syllabus.Rmd

---
title: "Time Series for Climatologists and Ecologists: Syllabus"
author: "Andrew C Parnell and Doug McNeall"
date: "26 April 2016"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# Module 1: Time series analysis in climatology and ecology: some examples and general goals (DM)
Tuesday 10th May 9:15-10:00

------

# Module 2: An introduction to Bayesian statistics (AP)
Tuesday 10th May 10:45-11:30

Learning outcomes:

- Understand the terms prior, likelihood and posterior
- Know what a posterior probability distribution is, and why we take samples from it
- Know how to formulate of a linear regression model in a Bayesian format
- Be able to suggest appropriate prior distributions for different situations

Slides:

1. Title
2. Who was Bayes?
3. Why Bayes?
4. Bayes theorem in maths and English
5. A very basic Bayesian model with R code
6. Understanding the different parts of a Bayesian model
7. Lots of probability distributions
8. Choosing a likelihood and a prior
9. An example: linear regression
10. Simulating from the prior and the likelihood
11. Posterior computation in JAGS
12. Calculating the posterior vs sampling from it
13. Things you can do with posterior samples
14. Summary so far: for and against Bayes
15. How to create Bayesian models: a general recipe (start with the data, fit it into a framework, LR, GLM, TS, then look at the parameters, and think of what priors are suitable)
16. Checking assumptions (e.g. residuals)
17. The danger of vague priors
18. Replication in Science and the horror of p-values
19. Bayesian time series, some general notation
20. General tips: build one model for all the data, use informative priors, check your model

------

# Module 3: The JAGS software with simple examples (AP)
Tuesday 10th May 12:00-12:45

Learning outcomes:

- Understand the code and the output from a JAGS model
- Be able to write and run JAGS models for linear and logistic regression
- Be able to change the prior distributions in a JAGS model
- Be comfortable with plotting and manipulating the output from JAGS

Relevant JAGS files:
```
jags_linear_regression.R
jags_logistic_regression.R
```

Slides:

1. Title
2. What is JAGS?
3. Why JAGS?
4. The general framework for running jags
5. Step 1: model code
6. Step 2: setting up the data
7. Step 3: running jags
8. Setting the number of iterations, the size of the burn-in and the amount of thinning
9. Exploring the posterior distribution and checking convergence
10. Example 1. linear regression
11. Ex. model code
12. Ex. running the model
13. Getting the posterior simulations
14. Creating plots
15. Example 2. logistic regression
16. Model code. The likelihood and the link function
17. Running the model
18. Exploring the posterior
19. Checking the assumptions
20. Summary and conclusions

------

# Practical 1: Revision of R and introduction to JAGS (DM)
Tuesday 10th May 14:00-15:30

------

# Module 4: AR(1) models and Random walks (DM)
Wednesday 11th May 9:15-10:00

------

# Module 5: MA and ARIMA models (AP)
Wednesday 11th May 10:45-11:30

Learning outcomes:

- Recognise and understand the basic theory behind MA(1) and MA(q) models
- Recognise the basic ARMA(p,q) formulation
- Understand what an ARIMA(p,d,q) model is
- How to fit all of the above in JAGS

Relevant JAGS file:
```
jags_moving_average.R
jags_ARMA.R
jags_ARIMA.R
```

Slides:

1. Title
2. Learning outcomes
3. Reminder: AR models
4. Intro to Moving Average Models
5. Moving average models in the ACF/PACF
6. Example 1: MA(1)
7. Simulating from the MA(1) process
8. JAGS code
9. Extending to MA(q)
10. JAGS code
11. Combining AR and MA into ARMA
12. Combining ARMA with the random walk to produce ARIMA
13. Example: the ARIMA(1,0,1) model
14. Example: the ARIMA(1,1,1) model
15. General format: the ARIMA(p,d,q) model
16. Priors for ARIMA models
17. Example: hadcrut data
18. Predicting the future
19. Summary

------

# Module 6: ARIMAX, model choice, and forecasting (AP)
Wednesday 11th May 12:00-12:45

Learning outcomes:

- Be able to add on components to ARIMA models
- Understand the issues with fitting ARIMAX and other extensions to ARIMA models
- Understand how to create forecasts with JAGS via the NA method
- Know how to perform model choice with DIC
- Know how to perform model choice with cross-validation
- Know the basics of forecast calibration and scoring rules

Relevant JAGS file:
```
jags_ARIMAX.R
```

Slides:

1. Title
2. Learning outcomes
3. The great advantage of Bayes: bolting together models
4. Mixing up GLMs with time series models: some basic ideas
5. The ARIMAX framework
6. JAGS code for an ARIMAX model
7. Example: ARIMAX applied to the hadcrut data
8. JAGS and the NA trick
9. The NA trick for producing fitted values
10. The NA trick for producing forecasts
11. Bayesian model choice
12. The Deviance Information Criterion
13. The components of DIC 
14. Example: choosing the order of an ARIMAX model 
15. Using cross-validation to choose between models
16. Example: Estimating AR performance
17. Measuring the quality of a forecast; scoring rules
18. Summary

------

# Practical 2: Fitting ARIMA models in JAGS (DM)
Wednesday 11th May 14:00-15:30

------

# Module 7: Models with changing variance and frequency models (AP)
Thursday 12th May 9:15-10:00

Learning outcomes: 

- Understand how to fit ARCH, GARCH and SVM models in JAGS
- Know how to check assumptions for these methods
- Understand the basis of seasonal models
- Know the difference between time and frequency domain models and be able to implement a Fourier model

Relevant JAGS file:
```
jags_ARCH.R
jags_GARCH.R
jags_SVM.R
jags_Fourier.R
```

Slides:

1. Title
2. Learning outcomes
3. General principles of models for changing variance
4. Extension 1: ARCH 
5. JAGS code for ARCH models
6. Example model run
10. Example forecasts for ARCH models
11. From ARCH to GARCH
12. Example of using the GARCH model
13. Using DIC to compare ARCH and GARCH
14. Stochastic Volatility Modelling
15. JAGS code for SVMs
16. Example of SVMs and comparison of DIC
17. Time series in the frequency domain
18. Methods for estimating seasonal models
19. Estimating frequencies via a Fourier model
20. JAGS code for a Fourier model
21. Example: the Lynx data
22. Individual vs joint frequency models
23. Plotting the periodogram
24. Bayesian vs traditional frequency analysis
25. Summary

------

# Module 8: Gaussian processes for time series (DM)
Thursday 12th May 10:45-11:30

------

# Practical 3: Fitting Gaussian processes in JAGS (DM)
Thursday 12th May 12:00-12:45 and 14:00-15:00

------

# Module 9: Models for continuous time series (AP)
Friday 13th May 9:15-10:00

Learning outcomes: 

- Understand the differences between continuous and discrete time series
- Understand the basics of Brownian motion
- Extend the AR model to the Ornstein Uhlenbeck process
- Ito formulation vs Euler-Maruyama methods
- Forecasting and interpolation for continuous time models
- Fit different types of change point models

Relevant JAGS file:
```
jags_autoregressive.R (revision)
jags_BM.R
jags_OU.R
jags_changepoint.R
```

Slides:

1. Title
2. Learning outcomes
3. Discrete time vs continuous time
4. Some models for continuous time we have already met
5. Brownian motion
6. JAGS code for simple Brownian Motion
7. Example: ice core data
8. Revision of the AR(1) process
9. The Ornstein Uhlenbeck process
10. JAGS code for the OU process
11. Example: ice core data
12. Comparing between BM and OU
13. Ito vs Euler-Marayuma forms
14. Forecasting and interpolation for the BM and OU processes
15. Introduction to change point models
16. Dscontinuous change point models
17. JAGS code
18. Continuous change point models
19. JAGS code 
20. Example: change points of global temperature
21. Plotting the change point
22. Summary


------

# Module 10: Extensions: state space models, and multivariate time series (AP)
Friday 13th May 10:45-11:30

Learning outcomes: 

- Learn the basics of parameter and state estimation for simple state space models
- Create and fit some basic multivariate time series models
- Understand the state of the art in time series models, including latent factor models, dynamic parameter models,

Relevant JAGS file:
```
jags_statespace.R
jags_multivariate_AR.R (not written yet)
jags_multivariate_statespace.R (not written yet)
```

Slides:

1. Title
2. Learning outcomes
3. Introduction to state space models
4. JAGS code for a linear state space model
5. Priors for state space models
6. Example: palaeoclimate reconstruction
7. Palaeoclimate reconstruction results
8. Introduction to multivariate models
9. The Vector AR model
10. JAGS code for the VAR model
11. Example
12. Multivariate state-space models
13. Example: Palaeoclimate reconstruction
14. Mixing up Gaussian processes and state space models
15. Co-integrated models
16. Other types of multivariate model
17. Multivariate Gaussian processes
18. Dynamic linear models
19. Latent factor time series models
20. Summary



------

# Practical 4: Bring your own data (AP and DM)
Friday 13th May 12:00-13:30