linear algebra

This post comes out of the blue, nearly 2 years since my last one. I realize I’ve been lazy, so here’s hoping I move from an inertia of rest to that of motion, implying, regular and (hopefully) relevant posts. I also chanced upon some wisdom while scrolling through my Twitter feed:

Don’t worry about how many people you think will want to read your blog posts. Writing has value beyond its audience. It can help to crystallize your thinking and also serve as a milestone in your learning journey.

Do some writing this weekend!

— Coraline Ada Ehmke (@CoralineAda) November 9, 2019

This blog post in particular was meant to be a reminder to myself and other R users that the much used lm() function in R (for fitting linear models) can be replaced with some handy matrix operations to obtain regression coefficients, their standard errors and other goodness-of-fit stats printed out when summary() is called on an lm object.

Linear regression can be formulated mathematically as follows:
$\mathbf{y} = \mathbf{X} \mathbf{\beta} + \mathbf{\epsilon}$ ,
$\mathbf{\epsilon} \sim N(0, \sigma^2 \mathbf{I})$

$\mathbf{y}$ is the $\mathbf{n}\times \mathbf{1}$ outcome variable and $\mathbf{X}$ is the $\mathbf{n}\times \mathbf{(\mathbf{k}+1)}$ data matrix of independent predictor variables (including a vector of ones corresponding to the intercept). The ordinary least squares (OLS) estimate for the vector of coefficients $\mathbf{\beta}$ is:

$\hat{\mathbf{\beta}} = (\mathbf{X}^{\prime} \mathbf{X})^{-1} \mathbf{X}^{\prime} \mathbf{y}$

The covariance matrix can be obtained with some handy matrix operations:
$\textrm{Var}(\hat{\mathbf{\beta}}) = (\mathbf{X}^{\prime} \mathbf{X})^{-1} \mathbf{X}^{\prime} \;\sigma^2 \mathbf{I} \; \mathbf{X} (\mathbf{X}^{\prime} \mathbf{X})^{-1} = \sigma^2 (\mathbf{X}^{\prime} \mathbf{X})^{-1}$
given that $\textrm{Var}(AX) = A \times \textrm{Var}X \times A^{\prime}; \textrm{Var}(\mathbf{y}) = \mathbf{\sigma^2}$

The standard errors of the coefficients are basically $\textrm{Diag}(\sqrt{\textrm{Var}(\hat{\mathbf{\beta}})}) = \textrm{Diag}(\sqrt{\sigma^2 (\mathbf{X}^{\prime} \mathbf{X})^{-1}})$ and with these, one can compute the t-statistics and their corresponding p-values.

Lastly, the F-statistic and its corresponding p-value can be calculated after computing the two residual sum of squares (RSS) statistics:

$\mathbf{RSS}$ – for the full model with all predictors
$\mathbf{RSS_0}$ – for the partial model ( $\mathbf{y} = \mathbf{\mu} + \mathbf{\nu}; \mathbf{\mu} = \mathop{\mathbb{E}}[\mathbf{y}]; \mathbf{\nu} \sim N(0, \sigma_0^2 \mathbf{I})$ ) with the outcome observed mean as estimated outcome

$\mathbf{F} = \frac{(\mathbf{RSS_0}-\mathbf{RSS})/\mathbf{k}}{\mathbf{RSS}/(\mathbf{n}-\mathbf{k}-1)}$

I wrote some R code to construct the output from summarizing lm objects, using all the math spewed thus far. The data used for this exercise is available in R, and comprises of standardized fertility measures and socio-economic indicators for each of 47 French-speaking provinces of Switzerland from 1888. Try it out and see for yourself the linear algebra behind linear regression.

	### Linear Regression Using lm() ----------------------------------------
	data("swiss")
	dat <- swiss
	linear_model <- lm(Fertility ~ ., data = dat)
	summary(linear_model)


	# Call:
	# lm(formula = Fertility ~ ., data = dat)
	#
	# Residuals:
	# Min 1Q Median 3Q Max
	# -15.2743 -5.2617 0.5032 4.1198 15.3213
	#
	# Coefficients:
	# Estimate Std. Error t value Pr(>\|t\|)
	# (Intercept) 66.91518 10.70604 6.250 1.91e-07 ***
	# Agriculture -0.17211 0.07030 -2.448 0.01873 *
	# Examination -0.25801 0.25388 -1.016 0.31546
	# Education -0.87094 0.18303 -4.758 2.43e-05 ***
	# Catholic 0.10412 0.03526 2.953 0.00519 **
	# Infant.Mortality 1.07705 0.38172 2.822 0.00734 **
	# ---
	# Signif. codes: 0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
	#
	# Residual standard error: 7.165 on 41 degrees of freedom
	# Multiple R-squared: 0.7067, Adjusted R-squared: 0.671
	# F-statistic: 19.76 on 5 and 41 DF, p-value: 5.594e-10


	### Using Linear Algebra ------------------------------------------------

	y <- matrix(dat$Fertility, nrow = nrow(dat))
	X <- cbind(1, as.matrix(x = dat[,-1]))
	colnames(X)[1] <- "(Intercept)"

	# N x k matrix
	N <- nrow(X)
	k <- ncol(X) - 1 # number of predictor variables (ergo, excluding Intercept column)

	# Estimated Regression Coefficients
	beta_hat <- solve(t(X)%%X)%%(t(X)%*%y)

	# Variance of outcome variable = Variance of residuals
	sigma_sq <- residual_variance <- (N-k-1)^-1 * sum((y - X %*% beta_hat)^2)
	residual_std_error <- sqrt(residual_variance)

	# Variance and Std. Error of estimated coefficients of the linear model
	var_betaHat <- sigma_sq * solve(t(X) %*% X)
	coeff_std_errors <- sqrt(diag(var_betaHat))

	# t values of estimates are ratio of estimated coefficients to std. errors
	t_values <- beta_hat / coeff_std_errors

	# p-values of t-statistics of estimated coefficeints
	p_values_tstat <- 2 * pt(abs(t_values), N-k, lower.tail = FALSE)

	# assigning R's significance codes to obtained p-values
	signif_codes_match <- function(x){
	ifelse(x <= 0.001,"***",
	ifelse(x <= 0.01,"**",
	ifelse(x < 0.05,"*",
	ifelse(x < 0.1,"."," "))))

	}
	signif_codes <- sapply(p_values_tstat, signif_codes_match)

	# R-squared and Adjusted R-squared (refer any econometrics / statistics textbook)
	R_sq <- 1 - (N-k-1)residual_variance / (Nmean((y - mean(y))^2))
	R_sq_adj <- 1 - residual_variance / ((N/(N-1))*mean((y - mean(y))^2))

	# Residual sum of squares (RSS) for the full model
	RSS <- (N-k-1)*residual_variance
	# RSS for the partial model with only intercept (equal to mean), ergo, TSS
	RSS0 <- TSS <- sum((y - mean(y))^2)

	# F statistic based on RSS for full and partial models
	# k = degress of freedom of partial model
	# N - k - 1 = degress of freedom of full model
	F_stat <- ((RSS0 - RSS)/k) / (RSS/(N-k-1))

	# p-values of the F statistic
	p_value_F_stat <- pf(F_stat, df1 = k, df2 = N-k-1, lower.tail = FALSE)

	# stitch the main results toghether
	lm_results <- as.data.frame(cbind(beta_hat, coeff_std_errors,
	t_values, p_values_tstat, signif_codes))
	colnames(lm_results) <- c("Estimate","Std. Error","t value","Pr(>\|t\|)","")


	### Print out results of all relevant calcualtions -----------------------


	print(lm_results)
	cat("Residual standard error: ",
	round(residual_std_error, digits = 3),
	" on ",N-k-1," degrees of freedom",
	"\nMultiple R-squared: ",R_sq," Adjusted R-squared: ",R_sq_adj,
	"\nF-statistic: ",F_stat, " on ",k-1," and ",N-k-1,
	" DF, p-value: ", p_value_F_stat,"\n")

	# Estimate Std. Error t value Pr(>\|t\|)
	# (Intercept) 66.9151816789654 10.7060375853301 6.25022854119771 1.73336561301153e-07 ***
	# Agriculture -0.172113970941457 0.0703039231786469 -2.44814177018405 0.0186186100433133 *
	# Examination -0.258008239834722 0.253878200892098 -1.01626779663678 0.315320687313066
	# Education -0.870940062939429 0.183028601571259 -4.75849159892283 2.3228265226988e-05 ***
	# Catholic 0.104115330743766 0.035257852536169 2.95296858017545 0.00513556154915653 **
	# Infant.Mortality 1.07704814069103 0.381719650858061 2.82156849475775 0.00726899472564356 **

	# Residual standard error: 7.165 on 41 degrees of freedom
	# Multiple R-squared: 0.706735 Adjusted R-squared: 0.670971
	# F-statistic: 19.76106 on 4 and 41 DF, p-value: 5.593799e-10

view raw lm_linear_algebra.R hosted with ❤ by GitHub

Hope this was useful and worth your time!

I have been doing Gilbert Strang’s linear algebra assignments, some of which require you to write short scripts in MatLab, though I use GNU Octave (which is kind of like a free MatLab). I was trying out this problem:

To solve this quickly, it would have been nice to have a function that would give a list of permutation matrices for every n-sized square matrix, but there was none in Octave, so I wrote a function permMatrices which creates a list of permutation matrices for a square matrix of size n.

	% function to generate permutation matrices given the size of the desired permutation matrices
	function x = permMatrices(n)
	x = zeros(n,n,factorial(n));
	permutations = perms(1:n);
	for i = 1:size(x,3)
	x(:,:,i) = eye(n)(permutations(i,:),:);
	end
	endfunction

view raw permMatrices.m hosted with ❤ by GitHub

For example:

The MatLab / Octave code to solve this problem is shown below:

	% Solution for part (a)
	p = permMatrices(3);
	n = size(p,3); % number of permutation matrices
	v = zeros(n,1); % vector of zeros with dimension equalling number of permutation matrices
	% check for permutation matrices other than identity matrix with 3rd power equalling identity matrix
	for i = 1:n
	if p(:,:,i)^3 == eye(3)
	v(i,1) = 1;
	end
	end
	v(1,1) = 0; % exclude identity matrix
	ans1 = p(:,:,v == 1)

	% Solution for part (b)
	P = permMatrices(4);
	m = size(P,3); % number of permutation matrices
	t = zeros(m,1); % vector of zeros with dimension equalling number of permutation matrices
	% check for permutation matrices with 4th power equalling identity matrix
	for i = 1:m
	if P(:,:,i)^4 == eye(4)
	t(i,1) = 1;
	end
	end
	% print the permutation matrices
	ans2 = P(:,:,t == 0)