Recall the Multivariate Normal Density function below:

\(\phi(\textbf{x}) = \left(\frac{1}{2\pi}\right)^{p/2}|\Sigma|^{-1/2}\exp\{-\frac{1}{2}(\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu})\}\)

You will note that this density function, \(\phi(\textbf{x})\), only depends on x through the squared Mahalanobis distance:

\((\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu})\)

This is the equation for a hyper-ellipse centered at \(\mu\).

For a bivariate normal, where p = 2 variables, we have an ellipse as shown in the plot below:

Useful facts about the Exponent Component: \( (\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu})\)

• All values of x such that \( (\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu})=c\) for any specified constant value c have the same value of the density \(f(x)\) and thus have equal likelihood.
• As the value of \((\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu})\) increases, the value of the density function decreases. The value of \((\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu})\) increases as the distance between x and \(\mu\) increases.
• The variable \(d^2=(\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu})\) has a chi-square distribution with p degrees of freedom.
• The value of d ² for a specific observation \(\textbf{x}_j\) is called a squared Mahalanobis distance.

If we define a specific hyper-ellipse by taking the squared Mahalanobis distance equal to a critical value of the chi-square distribution with p degrees of freedom and evaluate this at \(α\), then the probability that the random value X will fall inside the ellipse is going to be equal to \(1 - α\).

\(\text{Pr}\{(\textbf{x}-\mathbf{\mu})'\Sigma^{-1}(\textbf{x}-\mathbf{\mu}) \le \chi^2_{p,\alpha}\}=1-\alpha\)

This particular ellipse is called the \((1 - α) \times 100%\) prediction ellipse for a multivariate normal random vector with mean vector \(\mu\) and variance-covariance matrix \(Σ\).