Lesson 4 - Part B Forecasting

Lesson 4 - Part BForecasting

• Trend Projection

• Trend and Seasonal Components

• Regression Analysis

Forecasting Methods

ForecastingForecastingMethodsMethods

ForecastingForecastingMethodsMethods

QuantitativeQuantitativeQuantitativeQuantitative QualitativeQualitativeQualitativeQualitative

CausalCausalCausalCausal Time SeriesTime SeriesTime SeriesTime Series

SmoothingSmoothingSmoothingSmoothing TrendTrendProjectionProjection

TrendTrendProjectionProjection

Trend ProjectionTrend ProjectionAdjusted forAdjusted for

Seasonal InfluenceSeasonal Influence

Trend ProjectionTrend ProjectionAdjusted forAdjusted for

Seasonal InfluenceSeasonal Influence

Trend Projection• If a time series exhibits a linear trend, the method

of least squares may be used to determine a trend line (projection) for future forecasts.

• Least squares, also used in regression analysis, determines the unique trend line forecast which minimizes the mean square error between the trend line forecasts and the actual observed values for the time series.

• The independent variable is the time period and the dependent variable is the actual observed value in the time series.

Trend Projection

• Using the method of least squares, the formula for the trend projection is:

where: Tt = trend forecast for time period t

b1 = slope of the trend line

b0 = trend line projection for time 0

TTtt = = bb00 + + bb11tt

Trend Projection

• For the trend projection equation Tt = b0 + b1t

0 1b Y b t 0 1b Y b t

tt= average time period for the = average time period for the nn observations observations

YY= average of the observed values for = average of the observed values for YYtt

1 2 2

( )/

( ) /t ttY t Y n

bt t n

1 2 2

( )/

( ) /t ttY t Y n

bt t n

where: where: YYtt = observed value of the time series at time = observed value of the time series at time period period tt

The number of plumbing repair jobs performedby Auger's Plumbing Service in each of thelast nine months is listed on the nextslide. Forecast the number of repairjobs Auger's will perform in Decemberusing the least squares method.

Trend Projection• Example: Auger’s Plumbing Service

Trend Projection

• Example: Auger’s Plumbing Service

MonthMonth JobsJobs

March 353March 353

May 342May 342April 387April 387

July 396July 396June 374June 374

August 409August 409

September 399September 399

October 412October 412

November 408November 408

MonthMonth JobsJobs

Months

Jobs

NovOctSepAugJulJunMayAprMar

420

410

400

390

380

370

360

350

340

Time Series Plot of Jobs

Although this graph shows some up and down movement over the past 9 months, the time series seems to have an overall increasingor upward trend.

Trend Projection

Sum 45 3480 17844 285Sum 45 3480 17844 285

(Nov.) 9 408 3672 81(Nov.) 9 408 3672 81(Oct.) 8 412 3296 64(Oct.) 8 412 3296 64(Sep.) 7 399 2793 49(Sep.) 7 399 2793 49(Aug.) 6 409 2454 36(Aug.) 6 409 2454 36(July) 5 396 1980 25(July) 5 396 1980 25(June) 4 374 1496 16(June) 4 374 1496 16(May) 3 342 1026 9(May) 3 342 1026 9(Apr.) 2 387 774 4(Apr.) 2 387 774 4(Mar.) 1 353 353 1(Mar.) 1 353 353 1

(month) (month) tt YYtt tYtYtt t t 22

T10 = 349.667 + (7.4)(10) = 423.667

3480/ 9 386.667Y 3480/ 9 386.667Y 45/ 9 5t 45/ 9 5t

Trend Projection

1 2 2 2

( )/ (9)(17844) (45)(3480)7.4

( ) / (9)(285) (45)t ttY t Y n

bt t n

1 2 2 2

( )/ (9)(17844) (45)(3480)7.4

( ) / (9)(285) (45)t ttY t Y n

bt t n

0 1 386.667 7.4(5) 349.667b Y b t 0 1 386.667 7.4(5) 349.667b Y b t

Forecast for December (Month 10) using a

three-period (n = 3) weighted moving

average with weights of .6, .3, and .1

for the newest to oldest data, respec-

tively. Then, compare this Month 10

weighted moving average forecast with

the Month 10 trend projection forecast.

• Example: Auger’s Plumbing Service

Trend Projection

• Three-Month Weighted Moving Average

Trend Projection

FF1010 = 423.7 (from earlier slide) = 423.7 (from earlier slide)• Trend Projection

FF1010 = .1 = .1YYSep.Sep. + .3 + .3YYOct.Oct. + .6 + .6YYNov.Nov.

= .1(399) + .3(412) + .6(408) = .1(399) + .3(412) + .6(408)

= 408.3= 408.3

The forecast for December will be the weightedThe forecast for December will be the weightedaverage of the preceding three months: September,average of the preceding three months: September,October, and November.October, and November.

Due to the positive trend component in the time series, the trend projection produced a forecast that is more in tune with the trend that exists. The weighted moving average, even with heavy (.6) weight placed on the current period, produced a forecast that is lagging behind the changing data.

• Conclusion

Trend Projection

Using MINITAB for trend projections

# of future forecasts needed # of data points

Dependentvariable

Can use different types of trend models

Months

Jobs

DecNovOctSepAugJulJunMayAprMar

430

420

410

400

390

380

370

360

350

340

Accuracy MeasuresMAPE 2.948MAD 11.096MSD 204.267

Variable

Forecasts

ActualFits

Trend Analysis Plot for JobsLinear Trend Model

Yt = 349.667 + 7.4*t

Interpretation of slope coefficient:In every month, # of repair jobs undertaken increased by 7.4, on average.

Obtaining detailed MINITAB output

Trend Analysis for Jobs

Fitted Trend Equation

Yt = 349.667 + 7.4*t

Time Jobs Trend Detrend(Jobs – Trend)

1 353 357.067 -4.0667

2 387 364.467 22.5333

3 342 371.867 -29.8667

4 374 379.267 -5.2667

5 396 386.667 9.3333

6 409 394.067 14.9333

7 399 401.467 -2.4667

8 412 408.867 3.1333

9 408 416.267 -8.2667

Forecasts

Period Forecast

10 423.667

Removing trend effect from a time series

Forecasting with Trend and 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 (St It ).

4. Determine the average seasonal factors.

5. Scale the seasonal factors (St ).

6. Determine the deseasonalized data.

7. Determine a trend line of the deseasonalized data.

8. Determine the deseasonalized predictions.

9. Take into account the seasonality.

Business at Terry's Tie Shop can beviewed as falling into three distinctseasons: (1) Christmas (November andDecember); (2) Father's Day (late May tomid June); and (3) all other times. Average weekly sales ($) during each of the threeseasons during the past four years are shown onthe next slide.


• Example: Terry’s Tie Shop

• Example: Terry’s Tie Shop


Determine a forecast for the average weeklyDetermine a forecast for the average weekly

sales in year 5 for each of the three seasons.sales in year 5 for each of the three seasons.

YearYearSeasonSeason

11 22 33 1856 2012 9851856 2012 9851995 2168 10721995 2168 10722241 2306 11052241 2306 11052280 2408 11202280 2408 1120

11223344

Year/ Season

Sale

s

321321321321

2400

2200

2000

1800

1600

1400

1200

1000

Time Series Plot of Sales

•In every year sales are lowest in season 3 and sales are increased inseasons 1 and 2. Thus, a seasonal pattern exits for these sales.

•Also, it seems overall sales are increasing (on average) over theseyears. Thus, a trend exits for these sales.


22ndnd CMA = (2012 + 985 + 1995)/3 = 1664.00 CMA = (2012 + 985 + 1995)/3 = 1664.00

11stst CMA = (1856 + 2012 + 985)/3 = 1617.67 CMA = (1856 + 2012 + 985)/3 = 1617.67

Step 1. Calculate the centered moving averages.

There are three distinct seasons in each year. Hence, take a three-season moving average to eliminate seasonal and irregular factors. For example:

Etc.Etc.


Step 2. Center the CMAs on integer-valued periods.

The first centered 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 center themselves on integer-valued periods because n is an odd number.

If n is even, we have to compute the average of consecutive moving averages to get centered moving averages.


YearYear SeasonSeasonDollarDollar

Sales (Sales (YYtt))MovingMovingAverageAverage

11

22

33

44

112233112233112233112233

(1856 + 2012 + (1856 + 2012 + 985)/3985)/3

1664.001664.001716.001716.001745.001745.001827.001827.001873.001873.001884.001884.001897.001897.001931.001931.001936.001936.00

1617.671617.67

199199552162168810710722

1851856620120122 985985224224112302306611011055

228228002402408811211200

Using MINITAB to get centered moving averages

You can specify this i.e. 4, 6, 12, … & its depend on the problem

Centered Moving Average for Sales

Moving Average: Length 3

Time Sales CMA Predict Error

1 1856 * * *

2 2012 1617.67 * *

3 985 1664.00 * *

4 1995 1716.00 1617.67 377.333

5 2168 1745.00 1664.00 504.000

6 1072 1827.00 1716.00 -644.000

7 2241 1873.00 1745.00 496.000

8 2306 1884.00 1827.00 479.000

9 1105 1897.00 1873.00 -768.000

10 2280 1931.00 1884.00 396.000

11 2408 1936.00 1897.00 511.000

12 1120 * 1931.00 -811.000

Year/ Season

Sale

s

321321321321

2400

2200

2000

1800

1600

1400

1200

1000

Moving AverageLength 3

Accuracy MeasuresMAPE 36MAD 554MSD 328008

VariableActualSmoothed

Centered Moving Average Plot for Sales

CMA values smooth out both the seasonal and irregular fluctuations in the time series.


SSt t IIt t = = YYt t /(Moving Average for period /(Moving Average for period t t ))

Step 3. Determine the seasonal & irregular factors (St It ).

Isolate the trend and cyclical components. For each period t, this is given by:




11

22

33

44

112233112233112233112233

1851856620120122 985985199199552162168810710722224224112302306611011055228228002402408811211200

1617.671617.671664.001664.001716.001716.001745.001745.001827.001827.001873.001873.001884.001884.001897.001897.001931.001931.001936.001936.00

SSttIItt

1.2441.2442012/1617.2012/1617.

6767.592.592

1.1631.1631.2421.242.587.587

1.1961.1961.2241.224.582.582

1.1811.1811.2441.244


Season 3: (.592 + .587 + .582) /3 = .587Season 3: (.592 + .587 + .582) /3 = .587Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.238Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.238Season 1: (1.163 + 1.196 + 1.181) /3 = 1.180Season 1: (1.163 + 1.196 + 1.181) /3 = 1.180

Step 4. Determine the average seasonal factors. Averaging all St It values corresponding to that season:

3.0053.005


Season 3: .587/1.002 = .586Season 3: .587/1.002 = .586Season 2: 1.238/1.002 = 1.236Season 2: 1.238/1.002 = 1.236Season 1: 1.180/1.002 = 1.178Season 1: 1.180/1.002 = 1.178

Step 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.

3.0003.000

Season 2 sales (on average) 23.6% above the average annual sales.Season 1 sales (on average) 17.8% above the average annual sales. Season 3 sales (on average) 42.4% below the average annual sales.




11

22

33

44

112233112233112233112233

1851856620120122 985985199199552162168810710722224224112302306611011055228228002402408811211200

1617.671617.671664.001664.001716.001716.001745.001745.001827.001827.001873.001873.001884.001884.001897.001897.001931.001931.001936.001936.00

SSttIItt

1.2441.244.592.592

1.1631.1631.2421.242.587.587

1.1961.1961.2241.224.582.582

1.1811.1811.2441.244

ScaledScaledSStt

1.1781.1781.2361.236

.586.5861.1781.1781.2361.236

.586.5861.1781.1781.2361.236

.586.5861.1781.1781.2361.236

.586.586

Step 6. Determine the deseasonalized data.


Divide the data point values, Yt , by St

.




11

22

33

44

112233112233112233112233

1851856620120122 985985199199552162168810710722224224112302306611011055228228002402408811211200

1617.671617.671664.001664.001716.001716.001745.001745.001827.001827.001873.001873.001884.001884.001897.001897.001931.001931.001936.001936.00

SSttIItt

1.2441.244.592.592

1.1631.1631.2421.242.587.587

1.1961.1961.2241.224.582.582

1.1811.1811.2441.244

ScaledScaledSStt

1.1781.1781.2361.236

.586.5861.1781.1781.2361.236

.586.5861.1781.1781.2361.236

.586.5861.1781.1781.2361.236

.586.586

YYtt//SStt

15761576

1856/1.11856/1.17878

1628162816811681169416941754175418291829190219021866186618861886193519351948194819111911

Year/ Season

Dese

aso

naliz

ed S

ale

s

321321321321

2000

1900

1800

1700

1600

Deseasonalized Sales Time Series

Although this shows some up and down movement over the past 12 seasons, the time series seems to have an upward linear trend.


TTtt = 1580.11 + 33.96 = 1580.11 + 33.96tt

Step 7. Determine a trend line of the deseasonalized data (similar to trend projections described before).

Using the least squares method for t = 1, 2, ..., 12, gives:


TT1313 = 1580.11 + (33.96)(13) = 2022 = 1580.11 + (33.96)(13) = 2022

TT1414 = 1580.11 + (33.96)(14) = 1580.11 + (33.96)(14) = 2056= 2056

TT1515 = 1580.11 + (33.96)(15) = 2090 = 1580.11 + (33.96)(15) = 2090

Step 8. Determine the deseasonalized predictions.

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


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

Season 2: (1.236)(2056) = 2541Season 2: (1.236)(2056) = 2541

Season 1: (1.178)(2022) = 2382Season 1: (1.178)(2022) = 2382

Step 9. Take into account the seasonality.

Multiply each deseasonalized prediction by its seasonal factor to give the following forecasts for year 5:

If you have detailed information (i.e. in Terry’s Tie Shop example,monthly sales over the 4 year period) we can use MINITAB to carryout all these 9 steps.

Depends on the problem:• # of months for a year• # of days for a week• # of hours of a working day, etc.

If both trend & seasonaleffects present in the time series (based on time series plot) select this.

Forecasting sales for 12 months in next year

# of data points in the time series

Lesson 4 - Part B Forecasting

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Transcript of Lesson 4 - Part B Forecasting