1. Time Series introduction in R - Iñaki Puigdollers
10th of March 2016

2. Quick Overview A time series is: Sequence of observations
TIME SERIES MODELLING AND FORECASTING
Quick Overview
A time series is:
Sequence of observations
Generated from a stochastic process (random variables)
Ordered in time
Model a time series: find a mathematical expression for a time series
Time series forecasting: use a model to predict its future values
Examples of models:
Linear Regression
SARIMA

3. Quick Overview A time series is: Sequence of observations
TIME SERIES MODELLING AND FORECASTING
Quick Overview
A time series is:
Sequence of observations
Generated from a stochastic process (random variables)
Ordered in time
Model a time series: find a mathematical expression for a time series
Time series forecasting: use a model to predict its future values
Examples of models:
Linear Regression
SARIMA

4. Linear regression Simple linear regression Dependent/Response variable
TIME SERIES MODELLING AND FORECASTING
Linear regression
Simple linear regression
Dependent/Response variable
Regression coefficient
Independent variable or regressor
Error term

5. Linear regression Multiple linear regression
TIME SERIES MODELLING AND FORECASTING
Linear regression
Multiple linear regression
Smaller n’s are preferable (parsimony principle, a.k.a. Occam’s razor)

6. Linear regression Assumptions:
TIME SERIES MODELLING AND FORECASTING
Linear regression
Assumptions:
Linear relation between Y and X (in case of multiple linear regression with each Xi)
In multiple linear regression, the regressors (Xi) are pairwise independent
The errors (𝜀t) are gaussian white noise: independent and normally distributed

7. SARIMA (Seasonal Autoregressive Integrated moving Average)
TIME SERIES MODELLING AND FORECASTING
SARIMA (Seasonal Autoregressive Integrated moving Average)
SARIMA (p,d,q)x(P,D,Q)s:
Season Period (12 for a year, 7 for a week, …)
COMPONENTS:
Non-seasonal
Seasonal
TERMS:
Autoregressive
Error
Moving average
Autoregressive
Moving average

8. TIME SERIES MODELLING AND FORECASTING
SARIMA
Assumptions:
Weak (2nd order) stationarity*: the time series behaviour is not varying over time
Errors are gaussian white noise: independent and normally distributed
*Stationarity:
A time series is called stationary if all their moments are constant over time
A time series is called 2nd order stationary if the mean (1st order moment) and variance (2nd order moment) are constant over time

9. Box & Jenkins Methodology
TIME SERIES MODELLING AND FORECASTING
Box & Jenkins Methodology
A methodology to model SARIMA time series
The methodology is composed by 3 different stages:
Model selection
Parameters estimation
Model Checking

10. B&J: Model Selection Detect stationarity
TIME SERIES MODELLING AND FORECASTING
B&J: Model Selection
Detect stationarity
To detect mean-sationarity: use stationarity tests like Kiwatowski-Phillips-Schmidt-Shin (KPSS) or Augmented Dickey-Fuller (ADF) or Autocorrelation function (ACF) or by simply looking at the plot of the data
Quite often you can make a time series stationary by applying a difference filter of order , i.e. to every point Xt in the time series apply the following transformation: Yt = Xt – Xt-1
To detect variance stationarity: use Priestly-Suba-Rao test (PSR) or the ACF, or looking at the plot of the data
You can try to achieve stationarity in variance by applying the Box-Cox transformation to the series (for positive time series) or Yeo-Johnson transformation (for non-positive time series). Please check the appendix for the closed-for formulae.
Remember: If you can’t make the time series stationary, it can’t be modelled as a SARIMA!!!!

11. Stationarity? Stationary time series Non-Stationary time series
TIME SERIES MODELLING AND FORECASTING
Stationarity?
Stationary time series
Non-Stationary time series

12. B&J: Model Selection Detect seasonality
TIME SERIES MODELLING AND FORECASTING
B&J: Model Selection
Detect seasonality
Use ACF and partial autocorrelation function (PACF) to detect if there is any seasonal pattern
If there is seasonality you have to seasonal-difference the time series before modelling it, i.e. difference the time series S times (where S is the order of the seasonality, e.g. if we have monthly data and the seasonality is quarterly you have to apply a order 4 difference filter. See appendix for a closed-for formula.

13. Seasonality? Seasonal time series Non-seasonal time series
TIME SERIES MODELLING AND FORECASTING
Seasonality?
Seasonal time series
Non-seasonal time series

14. EXPONENTIALLY DECREASES
TIME SERIES MODELLING AND FORECASTING
B&J: Model Selection
Identify the parameters p & q (same for P & Q but at seasonal level)
EXPONENTIALLY DECREASES

15. B&J: Parameter Estimation
TIME SERIES MODELLING AND FORECASTING
B&J: Parameter Estimation
Most common practice is to use numerical methods to estimate the parameters by finding the most likelihood estimator. In order to asses which parameter is best fitting the most common practice is minimizing some information criteria, either:
Akaike information criterion (AIC)
Akaike information criterion corrected (AICc), this one tends to be preferred as it is not dependent on the size of the sample
Bayesian information criterion (BIC)

16. TIME SERIES MODELLING AND FORECASTING
B&J: Model Checking
Once the model is built we need to make sure that the errors of the model are gaussian distributed (i.e. they are white noise). If they are not, this means that the the errors may have a SARIMA structure itself, so the most common practice is to apply the same methodology (Box-Jenkins)again to the errors of the series
In order to assess the normality of the errors you can use among other Ljung-Box normality test (LB) or Durbin-Watson (DW), but you can also check it by plotting the ACF/PACF of the errors (if they are normal all the values of the ACF/PACF will be inside the confidence interval)

17. TIME SERIES MODELLING AND FORECASTING
DEMO TIME !!!

18. TIME SERIES MODELLING AND FORECASTING
Appendix
Box-Cox transformation (for trying to make non-variance-stationary POSITIVE time series into variance-stationary time series)

19. TIME SERIES MODELLING AND FORECASTING
Appendix
Yeo-Johnson transformation (for trying to make non-variance-stationary time series into variance-stationary time series)

20. Appendix The difference filter
TIME SERIES MODELLING AND FORECASTING
Appendix
The difference filter
The difference operator (or order 1 difference filter) is
In general an order r difference filter can be calculated recursively as a sequence of difference operators