forecasting introduction quantitative models time series

FORECASTING

Introduction Quantitative ModelsTime Series

Forecasting Models

Forecasting

Quantitative Qualitative

Causal Model

Time series

Expert Judgment

Trend

Stationary

Trend

Trend + Seasonality

Delphi Method

Grassroots

Market Research

Jury Exec. Opinion

FORECASTING

Introduction toQuantitative

FORECASTING

Quantitative Forecasts

Quantitative forecasting models possess two important and attractive features:They are expressed in mathematical

notation. Thus, they establish an unambiguous record of how the forecast is made.

With the use of spreadsheets and computers, quantitative models can be based on an amazing quantity of data.


Two types of quantitative forecasting models :Causal modelsTime-Series models


- Forecasting based on data and models

Causal Models:

Time Series Models:

PricePopulationAdvertising

……

CausalModel

Year 2000 Sales

Sales1999

Sales1998

Sales1997

…………………

Time Series Model

Year 2000 Sales

FORECASTING

Time Series Forecasting

FORECASTING

Forecasting Models

Forecasting

Quantitative Qualitative

Causal Model

Time series

Expert Judgment

Trend

Stationary

Trend

Trend + Seasonality

Delphi Method

Grassroots

Market Research

Jury Exec. Opinion

Time Series Forecasting Models

Time-series forecasting models produce forecasts by extrapolating the historical behavior of the values of a particular single variable of interest.

Time-series data are historical data in chronological order, with only one value per time period.

Components of a Time Series

11

Time Series Components

Time

Trend

Randommovement

Time

Cycle

Time

Seasonalpattern

Dem

and

Time

Trend with seasonal pattern

12

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 Plot Techniques to try Ways to evaluate

Data is reasonably stationary (no trend or seasonality)

Heuristics - Averaging methods Naive Moving Averages Simple Exponential Smoothing

MAD MAPE Standard Error BIAS

Data shows a consistent trend

Regression Linear Non-linear Regressions (not covered in this course)

MAD MAPE Standard Error BIAS R-Squared

Data shows both a trend and a seasonal pattern

Classical decomposition Find Seasonal Index Use regression analyses to find the trend component

MAD MAPE Standard Error BIAS R-Squared

13

BIAS The arithmetic mean of the errors

n is the number of forecast errors

Mean Absolute Deviation - MAD Average of the absolute errors

Evaluation of Forecasting Model

nError

nForecast) - (Actual

BIAS

n|Error|

nForecast - Actual|

MAD |

14

Mean Square Error - MSE

Standard error Square Root of MSE

Mean Absolute Percentage Error - MAPE

Calculate the % error using the absolute error, then average the results

n(Error)

nForecast) - (Actual

MSE22

nActual

|Forecast - Actual|

MAPE

%100*

Evaluation of Forecasting Model

Time Series: Stationary Models Stationary Model Assumptions

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

Techniques will smooth out short-term irregularitiesThe forecast is revised only when new data becomes

available.

Stationary Model TypesNaïve ForecastMoving Average/Weighted Moving AverageExponential Smoothing

16

Stationary data forecasting

Naïve 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 (yesterday’s forecast) by adding adjusted error (α * error). This will compensate over (under) forecast of yesterday.

17

Naïve Model

The simplest time series forecasting model Idea: “what happened last time period (last

year, last month, yesterday) will happen again this time”

Naïve Model: Algebraic: Ft = Yt-1

Yt-1 : actual value in period t-1 Ft : forecast for period t

Spreadsheet: B3: = A2; Copy down

Naïve Forecast Wallace Garden SupplyForecasting

PeriodActual Value

Naïve Forecast Error

Absolute Error

Percent Error

Squared Error

January 10 N/AFebruary 12 10 2 2 16.67% 4.0March 16 12 4 4 25.00% 16.0April 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% 16.0August 20 15 5 5 25.00% 25.0September 22 20 2 2 9.09% 4.0October 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

Wallace Garden - Naive Forecast

0

5

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25

February March April May June July August September October November December

Period

Shed

s Actual Value

Naïve Forecast

Naïve Forecast

20

Moving Average Model

Simple n-Period Moving Average

Issues of MA Model Naïve model is a special case of MA with n = 1 Idea is to reduce random variation or smooth data All previous n observations are treated equally (equal

weights) Suitable for relatively stable time series with no trend or

seasonal pattern

nY

Y

nT

Tit

t

1

nntY2tY1tY

ˆ=tF

n periodsn previousin valuesactual of Sumˆ

tF

Moving Averages

Wallace Garden SupplyForecasting

PeriodActual Value Three-Month Moving Averages

January 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

Moving Averages Forecast

Wallace Garden SupplyForecasting 3 period moving average

Input Data Forecast Error Analysis

Period Actual Value Forecast ErrorAbsolute

errorSquared

errorAbsolute % 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

Moving Average Forecast

Three Period Moving Average

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5

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25

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

Time

Valu

e Actual Value

Forecast

Stability vs. Responsiveness

Should I use a 3-period moving average or a 5-period moving average? The larger the “n” the more stable the forecast. A 3-period model will be more responsive to

change. We don’t want to chase outliers. But we don’t want to take forever to correct for a

real change. We must balance stability with responsiveness.

25

Smoothing Effect of MA Model

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

Weighted Moving Average Model

Assumes data from some periods are more important than data from other periods (e.g. earlier periods).

Uses weights to place more emphasis on some periods and less on others

Historical values of the time series are assigned different weights when performing the forecast

27

Weighted Moving Average Model

Weighted n-Period Moving Average

Typically weights are decreasing: w1>w2>…>wn

Sum of the weights = wi = 1Flexible weights reflect relative importance

of each previous observation in forecasting

1ntYnw2tY2w1tY1 w=tF

iw

28

Weighted MA: An Illustration

Month Weight DataAugust 17% 130September 33% 110October 50% 90

November forecast:

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

= 103.4

Weighted Moving AverageWallace Garden SupplyForecasting

PeriodActual Value Weights Three-Month Weighted Moving Averages

January 10 0.222February 12 0.593March 16 0.185April 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 = 16.484August 20 3.8 + 11 + 2.8 / 1 = 17.814September 22 4.2 + 8.9 + 3.7 / 1 = 16.815October 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

Weighted Moving AverageWallace Garden Supply Forecasting 3 period weighted moving average


Period Actual value Weights Forecast ErrorAbsolute

errorSquared


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

Operational Problems With Moving Averages

The operational shortcoming of simple moving average models is that if n observations are to be included in the moving average, then (n-1) pieces of past data must be brought forward to be combined with the current (the nth) observation

All this data must be stored in some way, in order to calculate the forecast.

This may become a problem when a company needs to forecast the demand for thousands of individual products on an item-by-item basis.

The next weighting scheme addresses this problem.

Exponential Smoothing

• Moving average technique that requires little record keeping of past data.• Uses a smoothing constant α with a value between 0 and 1. (Usual range 0.1

to 0.3)• Applies alpha to most recent period, and applies one minus alpha distributed

to previous values• α = The weight assigned to the latest period (smoothing constant)• Forecast = α(Actual value in period t-1) + (1- α)(Forecast in period t-1)

• Can also be forecast for period t-1 plus α times the difference between the actual value and forecast in period t-1:

)Y)( -(1 )(YY=tF 1-T1-TT

)Y-(Y YY=tF 1-T1-T1-TT

valueactual periodLast :Yforecast periodLast :ˆF

T periodfor Forecast :ˆF

Constant Smoothing :

1-T

11-T

T

T

T

Y

Y

Exponential Smoothing Data

PeriodActual

Value(Yt) Ŷt-1 α Yt-1 Ŷt-1 Ŷt

January 10 = 10 0.1February 12 10 + 0.1 *( 10 - 10 ) = 10.000March 16 10 + 0.1 *( 12 - 10 ) = 10.200April 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 ) = 12.342August 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: α equals 0.1.)

Exponential Smoothing (Alpha = .419)

Wallace Garden SupplyForecasting Exponential smoothing


Period Actual value Forecast ErrorAbsolute

errorSquared


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



0

5

10

15

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25

Janu

ary

Februa

ryMar

chApr

ilMay

June Ju

ly

Augus

t

Septem

ber

Octobe

r

Novem

ber

Decem

ber

Shed

s Actual value

Forecast

36

Simple Exponential Smoothing

Properties of Simple Exponential Smoothing Widely used and successful model Requires very little data Larger , more responsive forecast Smaller , smoother forecast Suitable for relatively stable time series

Evaluating the Performance of Forecasting Techniques

Several forecasting methods have been presented.

Which one of these forecasting methods gives the “best” forecast?

Performance Measures – Sample Example• Find the forecasts and the errors for each forecasting

technique applied to the following stationary time series.

3-Period Moving Average 3-Period Weighted Moving Avg. (.5,.3,.2)Period Sales Forecast Error A. Error P. Error Forecast Error A. Error P. Error

1 1002 1103 904 80 100.00 -20.00 20.00 25.00 98.00 -18.00 18.00 22.505 105 93.33 11.67 11.67 11.11 89.00 16.00 16.00 15.246 115 91.67 23.33 23.33 20.29 94.50 20.50 20.50 17.83

18.33 18.80 18.17 18.52BIAS MAD MAPE BIAS MAD MAPE

MAPE for the moving average technique:

MAPE for the weighted moving average technique:

Performance Measures – MAPE for the Sample Example

80.1811533.23

10567.11

8020%100*

3nActual

|Forecast - Actual|

MAPE

52.1811550.20

10516

8018%100*

3nActual

|Forecast - Actual|

MAPE

Use the performance measures to select a good set of 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 ().

Excel Solver can be used to determine the values of the model parameters.

Performance Measures –Selecting Model Parameters

Excel Solver

EXCEL SOLVER EXAMPLE

Time Series Components Trend

persistent upward or downward pattern in a time series Seasonal

Variation dependent on the time of year Each year shows same pattern

Cyclical up & down movement repeating over long time frame Each year does not show same pattern

Noise or random fluctuations follow no specific pattern short duration and non-repeating

44

Time Series Components

Time

Trend

Randommovement

Time

Cycle

Time

Seasonalpattern

Dem

and

Time

Trend with seasonal pattern

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 of the

item in a season to the overall annual average value. Examples: demand for coal in winter months; demand for

soft drinks in the summer and over major holidays

Linear Trend AnalysisMidwestern Manufacturing Sales

Scatter Diagram

Actual value (or)

Y

Period number (or) X

74 199579 199680 199790 1998

105 1999142 2000122 2001

Sales(in units) vs. Time

0

20

40

60

80

100

120

140

160

1994 1995 1996 1997 1998 1999 2000 2001 2002

Least Squares for Linear RegressionMidwestern Manufacturing

Least Squares Method

Time

Valu

es o

f Dep

ende

nt V

aria

bles

Objective: Minimize the squared deviations!

Trend Analysis - Least Squares Method For Linear Regression

Y^

]Xn - XY[ __

Y

_

22 Xn -X

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,…

bt a Y^

Where

Y^

= predicted value of the dependent variable (demand)

a = Y- intercept

b = Slope of the regression line

t = independent variable (time period = 1, 2, 3, ….)

Trend Analysis - Least Squares Method For Linear Regression

50

Curve Fitting: Simple Linear Regression

One Independent Variable (X) is used to predict one Dependent Variable (Y): Y = a + b X

Given 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:

nX

bnY

a

nX

XnYX

YXb

ii

ii

iiii

22 )(

/

Regression formula is an optional learning objective

51

Find the regression line with ExcelUse Excel’s 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 Excel’s Tools | Data Analysis | Regression

Curve Fitting: Simple Linear Regression

Linear Trend Data & Error AnalysisMidwestern Manufacturing CompanyForecasting Linear trend analysis


PeriodActual value

(or) YPeriod number

(or) X Forecast ErrorAbsolute

errorSquared


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

Least Squares Graph

Trend Analysis

y = 10.536x + 56.714

0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7

Time

Valu

e

Actual values Linear (Actual values)

54

Pattern-based forecasting – Seasonal

The methods we have learned (Heuristic methods and Regression) are not suitable for data that has pronounced fluctuations.

If data seasonalized: Deseasonalize the data. Make forecast based on the deseasonalized data Reseasonalize the forecast

Good forecast should mimic reality. Therefore, it is needed to give seasonality back.

55


Deseasonalize

Forecast

Reseasonalize

Actual data Deseasonalized data

Example (SI + Regression)

56


Deseasonalization

Deaseasonalized Data =

Reseasonalization Reseasonalized Forecast = (Deseasonalized Forecast )* (Seasonal Index)

Index SeasonalValue Actual

Forecasting Seasonal Data With Trend

1. Calculate the seasonal indices.2. Calculate “deseasonalized” trend by divide the

actual value (Y) by the seasonal index for that period.

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

4. Now that we have the Seasonal Indices and Trend line, we can reseasonalize the data and generate the “seasonalized” forecast by multiplying the trend line values in the forecast period by the appropriate seasonal indices for each time period.

Forecasting Seasonal Data: Calculating Seasonal IndexesEichler Supplies

Year Month DemandAverage Demand Ratio

Seasonal Index

1 January 80 94 0.851 0.957February 75 94 0.798 0.851

March 80 94 0.851 0.904April 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





Seasonal Index – ratio of the average value of the item in a season to the overall average annual value.

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

Ratio = Demand / Average Demand

If Year 3 average monthly demand is expected 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

Forecasting Seasonal Data: Calculating Seasonal IndexesEichler Supplies

Year Month DemandAverage Demand Ratio

Seasonal Index









1. Take average of all Demand Values (Average Demand Column)

2. Get Ratio of “Actual Value: Average Value” for each period (Ratio Column)

3. Average the ratio for corresponding periods to get seasonal index

064.194

100 January :Y2

851.9480 January :Y1

957.2

064.1851.2

RatioJanuary :Y2 RatioJanuary :Y1January for Index Seasonal

Seasonal Forecasting

Seasonal Forecasting Example

61

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?

forecasting introduction quantitative models time series

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