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Matrix multiplication

Peter Ralph

1 December 2020 – Advanced Biological Statistics

Math minute: matrix multiplication

To simulate from: \[\begin{aligned} \mu_i &= b_0 + b_1 X_{i1} + \cdots + b_k X_{ik} \\ Y_i &\sim \Normal(\mu_i, \sigma) . \end{aligned}\]

either

coefs <- list(b0=1.0, b=c(3.0, -1.0, 0.0, 0.0),
              sigma=0.5)
n <- 200
X <- matrix(rnorm(4*n, mean=0, sd=3), ncol=4)
Y <- coefs$b0 
for (k in 1:ncol(X)) {
    Y <- Y + coefs$b[k] * X[,k]
}
Y <- Y + rnorm(n, mean=0, sd=coefs$sigma)

To simulate from: \[\begin{aligned} \mu_i &= b_0 + b_1 X_{i1} + \cdots + b_k X_{i1} \\ Y_i &\sim \Normal(\mu_i, \sigma) . \end{aligned}\]

coefs <- list(b0=1.0, b=c(3.0, -1.0, 0.0, 0.0),
              sigma=0.5)
n <- 200
X <- matrix(rnorm(4*n, mean=0, sd=3), ncol=4)
Y <- coefs$b0 + X %*% coefs$b + rnorm(n, mean=0, sd=coefs$sigma)

Because

In R, %*% is matrix multiplication: if

\(b\) is a \(k\)-vector
\(X\) is an \(n \times k\) matrix

then X %*% b (or, \(X b\) in math notation) is shorthand for the \(n\)-vector \[ (Xb)_i = \sum_{j=1}^k X_{ij} b_j . \]

In Stan, matrix multiplication is *.