Business Analysis- Time-Series Models -Measuring Forecast Error

8/3/2019 Business Analysis- Time-Series Models -Measuring Forecast Error

1/36

Slides 13b:Time-Series Models;

Measuring Forecast Error

MGS3100 Chapter 13Forecasting


2/36

Forecasting ModelsForecastingTechniques

QualitativeModels

Time SeriesMethods

CausalMethods

Delphi

Method

Jury ofExecutiveOpinion

Sales ForceComposite

Consumer MarketSurvey

Naive

MovingAverage

WeightedMoving Average

ExponentialSmoothing

Trend Analysis

SeasonalityAnalysisSimple

RegressionAnalysis

MultipleRegressionAnalysis

MultiplicativeDecomposition


3/36

Time Series Models General Form: Y = T * C * S , where

T = Trend - long term movement of mean C = (Business) Cycle - an upturn or downturn not

caused by seasonal variation; effect of theeconomy S = Seasonal Variation - repetitive pattern

observed over a specific time period = Error (random variation)

Practical Forecast Form: = T * S C is important, but difficult to forecast

Dont forecast an error!


4/36

A stationary time series

Linear trend time series

Linear trend and seasonality time series

Time

Time

series

value

Future

Components of a Time Series


5/36

Time Series: Stationary Models Stationary Model Assumptions

Assumes item forecasted will stay steady over time (constantmean; random variation only)

Techniques will smooth out short-term irregularities Forecast for period t+1 is equal to forecast for period t+k; the

forecast is revised only when new data becomes available.

Stationary Model Types Nave Forecast Moving Average Weighted Moving Average Exponential Smoothing


6/36

Stationary Time Series Models:

The Nave Model Whatever happened

last period willhappen again this

time The model is simple

and flexible Provides a baseline

to measure othermodels

Attempts to captureseasonal factors atthe expense of

ignoring trend

dataMonthly:

dataQuarterly:

12

4

!

!

tt

tt

YF

YF

1! tt YFor


7/36

Measures of Forecast Error

Bias - The arithmetic sum of theerrors

MAD - Mean Absolute Deviation MAPE Mean Absolute Percentage

Error

Mean Square Error (MSE) - Similarto simple sample variance

Standard Error - Standard deviation ofthe sampling distribution (the squareroot of the MSE)

Bias, MAD, and MAPE - typically

used for time series

)(tt

FYErrorForecast !

TFYMADtt

T

t

/||/T|errorforecast|1

T

1t

!! ! !

TYFYMAPEttt

T

t

/]/|[|1001

! !

TFY

Bias

tt

T

t

/)(

/Terror)(forecast

1

T

1t

!

!

!

!

TFY

MSE

tt

T

t

/)(

/T|errorforecast|

2

1

T

1t

2

!

!

!

!


8/36

Nave ForecastWallace Garden SupplyForecasting

Period

Actual

Value

Nave

Forecast Err or

Absolute

Error

Percent

Error

Squared

Error

January 10 N/A

February 12 10 2 2 16.67% 4.0

March 16 12 4 4 25.00% 16.0

April 13 16 -3 3 23.08% 9.0

May 17 13 4 4 23.53% 16.0

June 19 17 2 2 10.53% 4.0

July 15 19 -4 4 26.67% 16.0

August 20 15 5 5 25.00% 25.0

September 22 20 2 2 9.09% 4.0

October 19 22 -3 3 15.79% 9.0November 21 19 2 2 9.52% 4.0

December 19 21 -2 2 10.53% 4.0

0.818 3 17.76% 10.091

BIAS MAD MAPE MSE

Standard Error (Square Root of MSE) = 3.176619

Storage Shed Sales


9/36

Nave Forecast GraphWallace Garden - Naive Fore cast

0

5

10

15

20

25

February March April May June July August September October November December

Period

S

h

ed

s Actual Value

Nave Forecast


10/36

The Moving Average Method

The forecast is the average of the last nobservations of the time series.

n

YYYF

ntttt

111

...

!


Moving Averages


11/36

Moving AveragesWallace Garden SupplyForecasting

Period ActualValue Three-Month Moving AveragesJanuary 10February 12March 16April 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 = 16.33August 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


12/36

Moving Averages ForecastWallace Garden SupplyForecasting 3 period moving average

Input Data Forecast ErrorAnalysis

Period Actual Value Forecast Err or

Absolute

error

Squared

error

Absolute

% errorMonth 1 10

Month 2 12

Month 3 16

Month 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

Actual Value - Forecast


13/36

Moving Averages GraphThree Period Moving Average

0

5

10

15

20

25

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

Time

Valu

e Actual Value

Forecast


14/36

Stability vs. Responsiveness

Should I use a 2-period moving average or a3-period moving average?

The larger the n the more stable the forecast.A 2-period model will be more responsive tochange.

We dont want to chase outliers.

But we dont want to take forever to correct fora real change.

We must balance stability with responsiveness.


15/36

The Weighted Moving Average MethodHistorical values of the time series are assigneddifferent weights when performing the forecast

Stationary Time Series Models:Weighted Moving Averages

1tF

= w1Yt + w2Yt-1 +w3Yt-2 + + wnYt-n+17wi = 1


16/36

Weighted Moving AverageWallace Garden SupplyForecasting

Period

Actual

Value Weights Three-Month Weighted Moving Averages

January 10 0.222

February 12 0.593March 16 0.185

April 13 2.2+ 7.1 + 3 / 1 = 12.298

May 17 2.7 + 9.5 + 2.4 / 1 = 14.556

June 19 3.5 + 7.7 + 3.2 / 1 = 14.407

July 15 2.9 + 10 + 3.5 / 1 = 16.484

August 20 3.8+ 11 + 2.8 / 1 = 17.814

September 22 4.2 + 8.9 + 3.7 / 1 = 16.815

October 19 3.3 + 12 + 4.1 / 1 = 19.262November 21 4.4 + 13 + 3.5 / 1 = 21.000

December 19 4.9 + 11 + 3.9 / 1 = 20.036

Next period 20.185

Sum of weights = 1.000

Storage Shed Sales


17/36

Weighted Moving AverageWallace Garden Supply Forecasting 3 period weighted moving averageInput Data Forecast ErrorAnalysis

Period Actual value Weights Forecast Err or

Absolute

error

Squared

error

Absolute

% error

Month 1 10 0.222Month 2 12 0.593

Month 3 16 0.185

Month 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


18/36


Exponential SmoothingExponential Smoothing

Moving average technique that requires a minimumamount of past data

Uses a smoothing constant with a value between 0 and 1(Usual range 0.1 to 0.3)

Forecast for period t = Forecast for period t-1 plus timesthe difference between the actual value and forecast in

period t-1: t = t-1 + (Yt-1 - t-1), or Can also be expressed as: t = (Yt-1) + (1- )(t-1) =

(Actual value in period t-1) + (1- )(Forecast in period t-1)


19/36

Exponential Smoothing Data

PeriodActual

Value(Yt) t-1 Yt-1 t-1 t

January 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

Class Exercise: What is the forecast for January of the following year?How about March? Find the Bias, Mad & MAPE. (Note: equals 0.1.)


20/36

Exponential Smoothing(Alpha = .419)

Wallace Garden SupplyForecasting Exponential smoothing


Period Actual value Forecast Err or

Absolute

error

Squared

error

Absolute

% error

Month 1 10 10.000Month 2 12 10.000 2.000 2.000 4.000 16.67%

Month 3 16 10.838 5.162 5.162 26.649 32.26%

Month 4 13 13.000 0.000 0.000 0.000 0.00%

Month 5 17 13.000 4.000 4.000 16.000 23.53%

Month 6 19 14.675 4.325 4.325 18.702 22.76%

Month 7 15 16.487 -1.487 1.487 2.211 9.91%

Month 8 20 15.864 4.136 4.136 17.106 20.68%

Month 9 22 17.596 4.404 4.404 19.391 20.02%

Month 10 19 19.441 -0.441 0.441 0.194 2.32%

Month 11 21 19.256 1.744 1.744 3.041 8.30%

Month 12 19 19.987 -0.987 0.987 0.973 5.19%

Average 2.608 9.842 14.70%

Alpha 0.419 MAD MSE MAPE

Next period 19.573


21/36

Exponential SmoothingExponential Smoothing

0

5

10

15

20

25

Janu

ary

February Ma

rch April May June JulyAu

gust

Sept

ember

October

Nove

mber

December

Sheds Actual value

Forecast


22/36

Evaluating the Performanceof Forecasting Techniques

Several forecasting methods have been

presented.

Which one of these forecasting methods

gives the best forecast?


23/36

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 23.43-Period Weighted MA(.5, .3, .2) 98 89 85.5

Error for the 3-Period WMA - 18 16 29.5

Performance Measures

Sample Example Find the forecasts and the errors for each forecasting

technique applied to the following stationary time series.


24/36

MAD forthe moving averagetechnique:

MAD forthe weighted moving averagetechnique:

= 21.17

= 18.35|-20| + |11.67| + |23.4|

3MAD = =

7 `(t|n

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

3MAD = =

7 `(t|n


MAD for the Sample Example


25/36

MAPE forthe moving averagetechnique:

MAPE forthe weighted moving averagetechnique:

= .211

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

3MAPE= =

7 `(t|n

|-18|/80 + |16|/105 + |29.5|/115

3MAPE= =

7 `(t|n


MAPE for the Sample Example


26/36

Use the performance measures to select a good setof values for each model parameter.

For the moving average: the number of periods (n).For the weighted moving average:

The number of periods (n), The weights (wi).

For the exponential smoothing: The exponential smoothing factor (E).

Excel Solver can be used to determine the valuesof the model parameters.


Selecting Model Parameters


27/36

Trend & Seasonality

Trend analysisTechnique that fits a trend equation (or curve) to aseries of historical data pointsProjects the equation into the future for medium andlong term forecasts. Typically do not want to forecast

into the future more than half the number of timeperiods used to generate the forecast

Seasonality analysisAdjustment to time series data due to variations at

certain periods.Adjust with seasonal index - ratio of average value ofthe item in a season to the overall annual average value.Examples: demand for coal in winter months; demandfor soft drinks in the summer and over major holidays


28/36

Linear Trend Analysis

Midwestern Manufacturing Sales

ScatterDiagram

Actual

value (or)Y

Period

number(or) X

74 1995

79 1996

80 1997

90 1998

105 1999

142 2000

122 2001

Sales(in units) vs. Time

0

20

40

60

80

100

120

140

160

1994 1995 1996 1997 1998 1999 2000 2001 2002


29/36

Least Squares forLinear Regression

Midwestern ManufacturingLeast Squares Method

Time

Valueso

fDependentVariab

les

Objective: Minimizethe squared deviations!


30/36

Least Squares Method

bXaY

^

!

Where

Y^

= predicted value of the dependent variable (demand)

a = Y-axis intercept = - b*

b = Slope of the regression line =]Xn-XY[__

Y

-

_22 Xn-X

X = value of the independent variable (time)

XY


31/36

Linear Trend Data & Error Analysis

Midwestern Manufacturing CompanyForecasting Linear trend analysis


Period

Actual value

(or) Y

Period number

(or) X Forecast Err or

Absolute

error

Squared

error

Absolute

% error

Year 1 74 1 67.250 6.750 6.750 45.563 9.12%

Year 2 79 2 77.786 1.214 1.214 1.474 1.54%

Year 3 80 3 88.321 -8.321 8.321 69.246 10.40%

Year 4 90 4 98.857 -8.857 8.857 78.449 9.84%

Year 5 105 5 109.393 -4.393 4.393 19.297 4.18%

Year 6 142 6 119.929 22.071 22.071 487.148 15.54%

Year 7 122 7 130.464 -8.464 8.464 71.644 6.94%Average 8.582 110.403 8.22%

Intercept 56.714 MAD MSE MAPE

Slope 10.536

Next period 141.000 8

Enter the actual values in cells shaded YELLOW. Enter new time period at the bottom toforecast


32/36

Least Squares GraphTrend Analysis

y = 10.536x + 56.714

0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7

Time

Value

Actual values Linear (Actual values)


33/36

Run linear regression to testF1 in the modelYt=F0+F1t+It

Excel results:Coeff. Stand. Err t-Stat P-value Lower 95 Upper 95Intercept 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.

Another way to Determine Trend:

Use the Excel Regression Function


34/36

Forecasting Seasonal Data: Quick Method

Eichler Supplies

Year Month Demand

Average

Demand Ratio

Seasonal

Index

1 January 80 94 0.851 0.957

February 75 94 0.798 0.851

March 80 94 0.851 0.904

April 90 94 0.957 1.064

May 115 94 1.223 1.309

June 110 94 1.170 1.223

July 100 94 1.064 1.117 August 90 94 0.957 1.064

September 85 94 0.904 0.957

October 75 94 0.798 0.851

November 75 94 0.798 0.851

December 80 94 0.851 0.851

2 January 100 94 1.064 0.957

February 85 94 0.904 0.851

March 90 94 0.957 0.904

April 110 94 1.170 1.064May 131 94 1.394 1.309

June 120 94 1.277 1.223

July 110 94 1.170 1.117

August 110 94 1.170 1.064

September 95 94 1.011 0.957

October 85 94 0.904 0.851

November 85 94 0.904 0.851

December 80 94 0.851 0.851

Seasonal Index ratioof theaverage value of the item in aseason to the overall averageannual value.

Example: average ofyear 1January ratio to year 2 Januaryratio.(0.851 + 1.064)/2 = 0.957

Ratio = Demand / Average Demand

IfYear 3 average monthly demand isexpected to be 100 units.Forecast demand Year 3 January:

100 X 0.957 = 96 unitsForecast demand Year 3 May:

100 X 1.309 = 131 units


35/36

Forecasting Seasonal Data With Trend

1. Calculate the seasonal indices (as shown on theprevious slide)

2. Calculate deseasonalized treand by dividingthe actual value (Y) by the seasonal index forthat period:Deseasonalized Trend = Y / Seasonal index

(e.g., 80 units/ 0.957 = 83.595)

3. Find the trend line, and extend the trend line intothe desired forecast period.


36/36

Forecasting Seasonal Data With Trend:

Calculating the Seasonal Forecast4. Now that we have the Seasonal Indices and Trend

line, we can reseasonalize the data and generatethe seasonalized forecast by multiplying thetrend line values in the forecast period by theappropriate seasonal indices for each time period

as follows: = Trend x Seasonal Index

Business Analysis- Time-Series Models -Measuring Forecast Error

Documents

Transcript of Business Analysis- Time-Series Models -Measuring Forecast Error