Quadratic Form

This long post covers the quadratic form and the positive definiteness of matrices. The decomposition of symmetric matrices is slightly touched on, and the entire post is mainly to prepare for the next chapter -- eigenvalues and eigenvectors.

Decomposition and Smoothing Methods

Decomposition procedures to extract trend, seasonal and other components from a time series. Smoothing techniques like moving average and Lowess are often used, and exponential smoothing (Holt-Winters) is another powerful tool.

Determinant

The determinant is a very important concept for square matrices, and its properties are key to various other notions such as block matrices and matrix inverses.

Projection Matrix

We introduce idempotent matrices and the projection matrix. Both are very important concepts in statistical analyses such as linear regression.

Generalized Inverse

The generalized matrix inverse that applies to any $m \times n$ matrix.

Seasonal Time Series

We introduce seasonal differencing, seasonal ARMA models, and combine them to get SARIMA models.

Matrix Inverse

...for a nonsingular matrix. We talk about left and right inverses, *the* matrix inverse and orthogonal matrices.

Variability of Nonstationary Time Series

Using the Box-Cox power transformation to stabilize the variance. At the end of this section, the standard procedure for fitting an ARIMA model is discussed.

Unit Root Test

A test that helps us determine whether differencing is needed or not. We also talk about over-differencing (don't do it!) and model selection (AIC/BIC and MAPE).

Matrix Trace

Such a simple concept with so many properties and applications!