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:)
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.