04 Forecasting

8/20/2019 04 Forecasting

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King Saud UniversityCollege of Computer & Information Sciences

IS 466 Decision Support Systems

Lecture 4

Forecasting

Dr. Mourad YKHLEFThe slides content is derived and adopted from many references

IS 466 - Forecasting - Dr. Mourad Ykhlef 2

Outline• Definitions

• Forecasting types

• Time series

• Stationary forecasting models

• Performance of forecasting methods

• Linear trend time series

• Trend, Seasonal and Cyclical time series

• Associative forecasting


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Definitions

• Forecasting is the process of predicting the future.

• Forecasting is an integrated part of almost allbusiness enterprises.

• Examples:– Manufacturing firms forecast demand for their product, to schedule

manpower and raw material allocation.

– Service organizations forecast customer arrival patterns to maintain

adequate customer service.

– Firms consider economic forecasts of indicators (housing starts, changes

in gross national profit) before deciding on capital investments.


Definitions• Good forecasts can lead to

– Reduced inventory costs.

– Lower overall personnel costs.

– Increased customer satisfaction.

• The forecasting process can be based on:

– Educated guess.– Expert opinions.

– Past history of data values, known as a time series.


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Outline

• Definitions


• Time series• Stationary forecasting models






Type of Forecasts by Time Horizon• Short-range forecast

– Up to 1 year; gnerally less than 3 months

– Job scheduling, worker assignements

• Medium-range forecast

– 3 months to 3 years

– Sales and production planning, budgeting• Long-range forecast

– 3+ years

– New product planning, facility location or expansion


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Forecasting approaches

• Quantitive forecasts– Used when situation is

stable and historical

data exist• Existing products

• Current technology

– Use a variety ofmathematical modelsthat rely on historicaldata and/or causalvariables

• e.g., forecasting salesof color televisions

• Qualtitative forecasts

– Used when situation isvague & litle data exist

• New products

• New technology

– Involve intuition,experience

– e.g., forecasting saleson Internet


Overview of Qualitative methods• Consumer Market Survey

– Ask the customer

• Sales force composite

– Estimates from individual sales person are reviewedfor checking realistic, then aggregated

• Jury of executive opinion– Pool opinions of high-level executives, sometines

augment by statistical models

• Delphi method

– Panel of experts, queried iteratively


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Overview of Quantitive methods

• Time-series models

– (Weighted) Moving average

– Exponential Smoothing

– Exponential Smoothing with Trend Adjustment

– Seasonal and Cyclical

• Associative models

– Liner regression

– Multiple regression

– Logistic regression




• Time series







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What is a Time Series?

• Set of evenly spaced numerical data

– Obtained by observing response variable at regulartime periods

• Forecasting technique

– That uses a series of past data points to make a forecast

• Example

Year 2006 2007 2008 2009 2010

Sales 78.7 63. 5 89.7 93.2 92. 1


Time series components

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom


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Trend component

• Time series may be relativelystationary or it may exhibittrend over time

• Trend is the gradual upwardor downward movement ofdata over time

• Trend indicates that the timeseries is increasing ordecreasing

• Trend is typically modeled asa linear, quadratic or

exponential function

Mo., Qtr., Yr.

Response

© 1984-1994 T/Maker Co.


Seasonal component• When a repetitive pattern

is observed over some timehorizon, the series is said tohave seasonal behavior.

• Seasonality is a data

pattern that repeats itselfafter a period of days,weeks, months or quarters.

• Occurs within 1 year

Response

Mo., Qtr.

Summer

Period of

Pattern

“Season”

Length

Number of

“Seasons” inPattern

Week Day 7

Month Week 4 – 4 ½

Month Day 28 – 31

Year Quarter 4

Year Month 12

Year Week 52


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Cyclical component

• Cycles are patterns in the data that occur every several years.

– Usually tied into the business cycle and are of the majorimportance in short-term business analysis and planning.

• Cycles are upturn or downturn not tied to seasonal variation.

– Usually result from changes in economic conditions.

– Usually 2-10 years duration

Mo., Qtr., Yr.Mo., Qtr., Yr.

ResponseResponse

Cycle


Random component• Erratic, unsystematic fluctuations

• Due to random variations or unforeseen events

– Union strike

– Tornado

• Short duration and non repeating


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Stationary

Linear trend

Linear trend and seasonality

Time

Time

series

value

Future

Components of a Time Series

In Stationary, the mean value of the time series is assumed to be constant


Steps in the Time Series Forecasting • The goal of a time series forecast is to identify factors that can be

predicted.

• This is a systematic approach involving the following steps.– Step 1: Data collection and Hypothesization.

• Collect historical data.

• Graph the data vs. time.

• Hypothesize a form for the time series model.• Verify this hypothesis statistically.

– Step 2: Select a forecasting technique.

• Determination of input parameter values

• Performance evaluation on past data of each technique– Step 3: Prepare a forecast using the selected techniques


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Outline

• Definitions








Stationary Forecasting Models• In a stationary model the mean value of the time series is

assumed to be constant.

• No trend, seasonal, or cyclical components

• The general form of such a model is

Where:yt = the value of the time series at time period t.

β0 = the unchanged mean value of the time series.

εt = a random error term at time period t

• If a time series does not have a trend, seasonlity or cyclicalcomponents it will be stationary.

yt = ββββ0 + εεεεt

•The values of etare assumed to beindependent

• The values of etare assumed to

have a mean of 0.


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Checking the Stationary assumption(Homework)

• Checking for trend– Use Linear Regression if et is normally distributed.

– Use a nonparametric test if et is not normally distributed.

• Checking for seasonality component

– Autocorrelation measures the relationship between the values of the timeseries in different periods.

– Lag k autocorrelation measures the correlation between time seriesvalues which are k periods apart.

• Autocorrelation between successive periods indicates a possible trend.

• Lag 7 autocorrelation indicates one week seasonality (daily data).

• Lag 12 autocorrelation indicates 12-month seasonality (monthly data).

• Checking for Cyclical Components

• If a time series does not have a trend, seasonlity or cyclical

components it will be stationary.


Methods for a stationary time series

• The Last Period Method

• The Moving Average Method

• The Weighted Moving Average Method

• The Exponential Smoothing Method


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The Last Period Method

• The forecast for the next period is the lastobserved value.

• e.g., if May sales were 50 then June sales will be 50

• Sometimes cost effective & efficient

– At least it provides a starting point which moresophisticated models that follow can be compared

t1t yF =+


The (Weighted) Moving Average Method• The forecast is the average of the last n observations of the

time series.

• More recent values of the time series get larger weightsthan past values when performing the forecast.

n

y...yyF 1nt1tt1t

+−−+

+++=

1tF + = w1yt + w2yt-1 +w3yt-2 + …+ wnyt-n+1w1 ≥ w2 ≥ … ≥ wn

Σwi = 1


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Example (1/7)

• Galaxy Industries is interested in forecasting

weekly demand for its YoHo brand yo-yos.

• The yo-yo is a mature product. This year demandpattern is expected to repeat next year.

• To forecast next year demand, the past 52 weeksdemand records were collected.


Example (2 /7)

• Three forecasting methods were suggested:

– Last period technique - suggested by Ahmed.

– Four-period moving average - suggested by Karim.

– Four-period weighted moving average - suggested by

Omar.• Management wants to determine:

– If a stationary model can be used.

– What forecast will be obtained using each method?


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Example (3 /7)

• Collection of demand records

Week Demand Week Demand Week Demand Week Demand1 415 14 365 27 351 40 282

2 236 15 471 28 388 41 399

3 348 16 402 29 336 42 309

4 272 17 429 30 414 43 435

5 280 18 376 31 346 44 299

6 395 19 363 32 252 45 522

7 438 20 513 33 256 46 376

8 431 21 197 34 378 47 483

9 446 22 438 35 391 48 416

10 354 23 557 36 217 49 245

11 529 24 625 37 427 50 393

12 241 25 266 38 293 51 482

13 262 26 551 39 288 52 484

Week Demand Week Demand Week Demand Week Demand

1 415 14 365 27 351 40 2822 236 15 471 28 388 41 399

3 348 16 402 29 336 42 309

4 272 17 429 30 414 43 435

5 280 18 376 31 346 44 299

6 395 19 363 32 252 45 522

7 438 20 513 33 256 46 376

8 431 21 197 34 378 47 483

9 446 22 438 35 391 48 416

10 354 23 557 36 217 49 245

11 529 24 625 37 427 50 393

12 241 25 266 38 293 51 482

13 262 26 551 39 288 52 484


Example (4 /7)• Construct the time series plot

• Neither seasonality nor cyclical effects can beobserved

0

200

400

600

800

1 6 1 1 1 6 2 1 2 6 3 1 3 6 4 1 4 6 5 1

Weeks

D e m a n

d

Series1


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Example (5 /7) (Home work)

• Is the trend present?

• Run linear regression to test β1 in the model

yt=β0+β1t+εt• Excel results

Coeff. S tand. E rr t-S tat P -value Lower 95 U pper 95

Intercept 369.27 27.79436 13.2857 5E-18 313.44 425.094

Weeks 0.3339 0.912641 0.36586 0.71601 -1.49919 2.166990.71601

This large P-value indicates

that there is little evidence that trend exists

• Conclusion: A stationary model is appropriate.


Example (6 /7)• Forecast for Week 53

• Last period technique (Ahmed’s Forecast)

• Four-period moving average (Karim’s forecast)

• Four period weighted moving average (Omar’s forecast)

= 484 boxes.53 = y52$y

53 = (y52 + y51 + y50 + y49) / 4 =(484+482+393+245) / 4 = 401 boxes.

$y

$y53 =0.4y52 + 0.3y51 + 0.2y 50 + 0.1y49 =

0.4(484) + 0.3(482) + 0.2(393) + 0.1(245) = 441.3 boxes.


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Example (7 /7)

• Forecast for Weeks 54 and 55

• Since the time series is stationary, the forecasts forweeks 54 and 55 remain as the forecast for week53.

• These forecasts will be revised pendingobservation of the actual demand in week 53.


Drawbacks of (Weighted) Moving Average Methods

• Weighted Moving Average can put greater weight on themore recent observations, it uses only last n periods datavalues and ignores the history of the time series prior tothat time.

• Increasing n makes forecast less sensitive to changes

• Do not forecast trend well.

• Require much historical data.

• Solution: Exponential Smoothing Method


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Exponential Smoothing Method

• All the previous values of historical data affect theforecast.

• For each period create a smoothed value Lt of the timeseries, that represents all the information known by t.

• The smoothed value Lt is the weighted average of– The actual value for the current period (with weight of α).

– The forecast value for the current period (with weight of 1-α).

• The smoothed value Lt becomes the forecast for period t+1.


Exponential Smoothing Method• New forecast = last period’s forecast

+α(last period’s actual demand – last period’s forecast)

• An initial “forecast” is needed to start the process.

ttt1t F)1(yLF α−+α==+

Define:

Lt = smoothed value for time t

Ft+1 = the forecast value for time t+1

yt = the value of the time series at time tαααα = smoothing constant (weight) between 0 and 1


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Generating an initial forecast

– Approach 1:

Continue from t=3 with the recursive formula.

– Approach 2: (when large number of historical values exist)

• Average the initial “ n ” values of the time series.

• Use this average as the forecast for period n + 1

• Begin using exponential smoothing from that time period

onward and so on.

nn yF =+1

nnnnn y yF yF )1()1( 1112 α α α α −+=−+= ++++

112 y LF ==



Future Forecasts

• Since this technique deals with stationary timeseries, the forecasts for future periods does notchange.

• Assume N is the number of periods for which

data are available. Then

FN+1 = ααααyN + (1 – αααα)FN,FN+k = FN+1, for k = 2, 3, …


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Example 1 (1/3)

Period Series Forecast

1 415 #N/A

2 236 4153 348 397.1

4 272 392.19

50 393 368.8296268

51 482 371.2466641

52 484 382.3219977

53 392.4898

54 392.4898

55 392.4898


Example 1 (2/3)• An exponential smoothing forecast is suggested,

with α = 0.1.

• An Initial Forecast is created at t=2 byF2 = y1 = 415.

• The recursive formula is used from period 3

onward:F3 = .1y2 + .9F2 = .1(236) + .9(415) = 397.10

F4 = .1y3 + .9F3 = .1(348) + .9(397.10) = 392.19

and so on, until period 53 is reached (N+1 = 52+1 = 53).

F53 = .1y52 + .9F52 = .1(484) + .9(382.32) = 392.49

F54 = F55 = 392.49 ( = F53)


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Example 1 (3/3)

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60

Notice the amount of smoothing

Included in the smoothed series



Relationship with simple moving average• Simple moving average of length k

(1+2+3+…+k)/k = (k+1)/2

• Exponential smoothing(1) α +(2) α (1- α)+(3)α(1- α)2+(4)α(1- α)3…. = 1/α

• The two techniques will generate forecasts having thesame average age of information if

– Exponential smoothing with α=.10 is equivalent, in some sense,to moving average based on 19 periods

– Exponential smoothing with α=1 is equivalent, in some sense, tomoving average based on 1 period

α

α−=

2k


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Example 2 (1/6)

• If Drugs uses exponential smoothing to forecastsales, which value for the smoothing constant α, .1or .8, gives better forecasts?

WeekWeek SalesSales WeekWeek SalesSales

1 110 6 1201 110 6 120

2 115 7 1302 115 7 130

3 125 8 1153 125 8 115

4 120 9 1104 120 9 110

5 1255 125 1010 130130


Example 2 (2/6)• Exponential Smoothing: To evaluate the two

smoothing constants, determine how theforecasted values would compare with the actualhistorical values in each case.

Let: Y t= actual sales in week t

F t = forecasted sales in week tF 2 = Y 1 = 110

For other weeks, F t+1 = .1Y t + .9F t


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Example 2 (3/6)

• Exponential Smoothing (α = .1, 1 - α = .9)

F 2 = 110

F 3

= .1Y 2

+ .9F 2

= .1(115) + .9(110) = 110.5

F 4 = .1Y 3 + .9F 3 = .1(125) + .9(110.5) = 111.95

F 5 = .1Y 4 + .9F 4 = .1(120) + .9(111.95) = 112.76

F 6 = .1Y 5 + .9F 5 = .1(125) + .9(112.76) = 113.98

F 7 = .1Y 6 + .9F 6 = .1(120) + .9(113.98) = 114.58

F 8 = .1Y 7 + .9F 7 = .1(130) + .9(114.58) = 116.12

F 9 = .1Y 8 + .9F 8 = .1(115) + .9(116.12) = 116.01

F 10= .1Y 9 + .9F 9 = .1(110) + .9(116.01) = 115.41


Example 2 (4/6)• Exponential Smoothing (α = .8, 1 - α = .2)

F 2 = 110

F 3 = .8(115) + .2(110) = 114

F 4 = .8(125) + .2(114) = 122.80

F 5 = .8(120) + .2(122.80) = 120.56

F 6 = .8(125) + .2(120.56) = 124.11F 7 = .8(120) + .2(124.11) = 120.82

F 8 = .8(130) + .2(120.82) = 128.16

F 9 = .8(115) + .2(128.16) = 117.63

F 10= .8(110) + .2(117.63) = 111.53


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Example 2 (5/6)

• Mean Squared Error: In order to determine whichsmoothing constant gives the better performance,calculate, for each, the mean squared error for thenine weeks of forecasts, weeks 2 through 10 by:

[(Y 2-F 2)2 + (Y 3-F 3)

2 + (Y 4-F 4)2 + . . . + (Y 10-F 10)

2]/9

• Select the forecast with the smallest error value


Example 2 (6/6)α = .1 α = .8

Week Y t F t (Y t - F t)2 F t (Y t - F t)

2

1 1102 115 110.00 25.00 110.00 25.003 125 110.50 210.25 114.00 121.004 120 111.95 64.80 122.80 7.845 125 112.76 149.94 120.56 19.71

6 120 113.98 36.25 124.11 16.917 130 114.58 237.73 120.82 84.238 115 116.12 1.26 128.16 173.309 110 116.01 36.12 117.63 58.26

10 130 115.41 212.87 111.53 341.27

Sum 974.22 Sum 847.52MSE Sum/9 Sum/9108.25108.25 94.1794.17

Mean Squared Error


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Outline

• Definitions



• Performance of forecasting methods (for reading)





Performance of Forecasting Methods• Calculate the value of the evaluation measure

using the forecast error equation

Error = Actual – Forecast

• Select the forecast with the smallest error value• Types of forecast error equations:

– Mean Squared Error MSE

– Mean Absolute Deviation MAD

– Mean Absolute Percent Error MAPE

– Largest Absolute Deviation LAD

ttt Fy −=∆


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Forecast Error Equations

ΣΣΣΣ(∆∆∆∆ t)2n

MSE =

MAD =|∆|∆|∆|∆ t|

nΣΣΣΣ

MAPE =nΣΣΣΣ |∆|∆|∆|∆t|n

yt

LAD = max |∆∆∆∆ t|


Example (1/5)

Time 1 2 3 4 5 6Time series: 100 110 90 80 105 115

3-Period Moving average: 100 93.33 91.6

Error for the 3-Period MA: - 20 11.67 23.4

3-Period Weighted MA(.5, .3, .2) 98 89 85.5Error for the 3-Period WMA - 18 16 29.5


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= 361.24(-20)2+(11.67)2+(23.4)2

3MSE = =

ΣΣΣΣ(∆∆∆∆t)2222

n

Example (2/5)

MSE for the moving average technique:

MSE for the weighted moving average technique:

(-18)2 + (16)2 + (29.5)23

MSE = =ΣΣΣΣ(∆∆∆∆t)2222

n= 483.4


MAD for the moving average technique:

MAD for the weighted moving average technique:

= 21.17

=18.35

|-20| + |11.67| + |23.4|

3MAD = =

ΣΣΣΣ |∆|∆|∆|∆ t|

n

|-18| + |116| + |29.5|

3MAD = =

ΣΣΣΣ |∆|∆|∆|∆ t|n

Example (3/5)


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MAPE for the moving average technique:

MAPE for the weighted moving average technique:

= .211

= .188|-20|/80 + |11.67|/105+ |23.4|/115

3MAPE= =

ΣΣΣΣ |∆|∆|∆|∆ t|n

|-18|/80 + |16|/105 + |29.5|/1153

MAPE= =ΣΣΣΣ |∆|∆|∆|∆ t|n

Example (4/5)


LAD for the moving average technique:

LAD for the weighted moving average technique:

= 23.4

= 29.5

Example (5/5)

|-20|, |11.67|, |23.4|

LAD= max =|∆∆∆∆t

| max {|-18|, |16|, |29.5|}

LAD= max =|∆∆∆∆ t|


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Outline

• Definitions








Linear Trend Time Series• Two methods

– Expentional Smoothing with Trend Adjustment

– Trend Projections (Linear Regression)


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Expentional Smootthing with Trend Adjustment

• Ft = exponentially smoothed forecast of the dataseries in period t

• Tt = exponentially smoothed trend in periodt

• yt = times series value in period t

• α = smoothing constant for the average

• β = smoothing constant for the trend


Expentional Smootthing with Trend Adjustment

• Forecast including trend (FITt) =

exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

• Ft = α(value last period)

+ (1-α)(Forecast last period +Trend estimate Last period)

Ft = α yt-1 + (1– α) (Ft-1 + Tt-1)

• Tt = β(Forecast this period - Forecast last period)

+ (1- β)(Trend estimate last period)

Tt = β(Ft - Ft-1) + (1- β)Tt-1


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Example (1/4)

• A large Portland manufacturer uses exponentialsmoothing to forecast demand for a piece ofpollution control equipment. It appears that anincreasing trend is present.

• α =.2 and β =.4

• F1=11 and T1=2

Month Demand

1 12

2 17

3 20

4 19

5 24

6 217 31

8 28

9 3610 ?


Example (2/4)• Forecast for month 2

F2 = α y1 + (1– α) (F1 + T1)=.2(12)+.8(11+2)=12.8

T2 = β(F2 - F1) + (1- β)T1=.4(12.8-11)+0.6(2)=1.92

FIT2 = F2 +T2 = 14.72 units

• Forecast for month 3

F3 = α y2 + (1– α) (F2 + T2)=.2(17)+.8(12.8+1.92)=15.18

T3 = β(F3 – F2) + (1- β)T2=.4(15.18-12.8)+0.6(1.92)=2.10

FIT3 = F3 +T3 = 17.28 units


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Example (3/4)

Actual

Demand

Smoothed

Forecast

Smoothed

Trend

Forecast

including trend

12 11.00 2.00 13.00

17 12.80 1.92 14.72

20 15.18 2.10 17.28

19 17.82 2.32 20.14

24 19.91 2.23 22.14

21 22.51 2.38 24.89

31 24.11 2.07 26.18

28 27.14 1.45 28.59

36 29.28 2.32 31.60

32.48 2.68 35.16


Example (4/4)

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

D

e m

a n d

Actual

Demand

Smoothedforecast

Smoothed trend

Forecast including

trend


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Trend Projections

• This method files a trend line to a series of historicalpoints and then projects the line into the future formedium-to-long-range forecasts (other methods are

quadratic or exponential)

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

V a l u e s o f D e p e n d e n t V a r i a b l e

bxaY +=ˆ

Actual

observation

Point on

regression

line


Trend Projections• Least Squares Equations (from statisticians)

Equation:ii bxaŶ +=

Slope: 221

1

xn x

y xn y x

bi

n

i

ii

n

i

−∑

−∑

=

=

=

Y-Intercept: xbya −=

x Average of the values of x

n number of data points


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Example (1/4)

Year Demand

2006 74

2007 79

2008 80

2009 90

2010 105

2011 142

2012 122

The demand forelectrical power at

N.Y.Edison overthe years 1997 –2003 is given atthe left. Find theoverall trend.


Example (2/4)

Year TimePeriod

PowerDemand

x2 xy

2006 1 74 1 74

2007 2 79 4 158

2008 3 80 9 240

2009 4 90 16 3602010 5 105 25 525

2011 6 142 36 852

2012 7 122 49 854

Σx=28 Σy=692 Σx2=140 Σxy=3,063


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Example (3/4)

megawatts151.5610.54(9)56.702014inDemand

megawatts141.0210.54(8)56.702013inDemand

56.7010.54(4)-98.86xb-ya

10.5428

295

(7)(4)140

86)(7)(4)(98.3,063

xnΣx

yxn-Σxyb

98.867

692

n

Σyy 4

7

28

n

Σxx

222

=+=

=+=

===

==−

−=

−=

======


Example (4/4)Electric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year


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Outline

• Definitions








General Time Series Models• Components of a Time Series

– Tt = Trend of the time series at time t

– St = Seasonal

– Ct = Cyclical

– It = Irregular

• Any observed value in a time series is the product (or

sum) of time series components

• Multiplicative modelYt = Tt · St · Ct · It

• Additive modelYt = Tt + St + Ct + It


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Additive and multiplicative models

• Additive model

– The magnitude of seasonal component is constant overtime.

– For example the electric power consumption.

• Multiplicative model

– The magnitude of seasonal component grows inproportion to the trend of series.

– For example the cost of electric power consumption.


Forecasting with Trendand Seasonal Components

• Steps of Multiplicative Time Series Model:

1. Calculate the centered moving averages (CMAs).

2. Center the CMAs on integer-valued periods.

3. Determine the seasonal and irregular factors (StI t ).

4. Determine the average seasonal factors.5. Scale the seasonal factors (St ).

6. Determine the deseasonalized data (Y t/St).

7. Determine a trend line of the deseasonalized data.

8. Determine the deseasonalized predictions.

9. Take into account the seasonality.


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Example (1/11)

• Business at Cloths Shop can be viewed as falling into threedistinct seasons:

(1) season 1 (November-December);

(2) season 2 (late May - mid-June); and(3) all other times.

Average weekly sales (SR) during each of the threeseasons during the past four years are shown on

the next slide.

• Determine a forecast for the average weekly sales

in year 5 for each of the three seasons.


Example (2/11)• Past Sales (SR)

Year

Season 1 2 3 41 1856 1995 2241 2280

2 2012 2168 2306 24083 985 1072 1105 1120


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Example (3/11)

Moving ScaledYear Season Sales (Y t) Average StI t St Y t/St

1 1 1856 1.178 1576

2 2012 / 1617.67= 1.244 1.236 16283 985 1664.00 .592 .586 1681

2 1 1995 1716.00 1.163 1.178 16942 2168 1745.00 1.242 1.236 17543 1072 1827.00 .587 .586 1829

3 1 2241 1873.00 1.196 1.178 19022 2306 1884.00 1.224 1.236 18663 1105 1897.00 .582 .586 1886

4 1 2280 1931.00 1.181 1.178 1935

2 2408 1936.00 1.244 1.236 19483 1120 .586 1911

Remove trendand cyclical components


Example (4/11)• 1. Calculate the centered moving averages

– There are three distinct seasons in each year.

– Hence, take a three-season moving average to eliminateseasonal and irregular factors (smoothing).

– Moving Average keeps trend and cyclical factors

For example:

1st MA = (1856 + 2012 + 985)/3 = 1617.67

2nd MA = (2012 + 985 + 1995)/3 = 1664.00

etc.


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Example (5/11)

• 2. Center the CMAs on integer-valued periods

The first moving average computed in step 1(1617.67) will be centered on season 2 of year 1.

Note that the moving averages from step 1 centerthemselves on integer-valued periods because n=3is an odd number.

Step 1 = (1+3)/2 = 2


Example (6/11)• 3. Determine the seasonal & irregular factors (St It )

Isolate (or remove) the trend and cyclicalcomponents, for each period t, this is given by:

St It = Yt / (Moving Average for period t )

Yt = Tt· St· Ct· ItSt It = Yt / TtCtSt It = Yt / (Moving Average for period t )

Moving Average keeps Trend and Cycles

and eliminates Season and Irregular factor


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Example (7/11)

• 4. Determine the average seasonal factors

Averaging all (St It) values corresponding tothat season:

Season 1: ( 1.163 + 1.196 + 1.181) /3 = 1.180

Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.238

Season 3: ( .592 + .587 + .582 ) /3 = .587


Example (8/11)• 5. Scale the seasonal factors (St)

Average the seasonal factors =

(1.180 + 1.238 + .587)/3 = 1.002

Then, divide each seasonal factor by the average of the

seasonal factors.

Season 1: 1.180/1.002 = 1.178

Season 2: 1.238/1.002 = 1.236

Season 3: .587/1.002 = .586

Total = 3.000


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Example (9/11)

• 6. Determine the deseasonalized data (Yt /St )

Divide the data point values, Yt , by St .

• 7. Determine a trend line of the deseasonalized data

Using the least squares method (trend projection) for t = 1,2, ..., 12, gives:

Tt = 1580.11 + 33.96t


Example (10/11)• 8. Determine the deseasonalized predictions

Substitute t = 13, 14, and 15 into the least squaresequation:

T13 = 1580.11 + (33.96)(13) = 2022

T14 = 1580.11 + (33.96)(14) = 2056

T15 = 1580.11 + (33.96)(15) = 2090


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Example (11/11)

• 9. Take into account the seasonality.

Multiply each deseasonalized prediction by its

seasonal factor to give the following forecasts foryear 5:

Season 1: (1.178)(2022) = 2382

Season 2: (1.236)(2056) = 2541

Season 3: ( .586)(2090) = 1225




• Time series







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Associative Forecasting

• Regression analysis (Trend projection)

• Multiple regression analysis

• Logistic regression

Final ThoughtThe best way toThe best way to

predict the futurepredict the futureis to create it!is to create it!

Date post:	07-Aug-2018
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04 Forecasting

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