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# Modelling spatio-temporal processes
## 0. How do R markdown files work?
* load them with rstudio
* click the `knit` button
* debug/step through:
* load all code up to here-button
* ctrl-Enter: run this line
* output in console, objects appear in Environment browser
## 1. Course overview, time series
### 1.1 Literature
* C. Chatfield, The analysis of time series: an introduction. Chapman and Hall: chapters 1, 2 and 3
* Applied Spatial Data Analysis with R, by R. Bivand, E. Pebesma and V. Gomez-Rubio (Springer;
[first edition](http://www.springer.com/978-0-387-78170-9) or its [second edition](http://www.springer.com/statistics/life+sciences%2C+medicine+%26+health/book/978-1-4614-7617-7)):
* Ch 1, 2, 3
* Ch 4, 5
* 1st ed, Ch 6 (customizing classes for spatial data) or 2nd ed, Ch 6 (spatio-temporal data)
* Ch 8 (geostatistics)
### 1.2 Organization
Teachers:
* Christian Knoth (exercises, Wed 12-14)
* Edzer Pebesma (lectures)
Learnweb:
* subscribe
* no password
* Lecture+exercise is only one course.
Slides:
* html on http://edzer.github.io/mstp/
* Rmd sources on on http://github.com/edzer/mstp
* you can run the Rmd files in rstudio (http://www.rstudio.com/)
* pull requests with improvements are appreciated (and may be rewarded)
### 1.3 examen:
* multiple choice, 4 possibilities, 40 questions, 20 need to be correct.
### 1.4 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
## 2. Where we come from
+ introduction to geostatistics
+ mathematics, linear algebra
+ computer science
### 2.1 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?
### 2.2 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
## 3. 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
### 3.1 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?
### 3.2 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.
http://ifgi.uni-muenster.de/~j_jone02/publications/GIScience2014.pdf
+ sometimes this view is rather artificial, think cars, persons, buildings
### 3.3 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
### 3.4 aggregations
+ we can aggregate fields, or objects, but do this differently:
+ population can be summed, temperature cannot
## 3.5 Aims of modelling
... could be
+ curiousity
+ 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.
### 3.5 What is a model?
+ conceptual models (the water cycle: http://en.wikipedia.org/wiki/File:Water_cycle.png)
+ object models (e.g., UML: http://en.wikipedia.org/wiki/File:UML_diagrams_overview.svg)
+ mathematical models, such as Navier Stokes' equation,
### 3.6 What is a mathematical model?
A mathematical model is an abstract model that uses mathematical
language to describe the behaviour of a system.
> 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 (P. Eykhoff, 1974, System Identification, J. Wiley, London.)
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.