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Forecasting: principles and practice

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intended to provide a comprehen- sive introduction to forecasting methods and to present enough inform

Description

Forecasting: principles and practice Rob J Hyndman

George Athanasopoulos May 2012

Forecasting: principles and practice

Contents Foreword

2 The 2

forecaster’s toolbox Graphics

Numerical data summaries

Some simple forecasting methods Transformations and adjustments Evaluating forecast accuracy

Residual diagnostics

Prediction intervals

Exercises

Further reading

The forecast package in R

Forecasting: principles and practice 4

Forecasting with regression

Statistical inference

Non-linear functional forms

Regression with time series data

Summary of notation and terminology Exercises

Further reading

Forecasting: principles and practice

10 Data

11 Using R

Forecasting: principles and practice

List of Figures 1

Australian quarterly beer production: 1992Q1–2008Q3

Weekly economy passenger load on Ansett Airlines

Monthly sales of antidiabetic drugs in Australia

Seasonal plot of monthly antidiabetic drug sales in Australia

Seasonal plot of monthly antidiabetic drug sales in Australia

Carbon footprint and fuel economy for cars made in 2009

Scatterplot matrix of measurements on 2009 model cars

Examples of data sets with different levels of correlation

Plots with correlation coefficiente of 0

Lagged scatterplots for quarterly beer production

Autocorrelation function of quarterly beer production

A white noise time series

Autocorrelation function for the white noise series

Forecasts of Australian quarterly beer production

Forecasts based on 250 days of the Dow Jones Index

Power transformations for Australian monthly electricity data

Monthly milk production per cow

Source: Cryer (2006)

Forecasts of Australian quarterly beer production using data up to the end of 2005

Forecasts of the Dow Jones Index from 16 July 1994

The Dow Jones Index measured daily to 15 July 1994

Residuals from forecasting the Dow Jones Index with the naïve method

Histogram of the residuals from the naïve method applied to the Dow Jones Index

ACF of the residuals from the naïve method applied to the Dow Jones Index

Process for producing PBS forecasts

Long run annual forecasts for domestic visitor nights for Australia

An example of data from a linear regression model

Estimated regression line for a random sample of size N

Fitted regression line from regressing the carbon footprint of cars versus their fuel economy in city driving conditions

Forecasting: principles and practice 4

Australian quarterly beer production

Scatterplot matrix of the credit scores and the four predictors

Scatterplot matrix of the credit scores and the four predictors

Actual credit scores plotted against fitted credit scores using the multiple regression model

Time plot of beer production and predicted beer production

Actual beer production plotted against predicted beer production

Forecasts from the regression model for beer production

The residuals from the regression model for credit scores plotted against each of its predictors

The residuals from the credit score model plotted against the fitted values obtained from the model

Residuals from the regression model for beer production

Histogram of residuals from regression model for beer production

Piecewise linear trend to fuel economy data

Cubic regression spline fitted to the fuel economy data

Four time series exhibiting different types of time series patterns

Electrical equipment orders

The electricial equipment orders (top) and its three additive components

Seasonal sub-series plot of the seasonal component from the STL decomposition

Seasonally adjusted electrical equipment orders and the original data

Residential electricity sales for South Australia: 1989-2008

Residential electricity sales along with the 5-MA estimate of the trend-cycle

Different moving averages applied to the residential electricity sales data

A 2 × 12-MA applied to the electrical equipment orders index

The electrical equipment orders and its three additive components obtained from a robust STL decomposition

after an an STL decomposition of the data

114 115

Oil production in Saudi Arabia from 1996 to 2007

List of Tables 1

Fuel economy and carbon footprints for 2009 model cars

Multipliers to be used for prediction intervals

Summary of selected functional forms

First few years of the Australian quarterly beer production data

Forecasting: principles and practice

Foreword Welcome to our online textbook on forecasting

This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information about each method for readers to be able to use them sensibly

We don’t attempt to give a thorough discussion of the theoretical details behind each method,

although the references at the end of each chapter will fill in many of those details

The book is written for three audiences: (1) people finding themselves doing forecasting in business when they may not have had any formal training in the area

(2) undergraduate students studying business

(3) MBA students doing a forecasting elective

We use it ourselves for a second-year subject for students undertaking a Bachelor of Commerce degree at Monash University,

Australia

For most sections,

we only assume that readers are familiar with algebra,

and high school mathematics should be sufficient background

Readers who have completed an introductory course in statistics will probably want to skip some of Chapters 2 and 4

There are a couple of sections which require knowledge of matrices,

At the end of each chapter we provide a list of “further reading”

In general,

these lists comprise suggested textbooks that provide a more advanced or detailed treatment of the subject

Where there is no suitable textbook,

we suggest journal articles that provide more information

We use R throughout the book and we intend students to learn how to forecast with R

R is free and available on almost every operating system

It is a wonderful tool for all statistical analysis,

See Using R for instructions on installing and using R

The book is different from other forecasting textbooks in several ways

making it accessible to a wide audience

and extremely powerful software

• It is continuously updated

You don’t have to wait until the next edition for errors to be removed or new methods to be discussed

We will update the book frequently

• There are dozens of real data examples taken from our own consulting practice

We have worked with hundreds of businesses and organizations helping them with forecasting issues,

and this experience has contributed directly to many of the examples given here,

as well as guiding our general philosophy of forecasting

• We emphasise graphical methods more than most forecasters

We use graphs to explore the data,

analyse the validity of the models fitted and present the forecasting results

Use the table of contents on the right to browse the book

If you have any comments or suggestions on what is here so far,

feel free to add them on the book page

Happy forecasting

! Rob J Hyndman George Athanasopoulos May 2012

Forecasting: principles and practice

Chapter 1

Getting started Forecasting has fascinated people for thousands of years,

sometimes being considered a sign of divine inspiration,

and sometimes being seen as a criminal activity

The Jewish prophet Isaiah wrote in about 700 BC Tell us what the future holds,

so we may know that you are gods

(Isaiah 41:23) One hundred years later,

forecasters would foretell the future based on the distribution of maggots in a rotten sheep’s liver

By 300 BC,

people wanting forecasts would journey to Delphi in Greece to consult the Oracle,

who would provide her predictions while intoxicated by ethylene vapours

Forecasters had a tougher time under the emperor Constantine,

who issued a decree in AD357 forbidding anyone “to consult a soothsayer,

or a forecaster May curiosity to foretell the future be silenced forever

” A similar ban on forecasting occurred in England in 1736 when it became an offence to defraud by charging money for predictions

The punishment was three months’ imprisonment with hard labour

! The varying fortunes of forecasters arise because good forecasts can seem almost magical,

while bad forecasts may be dangerous

Consider the following famous predictions about computing

• I think there is a world market for maybe five computers

Not surprisingly,

you can no longer buy a DEC computer

Forecasting is obviously a difficult activity,

and businesses that do it well have a big advantage over those whose forecasts fail

In this book,

we will explore the most reliable methods for producing forecasts

The emphasis will be on methods that are replicable and testable,

What can be forecast

Forecasting is required in many situations: deciding whether to build another power generation plant in the next five years requires forecasts of future demand

scheduling staff in a call centre next week requires forecasts of call volumes

stocking an inventory requires forecasts of stock requirements

Forecasts can be required several years in advance (for the case of capital investments),

or only a few minutes beforehand (for telecommunication routing)

Whatever the circumstances or time horizons involved,

forecasting is an important aid to effective and efficient planning

Some things are easier to forecast than others

The time of the sunrise tomorrow morning can be forecast very precisely

On the other hand,

tomorrow’s lotto numbers cannot be forecast with any accuracy

The predictability of an event or a quantity depends on several factors including: 3

Forecasting: principles and practice 1

how well we understand the factors that contribute to it

whether the forecasts can affect the thing we are trying to forecast

For example,

forecasts of electricity demand can be highly accurate because all three conditions are usually satisfied

We have a good idea on the contributing factors: electricity demand is driven largely by temperatures,

with smaller effects for calendar variation such as holidays,

Provided there is a sufficient history of data on electricity demand and weather conditions,

and we have the skills to develop a good model linking electricity demand and the key driver variables,

the forecasts can be remarkably accurate

On the other hand,

when forecasting currency exchange rates,

only one of the conditions is satisfied: there is plenty of available data

However,

we have a very limited understanding of the factors that affect exchange rates,

and forecasts of the exchange rate have a direct effect on the rates themselves

If there are well-publicized forecasts that the exchange rate will increase,

then people will immediately adjust the price they are willing to pay and so the forecasts are selffulfilling

In a sense the exchange rates become their own forecasts

This is an example of the "efficient market hypothesis"

Consequently,

forecasting whether the exchange rate will rise or fall tomorrow is about as predictable as forecasting whether a tossed coin will come down as a head or a tail

In both situations,

you will be correct about 50 Often in forecasting,

a key step is knowing when something can be forecast accurately,

and when forecasts will be no better than tossing a coin

Good forecasts capture the genuine patterns and relationships which exist in the historical data,

but do not replicate past events that will not occur again

In this book,

we will learn how to tell the difference between a random fluctuation in the past data that should be ignored,

and a genuine pattern that should be modelled and extrapolated

Many people wrongly assume that forecasts are not possible in a changing environment

Every environment is changing,

and a good forecasting model captures the way in which things are changing

Forecasts rarely assume that the environment is unchanging

What is normally assumed is that the way in which the environment is changing will continue into the future

That is,

a highly volatile environment will continue to be highly volatile

a business with fluctuating sales will continue to have fluctuating sales

and an economy that has gone through booms and busts will continue to go through booms and busts

A forecasting model is intended to capture the way things move,

As Abraham Lincoln said,

"If we could first know where we are and whither we are tending,

we could better judge what to do and how to do it"

Forecasting situations vary widely in their time horizons,

factors determining actual outcomes,

Forecasting methods can be very simple such as using the most recent observation as a forecast (which is called the "naïve method”),

or highly complex such as neural nets and econometric systems of simultaneous equations

Sometimes,

there will be no data available at all

For example,

we may wish to forecast the sales of a new product in its first year,

but there are obviously no data to work with

In situations like this,

we use judgmental forecasting,

The choice of method depends on what data are available and the predictability of the quantity to be forecast

Forecasting,

Forecasting is a common statistical task in business,

where it helps to inform decisions about the scheduling of production,

and provides a guide to long-term strategic planning

However,

business forecasting is often done poorly,

and is frequently confused with planning and goals

They are three different things

Forecasting

Forecasting: principles and practice

is about predicting the future as accurately as possible,

given all of the information available,

including historical data and knowledge of any future events that might impact the forecasts

Goals are what you would like to have happen

Goals should be linked to forecasts and plans,

but this does not always occur

Too often,

goals are set without any plan for how to achieve them,

and no forecasts for whether they are realistic

Planning is a response to forecasts and goals

Planning involves determining the appropriate actions that are required to make your forecasts match your goals

Forecasting should be an integral part of the decision-making activities of management,

as it can play an important role in many areas of a company

Modern organizations require short-term,

medium-term and long-term forecasts,

depending on the specific application

Short-term forecasts are needed for the scheduling of personnel,

As part of the scheduling process,

forecasts of demand are often also required

Medium-term forecasts are needed to determine future resource requirements,

in order to purchase raw materials,

or buy machinery and equipment

Long-term forecasts are used in strategic planning

Such decisions must take account of market opportunities,

environmental factors and internal resources

An organization needs to develop a forecasting system that involves several approaches to predicting uncertain events

Such forecasting systems require the development of expertise in identifying forecasting problems,

applying a range of forecasting methods,

selecting appropriate methods for each problem,

and evaluating and refining forecasting methods over time

It is also important to have strong organizational support for the use of formal forecasting methods if they are to be used successfully

Determining what to forecast

In the early stages of a forecasting project,

decisions need to be made about what should be forecast

For example,

if forecasts are required for items in a manufacturing environment,

it is necessary to ask whether forecasts are needed for: 1

or for outlets grouped by region,

? It is also necessary to consider the forecasting horizon

Will forecasts be required for one month in advance,

? Different types of models will be necessary,

depending on what forecast horizon is most important

How frequently are forecasts required

? Forecasts that need to be produced frequently are better done using an automated system than with methods that require careful manual work

Forecasting: principles and practice

It is worth spending time talking to the people who will use the forecasts to ensure that you understand their needs,

and how the forecasts are to be used,

before embarking on extensive work in producing the forecasts

Once it has been determined what forecasts are required,

it is then necessary to find or collect the data on which the forecasts will be based

The data required for forecasting may already exist

These days,

a lot of data are recorded and the forecaster’s task is often to identify where and how the required data are stored

The data may include sales records of a company,

the historical demand for a product,

or the unemployment rate for a geographical region

A large part of a forecaster’s time can be spent in locating and collating the available data prior to developing suitable forecasting methods

Forecasting data and methods

The appropriate forecasting methods depend largely on what data are available

If there are no data available,

or if the data available are not relevant to the forecasts,

then qualitative forecasting methods must be used

These methods are not purely guesswork—there are well-developed structured approaches to obtaining good forecasts without using historical data

These methods are discussed in Chapter 3

Quantitative forecasting can be applied when two conditions are satisfied: 1

numerical information about the past is available

it is reasonable to assume that some aspects of the past patterns will continue into the future

There is a wide range of quantitative forecasting methods,

often developed within specific disciplines for specific purposes

Each method has its own properties,

and costs that must be considered when choosing a specific method

Most quantitative forecasting problems use either time series data (collected at regular intervals over time) or cross-sectional data (collected at a single point in time)

Cross-sectional forecasting With cross-sectional data,

we are wanting to predict the value of something we have not observed,

using the information on the cases that we have observed

Examples of cross-sectional data include: • House prices for all houses sold in 2011 in a particular area

We are interested in predicting the price of a house not in our data set using various house characteristics: position,

• Fuel economy data for a range of 2009 model cars

We are interested in predicting the carbon footprint of a vehicle not in our data set using information such as the size of the engine and the fuel efficiency of the car

Example 1

each of which has an automatic transmission,

four cylinders and an engine size under 2 liters

Model Chevrolet Aveo Chevrolet Aveo 5 Honda Civic

Engine (litres) 1

City (mpg) 25 25 25

Highway (mpg) 34 34 36

Carbon (tons CO2 per year) 6

Forecasting: principles and practice Honda Civic Hybrid 1

four cylinders and small engines

City and Highway represent fuel economy while driving in the city and on the highway

A forecaster may wish to predict the carbon footprint (tons of CO2 per year) for other similar vehicles that are not included in the above table

It is necessary to first estimate the effects of the predictors (number of cylinders,

and fuel economy) on the variable to be forecast (carbon footprint)

provided that we know the predictors for a car not in the table,

we can forecast its carbon footprint

Cross-sectional models are used when the variable to be forecast exhibits a relationship with one or more other predictor variables

The purpose of the cross-sectional model is to describe the form of the relationship and use it to forecast values of the forecast variable that have not been observed

Under this model,

any change in predictors will affect the output of the system in a predictable way,

assuming that the relationship does not change

Models in this class include regression models,

and some kinds of neural networks

These models are discussed in Chapters 4,

Some people use the term "predict" for cross-sectional data and "forecast" for time series data (see below)

In this book,

we will not make this distinction—we will use the words interchangeably

Time series forecasting Time series data are useful when you are forecasting something that is changing over time (e

Examples of time series data include: • Daily IBM stock prices • Monthly rainfall • Quarterly sales results for Amazon • Annual Google profits

Forecasting: principles and practice

Anything that is observed sequentially over time is a time series

In this book,

we will only consider time series that are observed at regular intervals of time (e

Irregularly spaced time series can also occur,

but are beyond the scope of this book

When forecasting time series data,

the aim is to estimate how the sequence of observations will continue into the future

The following figure shows the quarterly Australian beer production from 1992 to the third quarter of 2008

Figure 1

The blue lines show forecasts for the next two years

Notice how the forecasts have captured the seasonal pattern seen in the historical data and replicated it for the next two years

The dark shaded region shows 80% prediction intervals

That is,

each future value is expected to lie in the dark blue region with a probability of 80%

The light shaded region shows 95% prediction intervals

These prediction intervals are a very useful way of displaying the uncertainty in forecasts

In this case,

the forecasts are expected to be very accurate,

hence the prediction intervals are quite narrow

Time series forecasting uses only information on the variable to be forecast,

and makes no attempt to discover the factors which affect its behavior

Therefore it will extrapolate trend and seasonal patterns,

but it ignores all other information such as marketing initiatives,

changes in economic conditions,

Time series models used for forecasting include ARIMA models,

exponential smoothing and structural models

These models are discussed in Chapters 6,

Predictor variables and time series forecasting Predictor variables can also be used in time series forecasting

For example,

suppose we wish to forecast the hourly electricity demand (ED) of a hot region during the summer period

A model with predictor variables might be of the form ED = f (current temperature,

The relationship is not exact—there will always be changes in electricity demand that cannot be accounted for by the predictor variables

The “error” term on the right allows for random variation and the effects of relevant variables that are not included in the model

We call this an “explanatory model” because it helps explain what causes the variation in electricity demand

Forecasting: principles and practice

Because the electricity demand data form a time series,

we could also use a time series model for forecasting

In this case,

a suitable time series forecasting equation is of the form EDt+1 = f (EDt ,

EDt−1 ,

EDt−2 ,

EDt−3 ,

prediction of the future is based on past values of a variable,

but not on external variables which may affect the system

the "error" term on the right allows for random variation and the effects of relevant variables that are not included in the model

There is also a third type of model which combines the features of the above two models

For example,

it might be given by EDt+1 = f (EDt ,

These types of mixed models have been given various names in different disciplines

They are known as dynamic regression models,

and linear system models (assuming f is linear)

These models are discussed in Chapter 9

An explanatory model is very useful because it incorporates information about other variables,

rather than only historical values of the variable to be forecast

However,

there are several reasons a forecaster might select a time series model rather than an explanatory model

the system may not be understood,

and even if it was understood it may be extremely difficult to measure the relationships that are assumed to govern its behavior

Second,

it is necessary to know or forecast the various predictors in order to be able to forecast the variable of interest,

the main concern may be only to predict what will happen,

Finally,

the time series model may give more accurate forecasts than an explanatory or mixed model

The model to be used in forecasting depends on the resources and data available,

the accuracy of the competing models,

and how the forecasting model is to be used

Notation For cross-sectional data,

we will use the subscript i to indicate a specific observation

For example,

yi will denote the ith observation in a data set

We will also use N to denote the total number of observations in the data set

For time series data,

we will use the subscript t instead of i

For example,

yt will denote the observation at time t

We will use T to denote the number of observations in a time series

When we are making general comments that could be applicable to either cross-sectional or time series data,

Some case studies

The following four cases are from our consulting practice and demonstrate different types of forecasting situations and the associated problems that often arise

Case 1 The client was a large company manufacturing disposable tableware such as napkins and paper plates

They needed forecasts of each of hundreds of items every month

The time series data showed a range of patterns,

At the time,

they were using their own software,

but it often produced forecasts that did not seem sensible

The methods that were being used were the following: 1

average of the last 12 months data

Forecasting: principles and practice 2

average of the last 6 months data

prediction from a straight line regression over the last 12 months

prediction from a straight line regression over the last 6 months

prediction obtained by a straight line through the last observation with slope equal to the average slope of the lines connecting last year’s and this year’s values

prediction obtained by a straight line through the last observation with slope equal to the average slope of the lines connecting last year’s and this year’s values,

where the average is taken only over the last 6 months

They required us to tell them what was going wrong and to modify the software to provide more accurate forecasts

The software was written in COBOL making it difficult to do any sophisticated numerical computation

Case 2 In this case,

the client was the Australian federal government who needed to forecast the annual budget for the Pharmaceutical Benefit Scheme (PBS)

The PBS provides a subsidy for many pharmaceutical products sold in Australia,

and the expenditure depends on what people purchase during the year

The total expenditure was around A$7 billion in 2009 and had been underestimated by nearly $1 billion in each of the two years before we were asked to assist with developing a more accurate forecasting approach

In order to forecast the total expenditure,

it is necessary to forecast the sales volumes of hundreds of groups of pharmaceutical products using monthly data

Almost all of the groups have trends and seasonal patterns

The sales volumes for many groups have sudden jumps up or down due to changes in what drugs are subsidised

The expenditures for many groups also have sudden changes due to cheaper competitor drugs becoming available

Thus we needed to find a forecasting method that allowed for trend and seasonality if they were present,

and at the same time was robust to sudden changes in the underlying patterns

It also needed to be able to be applied automatically to a large number of time series

Case 3 A large car fleet company asked us to help them forecast vehicle re-sale values

They purchase new vehicles,

lease them out for three years,

Better forecasts of vehicle sales values would mean better control of profits

understanding what affects resale values may allow leasing and sales policies to be developed in order to maximize profits

At the time,

the resale values were being forecast by a group of specialists

Unfortunately,

they saw any statistical model as a threat to their jobs and were uncooperative in providing information

Nevertheless,

the company provided a large amount of data on previous vehicles and their eventual resale values

Case 4 In this project,

we needed to develop a model for forecasting weekly air passenger traffic on major domestic routes for one of Australia’s leading airlines

The company required forecasts of passenger numbers for each major domestic route and for each class of passenger (economy class,

business class and first class)

The company provided weekly traffic data from the previous six years

Air passenger numbers are affected by school holidays,

School holidays often do not coincide in different Australian cities,

and sporting events sometimes move from one city to another

During the period of the

Forecasting: principles and practice

there was a major pilots’ strike during which there was no traffic for several months

A new cut-price airline also launched and folded

Towards the end of the historical data,

the airline had trialled a redistribution of some economy class seats to business class,

and some business class seats to first class

After several months,

the seat classifications reverted to the original distribution

The basic steps in a forecasting task

A forecasting task usually involves five basic steps

Step 1: Problem definition

Often this is the most difficult part of forecasting

Defining the problem carefully requires an understanding of the way the forecasts will be used,

and how the forecasting function fits within the organization requiring the forecasts

A forecaster needs to spend time talking to everyone who will be involved in collecting data,

and using the forecasts for future planning

Step 2: Gathering information

There are always at least two kinds of information required: (a) statistical data,

and (b) the accumulated expertise of the people who collect the data and use the forecasts

it will be difficult to obtain enough historical data to be able to fit a good statistical model

However,

very old data will be less useful due to changes in the system being forecast

Step 3: Preliminary (exploratory) analysis

Always start by graphing the data

Are there consistent patterns

? Is there a significant trend

? Is there evidence of the presence of business cycles

? Are there any outliers in the data that need to be explained by those with expert knowledge

? How strong are the relationships among the variables available for analysis

? Various tools have been developed to help with this analysis

These are discussed in Chapters 2 and 6

Step 4: Choosing and fitting models

The best model to use depends on the availability of historical data,

the strength of relationships between the forecast variable and any explanatory variables,

and the way the forecasts are to be used

It is common to compare two or three potential models

Each model is itself an artificial construct that is based on a set of assumptions (explicit and implicit) and usually involves one or more parameters which must be "fitted" using the known historical data

We will discuss regression models (Chapters 4 and 5),

exponential smoothing methods (Chapter 7),

Box-Jenkins ARIMA models (Chapter 8),

and a variety of other topics including dynamic regression models,

and vector autoregression in Chapter 9

Step 5: Using and evaluating a forecasting model

Once a model has been selected and its parameters estimated,

the model is used to make forecasts

The performance of the model can only be properly evaluated after the data for the forecast period have become available

A number of methods have been developed to help in assessing the accuracy of forecasts

There are also organizational issues in using and acting on the forecasts

A brief discussion of some of these issues is in Chapter 2

Forecasting: principles and practice

The statistical forecasting perspective

The thing we are trying to forecast is unknown (or we wouldn’t be forecasting it),

and so we can think of it as a random variable

For example,

the total sales for next month could take a range of possible values,

and until we add up the actual sales at the end of the month we don’t know what the value will be

until we know the sales for next month,

Because next month is relatively close,

we usually have a good idea what the likely sales values could be

On the other hand,

if we are forecasting the sales for the same month next year,

the possible values it could take are much more variable

In most forecasting situations,

the variation associated with the thing we are forecasting will shrink as the event approaches

In other words,

the further ahead we forecast,

When we obtain a forecast,

we are estimating the middle of the range of possible values the random variable could take

Very often,

a forecast is accompanied by a prediction interval giving a range of values the random variable could take with relatively high probability

For example,

a 95% prediction interval contains a range of values which should include the actual future value with probability 95%

A forecast is always based on some observations

Suppose we denote all the information we have observed as I and we want to forecast yi

We then write yi|I meaning "the random variable yi given what we know in I"

The set of values that this random variable could take,

along with their relative probabilities,

is known as the "probability distribution" of yi|I

In forecasting,

we call this the "forecast distribution"

When we talk about the "forecast",

we usually mean the average value of the forecast distribution,

and we put a "hat" over yy to show this

we write the forecast of yi as yˆi ,

meaning the average of the possible values that yi could take given everything we know

Occasionally,

we will use yˆi to refer to the median (or middle value) of the forecast distribution instead

With time series forecasting,

it is often useful to specify exactly what information we have used in calculating the forecast

Then we will write,

yˆt|t−1 to mean the forecast of yt taking account of all previous observations (y1 ,

Similarly,

yˆT +h|T means the forecast of yT +h taking account of y1 ,

an h-step forecast taking account of all observations up to time T )

Exercises

For each of the four case studies in Section 1

what sort of data is involved: time series or cross-sectional data

For cases 3 and 4 in Section Section 1

list the possible predictor variables that might be useful,

assuming that the relevant data are available

For case 3 in Section Section 1

describe the five steps of forecasting in the context of this project

Further reading

Principles of forecasting: a handbook for researchers and practitioners

Boston,

MA: Kluwer Academic Publishers

Fildes (2012)

Principles of business forecasting

South-Western College Pub

Chapter 2

The forecaster’s toolbox Before we discuss any forecasting methods,

it is necessary to build a toolbox of techniques that will be useful for many different forecasting situations

Each of the tools discussed in this chapter will be used repeatedly in subsequent chapters as we develop and explore a range of forecasting methods

Graphics

The first thing to do in any data analysis task is to plot the data

Graphs enable many features of the data to be visualized including patterns,

and relationships between variables

The features that are seen in plots of the data must then be incorporated,

into the forecasting methods to be used

Just as the type of data determines what forecasting method to use,

it also determines what graphs are appropriate

Time plots For time series data,

the obvious graph to start with is a time plot

That is,

the observations are plotted against the time of observation,

with consecutive observations joined by straight lines

The figure below shows the weekly economy passenger load on Ansett Airlines between Australia’s two largest cities

Figure 2

Forecasting: principles and practice Listing 2

C l'a s's " ] ,

main=" Economy␣ c'l'a s's ␣ p a s's e n g e r s': ␣ Melbourne−Sydney " ,

The time plot immediately reveals some interesting features

• There was a period in 1989 when no passengers were carried — this was due to an industrial dispute

• There was a period of reduced load in 1992

This was due to a trial in which some economy class seats were replaced by business class seats

• A large increase in passenger load occurred in the second half of 1991

• There are some large dips in load around the start of each year

These are due to holiday effects

• There is a long-term fluctuation in the level of the series which increases during 1987,

decreases in 1989 and increases again through 1990 and 1991

• There are some periods of missing observations

Any model will need to take account of all these features in order to effectively forecast the passenger load into the future

A simpler time series is shown in Figure 2

Figure 2

y l'a b=" \$␣ m i l'l i o n " ,

main=" A n t i d'i a b e t i c'␣ drug ␣ s'a l'e s'" )

Here there is a clear and increasing trend

There is also a strong seasonal pattern that increases in size as the level of the series increases

The sudden drop at the end of each year is caused by a government subsidisation scheme that makes it cost-effective for patients to stockpile drugs at the end of the calendar year

Any forecasts of this series would need to capture the seasonal pattern,

and the fact that the trend is changing slowly

Forecasting: principles and practice

Time series patterns In describing these time series,

we have used words such as "trend" and "seasonal" which need to be more carefully defined

• A trend exists when there is a long-term increase or decrease in the data

There is a trend in the antidiabetic drug sales data shown above

• A seasonal pattern occurs when a time series is affected by seasonal factors such as the time of the year or the day of the week

The monthly sales of antidiabetic drugs above shows seasonality partly induced by the change in cost of the drugs at the end of the calendar year

• A cycle occurs when the data exhibit rises and falls that are not of a fixed period

These fluctuations are usually due to economic conditions and are often related to the "business cycle"

The economy class passenger data above showed some indications of cyclic effects

It is important to distinguish cyclic patterns and seasonal patterns

Seasonal patterns have a fixed and known length,

while cyclic patterns have variable and unknown length

The average length of a cycle is usually longer than that of seasonality,

and the magnitude of cyclic variation is usually more variable than that of seasonal variation

Cycles and seasonality are discussed further in Section 6

Many time series include trend,

When choosing a forecasting method,

we will first need to identify the time series patterns in the data,

and then choose a method that is able to capture the patterns properly

Seasonal plots A seasonal plot is similar to a time plot except that the data are plotted against the individual "seasons" in which the data were observed

An example is given below showing the antidiabetic drug sales

Figure 2

Forecasting: principles and practice Listing 2

y l'a b=" \$␣ m i l'l i o n " ,

main=" S e a s'o n a l'␣ p l'o t : ␣ a n t i d'i a b e t i c'␣ drug ␣ s'a l'e s'" ,

These are exactly the same data shown earlier,

but now the data from each season are overlapped

A seasonal plot allows the underlying seasonal pattern to be seen more clearly,

and is especially useful in identifying years in which the pattern changes

In this case,

it is clear that there is a large jump in sales in January each year

Actually,

these are probably sales in late December as customers stockpile before the end of the calendar year,

but the sales are not registered with the government until a week or two later

The graph also shows that there was an unusually low number of sales in March 2008 (most other years show an increase between February and March)

The small number of sales in June 2008 is probably due to incomplete counting of sales at the time the data were collected

Seasonal subseries plots An alternative plot that emphasises the seasonal patterns is where the data for each season are collected together in separate mini time plots

Figure 2

Listing 2

y l'a b=" \$␣ m i l'l i o n " ,

main=" S e a s'o n a l'␣ d'e v i a t i o n ␣ p l'o t : ␣ a n t i d'i a b e t i c'␣ drug ␣ s'a l'e s'" ) axis ( 1 ,

The horizontal lines indicate the means for each month

This form of plot enables the underlying seasonal pattern to be seen clearly,

and also shows the changes in seasonality over time

It is especially useful in identifying changes within particular seasons

In this example,

the plot is not particularly revealing

this is the most useful way of viewing seasonal changes over time

Scatterplots The graphs discussed so far are useful for time series data

Scatterplots are most useful for exploring relationships between variables in cross-sectional data

Forecasting: principles and practice

The figure below shows the relationship between the carbon footprint and fuel economy for small cars (using an extension of the data set shown in Section 1

Each point on the graph shows one type of vehicle

The points are slightly "jittered" to prevent overlapping points

Figure 2

Listing 2

5 ] ) ,

8 ] ) ,

y l'a b=" Carbon ␣ f o o t p r i n t " )

There is a strong non-linear relationship between the size of a car’s carbon footprint and its city-based fuel economy

Vehicles with better fuel-economy have a smaller carbon-footprint than vehicles that use a lot of fuel

However,

the relationship is not linear — there is much less benefit in improving fuel-economy from 30 to 40 mpg than there was in moving from 20 to 30 mpg

The strength of the relationship is good news for forecasting: for any cars not in this database,

knowing the fuel economy of the car will allow a relatively accurate forecast of its carbon footprint

The scatterplot helps us visualize the relationship between the variables,

and suggests that a forecasting model must include fuel-economy as a predictor variable

Some of the other information we know about these cars may also be helpful in improving the forecasts

Scatterplot matrices When there are several potential predictor variables,

it is useful to plot each variable against each other variable

These plots can be arranged in a scatterplot matrix,

Listing 2

7 ) ] ,

For each panel,

the variable on the vertical axis is given by the variable name in that row,

and the variable on the horizontal axis is given by the variable name in that column

For example,

the graph of carbon-footprint against city mpg is shown on the bottom row,

The value of the scatterplot matrix is that it enables a quick view of the relationships between all pairs of variables

Outliers can also be seen

In this example,

there are two vehicles that have very high highway mileage,

small engines and low carbon footprints

These are hybrid vehicles: Honda Civic and Toyota Prius

Forecasting: principles and practice