Part 3: Time Series I

Purpose of Time Series Analysis

Harmonic Analysis

What Does Each Harmonic Mean?

Orthogonality

Multiple Regression (shown before)

Fourier Coefficients

Amplitude of Harmonics

Fraction of Variance Explained by Harmonics

How Many Harmonics Do We Need?

Power Spectrum

Problems with Line Spectrum

Continuous Spectrum

Aliasing

How to Calculate Continuous Spectrum

Examples

Time-Lag Correlation Method

Resolution of Spectrum - Bandwidth

Bandwidth in Time-Lag Correlation Method

Autocorrelation Function

Autocorrelation Function – cont.

Example for Periodic Time Series

Example – Red Noise

Example – White Noise

Example – Noise + Periodic

Typical Autocorrelation Function

Degree of Freedom

An Example

Parseval’s Theorem

Example – Spectrum of Red Noise

Power Spectrum of Red Noise

How To Plot Power Spectrum?

Data Window

Power Spectrum of Finite Sample

Square Data Windows

The Weighting Effect of Square Window

Tapered Data Window

We Wish the Data Window Can…

Bartlett Window

Hanning Window (Cosine Bell)

Significance Test of Spectral Peak

Filtering of Time Series

Response Function

An Perfect Filter

A Sharp-Cutoff Filter