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DSCI 574: Spatial and Temporal Models

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

Assessments

Labs

# Lab topic Due Date
1 Use R to perform certain time series analyses including plots, smoothing, decomposition, computation of the sample autocorrelation function, lagging and differencing. 2019-02-09
2 Stochastic models for time series: simulation and model selection. 2019-02-16
3 Use R to select, fit and forecast based on the ARMA family of time series models. 2019-03-02
4 Explore visualization techniques for spatial data (R package gstat). Fit popular variogram models to spatial data; perform prediction using classical kriging. 2019-03-09

Quizzes

Time Date Location
1 15:00 - 15:30 2019-02-27 In your lab section
2 10:00 - 10:30 2019-03-13 DPM 301

Lecture

# Date Day Topic
1 2019-02-05 Tue Intro to time series; Chapter 1: Exploratory techniques in time series analysis
2 2019-02-07 Thur Continuing Chapter 1: Exploratory techniques in time series analysis
3 2019-02-12 Tue Chapter 2: Stochastic models for time series
4 2019-02-14 Thur Continuing Chapter 2: Stochastic models for time series
5 2019-02-26 Tue Chapter 3: Estimation and Model Fitting for Time Series
6 2019-02-28 Thur Chapter 4: Prediction for Time Series
7 2019-03-05 Tue Chapter 5: Spatial data and spatial processes
8 2019-03-07 Thur Chapter 6: Methods for spatial prediction

Course Learning Objectives

  • 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.

Reference Material

  • 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.