ARIMA Models

Combining differencing and ARMA models and we get ARIMA. The procedures of estimation, diagnosis and forecasting are very similar as that of ARMA models.

Linear Space of Matrices

The column space, row space and rank of a matrix and their properties.

Mean Trend

We introduce detrending and differencing, two methods that aim to remove the mean trends in time series.

Orthogonalization

Introducing the Gram-Schmidt process, a method for constructing an orthogonal basis given a non-orthogonal basis.

Projection

Geometrically speaking, what is the projection of a vector onto another vector, and the projection of a vector onto a subspace?

Definitions in Arbitrary Linear Space

This chapter provides an introduction to some fundamental geometrical ideas and results. We start by giving definitions for norm, distance, angle, inner product and orthogonality. The Cauchy-Schwarz inequality comes useful in many settings.

Model Fitting and Forecasting

This model-building strategy consists of three steps: model specification (identification), model fitting, and model diagnostics.

ARMA Model

The mean, variance, ACF and PACF of ARMA models. The backshift operator is introduced, and the stationarity and invertibility of the general ARMA(p, q) model is discussed.

Moving Average Model

The mean, variance, ACF and PACF of moving average models. Instead of stationarity, a new property called invertibility is introduced.

Vector Space

We introduce some basic terminology - vector space, subspace, span, basis, and dimension. These concepts lay the foundation for future discussions on matrices and matrix properties.