016).

The basic SWP operator is parameterized by a matrix and a set of indices associated with rows and columns of the matrix being swept. E.g. to sweep the matrix < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 722.5 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-3A3"> by the < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.42ex" height="2.676ex" style="vertical-align: -0.338ex;" viewBox="0 -1006.6 1041.8 1152.1" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-6C">< g transform="translate(278,422)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMATHI-74">< use transform="scale(0.707)" x="361" y="0" xlink:href="#MJMATHI-68"> index, such that < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.86ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 3384.1 1223.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-41">< use x="1028" y="0" xlink:href="#MJMAIN-3D">< use x="2084" y="0" xlink:href="#MJMAIN-7B">< use x="2585" y="0" xlink:href="#MJMATHI-6C">< use x="2883" y="0" xlink:href="#MJMAIN-7D">, the operator has the form < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.788ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 5075.5 1223.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use xlink:href="#MJMAIN-53">< use x="556" y="0" xlink:href="#MJMAIN-57">< use x="1585" y="0" xlink:href="#MJMAIN-50">< use x="2266" y="0" xlink:href="#MJMAIN-5B">< use x="2545" y="0" xlink:href="#MJMATHI-41">< use x="3295" y="0" xlink:href="#MJMAIN-5D">< use x="3574" y="0" xlink:href="#MJMAIN-28">< use x="3963" y="0" xlink:href="#MJMAIN-3A3">< use x="4686" y="0" xlink:href="#MJMAIN-29">. The operator may also be written more descriptively by listing the elements of < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 750.5 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-41"> as < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.738ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 4623.5 1223.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use xlink:href="#MJMAIN-53">< use x="556" y="0" xlink:href="#MJMAIN-57">< use x="1585" y="0" xlink:href="#MJMAIN-50">< use x="2266" y="0" xlink:href="#MJMAIN-5B">< use x="2545" y="0" xlink:href="#MJMATHI-6C">< use x="2843" y="0" xlink:href="#MJMAIN-5D">< use x="3122" y="0" xlink:href="#MJMAIN-28">< use x="3511" y="0" xlink:href="#MJMAIN-3A3">< use x="4234" y="0" xlink:href="#MJMAIN-29">. The application of the SWP operator to the < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.42ex" height="2.676ex" style="vertical-align: -0.338ex;" viewBox="0 -1006.6 1041.8 1152.1" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-6C">< g transform="translate(278,422)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMATHI-74">< use transform="scale(0.707)" x="361" y="0" xlink:href="#MJMATHI-68"> row and column of < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 722.5 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-3A3"> produces a new matrix with elements

As an example, consider the application of the SWP operator to the first row and column of the < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.165ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 2223.9 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-33">< use x="722" y="0" xlink:href="#MJMAIN-D7">< use x="1723" y="0" xlink:href="#MJMAIN-33"> covariance matrix for the random variables < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.328ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 5738.6 1223.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-7B">< g transform="translate(500,0)">< use x="0" y="0" xlink:href="#MJMATHI-58">< use transform="scale(0.707)" x="1171" y="-213" xlink:href="#MJMAIN-31">< use x="1782" y="0" xlink:href="#MJMAIN-2C">< g transform="translate(2228,0)">< use x="0" y="0" xlink:href="#MJMATHI-58">< use transform="scale(0.707)" x="1171" y="-213" xlink:href="#MJMAIN-32">< use x="3510" y="0" xlink:href="#MJMAIN-2C">< g transform="translate(3955,0)">< use x="0" y="0" xlink:href="#MJMATHI-58">< use transform="scale(0.707)" x="1171" y="-213" xlink:href="#MJMAIN-33">< use x="5238" y="0" xlink:href="#MJMAIN-7D">,

resulting in

How is this useful? Well, if we substitute the < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.165ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 2223.9 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-33">< use x="722" y="0" xlink:href="#MJMAIN-D7">< use x="1723" y="0" xlink:href="#MJMAIN-33"> covariance matrix < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 722.5 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-3A3"> with the sample covariance matrix, where < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.576ex" height="3.176ex" style="vertical-align: -1.005ex;" viewBox="0 -934.9 1109.1 1367.4" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-53">< use transform="scale(0.707)" x="926" y="488" xlink:href="#MJMAIN-32">< use transform="scale(0.707)" x="867" y="-463" xlink:href="#MJMATHI-6B"> is an estimator of < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.273ex" height="2.009ex" style="vertical-align: -0.671ex;" viewBox="0 -576.1 1409 865.1" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-3C3">< g transform="translate(571,-150)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMATHI-6B">< use transform="scale(0.707)" x="521" y="0" xlink:href="#MJMATHI-6B"> and

is an estimator of < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.094ex" height="2.343ex" style="vertical-align: -1.005ex;" viewBox="0 -576.1 1331.9 1008.6" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-3C3">< g transform="translate(571,-150)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMATHI-6A">< use transform="scale(0.707)" x="412" y="0" xlink:href="#MJMATHI-6B">, then by sweeping by the < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.682ex" height="2.509ex" style="vertical-align: -0.338ex;" viewBox="0 -934.9 1154.9 1080.4" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-31">< g transform="translate(500,393)">< use transform="scale(0.707)" xlink:href="#MJMAIN-73">< use transform="scale(0.707)" x="394" y="0" xlink:href="#MJMAIN-74"> index, < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.329ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 3586.1 1223.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-41">< use x="1028" y="0" xlink:href="#MJMAIN-3D">< use x="2084" y="0" xlink:href="#MJMAIN-7B">< use x="2585" y="0" xlink:href="#MJMAIN-31">< use x="3085" y="0" xlink:href="#MJMAIN-7D">, we can get the simple linear regression (SLR) estimates for < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.506ex" height="2.509ex" style="vertical-align: -0.671ex;" viewBox="0 -791.3 6676.3 1080.4" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-58">< use transform="scale(0.707)" x="1171" y="-213" xlink:href="#MJMATHI-6B">< use x="1575" y="0" xlink:href="#MJMAIN-3D">< g transform="translate(2631,0)">< use x="0" y="0" xlink:href="#MJMATHI-3B2">< g transform="translate(566,-150)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMAIN-31">< use transform="scale(0.707)" x="500" y="0" xlink:href="#MJMATHI-6B">< g transform="translate(4020,0)">< use x="0" y="0" xlink:href="#MJMATHI-78">< use transform="scale(0.707)" x="809" y="-213" xlink:href="#MJMAIN-31">< use x="5269" y="0" xlink:href="#MJMAIN-2B">< use x="6269" y="0" xlink:href="#MJMATHI-3F5"> for all < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.736ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 4191.7 1223.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-6B">< use x="799" y="0" xlink:href="#MJMAIN-2208">< use x="1744" y="0" xlink:href="#MJMAIN-7B">< use x="2245" y="0" xlink:href="#MJMAIN-32">< use x="2745" y="0" xlink:href="#MJMAIN-2C">< use x="3190" y="0" xlink:href="#MJMAIN-33">< use x="3691" y="0" xlink:href="#MJMAIN-7D">: where < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.72ex" height="3.676ex" style="vertical-align: -1.171ex;" viewBox="0 -1078.4 2032.3 1582.7" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-3B2">< use x="124" y="283" xlink:href="#MJMAIN-5E">< g transform="translate(624,-231)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMATHI-6C">< use transform="scale(0.707)" x="298" y="0" xlink:href="#MJMATHI-6B">< use transform="scale(0.707)" x="820" y="0" xlink:href="#MJMAIN-7C">< use transform="scale(0.707)" x="1098" y="0" xlink:href="#MJMATHI-41"> is the simple linear model estimator of the regression coefficient for the linear model < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.843ex" height="2.509ex" style="vertical-align: -0.671ex;" viewBox="0 -791.3 6390.7 1080.4" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-58">< use transform="scale(0.707)" x="1171" y="-213" xlink:href="#MJMATHI-6B">< use x="1575" y="0" xlink:href="#MJMAIN-3D">< g transform="translate(2631,0)">< use x="0" y="0" xlink:href="#MJMATHI-3B2">< g transform="translate(566,-150)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMATHI-6C">< use transform="scale(0.707)" x="298" y="0" xlink:href="#MJMATHI-6B">< g transform="translate(3877,0)">< use x="0" y="0" xlink:href="#MJMATHI-78">< use transform="scale(0.707)" x="809" y="-213" xlink:href="#MJMATHI-6C">< use x="4983" y="0" xlink:href="#MJMAIN-2B">< use x="5984" y="0" xlink:href="#MJMATHI-3F5"> for < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.736ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 4191.7 1223.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-6B">< use x="799" y="0" xlink:href="#MJMAIN-2208">< g transform="translate(1744,0)">< use x="0" y="0" xlink:href="#MJMAIN-7B">< g transform="translate(500,0)">< use x="0" y="0" xlink:href="#MJMAIN-32">< use x="500" y="0" xlink:href="#MJMAIN-2C">< use x="945" y="0" xlink:href="#MJMAIN-33">< use x="1946" y="0" xlink:href="#MJMAIN-7D">, < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.078ex" height="3.176ex" style="vertical-align: -0.671ex;" viewBox="0 -1078.4 1325.2 1367.4" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMAIN-39B">< use x="97" y="294" xlink:href="#MJMAIN-5E">< use transform="scale(0.707)" x="982" y="-230" xlink:href="#MJMATHI-41"> is the inverse of the covariance matrix of < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.693ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 298.5 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-6C"> unscaled by < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.398ex" height="2.343ex" style="vertical-align: -0.505ex;" viewBox="0 -791.3 2323.9 1008.6" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-6E">< use x="822" y="0" xlink:href="#MJMAIN-2212">< use x="1823" y="0" xlink:href="#MJMAIN-31">, and < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.786ex" height="3.009ex" style="vertical-align: -1.171ex;" viewBox="0 -791.3 2060.6 1295.7" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-3C3">< use x="36" y="9" xlink:href="#MJMAIN-5E">< g transform="translate(572,-187)">< use transform="scale(0.707)" x="0" y="0" xlink:href="#MJMATHI-6A">< use transform="scale(0.707)" x="412" y="0" xlink:href="#MJMATHI-6B">< use transform="scale(0.707)" x="934" y="0" xlink:href="#MJMAIN-7C">< use transform="scale(0.707)" x="1212" y="0" xlink:href="#MJMATHI-41"> is the sample covariance of the residuals from the regression models on < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.693ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 298.5 936.9" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-6C"> unscaled by < svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.398ex" height="2.343ex" style="vertical-align: -0.505ex;" viewBox="0 -791.3 2323.9 1008.6" role="img" focusable="false">< g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">< use x="0" y="0" xlink:href="#MJMATHI-6E">< use x="822" y="0" xlink:href="#MJMAIN-2212">< use x="1823" y="0" xlink:href="#MJMAIN-31">.

Example

Lets try this with some real data. I have used the SWP function included with the ISR3 package.

library(ISR3)

set.seed(12)
n <- 100
p <- 3

# generate a positive definite matrix for covariance
Sigma <- rWishart(1,p+1,diag(p))[,,1]
Sigma_inv <- chol2inv(chol(Sigma))
Sigma_inv_chol <- chol(Sigma_inv)

# generate 'n' multivate normal deviates
X <- Sigma_inv_chol %*% matrix(rnorm(n*p),nrow=p)
X <- t(X)
colnames(X) <- sprintf("X_%d",1:p)

XX <- t(X) %*% X

#Sweep by the first row/column of XX
SWP_1 <- SWP(XX,1)

SWP_1

##              X_1        X_2         X_3
## X_1 -0.002381092  0.1375626 -0.01674788
## X_2  0.137562633 50.0987932 -4.95181624
## X_3 -0.016747875 -4.9518162 41.73979419

# linear models
fit_21 <- lm(X_2 ~ -1 + X_1, data=as.data.frame(X))
fit_31 <- lm(X_3 ~ -1 + X_1, data=as.data.frame(X))

# SLR coefficents for X_1
print(fit_21$coefficients)

##       X_1 
## 0.1375626

print(fit_31$coefficients)

##         X_1 
## -0.01674788

# sum of squares
print(sum(fit_21$residuals^2))

## [1] 50.09879

print(sum(fit_31$residuals^2))

## [1] 41.73979

# covariance between X_2 and X_3 given X_1
print(sum(fit_21$residuals * fit_31$residuals))

## [1] -4.951816

# negative inverse of the covariance matrix of X_1
print( -1/((n-1)*var(X[,1])) )

## [1] -0.002381293

SWP(XX,1:3) + chol2inv(chol(XX))

##              X_1          X_2          X_3
## X_1 0.000000e+00 0.000000e+00 1.355253e-20
## X_2 0.000000e+00 3.469447e-18 8.673617e-19
## X_3 1.355253e-20 8.673617e-19 0.000000e+00