Time series /Applied Forecasting 7005 Hilary term 2016 Prof. Rozenn Dahyot Room 128 Lloyd Institute School of Computer Science and Statistics Trinity College Dublin or

Lecture notes available In the ‘teaching’ section. Possibly some materials will be on blackboard

Timetable 6-8pm LB04 Tuesday Thursday

Organization of the course Lectures-tutorials only: No labs but information using R for Forecasting will be provided. Exam 100% No assignments

Software R

Content

Introduction to forecasting; ARIMA models, GARCH models, Kalman Filters,data transformations, seasonality, exponential smoothing and Holt Winters algorithms, performance measures. Use of transformations and differences.

Textbook Forecasting: Methods and Applications by Makridakis, Wheelwright and Hyndman, published by Wiley Many more books in the libraries in Trinity on Forecasting, time series covering the content of this course.

Who Forecast?

Why Forecast?

How to Forecast? In this course we will use maths/stats techniques for forecasting

Steps in a Forecasting Procedure? Problem definition Exploratory Analysis Gathering information Selecting and fitting models to make forecast Using and evaluating the forecast

Examples com/media/research.google.co m/en//archive/papers/detecting -influenza-epidemics.pdf com/media/research.google.co m/en//archive/papers/detecting -influenza-epidemics.pdf

Examples…. Warnings Epidemiological modeling of online social network dynamics

Quantitative Forecasting

Quantitative methods

Time series models Vs Explanatory models Time series

What is the nature of the data to analyse? Examples from fma packages in R airpass beer internet cowtemp Dowjones mink Can you predict how these time series look like ?

Visualization tools Numerical values Time plot Season plot

Patterns to identify Trends Seasonal Error/noise Visualize and identify patterns: airpass beer internet cowtemp Dowjones mink

Time series Definition Sampling rate & Unit of time Preparation of Data before analysis

Limitations in this module 1D time series No outliers No missing data

Notations Variables Vs numerical values Time series

Auto-Correlation Function (ACF) Mean value of the time series Autocorrelation at lag k

Auto Correlation Function (ACF) Lag k r1r1 r2r2 r3r

> plot(beer-mean(beer),lwd="3") > lines(lag(beer-mean(beer),1),col="red",lwd=3) In red, The lag series beer (lag 1 ). The two time series overlap well.

In red, The lag series beer (lag 6 ). The two time series do not overlap well. > plot(beer-mean(beer),lwd="3") > lines(lag(beer-mean(beer),6),col="red",lwd=3)

In red, The lag series beer (lag 12 ). The two time series do overlap well. > plot(beer-mean(beer),lwd="3") > lines(lag(beer-mean(beer),12),col="red",lwd=3)

For the airpass time series Lag 1 Lag 6 Lag 12

Partial AutoCorrelation Function (PACF)

Holt-Winters Algorithms Part I

Algo I: Simple Exponential Smoothing (SES)

What does SES do? What happens when  =1 or  =0 ? SES is an algorithm suitable for a time series with … Algo I: Simple Exponential Smoothing (SES)

Algo II: Double Exponential Smoothing (DES)

SES(  ) DES(  )

SHW + (  )

SHW x (  )

SHW + (  ): Exercise

Linear Regression

Useful formulas

Auto-Regressive Models – AR(1) Explanatory variable Parameters to estimate

Auto-Regressive Models – AR(2) Explanatory variables Parameters to estimate

Auto-Regressive Models – AR(p) Parameters to estimate Explanatory variables

AR(1): Least Squares estimates of the parameters model Write the least squares solution.

AR(1): Least Squares estimates of the parameters model

AR(1): Least Squares estimates of the parameters

Estimate of  Estimate the standard deviation of the noise

Example: dowjones

Auto-Regressive Models – AR(p) Parameters to estimate Explanatory variables

Moving Average MA(1) Explanatory variable Parameters to estimate Can Least Squares Algorithm be used to estimate the parameters?

Moving average MA(q) Parameters to estimate Explanatory variables

Exercises

Remark

Expectation

Summary 17/11/2014 Using ACF and PACF to identify AR(p) and MA(q) Procedure to fit an ARIMA(p,d,q) Definition of BIC/AIC

Fitting ARIMA(p,d,q)

To avoid overfitting choose p ≤ 3 q ≤ 3 d ≤ 3

PACF for AR(1) Maths

ACF for MA(1) Maths

MA(1) as an AR(∞) For MA(1) the Damped sine wave/exponential decay in the PACF corresponds to these coefficients vanishing towards 0

AR(1) as an MA(∞)

Criteria to select the best ARIMA model

Exercise: Show

Hirotugu Aikaike ( ) 1970s: proposed model selection with an information Criterion (AIC)

Bayesian information Criterion Thomas Bayes ( ) The BIC was developed by Gideon E. Schwarz, who gave a Bayesian argument for adopting it.Bayesian

Seasonal ARIMA(p,d,q)(P,D,Q) s

Choose your criterion AIC or BIC (and stick to it). Select the ARIMA model with the lowest AIC or BIC with m=p+q+P+Q

ARIMA(0,0,0)(P=1,0,0) s Vs ARIMA(0,0,0)(0,D=1,0) s

Summary

s1950s 1970s 1980s1990s SES DES SHW+ SHWx ARIMAAIC BIC Holt Winters

Other time series models ARCH (1982): autoregressive conditional heteroskedasticity GARCH (1986): generalized autoregressive conditional heteroskedasticity … More at

Concluding Remarks time

Concluding remarks The Prediction – Update loop Combining experts