1 chapter 7: forecasting 7.1 introduction 7.2 time series characteristics and components 7.3...

Chapter 7: Forecasting

7.1 Introduction

7.2 Time Series Characteristics and Components

7.3 Introduction to SAS Forecast Studio

7.4 Time Series Regression Models

7.5 Time Series Data and Hierarchical Data Structure

7.6 Recommended Reading

7.1 Introduction7.1 Introduction

Objectives Name some business applications of time series

modeling. Define basic time series concepts and approaches.

Prediction in Time: ForecastingYou learned about statistical models to predict some outcome variable (buy/no buy, revenue, payoff/default).

Sometimes it is more important to predict what a particular variable will be in the future and at different points in time.

In these situations, the data could be based on an accumulation of measurements from customers (revenue), a physical process (customer service wait times), or even a technological phenomenon (demand on a server).

Prediction in Time: ForecastingThe data for forecasting is known as a time series. Hence, the name time series forecasting refers to the general modeling approach.

Time series forecasting involves the prediction of future values of a response or the interpretation of what produced changes that were observed in a series over time.

Examples of Forecasting in BusinessUnits in inventory

Demand on a system

Customer activity

Revenue

Transactions

Manufacturing supplies

Pricing

How many big-screen TVs are you likely to sell this week?

Demand on a system

Customer activity

Revenue

Transactions

Pricing

Should you use this shelf space for more peanut butter or for more salsa?

Demand on a system

Customer activity

Revenue

Transactions

Pricing

What time of day produces peak server demand? Can you allocate more resources at that time?

Demand on a system

Customer activity

Revenue

Transactions

Pricing

How many tables can you expect to fill at your restaurant on Valentine’s Day?

Demand on a system

Customer activity

Revenue

Transactions

Pricing

If you put the item on sale this week, will demand go down next week?

Demand on a system

Customer activity

Revenue

Transactions

Pricing

How much in ticket sales can you expect on Thursday?

Demand on a system

Customer activity

Revenue

Transactions

Pricing

Is there a cyclical pattern to the number of purchases made on your Web site over a week?

Demand on a system

Customer activity

Revenue

Transactions

PricingWhen should you reorder raw materials?

Demand on a system

Customer activity

Revenue

Transactions

Pricing

What are the pricing trends in the past quarter, compared to the previous three years?

Idea ExchangeName a specific example variable that you might want to forecast. Would you want to have daily forecasts? Weekly? Yearly?

What type of data would you be able to access to obtain forecasts?

Example: Singapore Unemployment

Example: Singapore UnemploymentThe time series includes information that is gathered over time is for equally spaced time intervals uses the same measurement at each time enables the visualization of a pattern over time enables the quantification of such patterns enables forecasts to be made for future time points

based on past behavior.

Example: Ice CreamYou are CEO of a large ice cream producer.

Profitability depends on accomplishing goals in three key areas: Supply chain activities for the coming year must be

coordinated and started. Production schedules must be set for the next three months. Strategic pricing initiatives and

promotional campaigns need to be assessed and approved.

Consider the key components of a time series.

Ice Cream Demand

Ice Cream Demand: Seasonal Cycle

Ice Cream Demand: Trend

Ice Cream Demand: Effects of Independent Variables

Ice Cream DemandBy evaluating and quantifying the effects described for the ice cream demand example, you can do the following: forecast with some confidence how much ice cream you are likely

to sell each month (and hence how much you should produce) make choices about when you should make a special promotion

(to uplift an anticipated sales slump) estimate how much difference

those promotions are likely to make

identify when something unexpected occurred (such as an undiscovered competitor taking market share)

Two Basic Approaches in Time Series Analysis Inference-based: what happened

– Policy or intervention evaluation– Marketing mix evaluation– Scenario evaluation or sensitivity analysis

Forecasting-based: what is likely to happen– Logistical decisions– Tactical decisions– Strategic decisions

Two Basic Approaches in Time Series Analysis Inference-based: what happened

– Policy or intervention evaluation– Marketing mix evaluation– Scenario evaluation or sensitivity analysis

Forecasting-based: what is likely to happen– Logistical decisions– Tactical decisions– Strategic decisions

7.01 PollTime series forecasting is concerned only with obtaining good forecasts of future values and not with understanding why a series changed.

7.01 Poll – Correct AnswerTime series forecasting is concerned only with obtaining good forecasts of future values and not with understanding why a series changed.

Idea ExchangeConsider the example that you gave of a forecasting variable earlier. Are you more interested in inference-based analysis, or forecast-based analysis? Or both? Why? Give specific examples.

7.1 Introduction

7.2 Time Series Characteristics and Components7.2 Time Series Characteristics and Components

Objectives Explain basic time series concepts and approaches. List the elements of a time series. Provide examples of time series models.

The Universal Time Series Model

),,,( ttttt EXSTfY

SEASONAL ERROR (Irregular)

Airline Passengers 1990-2004

Statistical Time Series

Interval=1 month

A statistical time series isan indexed set of numbers. The index can consist of dates or other numbers.Many business time series are equally spaced.

A time series is equally spaced if any two consecutive indices havethe same interval time difference.

Equally Spaced Time Series

Equally spaced time series

Unequally spaced time series

Equally spaced time serieswith missing values

7.02 Multiple Choice PollTime series data must be

a. equally spaced data with no missing times

b. equally spaced, with missing values padded in the response column, if necessary

c. equally or unequally spaced, as long as missing intervals are indicated by either missing values or a skip in the time index column.

7.02 Multiple Choice Poll – Correct AnswerTime series data must be

a. equally spaced data with no missing times.

b. equally spaced, with missing values padded in the response column, if necessary.

c. equally or unequally spaced, as long as missing intervals are indicated by either missing values or a skip in the time index column.

),,,( ttttt EXSTfY

Airline Passengers 1994–1997: Trend

Time Series TrendTrend usually refers to a deterministic function of time. Time series can be made of deterministic and

stochastic components. A stochastic component is subject to random variation

and can never be predicted perfectly except by chance.

A deterministic component exhibits no random variation and can be predicted perfectly. Common deterministic trend functions include linear trend, curvilinear trend, logarithmic trend, and exponential trend.

Deterministic Trend ModelsLinear Trend

Quadratic Trend

tYt 10

2210 ttYt Y

Notation

tYt 10

Time series

Time index Parameters Time index

Stochastic Trend ModelsRandom Walk

Random Walk with Drift

continued...

ttt EYY 1

Accommodating Stochastic Trend: Differencing

A First Difference of the Random Walk Process

First Difference

7.03 PollDeterministic trends are accommodated in time series models through differencing.

7.03 Poll – Correct AnswerDeterministic trends are accommodated in time series models through differencing.

No. Stochastic trends are accommodated through differencing.

),,,( ttttt EXSTfY

Airline Passengers 1994–1997: Seasonal

August

February

SeasonalityThe seasonal component of a time series represents the effects of seasonal variation. The foundation of seasonal variation is one or more

of the cycles produced by the motion of the celestial bodies in the solar system, dominated by the earth circling the sun every year. Another influential activity is the moon circling the earth approximately every 28 days.

The most general meaning of seasonality is a component that describes repetitive behavior at known seasonal periods. If the seasonal period is integer S, then seasonal factors are factors that repeat every S units of time.

Accommodating Seasonal Components Trigonometric functions (sine waves) Seasonal dummy variables Seasonal differences (Box-Jenkins modeling) Seasonal model components (ESM models)

Dummy Variables A dummy variable is an indicator variable. To indicate a specific time point, a dummy variable

takes one as the value for that time point. At all other time points, it takes zero as the value.

otherwise 0

2001 Sep when 1 tI t

Seasonal Dummy VariablesFor a time series with S seasons, there are S dummy variables, one for each season.

Monthly Data: IJAN , IFEB ,…, IDEC

Daily Data: ISUN , IMON ,…, ISAT

Quarterly Data: IQ1 , IQ2 , IQ3 , IQ4

Stochastic Seasonal Functions: Seasonal DifferencingFor seasonal data with period S, express the current value as a function that includes the value S time units in the past.

Yt = Yt-S + TRENDt + IRREGULARt

SYt = Yt Yt-S is called a difference of order S.

Examples:

Monthly: This January is a function of last January and so on.

Daily: This Sunday is a function of last Sunday and so on.

7.04 PollSeasonal data can be accommodated through differencing.

7.04 Poll – Correct AnswerSeasonal data can be accommodated through differencing.

),,,( ttttt EXSTfY

The Irregular Component The irregular component of a time series is what

remains when trend, seasonal, and input effects are removed.

The irregular component need not represent a random sequence of uncorrelated values. However, most models specify that the irregular component must be stationary.

A stationary time series has a constant mean and variance at all time points.

Additive Decomposition of the Airline Data

T: LinearTrend

S: SeasonalAverage

I: IrregularComponent

Analysis of Residuals (Forecast Error)

Residuals for Additive Decomposition Model

-3000000

-2000000

-1000000

1000000

2000000

3000000

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

Time Index

),,,( ttttt EXSTfY

Events

Event Examples Retail promotions Advertising campaigns Negative article in a major publication Nickel Beer Thursday Mergers and acquisitions Government legislated policy changes Organizational personnel and/or policy changes Christmas Strikes Scandal Injury, illness, or death of a key player

(such as a CEO, CFO, or chief scientist)

How Do Event Variables Improve Accuracy? Event variables enable the forecast model to

accommodate discrete shifts, also called jumps or bangs, in time series data.

Event variables in time series models are primarily intercept shifters.

Intercept shifters are included in the model as explanatory variables and are based on columns of 0s and 1s in the data set.

How Do Event Variables Improve Accuracy?The data is fit with a linear model: tt trendsales

Data jump at Date = T* causesa large residual.

How Do Event Variables Improve Accuracy?The linear model can be refined by modifying the intercept term as follows:

When Date = T*, the model’s intercept = ( + ), and when Date ≠ T*, the model’s intercept = .

otherwse0andTDateif1

trendDsales tt

Event Variable Creation

BigStormEventT* = '01AUG2003'd

Resulting DummyColumn = D

00…010…0

Demand HistoryFor Sales

01JAN200201FEB2002

…01JUL200301AUG200301SEP2003

…01JUN2003

The temporary intercept shift is accomplished by adding a 0-1 or dummy column to the data.

How Do Event Variables Improve Accuracy?The data is fit with a linear model and a pulse event variable.

Less biased forecast

The residual is much smaller.

trendDsalest )(

Event Variable Qualifiers The event variable discussed above is a pulse type.

The pulse event variable qualifies variation in the data as follows:– There is a discrete shift in the data at Date = T*.

Before and after Date = T*, the series is at its steady-state intercept and slope.

– That is, the series is impacted only for one time interval: Date = T*.

How might the linear model be refined if the shift in the data resembles what is depicted on the next slide?

How Do Event Variables Improve Accuracy?The data is fit with a linear model.

trendsalest

A permanent Intercept shift at this date

How Do Event Variables Improve Accuracy?The linear model can be refined by modifying the intercept term as follows:

otherwse0andTDateif1

trendDsales tt

This is the same model specification as before, but the dummy column is changed as shown on the nextslide.

When Date => T*, the model’s intercept = ( + ), and when Date < T*, the model’s intercept = .

Event Variable Creation

New Law EnactedT* = '01AUG2003'd

Resulting DummyColumn = D

00…011…1

Demand HistoryFor Sales

01JAN200201FEB2002

…01JUL200301AUG200301SEP2003

…01JUN2003

The permanent intercept shift is accomplished by adding a 0-1 or dummy column to the data.

How Do Event Variables Improve Accuracy?The data is fit with a linear model and a step event variable.

The permanent shift is accommodated in the model.

trendDsalest )(

Event Variable Qualifiers The event variable discussed above is a step type. The step event qualifies variation in the data as

follows: – There is a discrete shift in the data at Date = T*.

Before Date = T*, the series is at its pre-event, steady-state intercept. When Date => T*, the series it at a new, steady state intercept.

– That is, the series is impacted permanently; on and after Date = T*, the series has a new intercept.

Basic Event Variable Types

Idea ExchangeCompare the use of event variables for events that can be foreseen (such as Christmas holiday) to events that cannot be foreseen (such as a major storm).

How would these two types of events change the usefulness of your forecasts?

How would they change how you would make business decisions based on the forecasts?

7.1 Introduction

7.3 Introduction to SAS Forecast Studio7.3 Introduction to SAS Forecast Studio

Objectives Create a project and generate forecasts for a single

series. Explain the basics of navigating SAS Forecast Studio

and interpreting results.

SAS Forecast Studio

SAS Forecast Studio: Interface Tour

SAS Forecast Studio: Interface Tour Menu Bar and Shortcut Buttons

Menu Bar andShortcut Buttons

The Active Series

OverviewPanel

The Four View Tabs

The Forecasting View

The Modeling View

The Model Selection List (MSL) is associatedwith the highlighted series.

The Modeling View

The Series View

The ScenarioAnalysis View

The Forecasting Workflow

Forecasting workflow

Large-Scale Forecasting Scenario

Time Series Data

80% can be forecast automatically.

10% requires extra effort.

10% cannot be forecast accurately.

Large-Scale Forecasting Scenario

Time Series Data

80% can be forecast automatically.

10% requires extra effort.

10% cannot be forecast accurately.

SuperToys Inc.SuperToys Inc. ran a campaign to promote sales of a classic line of dolls. Your job is to measure the impact of the promotion

and determine whether the promotion should be run in the future, perhaps to promote Christmas sales.

Forecast the sales for the next several weeks and quantify the effect of the discount promotion on weekly sales.

The DataREG2_GBTOYS is a time series data set with the following: weekly data for a popular doll in all toy

stores averaged over four sales regions the number of units sold per region

each week (units) information about when special

discount promotions were implemented (pctpromo)

Generating Forecasts Automatically

Toy Case Study

Task: Use the REG2_GBTOYS data to forecast a single series with a discount promotion.

Use of Accuracy CriteriaTwo ways to judge accuracy: Accuracy can be calculated for one-step-ahead

forecasts over the entire range of the data. Accuracy can be calculated for a holdout sample of

data at the end of each time series that was not used to construct models. A time series might be too short to enable use of a holdout sample. This method is preferred, but it is often not feasible. Using a holdout sample to judge accuracy is often referred to as honest assessment because it simulates fitting and deploying a model and then judging accuracy in the live environment.

Assessing Generated Models

Toy Case Study

Tasks: Compare different candidate models to select the best one, and evaluate the parameter estimates produced by the model.

Creating a Scenario Analysis

Toy Case Study

Task: Create a scenario analysis that evaluates the effect of a promotional campaign on future forecasts.

Idea ExchangeYou could do more with scenario analysis. For example, if management wants to clear 250 extra dolls from inventory, how many weeks of sales promotion would be needed to sell 250 above and beyond the baseline forecast?

Exercise

This exercise reinforces the concepts discussed previously.

7.1 Introduction

7.4 Time Series Regression Models7.4 Time Series Regression Models

Objectives Describe the basic assumptions of static regression

modeling. Explain the basics of dynamic regression models. Define the role of transfer functions in time series

regression models.

Static RegressionIn static (non-time series) regression analysis, the predictors and the response are assumed to occur together in time.

DISCOUNT(today) INCREASED SALES(today)

Static RegressionHowever, it might be more important to understand how predictors affect a response at a different time.

DISCOUNT(today) INCREASED SALES (today) and DECREASED SALES (tomorrow)

Static Linear RegressionAssumptions The predictor variables are known and measured

without error. The functional relationship between inputs and target

is linear. The error term represents a set of random variables

that are independent and identically distributed with a normal distribution having a mean of 0 and a variance of σ2.

From Static Regression to Time Series RegressionThe time series regression model is an extension of the static regression model in which variables are observed in time autocorrelation is allowed the target variable can be influenced by past values of

inputs.

Time Series Regression TerminologyOrdinary Regressor An input variable that has only a concurrent influence

on the target variable: X at time t is correlated with Y at time t. Variation in X at times before and after t is uncorrelated with Y at time t.

Dynamic Regressor An input variable that influences the target variable at

current and future values: variation in X at time t can influence Y at time t, t + 1, t + 2, ….

Transfer Function A function that provides the mathematical relationship

between a dynamic regressor and the target variable.

7.05 Multiple Choice PollDynamic regressors require special treatment because they

a. change value frequently.

b. relate to the response at time t, and at subsequent times as well.

c. cannot be known in advance.

7.05 Multiple Choice Poll – Correct AnswerDynamic regressors require special treatment because they

a. change value frequently.

b. relate to the response at time t, and at subsequent times as well.

c. cannot be known in advance.

Types of Regressors: Measurement ScaleBinary (Dummy) Variables Take the value 0 or 1 Can be used to quantify nominal data

Categorical Variables Nominal scaled nonquantitative categories Ordinal scaled variables can be treated as categorical Must be coded into a quantitative input, usually using

a form of dummy coding for each level (less one if a constant term is used in the model)

Quantitative Variables Interval or ratio scaled Can be transformed

Types of Regressors: RandomnessDeterministic Controlled by experimenter Can be perfectly predicted without error

Stochastic Governed by unknown probability distributions Cannot be perfectly predicted

Advertising Case StudyAn online retail firm advertises in several media channels. With an ever-changing electronic marketplace, you must determine how to best allocate your advertising budget.

Find the optimal mix of advertising spending across

Internet Television Radio

Print media Direct mail

The Data Predictor Variable Description

DirectMail Weekly direct mail advertising (x$1000)

Internet Weekly Internet advertising (x$1000)

PrintMedia Weekly print media advertising (x$1000)

SalesRatio Ratio of competitor sales to total known sales

TVRadio Weekly TV/radio advertising (x$1000)

Response Variable Description

SalesAmount Total sales revenue for all customers (x$1000)

Comparing Advertising Effectiveness

Advertising Case Study

Task: Compare the effectiveness of different advertising channels, estimating the increase per dollar spent on direct mailings, Internet ads, print media ads, and TV/radio airtime.

7.1 Introduction

7.5 Time Series Data and Hierarchical Data 7.5 Time Series Data and Hierarchical Data StructureStructure

Objectives Explain the process of converting time-stamped data

into time series data: accumulation. Describe various accumulation options in the software. Explain the process of building the data hierarchy. Describe various aggregation options in the software.

Transactional Data Time BinningMonthly Time Bins – Only the past two years are shown.

Some time bins have no recorded observations.

Some time bins have more than one observation.

Producing Time Series DataData Accumulation Accumulates transactional data to the specified time

interval of the data

Data Aggregation Generates time series

data for upper levels (by groups) of the data hierarchy

Accumulating Unequally Spaced Data Time-stamped transactional data is rarely spaced

equally. Transactional data can be accumulated to make it

spaced equally. There are several accumulation options, including the

following:– Total (sum over accumulated period)– Average– Median– Minimum or Maximum– First or Last– Others based on summary statistics (STD, CSS,

USS, N, NOBS, NMISS)

Transactional Data AccumulationFor this time-stamped data, accumulating on a monthly average basis is different from accumulating on a total basis.

Choose an accumulation method that makes sense given your series.

ACCUMULATE=TOTAL ACCUMULATE=AVERAGE

Producing Time Series DataData Accumulation Accumulates transactional data to the specified time

interval of the data

Data Aggregation Generates time series

data for upper levels (by groups) of the datahierarchy

7.06 Multiple Answer PollChoose the statements below that are correct.

a. Data aggregation entails combining different series to form a hierarchy.

b. Data accumulation entails combining different series to form a hierarchy.

c. Data aggregation entails rolling up more frequent time intervals to produce fewer intervals.

d. Data accumulation entails rolling up more frequent time intervals to produce fewer intervals.

7.06 Multiple Answer Poll – Correct AnswersChoose the statements below that are correct.

a. Data aggregation entails combining different series to form a hierarchy.

b. Data accumulation entails combining different series to form a hierarchy.

c. Data aggregation entails rolling up more frequent time intervals to produce fewer intervals.

d. Data accumulation entails rolling up more frequent time intervals to produce fewer intervals.

Data Hierarchies: Aggregation

Data in the middle and upper levels of the hierarchy is constructed from data in the base level of the hierarchy. Above, group-level data is created by adding together department-level series. The top level (for example, total sales) is created by adding all base-level series.

BY Variables in Hierarchical Data BY variables group observations that have the same

value for the BY variable.

Assigning a BY variable enables you to obtain separate analyses for groups of observations.

For hierarchical time series, the order of the BY variables describes the structure of the hierarchy.

Aggregated DataThe group and value series are constructed from the accumulated dept series.

The chart is an abstract representation of the hierarchy.

Wine Case StudyWINECO has, until recently, focused on value (or jug) wines. The CEO thinks that there is room in the market for higher-end wine sellers and seeks to change WINECO’s focus from value wines to small, vintage wines.

Problem: How to price vintage wines There is not a long history, and demand for

small wines is more volatile. Can the CEO find a reasonable pricing

structure that the market can bear and that WINECO can profit from?

What types of promotions attract buyers without WINECO taking a huge financial hit?

Forecasting Objectives and Analytical Tasks Accumulate base-level transactional series to time

series data. Aggregate base level time series data to create the

wine data hierarchy. Automatically generate candidate models

for each series and select the best one as forecast specification.

Generate forecasts for each series. Assess price, holiday, and promotional

effects on selected series.

The Data The data is weekly case sales of wine from a wine

distribution company from January 17, 2004, to May 26, 2007.

The data hierarchy has three levels: Type (base level), Region, and Total Sales.

There are four aggregate wine types: tblred (table red), tblwt (table white), value, and vintage (limited production).

There are four regions: reg1 through reg4.

The Business Problem: Maximize Profit It is assumed that profit is maximized when the

following conditions are met: wine sales are maximized and inventory costs are minimized.

Wine sales are maximized when there are no lost sales due to wine being out of stock.

Inventory costs are minimized when inventories are kept as small as possible while still satisfying demand.

Accurate forecasts of wine demand over wine types and distribution regions are essential components in a profit-oriented business strategy.

Creating the Project for Hierarchical Forecasting

Wine Case Study

Tasks: Create the project for the WINECO data and begin the process of performing hierarchical forecasting.

Retail Forecasting Reconciliation Approaches

Top-down

Middle-out

Bottom-up

Company

Warehouse

Company

Warehouse

Company

Generating Forecasts of Wine Demand

Wine Case Study

Task: Perform middle-out reconciliation of the WINECO forecasts.

Exercise

This exercise reinforces the concepts discussed previously.

Disaggregation: Forecast ProportionsReconcile Bottom to Middle

Region

8 4 8 12

+5 +3 +2 +3

The Reconciled Forecasts

Region

Reconciling Forecasts

Wine Case Study

Task: Update the forecasts in each series to account for the effect of reconciliation.

Assessing Price and Promotional Effects on Vintage Type Wines

Wine Case Study

Task: Interpret the parameter estimates for the final model with respect to the effects of various promotions.

7.1 Introduction

7.6 Recommended Reading7.6 Recommended Reading

Recommended ReadingMay, Thornton. 2010. The New Know: Innovation Powered by Analytics. New York: Wiley. Chapters 6 through 8

1 chapter 7: forecasting 7.1 introduction 7.2 time series characteristics and components 7.3...

time series data

examples of forecasting

time series characteristics

inventory demand

time of day

time series regression

business units

basic time series concepts

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