4/16/2015 1 Ardavan Asef-Vaziri Variable of interest Time Series Analysis.

04/18/23 1Ardavan Asef-Vaziri

Variable of interest

Time Series Analysis

Components of an Observation

04/18/23 Ardavan Asef-Vaziri 2

Observed variable (O) =Systematic component (S) + Random component (R)

Level (current deseasonalized )

Trend (growth or decline)

Seasonality (predictable seasonal fluctuation)

• Systematic component: Expected value of the variable• Random component: The part of the forecast that deviates from the systematic component• Forecast error: difference between forecast and actual demand

At : Actual valued in period t

F(t+1) : Forecast for period t+1

F(t+1) = At

Naive Forecast

The naive forecast can also serve as an accuracy standard for other techniques.


Moving Average

Three period moving average in period 7 is the average of:

MA73 = (A7+ A6+ A5 )/3


MAt10 = (At+ At-1+ At-2 +At-3+ ….+ At-9 )/10

Ten period moving average in period t is the average of:

n period moving average in period t is the average of:

MAtn = (At+ At-1+ At-2 +At-3+ ….+ At-n+1 )/n

Forecast for period t+1 is equal to moving average for period t

Ft+1 =MAtn

MA421 = (A21+A20+A19+A18)/4

MA421 = (800+720+680+700)/4=725

4-Period Moving Average at period 20, and 21

It was used as forecast for period 21. The actual values in period 21 is 800

The Actual cost of a specific task type for periods 17-20 was 600, 700, 680, 720, respectively


MA420 = (A20+A19+A18+A17)/4

MA420 = (720+680+700+600)/4 = 675

MA421 = MA4

20 +(A21- A17)/4

MA421 = 675 +(800- 600) /4=725

Micro $oft Stock


AS n increases, we obtain a smoother curve

Exponential Smoothing

)(α1 tttt FAFF

ttt AFF α)α1(1

tttt FAFF αα1



α=.2

tAtFt

1100100

A1 F2

2

100

Since I have no information for F2, I just enter A1 which is 100

150

F3 =(1-α)F2 + α A2

F3 =.8(100) + .2(150)

F3 =80 + 30 = 110

3

110

F2 & A2 F3

A1 F2 A1 & A2 F3

F3 =(1-α)F2 + α A2



α=.2

tAtFt

1100100

F4 =(1-α)F3 + α A3

F4 =.8(110) + .2(120)

F4 =88 + 24 = 112

A3 & F3 F4

A1 & A2 F3 A1& A2 & A3 F4

2150100

3

110

4

112120

F4 =(1-α)F3 + α A3

Exponential SmoothingTakes into account

All pieces of actual data


.2

.05

Smoothing constant

The smaller the value of α, the smoother the curve.


t At Ft |At-Ft| MAD At-Ft SUM(At-Ft) TS1 7002 724 7003 710 712.0 2.0 2.0 -2.0 -2.0 -1.04 715 711.0 4.0 3.0 4.0 2.0 0.75 710 713.0 3.0 3.0 -3.0 -1.0 -0.36 710 711.5 1.5 2.6 -1.5 -2.5 -1.07 715 710.8 4.3 3.0 4.3 1.8 0.68 710 712.9 2.9 2.9 -2.9 -1.1 -0.49 720 711.4 8.6 3.7 8.6 7.4 2.010 730 715.7 14.3 5.1 14.3 21.7 4.3

α=0.5 25.3

Mean Absolute Deviation (MAD)

nsObservatioofNumber

ForecastActualMAD

||


The lower the MAD, The better the forecast

MAD is also an estimates of the Standard Deviation of forecast

1.25MAD

t At Ft |At-Ft| MAD At-Ft SUM(At-Ft) TS1 7002 724 7003 710 712.0 2.0 2.0 -2.0 -2.0 -1.04 715 711.0 4.0 3.0 4.0 2.0 0.75 710 713.0 3.0 3.0 -3.0 -1.0 -0.36 710 711.5 1.5 2.6 -1.5 -2.5 -1.07 715 710.8 4.3 3.0 4.3 1.8 0.68 710 712.9 2.9 2.9 -2.9 -1.1 -0.49 720 711.4 8.6 3.7 8.6 7.4 2.010 730 715.7 14.3 5.1 14.3 21.7 4.3

α=0.5 25.3

Mean Absolute Deviation (MAD)

MAD

ForecastActualTS

)(


Tracking Signal

Tracking Signal

UCL

LCL

Time

Detecting non-randomness in errors can be done using Control Charts (UCL and LCL)


Tracking Signal

UCL

LCL

Time

Tracking Signal



Other Measures of Forecast Error


ForecastActualMSE

2)(


ActualForecastMAPE

)/1(100

Mean Square Error (MSE)An estimate of the variance of the forecast error

Mean absolute percentage error (MAPE)


Measures of Forecast Error

Exponential SmoothingAlpha= 0.1 MAD=

Demand Forecast Error AbsErr SqAbsErr 10208t At Ft Et |Et| |Et|̂ 2 SumEt Sum|Et| Sum(|Et|̂ 2) MAD MSE %Error Sum%Err MAPE TS

Q12 1 8000 22083 -14083 14083 198340278 -14083 14083 198340278 14083 198340278 176 176 176 -1.00Q13 2 13000 20675 -7675 7675 58905625 -21758 21758 257245903 10879 128622951 59 235 118 -2.00Q14 3 23000 19908 3093 3093 9563556 -18666 24851 266809459 8284 88936486 13 249 83 -2.25Q21 4 34000 20217 13783 13783 189977981 -4883 38634 456787440 9659 114196860 41 289 72 -0.51Q22 5 10000 21595 -11595 11595 134445764 -16478 50229 591233204 10046 118246641 116 405 81 -1.64Q23 6 18000 20436 -2436 2436 5931989 -18913 52665 597165193 8777 99527532 14 419 70 -2.15Q24 7 23000 20192 2808 2808 7884804 -16105 55473 605049997 7925 86435714 12 431 62 -2.03Q31 8 38000 20473 17527 17527 307202401 1422 73000 912252397 9125 114031550 46 477 60 0.16Q32 9 12000 22226 -10226 10226 104561437 -8804 83225 1016813835 9247 112979315 85 562 62 -0.95Q33 10 13000 21203 -8203 8203 67288813 -17007 91428 1084102647 9143 108410265 63 625 63 -1.86Q34 11 32000 20383 11617 11617 134962165 -5389 103046 1219064812 9368 110824074 36 661 60 -0.58Q41 12 41000 21544 19456 19456 378519966 14066 122501 1597584777 10208 133132065 47 709 59 1.38

23490

23490

23490

23490

16

15

14

13

F

F

F

F

• Forecasts are rarely perfect because of randomness.

• Beside the average, we also need a measure of variation, which is called standard deviation

• Forecasts are more accurate for groups of items than for individuals.

• Forecast accuracy decreases as the time horizon increases.

I see that you willget an A this semester.

Four Basic Characteristics of Forecasts


4/16/2015 1 Ardavan Asef-Vaziri Variable of interest Time Series Analysis.

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