Aula 8 Forecast

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Forecasting Methods / Mtodos de Previso

Week 8 - Time Series Models

ISCTE - IUL, Gesto, Econ, Fin, Contab.

Diana Aldea Mendes

[email protected]

March 31, 2011

DMQ, ISCTE-IUL ([email protected] ) Forecasting Methods March 31, 2011 1 / 44


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Forecasting Methods - Time Series Models

Time Series Analysis techniquesinvolves consideration of historical data, and obtaining future estimatesbased on past valuesassume that what has occurred in the past will continue to occur in thefuturerelate the forecast to only one factortimeincludes: moving average and exponential smoothing modelsMoving Averages and Exponential Smoothing are short rangetechniques. They produce forecast for the next period

Trend equations are used for much longer time horizons.More than one forecasting techniques might be used to increasecondence



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A time series : time-ordered sequence of observations taken atregular intervals over a period of time

Y 1 , Y 2 , ..., Y t

Examples : daily stock price, monthly sales, annual revenue, etc.Components (Types of Variations) of a Time Series

Trend (T) long term upward or downward movementSeasonality (S) the pattern that occurs every year (short-term regularvariations in data)Cycles (C) the pattern that occurs over a period of years (wavelikevariations of long-term not caused by seasonal variation; eect of theeconomy)Random variations () caused by chance and unusual events



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A time series can be broken down into its individual components.Two approaches:

Multiplicative decomposition

Forecast = Trend x Seasonality x Cycles x Random

Additive decomposition

Forecast = Trend + Seasonality + Cycles + Random



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Stationary and Nonstationary Time Series DataIf a time series has an upward or downward trend, it isnonstationaryIf it has no trend, it is stationary

Stationary Model Assumptions

Assumes item forecasted will stay steady over time (constant mean;random variation only)Techniques will smooth out short-term irregularitiesForecast for period (t + 1) is equal to forecast for period (t + k ); theforecast is revised only when new data becomes available.

Stationary Model TypesNave ForecastMoving AverageWeighted Moving AverageExponential Smoothing



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A forecast is rarely completely accurate.Forecasts will usually deviate from the actual demand.The dierence between the forecast and the actual is the forecasterror.The objective of forecasting is to make the forecasting error as slightas possible.A large degree of error may indicate that either the forecasting

technique is the wrong one or it needs to be adjusted by changing itsparameters.



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Forecasting Performance (error): measures how accurate the forecastwasFor time period t:Forecast error = Actual value Forecast value

Fe = At FtMean Forecast Error (MFE or Bias ): the arithmetic sum of theerrors (average deviation of forecast from actual)Mean Absolute Deviation (MAD), Measures average absolute

deviation of forecast from actual (T - the number of time periods)

MAD =

T

t= 1j At Ft j

T



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Mean Square Error (MSE): measures variance of forecast error

MSE =

T

t= 1( At Ft)

2

T Mean Absolute Percentage Error (MAPE): measures absoluteerror as a percentage of the forecast

MAPE = 100

T

t= 1

At

Ft AT

T



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Bias, MAD, and MAPE - typically used for time seriesCompare the accuracy of alternative forecasting methods using MADand MSE (determine which method yields the lowest MAD or MSE

for a given set of data).Want MFE to be as close to zero as possible minimum biasA large positive (negative) MFE means that the forecast isundershooting (overshooting) the actual observations

Note that zero MFE does not imply that forecasts are perfect (noerror) only that mean is on target



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ExampleWa lla c e G a r d e n S u p p lyForecas t i ng

Period A c tua lValue

NaveF orec as t E rror

A bs o lu teError

Percen tError

SquaredError

January 10 N /AF ebruary 12 10 2 2 16 .67% 4 .0M arc h 16 12 4 4 25 . 00 % 16 .0 A pri l 13 16 -3 3 23 . 08 % 9 .0M ay 1 7 13 4 4 23 . 53% 16 .0Jun e 19 17 2 2 10 .53 % 4 .0Ju ly 15 19 -4 4 26 . 67 % 16 .0 A ugus t 20 15 5 5 25 . 00 % 25 .0

S ept em ber 22 20 2 2 9 . 09% 4 .0O c t ober 19 22 -3 3 15 .79 % 9 .0Novem ber 21 19 2 2 9 .52% 4 .0D ec em be r 19 21 -2 2 10 .53% 4 .0

0 .8 18 3 17 .76% 10 .091BI AS M AD M AP E M S E

Stan da rd E rro r (Squ a re Roo t o f MSE) = 3 .176619

S t o r a g e S h e d S a l e s



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Example

Wallace Garden - Naive Forecast

0

5

10

15

20

25

February March April May June July August September October November December

Period

S h e d s Actual Value

Nave Forecast



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Moving Average Method (MA): Nave methods just trace the

actual data with a lag of one period, Ft = At 1 , they dont smoothAveraging (over time) techniques are used to smooth variations in thedata.Average most current values to predict future outcomes. Thetrend-cycle can be estimated by smoothing the series to reducerandom variation.The forecast is the average of the last n observations of the timeseries.

Ft+ 1 =Y t + Y t 1 + ... + Y t n+ 1

nNote that the n past observations are equally weighted.Issues with moving average forecasts:

All n past observations treated equally;Observations older than n are not included at all;Requires that

npast observations be retained;



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Example

Wallace Garden SupplyForecasting

Period

Actual

Value Three-Month Moving AveragesJanuary 10February 12March 16

April 13 10 + 12 + 16 / 3 = 12.67May 17 12 + 16 + 13 / 3 = 13.67June 19 16 + 13 + 17 / 3 = 15.33July 15 13 + 17 + 19 / 3 = 16.33

August 20 17 + 19 + 15 / 3 = 17.00September 22 19 + 15 + 20 / 3 = 18.00October 19 15 + 20 + 22 / 3 = 19.00November 21 20 + 22 + 19 / 3 = 20.33December 19 22 + 19 + 21 / 3 = 20.67

Storage Shed Sales



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Example

Wallace Garden SupplyForecasting 3 period moving average

Input Data Forecast Error Analysis

Period Actual Value Forecast Error

Absolute

error

Squared

error

Absolute

% error Month 1 10Month 2 12Month 3 16Month 4 13 12.667 0.333 0.333 0.111 2.56%Month 5 17 13.667 3.333 3.333 11.111 19.61%Month 6 19 15.333 3.667 3.667 13.444 19.30%Month 7 15 16.333 -1.333 1.333 1.778 8.89%

Month 8 20 17.000 3.000 3.000 9.000 15.00%Month 9 22 18.000 4.000 4.000 16.000 18.18%Month 10 19 19.000 0.000 0.000 0.000 0.00%Month 11 21 20.333 0.667 0.667 0.444 3.17%Month 12 19 20.667 -1.667 1.667 2.778 8.77%

Average 1.333 2.000 6.074 10.61%Next period 19.667 BIAS MAD MSE MAPE

Act ual Value - Forecast



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Example

Three Period Moving Average

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Time

V a l u e Actual Value

Forecast



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MA Advantage = Easy to compute and easy to understandMA Disadvantage = All values in the average are weighted equallyThis technique derives its name from the fact that as each new actualvalues becomes available, the forecast is updated by adding thenewest value and dropping the oldest and the recalculating theaverage.



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Weighted Moving Average (WMA): In the moving averages

method, each observation in the MA calculation receives the same weight . One variation, known as weighted moving averages, involvesselecting a dierent weight for each data value and then computing aweighted average of the most recent m values as the forecast.In most cases, the most recent observation receives the most weight,and the weight decreases for older data values (most recentobservations must be better indicators of the future than olderobservations).Note that for the WMA the sum of each weights is equal to 1.

The larger the n the more stable the forecast.A 2-period model will be more responsive to change.We dont want to chase outliers. But we dont want to take foreverto correct for a real change. We must balance stability withresponsiveness



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The Weighted Moving Average Method: historical values of the timeseries are assigned dierent weights when performing the forecast

Ft+ 1 = Weighted sum of last n demands

= w1 At + w2 At 1 + ... + wn At n+ 1

where

Ft+ 1 = forecast for Period t + 1n = number of periods used in determining the moving averagew = weights assigned to Period i ( with wi = 1) At = actual demand in Period t



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Example

W allace Garden Su pplyForecas t i ng

Per iod A c tu a lVa lue W e igh ts Th ree -M o n th W e igh ted Mov ing Av e rages

Janua ry 10 0 .222F eb rua ry 12 0 .593M arch 16 0 .1 85

A pril 13 2 .2 + 7 .1 + 3 / 1 = 12.298M ay 17 2 .7 + 9 .5 + 2 .4 / 1 = 14.556June 1 9 3 .5 + 7 .7 + 3 .2 / 1 = 14.407Ju ly 15 2 .9 + 10 + 3 .5 / 1 = 16.484

A ug ust 20 3 .8+

11+

2 .8/

1=

17.814S ep tem be r 22 4 .2 + 8 .9 + 3 .7 / 1 = 16.815O ctobe r 19 3 .3 + 12 + 4 .1 / 1 = 19.262N ov em be r 21 4 .4 + 13 + 3 .5 / 1 = 21.000D ecem ber 19 4 .9 + 11 + 3 .9 / 1 = 20.036

N ext p erio d 20 .185

S u m o f w e ig h ts = 1 .000

S t o r a g e S h e d S a l e s



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Example

Wallace Garden SupplyForecasting 3 period weighted moving average

Input Data Forecast Error Analysis

Period Actual v alue Weights Forecast Error Absolute

error Squared

error Absolute

% error

Month 1 10 0.222Month 2 12 0.593Month 3 16 0.185Month 4 13 12.298 0.702 0.702 0.492 5.40%Month 5 17 14.556 2.444 2.444 5.971 14.37%Month 6 19 14.407 4.593 4.593 21.093 24.17%Month 7 15 16.484 -1.484 1.484 2.202 9.89%Month 8 20 17.814 2.186 2.186 4.776 10.93%

Month 9 22 16.815 5.185 5.185 26.889 23.57%Month 10 19 19.262 -0.262 0.262 0.069 1.38%Month 11 21 21.000 0.000 0.000 0.000 0.00%Month 12 19 20.036 -1.036 1.036 1.074 5.45%

Average 1.988 6.952 6.952 10.57%Next period 20.185 BIAS MAD MSE MAPE

Sum of weights = 1.000



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Exponential smoothing is one of the more popular and frequentlyused forecasting techniques, the reasons being:It requires minimal data.The mathematics of the technique are easy to understand bymanagement.

Most importantly, exponential smoothing has a good track record of success.

Simple Exponential Smoothing works well with data that is movingsideways (stationary)

Must be adapted for data series which exhibit a denite trendMust be further adapted for data series which exhibit seasonalpatternsThis technique is used for short term forecasting.



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Exponential Smoothing: Single Exponential Smoothing (SES o

EWMA)SES is a special case of the WMA method in which we select only oneweight, , the weight for the most recent observation.Estimate next outcome with a weighted combination of the forecastfor previous period and the most recent outcomeAssumptions: No trend and exponentially declining weight given topast observationsThe weights for the other data values are computed automaticallyand become exponentially smaller as the observations move farther

into the past.Ft+ 1 = Y t + (1 ) Ft = Ft + (Y t Ft)

Y t = time series value, Ft = tted (forecasted) value = weight, 0 1



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or:Forecast today = Forecast yesterday + (Forecast error yesterda

Each new forecast is equal to the previous forecast plus a percentage

of the previous error.The initial value for the smoothing recursive process can aect thequality of the forecasts for many observations. In practice, when thereare many leading observations prior to a crucial actual forecast, theinitial value will not aect that forecast by much, since its eect willhave long "faded" from the smoothed series.



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When applied recursively to each successive observation in the series,each new smoothed value (forecast) is computed as the weightedaverage of the current observation and the previous smoothedobservation.

Each smoothed value is the weighted average of the previousobservations, where the weights decrease exponentially depending onthe value of parameter .

If = 1 : Previous observations are ignored entirely (short memory).If = 0 : Current observation is ignored entirely (long memory).

The most straightforward way of evaluating the accuracy of theforecasts based on a particular value is to simply plot the observedvalues and the one-step-ahead forecasts.



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More complex ES models (double ES and Winters method), havebeen developed to accommodate time series with trend and seasonalcomponents.The general idea here is that forecasts are not only computed fromconsecutive previous observations (as in SES), but an independent

(smoothed) trend and seasonal component can be added.The quickness of forecast adjustment to error is determined by thesmoothing constant.The closer the alpha is to zero, the slower the forecast will be toadjust to forecast errors.Conversely, the closer the value of alpha is to 1.00, the greater theresponsiveness to the actual observations and the less the smoothingSelect a smoothing constant that balances the benets of respondingto real changes if and when they occur.



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Example:

Period Actual

Value(Y t) Y t-1 Yt-1 Yt-1 YtJanuary 10 = 10 0.1February 12 10 + 0.1 *( 10 - 10 ) = 10.000March 16 10 + 0.1 *( 12 - 10 ) = 10.200

April 13 10.2 + 0.1 *( 16 - 10.2 ) = 10.780May 17 10.78 + 0.1 *( 13 - 10.78 ) = 11.002June 19 11.002 + 0.1 *( 17 - 11.002 ) = 11.602July 15 11.602 + 0.1 *( 19 - 11.602 ) = 12.342

August 20 12.342 + 0.1 *( 15 - 12.342 ) = 12.607September 22 12.607 + 0.1 *( 20 - 12.607 ) = 13.347October 19 13.347 + 0.1 *( 22 - 13.347 ) = 14.212November 21 14.212 + 0.1 *( 19 - 14.212 ) = 14.691December 19 14.691 + 0.1 *( 21 - 14.691 ) = 15.322

Storage Shed Sales



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Expon ential Sm oothing

0

5

1 0

1 5

2 0

2 5

J a n u

a r y

F e b r

u a r y

M a r c h A p

r i l M a

y J u

n e J u l y

A u g u

s t

S e pt

e m b e

O c t o b

er

N o v e

m b e

D e c e

m b e

S h e d A ctua l va lue

F o r e c a s t


h d d l


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Example: The table below shows the number of cars sold by SpeedMotors in the last 10 days. You need to forecast the sales on day 11using exponential smoothing

Days 1 2 3 4 5 6 7 8 9 10Cars Sold 20 13 19 19 25 17 15 13 22 20

There are two problems with Exponential Smoothing:(a) What value of Alpha to use? Can be found by trial & error.(b) How to get the rst forecast? Choose a suitable value or Usewarm-up method


F i M h d Ti S i M d l


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For this problem we are going to use a warm-up period of 10 days,and therefore our rst proper forecast will be for day 11.Alpha values of 0.1 and 0.5 will be used for Speed Motors.Assume that the forecast for day 2 is the actual for day 1. Tocalculate the forecast for day 3 we use the formula

Next forecast = alpha*actual demand in the present period +(1-alpha)*previously determined forecast for the present period=0.1*13+(1-0.1)*20 = 1.3+18.0 = 19.30

Days Actual Sales Forecast1 20

2 13 20

3 19


F i M h d Ti S i M d l


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The simple forecasting and smoothing methods model

components in a series that are usually easy to see in a time seriesplot of the data.These methods decompose the data into its trend and seasonalcomponents , and then extend the estimates of the components intothe future to provide forecasts.

Static methods have patterns that do not change over time;dynamic methods have patterns that do change over time andestimates are updated using neighboring values.

STATIC (SIMPLE) METHODS

Trend Analysis

Decomposition

STATIC (SIMPLE) METHODS

Trend Analysis Decomposition

DYNAMIC (SMOOTHING) METHODS

Moving Average

Single Exponential Smoothing

Double Exponential Smoothing

Winters Method (Triple Exp. Smoothing)

DYNAMIC (SMOOTHING) METHODS

Moving Average Single Exponential Smoothing

Double Exponential Smoothing

Winters Method (Triple Exp. Smoothing)

You may use two methods in combination. That is, you may choose aDMQ, ISCTE-IUL ([email protected] ) Forecasting Methods March 31, 2011 33 / 44

F ti M th d Ti S i M d l


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Trend analysis :technique that ts a trend equation (or curve) to a series of historicaldata points.

Projects the curve into the future for medium and long term forecasts.forecast the future path of economic variables based on historical datause a regression model to model the trend as a function of time

Types of trend analysislinear trendnonlinear trend



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Forecasting Methods Time Series Models


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Common Nonlinear Trends

Parabolic

Exponential

Growth

Parabolic

Exponential

Growth




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Another possibility is anexponential trend , which can be modeled as

log(Y t) = a0 + a1 t + t , t = 1 , 2 , . . .

Another possibility is aquadratic trend , which can be modeled as

Y t = a0 + a1 t + a2 t2 + t , t = 1 , 2 , . . .

Adding a linear trend term to a regression is the same thing as usingdetrended series in a regressionDetrending a series involves regressing each variable in the model ontThe residuals form the detrended seriesAn advantage to actually detrending the data (vs. adding a trend)involves the calculation of goodness of t

Time-series regressions tend to have very high R2 , as the trend is wellexplainedThe R2 from a regression on detrended data better reects how wellthe xts explain yt




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Exponential Smoothing with a Trend (Holts Method: DoubleExponential Smoothing)Assumptions: Linear trend and Exponentially declining weights topast observations/trends

Model:

Ft+ 1 = Y t + (1 ) (Ft + T t) : Smooth the base forecastT t = (Ft Ft 1 ) + (1 ) T t 1 : Smooth the trend forecast

This time trend is also smoothed, note that previous trend (of t-1)and current trend (of t) appear in the smoothing formula: T t 1 andFt Ft 1




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Ideas behind smoothing with trend:De-trend time-series by separating base from trend eectsSmooth base in usual manner using Smooth trend forecasts in usual manner using

Forecast k periods into future Ft+ k with base and trend

Ft+ k = Ft + kT t




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Techniques for seasonality :When a seasonal pattern repeats yearly, this can be used for futureforecastsNeed monthly or quarterly dataA seasonal index is the ratio of the average value in that season, overthe annual averageExamples: demand for coal in winter months; demand for soft drinks inthe summer and over major holidaysSeasonality can be dealt with by adding a set of seasonal dummiesAs with trends, the series can be seasonally adjusted before running theregressionSeasonality is expressed as a percentage of the average amountseasonal percentages = seasonal relatives = seasonal indices




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Seasonal relative = 1.45 for the quantity of television sold in Augustat Circuit City, meaning that TV sales for that month are 45% abovethe monthly average.

Seasonal factor =0.60 for the number of notebooks sold at the UTDbookstore in April, meaning that notebook sales are 40% below themonthly average.Deseasonalize historical observations: Divide them by seasonal indicesMake the analysis = Generate forecastsSeasonalize forecasts: Multiply them by seasonal indices




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Winters Method: Exponential Smoothing w/ Trend andSeasonalityIdeas behind smoothing with trend and seasonality:

De-trend: and de-seasonalizetime-series by separating base fromtrend and seasonality eectsSmooth base in usual manner using Smooth trend forecasts in usual manner using Smooth seasonality forecasts using

Assume m seasons in a cycle

12 months in a year4 quarters in a month3 months in a quarteret cetera




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Based on 3 equations, each of which smooth a factor associated with

one of the three components of the pattern: randomness, trend, andseasonalitySmooth the base forecast Ft

Ft = Dt

St m + (1

) (Ft 1 T t 1 )Smooth the trend forecast T t

T t = (Ft Ft 1 ) + (1 ) T t 1

Smooth the seasonality forecast St

St = DtFt

+ (1 ) St m




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Forecast Ft with trend and seasonality

Ft+ k = (Ft 1 + kT t 1 ) St+ k m

Smooth the trend forecast T t

T t = (Ft Ft 1 ) + (1 ) T t 1

Smooth the seasonality forecast St

St = DtFt+ (1 ) St m


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