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Description

Forecasting: Principles & Practice

Leader: Rob J Hyndman 23-25 September 2014 University of Western Australia robjhyndman

Contents

8 ETS in R

Forecasting: principles and practice

Forecasting: principles and practice

Introduction

Brief bio • Director of Monash University’s Business & Economic Forecasting Unit • Editor-in-Chief,

International Journal of Forecasting How my forecasting methodology is used: • • • • •

Pharmaceutical Benefits Scheme Cancer incidence and mortality Electricity demand Ageing population Fertilizer sales

Poll: How experienced are you in forecasting

Guru: I wrote the book,

now I do the conference circuit

Expert: It has been my full time job for more than a decade

Skilled: I have been doing it for years

Comfortable: I understand it and have done it

Learner: I am still learning

Beginner: I have heard of it and would like to learn more

Unknown: What is forecasting

? Is that what the weather people do

(2013) Forecasting: principles and practice

org/fpp/ • Free and online • Data sets in associated R package • R code for examples

Forecasting: principles and practice

Poll: How proficient are you in using R

Guru: The R core team come to me for advice

Expert: I have written several packages on CRAN

Skilled: I use it regularly and it is an important part of my job

Comfortable: I use it often and am comfortable with the tool

User: I use it sometimes,

but I am often searching around for the right function

Learner: I have used it a few times

Beginner: I’ve managed to download and install it

Unknown: Why are you speaking like a pirate

Install required packages install

dependencies=TRUE) Getting help with R # Search for terms help

search("forecasting") # Detailed help help(forecast) # Worked examples example("forecast

ar") # Similar names apropos("forecast") #Help on package help(package="fpp") Approximate outline Day

Chapter

The forecaster’s toolbox Seasonality and trends Exponential smoothing

1,2 6 7

2 2 2 2 2

Time series decomposition Time series cross-validation Transformations Stationarity and differencing ARIMA models

6 2 2 8 8

3 3 3 3

State space models Dynamic regression Hierarchical forecasting Advanced methods

Forecasting: principles and practice

Assumptions • This is not an introduction to R

I assume you are broadly comfortable with R code and the R environment

• This is not a statistics course

I assume you are familiar with concepts such as the mean,

• This is not a theory course

I am not going to derive anything

I will teach you forecasting tools,

when to use them and how to use them most effectively

Some case studies

CASE STUDY 1: Paperware company Problem: Want forecasts of each of hundreds of items

Series can be stationary,

They currently have a large forecasting program written in-house but it doesn’t seem to produce sensible forecasts

They want me to tell them what is wrong and fix it

Additional information • Program written in COBOL making numerical calculations limited

It is not possible to do any optimisation

• Their programmer has little experience in numerical computing

• They employ no statisticians and want the program to produce forecasts automatically

CASE STUDY 1: Paperware company Methods currently used A C E G H

(Equivalent to differencing at lag 12 and taking mean

) I Same as H except over 6 months

K I couldn’t understand the explanation

CASE STUDY 2: PBS The Pharmaceutical Benefits Scheme (PBS) is the Australian government drugs subsidy scheme

• Many drugs bought from pharmacies are subsidised to allow more equitable access to modern drugs

• The cost to government is determined by the number and types of drugs purchased

Currently nearly 1% of GDP

• The total cost is budgeted based on forecasts of drug usage

under-forecasted by $800 million

Seasonal demand

• Subject to covert marketing,

Forecasting: principles and practice

• Although monthly data available for 10 years,

data are aggregated to annual values,

and only the first three years are used in estimating the forecasts

• All forecasts being done with the FORECAST function in MS-Excel

! Problem: How to do the forecasting better

First class passengers: Melbourne−Sydney

0 2 4 6 8

Business class passengers: Melbourne−Sydney

1991 Year

Economy class passengers: Melbourne−Sydney

1991 Year

Problem: how to forecast passenger traffic on major routes

Additional information • They can provide a large amount of data on previous routes

• Traffic is affected by school holidays,

special events such as the Grand Prix,

• They have a highly capable team of people who are able to do most of the computing

Time series data

Time series consist of sequences of observations collected over time

We will assume the time periods are equally spaced

Daily IBM stock prices Monthly rainfall Annual Google profits Quarterly Australian beer production

Forecasting is estimating how the sequence of observations will continue into the future

Forecasting: principles and practice

450 400

Australian beer production

Australian GDP ausgdp ausgdp Qtr1 1971 1972 4645 1973 4780 1974 4921 1975 4938 1976 5028 1977 5130 1978 5100 1979 5349 1980 5388

Qtr2 Qtr3 4612 4615 4645 4830 4887 4875 4867 4934 4942 5079 5112 5101 5072 5166 5244 5370 5388 5403 5442

Qtr4 4651 4722 4933 4905 4979 5127 5069 5312 5396 5482

Forecasting: principles and practice

Residential electricity sales

> elecsales Time Series: Start = 1989 End = 2008 Frequency = 1 [1] 2354

34 2379

71 2318

52 2468

99 2386

09 2569

72 2762

72 2844

50 3000

70 3108

10 3357

70 3180

60 3221

60 3176

20 3430

60 3527

89 3655

Main package used in this course > library(fpp) This loads: • • • • • •

some data for use in examples and exercises forecast package (for forecasting functions) tseries package (for a few time series functions) fma package (for lots of time series data) expsmooth package (for more time series data) lmtest package (for some regression functions)

Forecasting: principles and practice

Some simple forecasting methods

450 400

Australian quarterly beer production

Dow Jones index (daily ending 15 Jul 94)

Number of pigs slaughtered in Victoria

Forecasting: principles and practice

Average method • Forecast of all future values is equal to mean of historical data {y1 ,

• Forecasts: yˆT +h|T = y¯ = (y1 + · · · + yT )/T

Naïve method (for time series only)

• Forecasts equal to last observed value

• Consequence of efficient market hypothesis

Seasonal naïve method • Forecasts equal to last value from same season

• Forecasts: yˆT +h|T = yT +h−km where m = seasonal period and k = b(h − 1)/mc+1

Forecasts for quarterly beer production

Drift method • Forecasts equal to last value plus average change

yˆT +h|T = yT +

h X (yt − yt−1 ) T −1 t=2

T −1 T

• Equivalent to extrapolating a line drawn between first and last observations

Mean: meanf(x,

h=20) Seasonal naive: snaive(x,

Forecasting: principles and practice

Dow Jones Index (daily ending 15 Jul 94)

Lab Session 1

Before doing any exercises in R,

load the fpp package using library(fpp)

Use the Dow Jones index (data set dowjones) to do the following: (a) Produce a time plot of the series

(b) Produce forecasts using the drift method and plot them

(c) Show that the graphed forecasts are identical to extending the line drawn between the first and last observations

(d) Try some of the other benchmark functions to forecast the same data set

Which do you think is best

For each of the following series,

make a graph of the data with forecasts using the most appropriate of the four benchmark methods: mean,

(a) Annual bituminous coal production (1920–1968)

Data set bicoal

(b) Price of chicken (1924–1993)

Data set chicken

(c) Monthly total of people on unemployed benefits in Australia (January 1956–July 1992)

Data set dole

(d) Monthly total of accidental deaths in the United States (January 1973–December 1978)

Data set usdeaths

(e) Quarterly production of bricks (in millions of units) at Portland,

Australia (March 1956–September 1994)

Data set bricksq

(f) Annual Canadian lynx trappings (1821–1934)

Data set lynx

In each case,

do you think the forecasts are reasonable

Time series graphics

20 15 0

Thousands

Economy class passengers: Melbourne−Sydney

1990 Year

Class"])

Forecasting: principles and practice

Antidiabetic drug sales

Seasonal plots 30

Seasonal plot: antidiabetic drug sales 2008 ● 2007 ●

2006 ●

2005 2004

● ● ● ● ● ● ● ● ● ● ● ●

2000 ●

2004 ●

• Data plotted against the individual “seasons” in which the data were observed

(In this case a “season” is a month

) • Something like a time plot except that the data from each season are overlapped

• Enables the underlying seasonal pattern to be seen more clearly,

and also allows any substantial departures from the seasonal pattern to be easily identified

Forecasting: principles and practice

Seasonal subseries plots 30

Seasonal subseries plot: antidiabetic drug sales

• Data for each season collected together in time plot as separate time series

• Enables the underlying seasonal pattern to be seen clearly,

and changes in seasonality over time to be visualized

• In R: monthplot Quarterly Australian Beer Production beer lag

● ● ● ● ● ● ● ● ● ●

●● ● ● ● ●● ●● ● ●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●

● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●●●●

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450 beer

● ●● ● ● ● ● ●● ● ●● ● ● ● ●

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●● ● ●● ● ●● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ● ●● ● ● ● ● ● ● ●● ● ● ●

450 beer

● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●● ●● ● ● ●●● ●

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• Each graph shows yt plotted against yt−k for different values of k

• The autocorrelations are the correlations associated with these scatterplots

We denote the sample autocovariance at lag k by ck and the sample autocorrelation at lag k by rk

Then define ck =

T 1 X ¯ t−k − y) ¯ (yt − y)(y T

• r1 indicates how successive values of y relate to each other • r2 indicates how y values two periods apart relate to each other • rk is almost the same as the sample correlation between yt and yt−k

Forecasting: principles and practice

Results for first 9 lags for beer data:

126 −0

650 −0

863 −0

099 −0

642 −0

834 −0

• r4 higher than for the other lags

This is due to the seasonal pattern in the data: the peaks tend to be 4 quarters apart and the troughs tend to be 2 quarters apart

• r2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks

the autocorrelations at lags 1,

make up the autocorrelation or ACF

• The plot is known as a correlogram Recognizing seasonality in a time series If there is seasonality,

the ACF at the seasonal lag (e

• For seasonal monthly data,

a large ACF value will be seen at lag 12 and possibly also at lags 24,

• For seasonal quarterly data,

a large ACF value will be seen at lag 4 and possibly also at lags 8,

Forecasting: principles and practice

Australian monthly electricity production

12000 8000

Australian electricity production

Time plot shows clear trend and seasonality

The same features are reflected in the ACF

• The slowly decaying ACF indicates trend

indicate seasonality of length 12

Forecasting: principles and practice

Which is which

Accidental deaths in USA (monthly)

Daily morning temperature of a cow

Annual mink trappings (Canada)

International airline passengers

Forecasting: principles and practice

Forecast residuals

Residuals in forecasting: difference between observed value and its forecast based on all previous observations: et = yt − yˆt|t−1

Assumptions

If they aren’t,

then information left in residuals that should be used in computing forecasts

If they don’t,

Useful properties (for Forecast intervals) 3

{et } are normally distributed

Dow−Jones index

Forecasting Dow-Jones index

150 Day

Naïve forecast: yˆt|t−1 = yt−1 et = yt − yt−1 Note: et are one-step-forecast residuals

Forecasting: principles and practice

Dow−Jones index

Change in Dow−Jones index

150 Day

Frequency

Histogram of residuals

Forecasting: principles and practice