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Forecasting Principles Practice Leader Rob J Hyndman 23 25 September 2014 University of Western Australia robjhyndman uwa? Resources Slides Exercises Textbook Useful links robjhyndman uwa2017 Forecasting principles and practice Background 3? This

- Forecasting
- Principles & Practice
- principles and practice
- Basic Forecasting
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Forecasting: Principles & Practice

- com/uwa

- 1 Introduction to forecasting 1
- 1 Introduction
- 2 Some case studies
- 3 Time series data
- 4 Some simple forecasting methods 1
- 5 Lab Session 1
- 2 The forecaster’s toolbox 2
- 1 Time series graphics
- 2 Seasonal or cyclic
- 3 Autocorrelation
- 4 Forecast residuals
- 5 White noise
- 6 Evaluating forecast accuracy 2
- 7 Lab Session 2
- 5 5 7 8 11 13
- 14 14 17 20 24 26 29 32
- 3 Exponential smoothing 3
- 1 The state space perspective
- 2 Simple exponential smoothing
- 3 Trend methods
- 4 Seasonal methods
- 5 Lab Session 3
- 6 Taxonomy of exponential smoothing methods 3
- 7 Innovations state space models

8 ETS in R

- 9 Forecasting with ETS models
- 10 Lab Session 4
- 34 34 34 36 39 40 41 41 46 50 51
- 52 52 54 54 55 57
- 4 Time series decomposition 4
- 1 Example: Euro electrical equipment
- 2 Seasonal adjustment
- 3 STL decomposition
- 4 Forecasting and decomposition
- 5 Lab Session 5a

Forecasting: principles and practice

- 5 Time series cross-validation 58 5
- 1 Cross-validation
- 2 Example: Pharmaceutical sales
- 3 Lab Session 5b
- 62 6 Making time series stationary 6
- 1 Transformations
- 2 Stationarity
- 3 Ordinary differencing
- 4 Seasonal differencing
- 5 Unit root tests
- 6 Backshift notation
- 7 Lab Session 6
- 7 Non-seasonal ARIMA models 7
- 1 Autoregressive models
- 2 Moving average models
- 3 ARIMA models
- 4 Estimation and order selection 7
- 5 ARIMA modelling in R
- 6 Forecasting
- 7 Lab Session 7
- 63 63 65 67 68 69 70 71
- 72 72 73 73 77 78 81 83
- 8 Seasonal ARIMA models 8
- 1 Common ARIMA models
- 2 ACF and PACF of seasonal ARIMA models 8
- 3 Example: European quarterly retail trade 8
- 4 Example: Cortecosteroid drug sales
- 5 ARIMA vs ETS
- 6 Lab Session 8
- 84 85 85 85 89 92 93
- 94 94 96 97 99 100 103 104
- 105 105 107 109 110 112 113
- 9 State space models 9
- 1 Simple structural models
- 2 Linear Gaussian state space models 9
- 3 Kalman filter
- 4 ARIMA models in state space form 9
- 5 Kalman smoothing
- 6 Time varying parameter models
- 7 Lab Session 9
- 10 Dynamic regression models 10
- 1 Regression with ARIMA errors
- 2 Example: US personal consumption & income 10
- 3 Forecasting
- 4 Stochastic and deterministic trends
- 5 Periodic seasonality
- 6 Dynamic regression models

- 7 Rational transfer function models
- 8 Lab Session 10
- 115 11 Hierarchical forecasting 11
- 1 Hierarchical and grouped time series
- 2 Forecasting framework
- 3 Optimal forecasts
- 4 OLS reconciled forecasts
- 5 WLS reconciled forecasts
- 6 Application: Australian tourism
- 7 Application: Australian labour market 11
- 8 hts package for R
- 9 Lab Session 11
- 116 116 117 119 119 120 120 122 125 127
- 12 Vector autoregressions
- 13 Neural network models
- 14 Forecasting complex seasonality 134 14
- 1 TBATS model
- 2 Lab Session 12
- 1 Introduction to forecasting

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

- done it for decades,

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

Learner: I am still learning

? Is that what the weather people do

- ? Key reference Hyndman,
- & Athanasopoulos,

(2013) Forecasting: principles and practice

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

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

but I am often searching around for the right function

- packages("fpp",

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

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

- – 9 9 9

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,

- standard deviation,
- quantiles,
- regression,
- normal distribution,

• This is not a theory course

when to use them and how to use them most effectively

- trended or seasonal

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

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

- 12 month average 6 month average straight line regression over last 12 months straight line regression over last 6 months average slope between last year’s and this year’s values

(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

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

- • In 2001: $4
- 5 billion budget,

under-forecasted by $800 million

- • Thousands of products

Seasonal demand

• Subject to covert marketing,

- volatile products,
- uncontrollable expenditure

• 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

- ? CASE STUDY 3: Airline

First class passengers: Melbourne−Sydney

0 2 4 6 8

Business class passengers: Melbourne−Sydney

1991 Year

1991 Year

• Traffic is affected by school holidays,

special events such as the Grand Prix,

- advertising campaigns,
- competition behaviour,

• 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

- • • • •

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

Forecasting: principles and practice

450 400

- megaliters

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

Forecasting: principles and practice

- > plot(ausgdp)

Residential electricity sales

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

34 2379

71 2318

52 2468

99 2386

09 2569

- 47 [7] 2575

72 2762

72 2844

50 3000

70 3108

10 3357

- 50 [13] 3075

70 3180

60 3221

60 3176

20 3430

60 3527

- 48 [19] 3637

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)

450 400

- megaliters

- 100 90 80 1990

Dow Jones index (daily ending 15 Jul 94)

- thousands

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

• Forecasts equal to last observed value

- • Forecasts: yˆT +h|T = yT

• 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: T

h X (yt − yt−1 ) T −1 t=2

- h = yT + (y − y1 )

T −1 T

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

- • • • •

- h=20) Naive: naive(x,
- h=20) or rwf(x,

h=20) Seasonal naive: snaive(x,

- h=20) Drift: rwf(x,
- drift=TRUE,

Lab Session 1

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

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

- seasonal naive or drift

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

(b) Price of chicken (1924–1993)

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

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

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

Data set bricksq

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

In each case,

do you think the forecasts are reasonable

- ? If not,
- how could they be improved
- 2 The forecaster’s toolbox

20 15 0

Thousands

1990 Year

- plot(melsyd[,"Economy

Antidiabetic drug sales

- $ million
- > plot(a10)

2006 ●

- ● ● ●
- 2005 ● ●
- 2003 ● 2002 ● 15
- ● ●

2005 2004

- 1999 ● 1998 ●●● 1997 1996 1995 ● 1994 1993 ● 1992 ●
- ● ●
- ● ● ●

● ● ● ● ● ● ● ● ● ● ● ●

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2000 ●

- ● ● ●
- 2001 ● ● ●
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2004 ●

- $ million
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- 2001 2000 1999 ● 1998 ● 1997
- ● ● ● ●
- 1995 1993 1994 1992
- ● ●
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• 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

- • In R: seasonplot

Forecasting: principles and practice

Seasonal subseries plots 30

Seasonal subseries plot: antidiabetic drug sales

- $ million
- > monthplot(a10)

• 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

- plot(beer,lags=9,do
- lines=FALSE)
- ●● ● ●
- ● ● ● ●
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● ● ● ● ● ● ● ● ● ●

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- lag 8 450

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

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450 beer

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- lag 2 ● ● ● ●● ●
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450 beer

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

● ● ● ● ● ● ●● ● ●

- ● ●
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- 400 ● ●●
- 450 ●● ● ● ●

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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

- rk = ck /c0

• 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

- r1 r2 r3 r4 r5 r6 r7 r8 r9 −0

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

- • Together,

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

- 12 for monthly data) will be large and positive

• 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,

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

- • The ACF peaks at lags 12,

indicate seasonality of length 12

Forecasting: principles and practice

Accidental deaths in USA (monthly)

- thousands
- chirps per minute 40 60 80

Annual mink trappings (Canada)

- thousands
- thousands 200 300 400

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

- {et } uncorrelated

If they aren’t,

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

- {et } have mean zero

If they don’t,

- then forecasts are biased

Useful properties (for Forecast intervals) 3

- {et } have constant variance

{et } are normally distributed

- 3800 3700 3600

Dow−Jones index

150 Day

Forecasting: principles and practice

- 3800 3700 3600

- 0 −50 −100

150 Day

Frequency

Histogram of residuals

- 0 Change in Dow−Jones index