topic 5: forecasting

Topic 5: FORECASTING

I. Introduction

-- What is forecasting?• Scientific (educated) guess• Based on past data or experience• Rarely perfect• Group forecast more accurate• Shorter time horizon, more accurate

-- Why forecasting?

-- Forecasting time horizon• Short range – usually less than 3 months

• Medium range – 3 months to 3 years

• Long range – over 3 years– * Related to the product life cycle

-- Which forecasting method to use?

• Time horizon

• Costs vs. accuracy

• Easy to understand?

-- Pattern of Data• Trend – gradual upward or downward

movement

• Cycle – background pattern over long period of time

Trend

Irregularvariation

Cycles

Several years

-- Pattern of Data (continued)

• Seasonality – repeated pattern

• Randomness (irregular variation) – unexplainable, unpredictable, unknown fluctuations

Seasonal variations

Year 2

Year 1

Year 3

II. Qualitative Forecasting Methods

Qualitative Forecasting Methods- no past data is available, for managerial decision, long term forecast

• Jury of executive opinion– Method of combining and averaging views of

several executives regarding a specific decision or forecast.

– Drawback: one person’s view may prevail

Qualitative Forecasting Methods

• The Delphi technique – systematic survey of experts – Collaborative estimating or forecasting

technique that combines independent analysis with maximum use of feedback, for building consensus among experts who interact anonymously.

– Drawback: expensive and time consuming


• Sales force composite

– A method of developing a sales forecast that uses the opinions of each member of the field sales staff regarding how much the individual expects to sell in the period as input.

– Drawback: Overly influenced by recent experiences


• Consumer market survey

– The gathering and evaluation regarding consumers' preferences for products and services

– Drawback: like to do will do

III. Quantitative Forecasting Methods

• Time series data: a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals

tiYi ,,2,1 ,

period; timerepresent ,i

)1(,,2,1 , tiFi

(actual)nobservatio real theis Yalueforecast v is F

Quantitative Forecasting Methods

1. Smoothing Techniques: for time series data with no clear pattern, short term forecast

• Naïve: estimating technique in which the last period's actual is used as this period's forecast, without adjusting them or attempting to establish causal factors.

– It is used only for comparison with the forecasts generated by the better (sophisticated) techniques.

tt YF 1

Quantitative Forecasting Methods (Smoothing Techniques)

• n Period Moving Average (MA-n): arithmetical average n most recent period real observations as the forecast for next time period.

n

YYYF ntttt

111


• n Period Weighted Moving Average (WMA-n): flexible weight assignment

• For n most recent observations , assign flexible weights of , respectively

Forecast for next time period t +1 is

11 ,,, nttt YYY 11 ,,, nttt www

11

11111

nttt

ntntttttt www

YwYwYwF


• Exponential Smoothing with smoothing constant : very little data record required, more weight on recent data, weight decreases exponentially

10

tttttt FYFYFF 11


• Example: Forecast the score for test 5 by the above smoothing techniques

For exponential smoothing, use a = 0.1 and F1 = 77.

Test 1 2 3 4 5

Avg.Score

72 82 85 90 ?


• Relationship between smoothing techniques– average of past data, but differ in weight

distribution– all methods lag behind actual data change– more weight on most recent data, more

responsive, less smooth


• Thinking challenge:– Which method requires the least amount of

data?– What happens if the a in the exponential

smoothing method increases?

Quantitative Forecasting Methods (Trend Projection by Linear Regression)

2. Trend Projection by Linear Regression – for data with linear trend pattern, medium

term

• Step 1. Find a trend line

to fit the past data the best (minimizing mean squared errors)

• best slope:

n

ii

n

iii

xnx

yxnyxb

1

22

1

)(

)(

xbay


• best intercept:

and are the average of x’s and y’s.

• * Excel commands:– calculating b:“=slope(range of y’s, range of

x’s)”– calculating a:“=intercept(range of y’s, range of

x’s)”

xbya

x y


• Example continued:

x y xy x2 y2

1 72 72 1 5184 a=

2 82 164 4 6724 68

3 85 255 9 7225 b=

4 90 360 16 8100 5.7

sum 10 329 851 30 27233

avg. 2.5 82.25


• Step 2. Trend projection• Trend projection forecast:

• -- Example continues:

nextnext xbaF


• Step 3. Find forecast interval, if necessary• Forecast interval = • standard error

– * Excel commands:

calculating :”=steyx(range of y’s, range of x’s)”

• Example continues:

2

)()(111

2

n

yxbyayS

n

iii

n

ii

n

ii

xy

xynext SZF

xyS

Quantitative Forecasting Methods (Causal Model by Linear Regression)

• Causal Model by Linear Regression: same set up as trend projection, except that x is an independent variable (other than time), y is a dependent variable, medium term


• Example: The sales manager of a large apartment rental complex feels the demand for apartments may be related to the number of newspaper ads placed during the previous month. She has collected the data shown below. If the number of ads placed in this month is 30, what would be her estimate of rentals in the coming month?

Ads 15 9 40 20 25 25 15 35

Rental 6 4 16 6 13 9 10 16


• Correlation coefficient r: measures the strength of linear relationship between x and y

– * Excel command: calculating r:”=correl(range of y’s, range of x’s)”

2

11

22

11

2

111

)()(

)(

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

iii

yynxxn

yxyxnr

10 r


• -- Interpretation:• r > 0:• r = 0:• r < 0:• = coefficient of determination: % of variation in

the dependent variable (y) is explained by regression equation (linear relationship).– Example continued: How strong is the relationship

between the ads placed and the rentals?

2r

Quantitative Forecasting Methods(Decomposition by Time Series)

• Decomposition of Time Series -- for data with both trend and seasonality patterns, short to medium term

• Idea: – Decompose the past data (filter out the seasonal

influence from original data)– Forecast trend pattern and seasonality pattern

separately– Combine the forecasts using the multiplicative model:

ttt STF


• -- Example: Data of a popular brand of sweater sale (by quarters) over the past three years. Forecast the sale of each season in next year (Year 4).Year 1 Sale Year 2 Sale Year 3 Sale

(t) (Yt) (t) (Yt) (t) (Yt)

1 14 5 16 9 18

2 25 6 32 10 33

3 76 7 80 11 84

4 52 8 58 12 64


• Step 1. Draw the historical data diagram to check if there is an obvious seasonal pattern

(Yt)

0102030405060708090

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

(Yt)


• Step 2. Calculate the "seasonal indexes" (S) (quarter relatives) for each season.– Seasonal Index = Avg. Seasonal Sale / Total

Avg Sale

springY

summerY

fallY

erwY int

Y

springS

summerS

fallS

erwS int


• Step 3. Deseasonalize the original data:

ttt SYT /


(t) (Yt) (Tt)=(Yt/St)

1 14 40.2500 Y-bar_sp 16

2 25 38.3333 Y-bar_su 30

3 76 43.7000 Y-bar_f 80

4 52 41.2414 Y-bar_w 58

5 16 46.0000

6 32 49.0667 Y-double-bar 46

7 80 46.0000

8 58 46.0000

9 18 51.7500 S_sp 0.3478

10 33 50.6000 S_su 0.6522

11 84 48.3000 S_f 1.7391

12 64 50.7586 S_w 1.2609


0

10

20

30

40

50

60

70

80

90

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

(Yt)

(Tt)=(Yt/St)


• Step 4. Calculate the trend based on the deseasonalized demand by linear regression

a b

tbaTt


• Step 5. Use trend projection to forecast the demand with trend only.

T13 = T14 =

T15 = T16 =


• Step 6. Use seasonal indexes to modify the forecasts to reflect the seasonality patterns:

F13 = F14 =

F15 = F16 =

ttt STF


(t) (Yt) (Tt)=(Yt/St)

1 14 40.2500 Y-bar_sp 16

2 25 38.3333 Y-bar_su 30

3 76 43.7000 Y-bar_f 80

4 52 41.2414 Y-bar_w 58

5 16 46.0000

6 32 49.0667 Y-double-bar 46

7 80 46.0000

8 58 46.0000

9 18 51.7500 S_sp 0.3478

10 33 50.6000 S_su 0.6522

11 84 48.3000 S_f 1.7391

12 64 50.7586 S_w 1.2609

13 18.3958 52.8880

14 35.1833 53.9477

15 95.6650 55.0074 a= 39.1120

16 70.6932 56.0671 b= 1.0597


0

20

40

60

80

100

120

1 3 5 7 9 11 13 15

(Yt)

(Tt)=(Yt/St)

IV. Choosing a Forecasting Method

• Choosing a Forecasting Method - all criteria are a function of the forecast error

– forecast error (E) = actual realization (Y)– forecast (F)

Choosing a Forecasting Method

• Bias – measures the direction of forecast– Bias = ( forecast error) / n


• Mean Absolute Deviation (MAD)

- same penalty for small and large errors

MAD = ( |forecast error|) / n


• Mean Squared Error (MSE)

• – more penalty on large errors

• MSE = ( (forecast error)2 )/n


• Mean Absolute Percent Error (MAPE)– relative error, scale independent– MAPE = ( |forecast error|/Actual Data) / n


• Comparing different forecasting methods– Based on past performance– Measure dependent– Choose (for exp. Smoothing) and n (for MA)

to minimize MAD, MSE, or MAPE


• Example: Same data as earlier

• Bias• MAD• MSE• MAPE

Test Ave.Score

Naive 2yr. MA Exp. Smth. ( = .1)

Exp. Smth. ( = .8)

Trend

Project

1 72 77 77 73.7

2 82 72 76.5 73 79.4

3 85 82 77 77.05 80.2 85.1

4 90 85 83.5 77.845 84.04 90.8


Test Ave. Score

Naive 2yr. MA Exp. Smooth. ( = .1)

Exp. Smooth. ( = .8)

Trend

Project

1 72 -5 -5 -1.7

2 82 10 5.5 9 2.6

3 85 3 8 7.95 4.8 -0.1

4 90 5 6.5 12.155 5.96 -0.8

Bias= 6 7.25 5.15125 3.69 0

MAD= 6 7.25 7.65125 6.19 1.3

MSE= 44.66667 53.125 66.5491 41.1404 2.575

MAPE 0.070934 0.08317 0.09128 0.07547 0.01635

V. Control of the Forecasting Process

• Control of the Forecasting Process - to check if the performance of a forecasting method changes

Control of the Forecasting Process

• Tracking Signal control chart:– plotting statistic TS(t) = RSFE(t) / MAD(t)– RSFE(t) = running sum of the forecast error

through time t– MAD(t) = MAD through time t– 3 sigma control chart limits: UCL = 3,

LCL = -3, CL = 0


• -- Interpretation?


• Example: A forecasting method provides the following forecasts in the past 6 periods. Does its performance change over time?

T Y F Error RFSE(t) MAD(t) TS(t)

1 8 6

2 7 9

3 10 6

4 2 6

5 12 8

6 11 7

7 10 7

Homework for Forecasting

Additional Problems: • Problem 1. Demand for heart transplant surgery

at Washington General Hospital has increased steadily in the past few years, as seen in the following table:

Year 1 2 3 4 5 6

Heart Transplant Surgeries Performed

46 51 53 56 58 ?

Homework for Forecasting Problem 1 continued

• a. The director of medical services predicted 6 years ago that demand in year 1 would be 42 surgeries.

• b. Use naïve method to forecast demand for year 2 through year 6.• Use a 3-year moving average to forecast demand for years 4, 5, and

6.• c. Use a 3-year weighted moving average to forecast demand for year

6, assuming the weights chosen for year 3, year 4, and year 5 are 3, 4, and 5, respectively.

• d. Use exponential smoothing with a smoothing constant of 0.3 to develop forecasts for year 2 through year 6. (Notice that the forecast for year 1 is 42.)

• e. Use the trend projection method to forecast demand for year 1 through year 6. (Hint: find a and b by applying the formula in my lecture notes and then obtain forecasts of year 1 through year 6 by substituting x = 1, …,6 in y = a + bx.)

• f. As a follow-up of question (e), calculate 90% confident forecast interval for year 6.


• Problem 2. The annual sales tax collected by Washington State is closely related to the new car registration. The following data are obtained for the past 7 years:

New Car Registration (in Millions)

10 13 15 16 14 18 21

Annual Sales TaxCollected (in Billions)

1 1.5 2 2.1 1.9 2.2 2.4

Homework for ForecastingProblem 2 continued

• a. Find the estimated sales tax collections if new car registrations is projected at 25 millions this year.(Hint: use causal method, x is # of new car registrations.)

• b. Calculate the correlation between the sales tax and the new car registration as well as the coefficient of determination. What conclusion can you draw based on your calculation?


• Problem 3. Attendance at Orlando’s newest Disney-like attraction, Vocation World, has been as follows:

Forecast the # of guest in each quarter of next year (year 4) using the decomposition of time series method you learn in class. You should follow the lecture example.

Year 1 Year 2 Year 3

Winter 77 85 101

Spring 94 116 158

Summer 136 180 217

Fall 64 86 110


• Problem 4. You have the set of data given below. Make forecasts for all years possible up to year 6 using exponential smoothing with = .4 and F1 = 800Calculate Bias, MSE, and MAPE for the forecast. Calculate the tracking signal (TS) for each forecast. Plot the TS’s in the 3-sigma control chart and draw a conclusion on the forecast method. You may put your calculation results in the following table.

Homework for Forecasting Problem 4 continued

Year Actual Demand

ES(=.4) forecast

Forecast Error

| Fore. Err.|

RSFE(t) MAD(t) TS(t)

1 805 800

2 810

3 815

4 820

5 825

topic 5: forecasting

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