# Time series analysis

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## Contents

## Links

## Vocab/parlance

- Stochastic process
- Adjacent points in time are correlated
- Parsimonious explanation
- white noise - uncorrelated random
- dominated by oscillatory behavior
- random walk with drift
- signal-to-noise-ratio (SNR)

## General

- two approaches: time domain approach and frequency domain approach
- moving averages, smoothing, filtering
- autoregression - e.g. a prediction of the current value as a function of the last two values
- dependent on initial conditions

## Auto covariance and auto correlation

- Autocovariance
- Autocorrelation
- The lack of independence between two adjacent values x_s and x_t can be assessed numerically using the notions of covariance and correlation.

### Autocovariance function

- Auto covariance measures the linear depencence between two points on the same series observed at different times
- For s = t, the autocovariance reduces to the variance because
- Autovariance function is independent of time

### ACF: Autocorrelation function

- ACF measures linear predictability of the series at time t (i.e., x_t) using only the value x_s
- Because it is a correlation

### Cross-covariance function

- Cross-covariance function between two series x_t and y_t

### CCF: Cross-correlation function

### Stationarity

- Mean is the is constant (does not depend on time) and autocovariance depends only on time *difference*
- A random walk is not stationary

### Lag

- Let h = time shift or lag where s = t + h
- autocovariance of stationary time series
- autocorrelation function of stationary time series

### Trend stationarity

- e.g. for a function

### Autoregressive models example

- where w_t is white noise