Chi-Square Distribution
Definition
Chi-square distribution is often used to check state estimators for "consistency" \(-\) whether their actual errors are consistent with the variances calculated by the estimator. A scalar random variable \(q\) is chi-square distributed with \(n\) degrees of freedom and is written as:
\(q\) is sum of squares of random variables \(z_i \sim \mathcal{N}(0, 1)\) for \(i = 1, \ldots n\):
Consider \(\mathbf{x} \in \mathbb{R}^n\) with \(\mathbf{x} \sim \mathcal{N}(\bar{\mathbf{x}}, \mathbf{P})\). The whitened version of \(\mathbf{x}\) is \(\mathbf{z} \triangleq \mathbf{P}^{-\frac{1}{2}}(\mathbf{x} - \bar{\mathbf{x}})\), where \(\mathbf{P}^{-\frac{1}{2}}\) is a symmetric matrix satisfying \(\mathbf{P}^{-\frac{1}{2}} \mathbf{P} \mathbf{P}^{-\frac{1}{2}} = \mathbf{I}\). Since \(\mathbf{x} - \bar{\mathbf{x}}\) is Gaussian with zero mean and covariance \(\mathbf{P}\), applying the linear transformation \(\mathbf{P}^{-\frac{1}{2}}\) yields to:
Then the quadratic form can be described as:
Properties
| Chi-Square Distribution | |
|---|---|
| Mean | \(\begin{align*} \mathbb{E}\left[q \right] = \mathbb{E}\left[ \sum^{n}_{i = 1} z^2_i \right] = n \end{align*}\) |
| Variance | \(\begin{align*} \mathcal{V}\text{ar}(q) &= \mathbb{E}\left[\sum^{n}_{i = 1} (z^2_i - 1) \right]^2 = \sum^{n}_{i = 1} \mathbb{E}\left[(z^2_i - 1)^2 \right] = 2n \end{align*}\) |
| \(\begin{align*}p(q) = \frac{1}{2^{\frac{n}{2}} \Gamma\left( \frac{n}{2} \right)} q^{\frac{n - 2}{2}} e^{-\frac{q}{2}}, \quad q \geq 0 \end{align*}\), where \(\Gamma\) is the gamma function with the following properties: \(\begin{align*} \Gamma\left( \frac{1}{2} \right) = \sqrt{\pi}, \ \ \Gamma(1) = 1, \ \ \Gamma(m + 1) = m \Gamma(m) \end{align*}\) |
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| Addition Rule | Given the independent random variables \(q_1 \sim \chi^2_{n_1}\) and \(q_2 \sim \chi^2_{n_2}\), their sum \(q_3 = q_1 + q_2\) is chi-square distributed with \(n_3 = n_1 + n_2\) degrees of freedom |