Glossary

Notation and terms used in time series analysis.

Notation

Series Characteristics

  • $X_t$ - the response variable.
  • $x_t$ - the value of $X_t$ at a particular time $t$.
  • $a_t$ - the series white noise.
  • $\mu_t$ - the mean of all possible realizations of $X_t$ for a given $t$.
  • $\sigma_t$ - the varaince of all possible realizations of $X_t$ for a given $t$.
  • $\sigma_a$ - the white noise varaince of all possible realizations of $X_t$ for a given $t$.
  • $\gamma_k$ - the autocovariance of $X_t$ for lag of $k$.
  • $\rho_k$ - the autocorrelation of $X_t$ for lag of $k$.
  • $S_x(f)$ - series spectral desnsity.

Periodic Signals

  • $A$ - amplitude of the periodic signal.
  • $f$ - frequency of a signal periodic signal.
  • $\omega$ - angular frequency of a periodic signal.
  • $\phi$ - phase shift of a periodic signal.
$$ X_t = A cos \left( 2 \pi ft + \phi \right) $$

Filtering

  • $H \left( B \right)$ - transfer function

ARIMA Modeling

  • $\phi \left( B \right)$ - autoregressive polynomial.
  • $\phi \left( B \right)$ - moving average polynomial.
  • $\psi_k$ - coefficients of model in GLP form.
$$ X_t = \frac{\theta \left( B \right)}{\phi \left( B \right)} a_t $$

Forecasting

  • $l$ - a forecast step.
  • $t_0$ - the forecast horizon.
  • $\hat{X}_{t_0} \left( l \right)$ - the forecast of $X_t$ from $t_0$ at step $l$.
  • $e_{t_0} \left( l \right)$ - the forecast error; the error between $\hat{X}_{t_0} \left( l \right)$ and $X_{t_0 + l}$.

Terms

  • Alaising: A signal above the Niquist frequency that appears as a low frequency signal.
  • Autocovariance: covariance between the a series and $k$ lags of itself.
  • Autocorrelation: The autocovariance normalized by the 0th order lag autocovariance ($\gamma_0$).
  • Ensemble: The totality of all possible realizations of a time series.
  • Forecast: The extrapolation of a model from a given time horizon and steps from the horizon.
  • Frequency: The number of cycles per unit time.
  • Lag effect: the value of a variable at a previous time is correlated with the repsonse at the current time.
  • Niquist frequency: The highest observable frequency of a signal, which is half of the samlping frequency - $\frac{f_{sample}}{2}$
  • Period: The amout of time for one cycle to complete.
  • Poles: The roots of the autoregressive portion ($\phi \left( B \right)$) of an ARIMA model.
  • Psuedo-periodic Series having a similar shape that repeats in a consistent cycle.
  • Realization: The observed time series. The may be many or only one.
  • Spectral Density: The frequency components (in the frequency domain) that exist in a time series; the frequency transfrom of a time series.
  • Sample Spectral Density: An estimate of the Spectral Density based on periodically sampled data.
  • Transfer Function: A ratio of polynomials that transforms a series i.e. $X_t$ -> $Z_t$.
  • Zeros: The roots of the moving average portion ($\theta \left( B \right)$) of an ARIMA model.

References

  • [1] B. Salder, "Stationary", SMU, 2019
  • [2] B. Salder, "Frequency Domain", SMU, 2019