Model fitting and prediction in the presence of correlation due to temporal and/or spatial association. ARIMA models and Gaussian processes.

Labs

Quizzes

• Chapter 1: Exploratory techniques in time series analysis

  • Informally define and explain terminology used to describe time series, including trend, seasonal effects, cyclical effects, outlier and white noise.
  • Recognize when curve–fitting may be an appropriate method for modelling a series.
  • Describe models for seasonal variation, including additive and multiplicative models.
  • Apply a filter (that is, a smoother) to a time series, centering if necessary.
  • Use a filter to estimate the seasonal indices in a time series that has an additive seasonal component.
  • Recognize the role of transformations for time series, and identify possible transformations to address certain features of series, such as a non-constant variance and multiplicative seasonal effects.
  • Define the sample autocorrelation function and the correlogram.
  • Describe the behaviour of the correlogram for series that alternate, have a trend or show seasonal fluctuations.

• Chapter 2: Stochastic models for time series

  • Define the autocovariance and autocorrelation functions for a time series model.
  • Define and explain what it means to say that a process is (weakly) stationary.
  • Define what is meant by a white noise process.
  • Define a moving average process of order q, i.e., an MA(q) .
  • Derive the mean, variance and autocovariance function of a stationary MA(q) process.
  • Define the notion of invertibility of a process.
  • Define an autoregressive process of order p, i.e., an AR(p) .
  • Derive properties for an AR(1) , including the mean, variance and autocorrelation function.
  • Define when an AR(p) is 
stationary.
  • Define an ARMA(p, q) process and state conditions when an ARMA(p, q) process is stationary and/or invertible.

• Chapter 3: Estimation and model fitting for time series

  • Given a class of ARMA models, list the main steps for fitting a suitable model to the data.
  • Describe estimation of the mean of the process, and of its autocovariance and autocorrelation functions. Know their statistical properties.
  • Use the correlogram to decide which model from the ARMA family is suitable for the data.
  • For an AR model, describe how to select the order of the process using the partial autocorrelation function and how to fit the remaining model parameters.
  • For an MA process, be able to determine the order of the process and describe how to carry parameter estimation.
  • Use the correlogram to identify situations when an ARMA model is suitable, and use software to fit the model.
  • Apply model-selection criteria to choose among possible models.

• Chapter 4: Prediction for time series

  • Describe how exponential smoothing technique can be used to make forecasts for stationary time series data.
  • Outline the steps of Box-Jenkins forecasting procedure.
  • Compute Box-Jenkins forecasts using the model equation.
  • Use the MA representation of the model to construct prediction intervals for the forecasts.

• Chapter 5: Spatial data and spatial processes

  • Use tools in R package gstat to visualize point referenced spatial data and identify its features to guide subsequent modelling.
  • Define a spatial random field and describe its possible representation in terms of an overall spatial trend plus a process having a spatial structure.
  • Define second-order and isotropic stationarity.
  • Define a Gaussian random field.
  • Define a variogram and covariance function, and recognize their role as measures of dependence over space.

• Chapter 6: Methods for spatial prediction

  • Fit popular variogram models to spatial data and explain features such as the nugget, sill and range.
  • Understand when a linear spatial predictor is appropriate. Apply classical kriging to make spatial predictions.
  • Be familiar with modern methods of spatial prediction such as Bayesian kriging.

• Shaddick, Gavin and Zidek, James V. Spatio-Temporal Methods in Environmental Epidemiology. CRC Press, 2016.
• Chatfield, Chris. The Analysis of Time Series: An Introduction. CRC Press, 2003.