Quadratic form (statistics)

In multivariate statistics, if $\epsilon$ is a vector of $n$ random variables, and $\Lambda$ is an $n$ -dimensional symmetric matrix, then the scalar quantity $\epsilon^T\Lambda\epsilon$ is known as a quadratic form in $\epsilon$ .

Expectation

It can be shown that^[1]

\operatorname{E}\left[\epsilon^T\Lambda\epsilon\right]=\operatorname{tr}\left[\Lambda \Sigma\right] + \mu^T\Lambda\mu

where $\mu$ and $\Sigma$ are the expected value and variance-covariance matrix of $\epsilon$ , respectively, and tr denotes the trace of a matrix. This result only depends on the existence of $\mu$ and $\Sigma$ ; in particular, normality of $\epsilon$ is not required.

A book treatment of the topic of quadratic forms in random variables is ^[2]

Proof

Since the quadratic form is a scalar quantity $\operatorname{E}\left[\epsilon^T\Lambda\epsilon\right] = \operatorname{tr}(\operatorname{E}[\epsilon^T\Lambda\epsilon])$ . Note that both $\operatorname{E}$ and $\operatorname{tr}$ are linear operators, so $\operatorname{E} \circ \operatorname{tr} = \operatorname{tr} \circ \operatorname{E}$ . It follows that

\operatorname{E}\left[\epsilon^T\Lambda\epsilon\right] = \operatorname{E}[\operatorname{tr}(\epsilon^T\Lambda\epsilon)],

and that, by the cyclic property of the trace operator,

\operatorname{E}[\operatorname{tr}(\epsilon^T\Lambda\epsilon)] = \operatorname{E}[\operatorname{tr}(\Lambda\epsilon\epsilon^T)] = \operatorname{tr} (\Lambda(\Sigma + \mu\mu^T)) = \operatorname{tr}(\Lambda\Sigma) + \mu^T\Lambda\mu.

Variance

In general, the variance of a quadratic form depends greatly on the distribution of $\epsilon$ . However, if $\epsilon$ does follow a multivariate normal distribution, the variance of the quadratic form becomes particularly tractable. Assume for the moment that $\Lambda$ is a symmetric matrix. Then,

\operatorname{var}\left[\epsilon^T\Lambda\epsilon\right]=2\operatorname{tr}\left[\Lambda \Sigma\Lambda \Sigma\right] + 4\mu^T\Lambda\Sigma\Lambda\mu

In fact, this can be generalized to find the covariance between two quadratic forms on the same $\epsilon$ (once again, $\Lambda_1$ and $\Lambda_2$ must both be symmetric):

\operatorname{cov}\left[\epsilon^T\Lambda_1\epsilon,\epsilon^T\Lambda_2\epsilon\right]=2\operatorname{tr}\left[\Lambda _1\Sigma\Lambda_2 \Sigma\right] + 4\mu^T\Lambda_1\Sigma\Lambda_2\mu

Computing the variance in the non-symmetric case

Some texts incorrectly state that the above variance or covariance results hold without requiring $\Lambda$ to be symmetric. The case for general $\Lambda$ can be derived by noting that

\epsilon^T\Lambda^T\epsilon=\epsilon^T\Lambda\epsilon

\epsilon^T\tilde{\Lambda}\epsilon=\epsilon^T\left(\Lambda+\Lambda^T\right)\epsilon/2

But this is a quadratic form in the symmetric matrix $\tilde{\Lambda}=\left(\Lambda+\Lambda^T\right)/2$ , so the mean and variance expressions are the same, provided $\Lambda$ is replaced by $\tilde{\Lambda}$ therein.

Examples of quadratic forms

In the setting where one has a set of observations $y$ and an operator matrix $H$ , then the residual sum of squares can be written as a quadratic form in $y$ :

\textrm{RSS}=y^T\left(I-H\right)^T\left(I-H\right)y.

For procedures where the matrix $H$ is symmetric and idempotent, and the errors are Gaussian with covariance matrix $\sigma^2I$ , $\textrm{RSS}/\sigma^2$ has a chi-squared distribution with $k$ degrees of freedom and noncentrality parameter $\lambda$ , where

k=\operatorname{tr}\left[\left(I-H\right)^T\left(I-H\right)\right]

\lambda=\mu^T\left(I-H\right)^T\left(I-H\right)\mu/2

may be found by matching the first two central moments of a noncentral chi-squared random variable to the expressions given in the first two sections. If $Hy$ estimates $\mu$ with no bias, then the noncentrality $\lambda$ is zero and $\textrm{RSS}/\sigma^2$ follows a central chi-squared distribution.

References

↑ Douglas, Bates. "Quadratic Forms of Random Variables". STAT 849 lectures. Retrieved August 21, 2011.
↑ Mathai, A. M. and Provost, Serge B. (1992). Quadratic Forms in Random Variables. CRC Press. p. 424. ISBN 978-0824786915.