time-seres analysis 2

8/10/2019 Time-seres Analysis 2

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Slides 13b:Time-Series Models;

Measuring Forecast Error

MGS3100 Chapter 13Forecasting


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Forecasting ModelsForecastingTechniques

QualitativeModels

Time SeriesMethods

CausalMethods

DelphiMethod

Jury of ExecutiveOpinion

Sales ForceComposite

Consumer MarketSurvey

Naive

Moving Average

WeightedMoving Average

ExponentialSmoothing

Trend Analysis

Seasonality AnalysisSimple

Regression Analysis

MultipleRegression

Analysis

MultiplicativeDecomposition


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A stationary time series

Linear trend time series

Linear trend and seasonality time series

Time

Timeseriesvalue

Future

Components of a Time Series


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Time Series: Stationary ModelsStationary 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


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Stationary Time Series Models:The Nave Model

Whatever happenedlast period willhappen again thistimeThe model is simpleand flexibleProvides a baselineto measure othermodelsAttempts to captureseasonal factors atthe expense of

ignoring trend

dataMonthly:

dataQuarterly:

12

4

t t

t t

Y F

Y F

1t t Y F

or


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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 of

the sampling distribution (the squareroot of the MSE)

Bias, MAD, and MAPE - typicallyused for time series

)( t t F Y Error Forecast

T F Y MADt t

T

t

/||/T|error forecast|1

T

1t

T Y F Y MAPE t t t

T

t

/]/|[|1001

T F Y

Bias

t t

T

t

/)(

/Terror)(forecast

1

T

1t

T F Y

MSE

t t

T

t

/)(

/T|error forecast|

2

1

T

1t

2


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Nave Forecast

Wallace Garden SupplyForecasting

Period Actual

ValueNave

Forecast Error Absolute

Error Percent

Error Squared

Error January 10 N/A

February 12 10 2 2 16.67% 4.0March 16 12 4 4 25.00% 16.0 April 13 16 -3 3 23.08% 9.0May 17 13 4 4 23.53% 16.0June 19 17 2 2 10.53% 4.0July 15 19 -4 4 26.67% 16.0

August 20 15 5 5 25.00% 25.0September 22 20 2 2 9.09% 4.0

October 19 22 -3 3 15.79% 9.0November 21 19 2 2 9.52% 4.0December 19 21 -2 2 10.53% 4.0

0.818 3 17.76% 10.091BIAS MAD MAPE MSE

Standard Error (Square Root of MSE) = 3.176619

Storage Shed Sales


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Nave Forecast GraphWallace 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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The Moving Average Method

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

n

Y Y Y F nt t t t 111

...

Stationary Time Series Models:

Moving Averages


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Moving AveragesWallace 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 = 15.33

July 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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Moving Averages ForecastWallace 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

Actual Value - Forecast


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Moving Averages GraphThree Pe riod 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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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 to

change. We dont want to chase outliers. But we dont want to take forever to correct for

a real change. We must balance stability with responsiveness.


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The Weighted Moving Average Method Historical values of the time series are assigned

different weights when performing the forecast

Stationary Time Series Models:Weighted Moving Averages

1tF

= w1Yt + w2Yt-1 +w3Yt-2 + + w nYt-n+1Swi = 1


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Weighted Moving AverageWallace Garden SupplyForecasting

Period Actual

Value Weights Three-Month Weighted Moving AveragesJanuary 10 0.222

February 12 0.593March 16 0.185 April 13 2.2 + 7.1 + 3 / 1 = 12.298May 17 2.7 + 9.5 + 2.4 / 1 = 14.556June 19 3.5 + 7.7 + 3.2 / 1 = 14.407July 15 2.9 + 10 + 3.5 / 1 = 16.484

August 20 3.8 + 11 + 2.8 / 1 = 17.814September 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 = 21.000December 19 4.9 + 11 + 3.9 / 1 = 20.036

Next period 20.185

Sum of weights = 1.000

Storage Shed Sales


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Weighted Moving AverageWallace Garden Supply Forecasting 3 period weighted moving average


Period Actual value 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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Stationary Time Series Models:Exponential Smoothing

Exponential Smoothing Moving average technique that requires a minimum

amount 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 + (Y t-1 - t-1), or Can also be expressed as: t = (Y t-1) + (1- )(t-1) =(Actual value in period t-1) + (1- )(Forecast in period t-1)


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Exponential Smoothing Data

Period Actual

Value(Y t) t-1 Y t-1 t-1 tJanuary 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.)


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Exponential Smoothing(Alpha = .419)

Wallace Garden SupplyForecasting Exponential smoothing


Period Actual value Forecast Error 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


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Exponential SmoothingExponential Smoothing

0

5

10

15

20

25

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 p t e

m b e r

O c t o b e

r

N o v e

m b e r

D e c e

m b e r

S h e

d s Actual value

Forecast


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Evaluating the Performanceof Forecasting Techniques

Several forecasting methods have been presented.

Which one of these forecasting methodsgives the best forecast?


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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|/1153MAPE= =t|

n

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

t| n

Performance Measures

MAPE for the Sample Example


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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 (w i).

For the exponential smoothing: The exponential smoothing factor ( a ). Excel Solver can be used to determine the values

of the model parameters.

Performance Measures

Selecting Model Parameters


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Trend & Seasonality

Trend analysis Technique that fits a trend equation (or curve) to a

series of historical data points Projects the equation into the future for medium and

long term forecasts. Typically do not want to forecast

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

Seasonality analysis Adjustment 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


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Linear Trend AnalysisMidwestern Manufacturing Sales

Scatter Diagram

Actualvalue (or)

Y

Periodnumber

(or) X74 199579 199680 199790 1998

105 1999142 2000122 2001

Sales(in units) vs. Time

0

20

40

60

80

100120

140

160

1994 1995 1996 1997 1998 1999 2000 2001 2002


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Least Squares for Linear Regression

Midwestern ManufacturingLeast Squares Method

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 s

Objective: Minimizethe squared deviations!


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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)

X Y


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Linear Trend Data & Error Analysis

Midwestern Manufacturing CompanyForecasting Linear trend analysis


Period

Actual value

(or) Y

Period number

(or) X Forecast Error

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 MAPESlope 10.536

Next period 141.000 8

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


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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

V a

l u e

Actual values Linear (Actual values )


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Forecasting Seasonal Data: Quick MethodEichler Supplies

Year Month Demand AverageDemand Ratio

SeasonalIndex

1 January 80 94 0.851 0.957February 75 94 0.798 0.851

March 80 94 0.851 0.904 April 90 94 0.957 1.064May 115 94 1.223 1.309June 110 94 1.170 1.223July 100 94 1.064 1.117

August 90 94 0.957 1.064September 85 94 0.904 0.957

October 75 94 0.798 0.851November 75 94 0.798 0.851December 80 94 0.851 0.851

2 January 100 94 1.064 0.957February 85 94 0.904 0.851

March 90 94 0.957 0.904 April 110 94 1.170 1.064May 131 94 1.394 1.309June 120 94 1.277 1.223July 110 94 1.170 1.117

August 110 94 1.170 1.064September 95 94 1.011 0.957

October 85 94 0.904 0.851November 85 94 0.904 0.851December 80 94 0.851 0.851

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

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

Ratio = Demand / Average Demand

If Year 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


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Forecasting Seasonal Data With Trend

1. Calculate the seasonal indices (as shown on the previous 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 into

the desired forecast period.


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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 generate

the 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

time-seres analysis 2

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