Time series analysis

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  • 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)


  • 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


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


  • 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