Time series analysis
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Contents
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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