Modelling Spatiotemporal Processes

Course overview

Literature

• C. Chatfield, The analysis of time series: an introduction. Chapman and Hall: chapters 1, 2 and 3 (Chatfield 2003)
• R. Hyndman, G. Athanasopoulos: Foreasting: Principles and Practice
• Spatial Data Science, with applications in R (Pebesma and Bivand 2022):
  • Ch 1 (intro), 7 (sf, stars)
  • Ch 12 (interpolation)

Organization

Teachers:

• Christian Knoth (exercises, Wed 12-14)
• Edzer Pebesma (lectures)

Learnweb:

• subscribe
• no password
• Lectures + exercises is only one course.

Slides:

• rendered: https://edzer.github.io/mstp/
• qmd (quarto) sources on on http://github.com/edzer/mstp
• you can load and run the individual qmd files in rstudio
• pull requests with improvements are appreciated (and may be rewarded):
  • fork the repository on GitHub,
  • click “Edit this page” on the right-hand-side

Examen:

• multiple choice, 4 possibilities, 40 questions, 20 need to be correct.

Overview of the course

Topics:

• Time series data
• Time series models: AR(p), MA(q), partial correlation, AIC, forecasting
• Optimisation:
  • Linear models, least squares: normal equations
  • Non-linear:
    • One-dimensional: golden search
    • Multi-dimensional least squares: Newton
    • Multi-dimensional stochastic search: Metropolis
    • Multi-dimensional stochastic optimisation: Metropolis
• Spatial models:
  • Simple, heuristic spatial interpolation approaches
  • Spatial correlation
  • Regression with spatially correlated data
  • Kriging: best linear (unbiased) prediction
  • Stationarity, variogram
  • Kriging varieties: simple, ordinary, universal kriging
  • Kriging as a probabilistic spatial predictor
• Spatio-temporal variation modelled by partial differential equations
  • Initial and boundary conditions
  • Example
  • Calibration: Kalman filter

Where we come from

• introduction to geostatistics
• mathematics, linear algebra
• computer science

Introduction to geostatistics

• types of variables: Stevens’ measurement scales – nominal, ordinal, interval, ratio
• … or: discrete, continuous
• t-tests, ANOVA
• regression, multiple regression (but not how we compute it)
• assumption was: observations are independent
• what does independence mean?

In this course

• we will study dependence in observations, in
  • space
  • time
  • or space-time
• in space and/or time, Stevens’ measurement scales are not enough! Examples:
  • linear time, cyclic time
  • space: functions, fields
• we will study how we can represent phenomena, by
  • mathematical representations (models)
  • computer representations (models)
• we will consider how well these models correspond to our observations

Spatio-temporal phenomena are everywhere

• if we think about it, there are no data that can be non-spatial or non-temporal.
• in many cases, the spatial or temporal references are not essential
  • think: brain image of a person: time matters, but mostly referenced with respect to the age of the person, spatial location of the MRI scanner does not
  • but: ID of the patient does!
  • and: time of scan matters too!
• we will “pigeon-hole” (classify) phenomena into: fields, objects, aggregations

Fields

• many processes can be represented by fields, meaning they could be measured everywhere
• think: temperature in this room
• typical problems: interpolation, patterns, trends, temporal development, forecasting?

Objects and events

• objects can be identified
• objects are identified within a frame (or window) of observation
• within this window, between objects, there are no objects (no point of interpolation)
• objects can be moving (people), or static (buildings)
• objects or events are sometimes obtained by thresholding fields, think heat wave, earthquake, hurricane, (see e.g. Camara et al. 2014)
• sometimes this view is rather artificial, think cars, persons, buildings

Fields - objects/events conversions

• we can convert a field into an object by thresholding (wind field, storm or hurricane)
• we can convert objects into a field e.g. by computing the density as a continuous function

Aggregations

• we can aggregate fields, or objects, but do this differently:
• population can be summed, temperature cannot (see intensive/extensive properties)

Aims of modelling

… could be

• curiousity
• fun: studying models is easier than measuring the world around us

More scientific aims of modelling are

• to learn about the world around us
• to predict the past, current or future, in case where measurement is not feasible.

What is a model?

• conceptual models, e.g. the water cycle (wikipedia:)

the water cycle, updated

• object models, such as UML (wikipedia:)
• mathematical models, such as Navier Stokes’ equation, (wikipedia:)

What is a mathematical model?

A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system, quoting (Eykhoff 1974) a mathematical model is:

a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form

In the natural sciences, a model is always an approximation, a simplification of reality. If degree of approximation meets the required accuracy, the model is useful, or valid (of value). A validated model does not imply that the model is “true”; more than one model can be valid at the same time.

Camara, Gilberto, Max J Egenhofer, Karine Ferreira, Pedro Andrade, Gilberto Queiroz, Alber Sanchez, Jim Jones, and Lubia Vinhas. 2014. “Fields as a Generic Data Type for Big Spatial Data.” In International Conference on Geographic Information Science, 159–72. Springer. https://link.springer.com/content/pdf/10.1007/978-3-319-11593-1_11.pdf.

Chatfield, Chris. 2003. The Analysis of Time Series: An Introduction. Chapman; hall/CRC.

Eykhoff, Pieter. 1974. System Identification. Vol. 14. Wiley London.

Pebesma, Edzer, and Roger Bivand. 2022. Spatial Data Science, with Applications in R. online. https://r-spatial.org/book/.