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Module 4. Forecasting

MGS3100

Forecasting

Quantitative Forecasting

Casual Models:

Causal

Model

Year 2000

Sales

Price

Population

Advertising

Time Series Models:

Time Series

Model

Year 2000

Sales

Sales1999

Sales1998

Sales1997

--Forecasting based on data and models

Causal forecasting

Regression

Find a straight line that fits the data best.

y = Intercept + slope * x (= b0 + b1x)

Slope = change in y / change in x

Best line!

Intercept

Chart3

5

6

5

7.5

6

8.5

8

10

7

8

11

8

11

Shoe Size (Y)

Raw Data

Example of Simple Regression - Does Shoe Size among teenagers depend on Age?

(Can you predict the shoe size if you know the age?)

AgeShoe Size

115

126

125

137.5

136

138.5

148

1510

157

178

1811

188

1911

Raw Data

5

6

5

7.5

6

8.5

8

10

7

8

11

8

11

Shoe Size (Y)

Simple



Deviations fromSquared

Age (X)Shoe Size (Y)the MeanDeviations

115-2.76923076927.6686390533

126-1.76923076923.1301775148

125-2.76923076927.6686390533

137.5-0.26923076920.0724852071

136-1.76923076923.1301775148

138.50.73076923080.5340236686

1480.23076923080.0532544379

15102.23076923084.9763313609

157-0.76923076920.5917159763

1780.23076923080.0532544379

18113.230769230810.4378698225

1880.23076923080.0532544379

19113.230769230810.4378698225

Mean Shoe Size =7.769230769248.8076923077

Sum of Squared Deviations, shown in the ANOVA

SUMMARY OUTPUTTable as the SS Total.

Regression Statistics

Multiple R0.798497882

R Square0.6375988676R-Squared is = SSR/SST = 31.119/48.807, from the ANOVA table below.

Adjusted R Square0.6046533101

Standard Error1.2680680711Std. Error is the square root of the Mean Squared Error

Observations13n is the number of obs., which is 13 in this case.

ANOVADegrees of FreedomThe Mean Squares (MS) are computed by dividing SS (Sum of Squares by the degrees of freedom)

dfSSMSFSignificance F

Regression31.119729344731.119729344719.35310603650.0010645821

Residual (Error)17.6879629631.607996633Shows that the overall model is significant. There is a 0.1% chance

Total48.8076923077that the relationship is non-existent, and that we falsely believe the model.

k is the number of independent (predictor) variables, in this case just 1 (age)

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept-1.17592592592.0635438706-0.56985748770.5802261973-5.71775765933.3659058075

Age0.6120370370.13912409944.39921652530.00106458210.3058268040.91824727

These coefficients are computed using a formula that guarantees that this is the best fitting line.

RESIDUAL OUTPUTYou do not have to know the formulas. They are available in every basic stat book

Squared

ObservationPredicted Shoe SizeResidualsResiduals

15.55648148-0.556481480.3096716392

26.16851852-0.168518520.0283984911

36.16851852-1.168518521.3654355281

46.780555560.719444440.5176003086

56.78055556-0.780555560.6092669753

66.780555561.719444442.9564891975

77.392592590.607407410.3689437586

88.004629631.995370373.981502915

98.00462963-1.004629631.0092806927

109.22870370-1.228703701.5097127915

119.840740741.159259261.3438820302

129.84074074-1.840740743.3883264746

1310.452777780.547222220.2994521605

17.687962963Sum of Squared Residuals(Errors), shown in the

ANOVA table.

How and Why are the Sum of Squares shown in the ANOVA table calculated?

The basic idea behind regression is to see if there is a relationship between X and Y, and if so, how well does X help predict Y.

If there was no info on X (Age), but all we had was a sample of shoe sizes, our best estimate of shoe sizes would be the mean

shoe size of about 7.7, shown at the top. However, this estimate would have a lot of error, since actual sizes deviate quite a bit from

the mean. These deviations are computed and squared (to avoid + and - cancelling each other) and summed, to get a SST (Sum

of Squares Total) value of 48.807.

Now, when we consider the info provided by age, we can better estimate shoe size than simply using the mean size. We can now say

that shoe size depends on Age according to the equation Y= 0.612 X - 1.1759. Now, our new estimates are better, but they are still

not perfect. There are still errors (residuals) shown in the excel output at the bottom. If we square each of the errors again and add them,

we get the SSE (Sum of Squared Errors) value of 17.687.

This means that by using Age to do the regression, we reduced our error squares by 48.807-17.687, or by a value of 31.119,

shown in the ANOVA table as SSR (Sum of Squares Regression). SSR is thus the reduction in SST brought about by the regression.

In other words, the regression helped to explain away 31.119 out of the total of 48.807 of error. Thus, the proportion of variability in

Y that is explained by the regression is 31.119/48.807 = 0.6376, which is the R-Squared value shown at the top.

What are Degrees of Freedom?

Once the SS are computed, the Mean Squares are computed by dividing by the degrees of freedom. Normally, a mean is simply

the sum of n numbers divided by n. Here, however, when we find the mean, we must compensate for the fact that we are averaging

errors, and even though there are n numbers, not all of them contribute to the error.

For example, if there is only 1 data point, there is no chance (freedom) for any variation at all to occur. Hence, total degrees of

freedom are always n-1. Thus, if there are 2 data points, there is one degree of freedom for variation to occur.

Next, suppose there are 2 points of data. Even though they could be different values of Y, the process of using a variable X to do

a regression means that we draw the best line through them. Now no matter what the points are, we can always draw a straight

line perfectly through those points. Thus, there is no freedom for error to occur, since the variable X "used up" the single degree of

freedom that Y had. In general, the number of independent variables used (K) is the number of degrees of freedom that are used up

from the total available (n-1), leaving n-k-1 degrees available for error to occur. Thus the SS Error is divided by n-k-1 to find the mean

squared error, instead of dividing by n.

What is F-value? When is a model significant?

F-value is the ratio of MSR/MSE = 31.119/1.6079. This shows the ratio of the average error that is explained by the regression to the average

error that is still unexplained. Thus, the higher the F, the better the model, and the more confidence we have that the model that we

derived from sample data actually applies to the whole population, and is not just an aberration found in the sample.

In this case, the level of confidence is around 99.9%, reflected in the significance value of 0.00106 shown in the ANOVA table.

That value was computed by looking at standardized tables that consider the F-value and your sample size to make that determination.

Simple

0

0

0

0

0

0

0

0

0

0

0

0

0

Shoe Size (Y)

Age in Years

Shoe Size

Shoe Sizes of Teens

Multiple-initial

Example of Multiple Regression: Can Shoe Size (Y) be predicted by

the independent variables X1 through X4?

YX1X2X3X4

Shoe SizeAgeWeightSexIQ Score

511750100

61285180

51288050

7.5131351120

613800115

8.5131800106

814140096

1015200088

715110078

817120065

11181501101

8181250105

11191651130

Female=0

Male=1

SUMMARY OUTPUT



R Square0.9614468824


Standard Error0.484985657

Observations13

ANOVA


Regression446.92600360811.73150090249.87647915960.0000107054

Residual81.88168869970.2352110875

Total1248.8076923077


Intercept-1.7743425060.9297924371-1.9083210780.092770928-3.91844909750.3697640855

Age0.34626069830.06232644985.55559797250.00053743450.20253555430.4899858422

Weight0.03038034870.00418029997.26750462880.00008656980.02074055370.0400201438

Sex0.79189071910.32269621242.45398206910.03968948280.04775143761.5360300006

IQ Score0.00396324190.00714713770.55452155650.5943806023-0.01251809790.0204445817

IQ not significantly related to Shoe Size

Multiple-revised

Multiple Regression with the insignificant variable (IQ) dropped

SUMMARY OUTPUT



R Square0.9599650251



Observations13

ANOVA


Regression346.85367757315.617892524371.93447942010.0000013077

Residual91.95401473470.2171127483

Total1248.8076923077

CoefficientsStandard Errort StatP-valueLower 95%

Intercept-1.51044939720.767427851-1.96819726480.0805754213-3.2464931304

Age0.34745099110.05984507885.8058406470.00025758330.2120719142

Weight0.03096629490.00388582827.96903343470.00002283110.0221759341

Sex0.85821709040.28794892222.9804490460.01543832690.2068308771

RESIDUAL OUTPUT

ObservationPredicted Shoe SizeResiduals

14.63398362350.3660163765

26.1493146541-0.1493146541

35.3839964485-0.3839964485

48.0450803909-0.5450803909

55.48371708030.5162829197

68.5803465716-0.0803465716

77.68914576620.3108542338

89.89457445210.1054255479

97.1076079099-0.1076079099

108.1121728412-0.1121728412

1110.24682977010.7531702299

128.6144553069-0.6144553069

1311.0587751849-0.0587751849

Causal Forecasting Models

Curve Fitting: Simple Linear RegressionOne Independent Variable (X) is used to predict one Dependent Variable (Y): Y = a + b XGiven n observations (Xi, Yi), we can fit a line to the overall pattern of these data points. The Least Squares Method in statistics can give us the best a and b in the sense of minimizing (Yi - a - bXi)2:

Regression formula is an optional learning objective

Curve Fitting: Simple Linear Regression Find the regression line with ExcelUse Function:

a = INTERCEPT(Y range; X range)

b = SLOPE(Y range; X range)

Use SolverUse Excels Tools | Data Analysis | Regression Curve Fitting: Multiple RegressionTwo or more independent variables are used to predict the dependent variable:

Y = b0 + b1X1 + b2X2 + + bpXp

Use Excels Tools | Data Analysis | Regression

Time Series Forecasting Process

Look at the data (Scatter Plot)

Forecast using one or more techniques

Evaluate the technique and pick the best one.

Observations from the scatter PlotTechniques to tryWays to evaluateData is reasonably stationary (no trend or seasonality)Heuristics - Averaging methods Naive Moving Averages Simple Exponential Smoothing MAD MAPE Standard Error BIASData shows a consistent trendRegression Linear Non-linear Regressions (not covered in this course) MAD MAPE Standard Error BIAS R-SquaredData shows both a trend and a seasonal patternClassical decomposition Find Seasonal Index Use regression analyses to find the trend component MAD MAPE Standard Error BIAS R-Squared

Evaluation of Forecasting Model

BIAS - The arithmetic mean of the errors

n is the number of forecast errors

Excel: =AVERAGE(error range)

Mean Absolute Deviation - MAD

No direct Excel function to calculate MAD

Evaluation of Forecasting Model

Mean Square Error - MSE

Excel: =SUMSQ(error range)/COUNT(error range)Standard error is square root of MSEMean Absolute Percentage Error - MAPE

R2 - only for curve fitting model such as regressionIn general, the lower the error measure (BIAS, MAD, MSE) or the higher the R2, the better the forecasting model

Stationary data forecasting

Nave

I sold 10 units yesterday, so I think I will sell 10 units today.

n-period moving average

For the past n days, I sold 12 units on average. Therefore, I think I will sell 12 units today.

Exponential smoothing

I predicted to sell 10 units at the beginning of yesterday; At the end of yesterday, I found out I sold in fact 8 units. So, I will adjust the forecast of 10 (yesterdays forecast) by adding adjusted error ( * error). This will compensate over (under) forecast of yesterday.

Nave Model

The simplest time series forecasting model Idea: what happened last time (last year, last month, yesterday) will happen again this time Nave Model:

Algebraic: Ft = Yt-1

Yt-1 : actual value in period t-1

Ft : forecast for period t

Spreadsheet: B3: = A2; Copy down

Moving Average Model

Simple n-Period Moving Average

Issues of MA Model Nave model is a special case of MA with n = 1 Idea is to reduce random variation or smooth dataAll previous n observations are treated equally (equal weights)Suitable for relatively stable time series with no trend or seasonal pattern

Smoothing Effect of MA Model

Longer-period moving averages (larger n) react to actual changes more slowly

Moving Average Model

Weighted n-Period Moving Average

Typically weights are decreasing: w1>w2>>wnSum of the weights = wi = 1Flexible weights reflect relative importance of each previous observation in forecastingOptimal weights can be found via Solver

Weighted MA: An Illustration

Month Weight Data

August 17%130

September 33%110

October 50%90

November forecast:

FNov = (0.50)(90)+(0.33)(110)+(0.17)(130)

= 103.4

Exponential Smoothing

Concept is simple!Make a forecast, any forecastCompare it to the actualNext forecast isPrevious forecast plus an adjustmentAdjustment is fraction of previous forecast errorEssentiallyNot really forecast as a function of timeInstead, forecast as a function of previous actual and forecasted value

Simple Exponential Smoothing

A special type of weighted moving average Include all past observationsUse a unique set of weights that weight recent observations much more heavily than very old observations:

Simple ES: The Model

New forecast = weighted sum of last period actual value and last period forecast

: Smoothing constant Ft :Forecast for period t Ft-1:Last period forecast Yt-1:Last period actual value

Simple Exponential Smoothing

Properties of Simple Exponential SmoothingWidely used and successful modelRequires very little dataLarger , more responsive forecast; Smaller , smoother forecast (See Table 13.2)best can be found by Solver Suitable for relatively stable time series

Time Series Components

Trendpersistent upward or downward pattern in a time seriesSeasonal Variation dependent on the time of year Each year shows same patternCyclical up & down movement repeating over long time frameEach year does not show same patternNoise or random fluctuations follow no specific pattern short duration and non-repeating

Time Series Components

Time

Trend

Random

movement

Time

Cycle

Time

Seasonal

pattern

Demand

Time

Trend with

seasonal pattern

Trend Model

Curve fitting method used for time series data (also called time series regression model) Useful when the time series has a clear trend Can not capture seasonal patterns Linear Trend Model: Yt = a + bt t is time index for each period, t = 1, 2, 3,

Chart2

0.6

1.2

1.8

2.4

3

3.6

4.2

4.8

5.4

6

Sheet1

10.811600.6

29.05096679927401.2

337.41229744354401.8

4102.42602.4

5223.606797752003

6423.27182755292603.6

7725.99415975614404.2

81158.5237502967404.8

91749.611605.4

102529.822128134717006

111324.32827878891080

121795.79027728741470

132376.41884565831920

143080.13236079232430

153921.3956380353000

164915.23630

176077.04538159794320

187422.92414618395070

198969.30635612365880

2010733.1262919996750

Sheet1

Sheet2

Sheet3

Sheet4

Sheet5

Pattern-based forecasting - Trend

Regression Recall Independent Variable X, which is now time variable e.g., days, months, quarters, years etc.

Find a straight line that fits the data best.

y = Intercept + slope * x (= b0 + b1x)

Slope = change in y / change in x

Best line!

Intercept

Chart3

5

6

5

7.5

6

8.5

8

10

7

8

11

8

11

Shoe Size (Y)

Raw Data



AgeShoe Size

115

126

125

137.5

136

138.5

148

1510

157

178

1811

188

1911

Raw Data

5

6

5

7.5

6

8.5

8

10

7

8

11

8

11

Shoe Size (Y)

Simple



Deviations fromSquared

Age (X)Shoe Size (Y)the MeanDeviations

115-2.76923076927.6686390533

126-1.76923076923.1301775148

125-2.76923076927.6686390533

137.5-0.26923076920.0724852071

136-1.76923076923.1301775148

138.50.73076923080.5340236686

1480.23076923080.0532544379

15102.23076923084.9763313609

157-0.76923076920.5917159763

1780.23076923080.0532544379

18113.230769230810.4378698225

1880.23076923080.0532544379

19113.230769230810.4378698225

Mean Shoe Size =7.769230769248.8076923077

Sum of Squared Deviations, shown in the ANOVA

SUMMARY OUTPUTTable as the SS Total.



R Square0.6375988676R-Squared is = SSR/SST = 31.119/48.807, from the ANOVA table below.


Standard Error1.2680680711Std. Error is the square root of the Mean Squared Error

Observations13n is the number of obs., which is 13 in this case.

ANOVADegrees of FreedomThe Mean Squares (MS) are computed by dividing SS (Sum of Squares by the degrees of freedom)


Regression31.119729344731.119729344719.35310603650.0010645821

Residual (Error)17.6879629631.607996633Shows that the overall model is significant. There is a 0.1% chance

Total48.8076923077that the relationship is non-existent, and that we falsely believe the model.

k is the number of independent (predictor) variables, in this case just 1 (age)


Intercept-1.17592592592.0635438706-0.56985748770.5802261973-5.71775765933.3659058075

Age0.6120370370.13912409944.39921652530.00106458210.3058268040.91824727

These coefficients are computed using a formula that guarantees that this is the best fitting line.

RESIDUAL OUTPUTYou do not have to know the formulas. They are available in every basic stat book

Squared

ObservationPredicted Shoe SizeResidualsResiduals

15.55648148-0.556481480.3096716392

26.16851852-0.168518520.0283984911

36.16851852-1.168518521.3654355281

46.780555560.719444440.5176003086

56.78055556-0.780555560.6092669753

66.780555561.719444442.9564891975

77.392592590.607407410.3689437586

88.004629631.995370373.981502915

98.00462963-1.004629631.0092806927

109.22870370-1.228703701.5097127915

119.840740741.159259261.3438820302

129.84074074-1.840740743.3883264746

1310.452777780.547222220.2994521605

17.687962963Sum of Squared Residuals(Errors), shown in the

ANOVA table.

How and Why are the Sum of Squares shown in the ANOVA table calculated?

The basic idea behind regression is to see if there is a relationship between X and Y, and if so, how well does X help predict Y.

If there was no info on X (Age), but all we had was a sample of shoe sizes, our best estimate of shoe sizes would be the mean

shoe size of about 7.7, shown at the top. However, this estimate would have a lot of error, since actual sizes deviate quite a bit from

the mean. These deviations are computed and squared (to avoid + and - cancelling each other) and summed, to get a SST (Sum

of Squares Total) value of 48.807.

Now, when we consider the info provided by age, we can better estimate shoe size than simply using the mean size. We can now say

that shoe size depends on Age according to the equation Y= 0.612 X - 1.1759. Now, our new estimates are better, but they are still

not perfect. There are still errors (residuals) shown in the excel output at the bottom. If we square each of the errors again and add them,

we get the SSE (Sum of Squared Errors) value of 17.687.

This means that by using Age to do the regression, we reduced our error squares by 48.807-17.687, or by a value of 31.119,

shown in the ANOVA table as SSR (Sum of Squares Regression). SSR is thus the reduction in SST brought about by the regression.

In other words, the regression helped to explain away 31.119 out of the total of 48.807 of error. Thus, the proportion of variability in

Y that is explained by the regression is 31.119/48.807 = 0.6376, which is the R-Squared value shown at the top.

What are Degrees of Freedom?

Once the SS are computed, the Mean Squares are computed by dividing by the degrees of freedom. Normally, a mean is simply

the sum of n numbers divided by n. Here, however, when we find the mean, we must compensate for the fact that we are averaging

errors, and even though there are n numbers, not all of them contribute to the error.

For example, if there is only 1 data point, there is no chance (freedom) for any variation at all to occur. Hence, total degrees of

freedom are always n-1. Thus, if there are 2 data points, there is one degree of freedom for variation to occur.

Next, suppose there are 2 points of data. Even though they could be different values of Y, the process of using a variable X to do

a regression means that we draw the best line through them. Now no matter what the points are, we can always draw a straight

line perfectly through those points. Thus, there is no freedom for error to occur, since the variable X "used up" the single degree of

freedom that Y had. In general, the number of independent variables used (K) is the number of degrees of freedom that are used up

from the total available (n-1), leaving n-k-1 degrees available for error to occur. Thus the SS Error is divided by n-k-1 to find the mean

squared error, instead of dividing by n.

What is F-value? When is a model significant?

F-value is the ratio of MSR/MSE = 31.119/1.6079. This shows the ratio of the average error that is explained by the regression to the average

error that is still unexplained. Thus, the higher the F, the better the model, and the more confidence we have that the model that we

derived from sample data actually applies to the whole population, and is not just an aberration found in the sample.

In this case, the level of confidence is around 99.9%, reflected in the significance value of 0.00106 shown in the ANOVA table.

That value was computed by looking at standardized tables that consider the F-value and your sample size to make that determination.

Simple

0

0

0

0

0

0

0

0

0

0

0

0

0

Shoe Size (Y)

Age in Years

Shoe Size

Shoe Sizes of Teens

Multiple-initial

Example of Multiple Regression: Can Shoe Size (Y) be predicted by

the independent variables X1 through X4?

YX1X2X3X4

Shoe SizeAgeWeightSexIQ Score

511750100

61285180

51288050

7.5131351120

613800115

8.5131800106

814140096

1015200088

715110078

817120065

11181501101

8181250105

11191651130

Female=0

Male=1

SUMMARY OUTPUT



R Square0.9614468824



Observations13

ANOVA


Regression446.92600360811.73150090249.87647915960.0000107054

Residual81.88168869970.2352110875

Total1248.8076923077


Intercept-1.7743425060.9297924371-1.9083210780.092770928-3.91844909750.3697640855

Age0.34626069830.06232644985.55559797250.00053743450.20253555430.4899858422

Weight0.03038034870.00418029997.26750462880.00008656980.02074055370.0400201438

Sex0.79189071910.32269621242.45398206910.03968948280.04775143761.5360300006

IQ Score0.00396324190.00714713770.55452155650.5943806023-0.01251809790.0204445817

IQ not significantly related to Shoe Size

Multiple-revised

Multiple Regression with the insignificant variable (IQ) dropped

SUMMARY OUTPUT



R Square0.9599650251



Observations13

ANOVA


Regression346.85367757315.617892524371.93447942010.0000013077

Residual91.95401473470.2171127483

Total1248.8076923077

CoefficientsStandard Errort StatP-valueLower 95%

Intercept-1.51044939720.767427851-1.96819726480.0805754213-3.2464931304

Age0.34745099110.05984507885.8058406470.00025758330.2120719142

Weight0.03096629490.00388582827.96903343470.00002283110.0221759341

Sex0.85821709040.28794892222.9804490460.01543832690.2068308771

RESIDUAL OUTPUT

ObservationPredicted Shoe SizeResiduals

14.63398362350.3660163765

26.1493146541-0.1493146541

35.3839964485-0.3839964485

48.0450803909-0.5450803909

55.48371708030.5162829197

68.5803465716-0.0803465716

77.68914576620.3108542338

89.89457445210.1054255479

97.1076079099-0.1076079099

108.1121728412-0.1121728412

1110.24682977010.7531702299

128.6144553069-0.6144553069

1311.0587751849-0.0587751849

Pattern-based forecasting Seasonal

Once data turn out to be seasonal, deseasonalize the data.The methods we have learned (Heuristic methods and Regression) is not suitable for data that has pronounced fluctuations.Make forecast based on the deseasonalized dataReseasonalize the forecastGood forecast should mimic reality. Therefore, it is needed to give seasonality back.


Deseasonalize

Forecast

Reseasonalize

Actual data

Deseasonalized data

Example (SI + Regression)


Deseasonalization

Deseasonalized data = Actual / SI

Reseasonalization

Reseasonalized forecast

= deseasonalized forecast * SI

Seasonal Index

Whats an index?RatioSI = ratio between actual and average demandSupposeSI for quarter demand is 1.20Whats that mean?Use it to forecast demand for next fallSo, where did the 1.20 come from?!

Calculating Seasonal Indices

Quick and dirty method of calculating SIFor each year, calculate average demandDivide each demand by its yearly averageThis creates a ratio and hence a raw indexFor each quarter, there will be as many raw indices as there are yearsAverage the raw indices for each of the quartersThe result will be four values, one SI per quarter

Classical decomposition

Start by calculating seasonal indicesThen, deseasonalize the demandDivide actual demand values by their SI values

y = y / SI

Results in transformed data (new time series)Seasonal effect removedForecastRegression if deseasonalized data is trendyHeuristics methods if deseasonalized data is stationaryReseasonalize with SI

Causal or Time series?

What are the difference?

Which one to use?

Can you

describe general forecasting process?compare and contrast trend, seasonality and cyclicality?describe the forecasting method when data is stationary?describe the forecasting method when data shows trend?describe the forecasting method when data shows seasonality?

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forecasting - georgia state university - georgia state …dscsss/teaching/mgs3100… · ppt...

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