dsc 3120 generalized modeling techniques with applications part ii. forecasting
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DSc 3120
Generalized Modeling Techniques with Applications
Part II. Forecasting
Forecasting Models 2
Forecasting
Why Forecasting?
Characteristics of Forecasts Forecasts are usually wrong or seldom correct Aggregate forecasts are usually more accurate Less accurate further into the future
Assumptions of Forecasting Models Information (data) about the past is available The pattern of the past will continue into the
future.
Forecasting Models 3
Qualitative Forecasting
Sales force composites (field sales force)
Consumer market survey (users’ expectations)
Jury of executive
The Delphi method
--Forecasting based on experience, judgement, and knowledge
Forecasting Models 4
Quantitative Forecasting
Casual Models:
Causal Model
Year 2000 Sales
Price PopulationAdvertising
……
Time Series Models:
Time Series Model
Year 2000 Sales
Sales1999 Sales1998
Sales1997……
--Forecasting based on data and models
Forecasting Models 5
Forecasting Models
Qualitative Quantitative
Causal Time series
Regression EconometricsMoving average Decomposition
Exponential smoothing
Sales force composite
Consumer survey
Jury of executive
Delphi method
ARIMA Neural networks
Overview of Forecasting Models
Forecasting Models 6
Causal Forecasting Models 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:
n
Xb
n
Ya
n
XX
n
YXYXb
ii
ii
iiii
22 )(
/
Forecasting Models 7
Curve Fitting: Simple Linear Regression Find the regression line with Excel
Use Function:
a = INTERCEPT(Y range; X range)
b = SLOPE(Y range; X range) Use Solver Use Excel’s Tools | Data Analysis | Regression
Curve Fitting: Multiple Regression Two or more independent variables are used to
predict the dependent variable:
Y = b0 + b1X1 + b2X2 + … + bpXp
Use Excel’s Tools | Data Analysis | Regression
Forecasting Models 8
BIAS - The arithmetic mean of the errors
n is the number of forecast errors Excel: =AVERAGE(error range)
Mean Absolute Deviation - MAD
No direct Excel function to calculate MAD
Evaluation of Forecasting Model
n
Error
n
Forecast) - (Actual BIAS
n
|Error|
n
Forecast - Actual| MAD
|
Forecasting Models 9
Mean Square Error - MSE
Excel: =SUMSQ(error range)/COUNT(error range)
Mean Absolute Percentage Error - MAPE
R2 - only for curve fitting model such as regression In general, the lower the error measure (BIAS,
MAD, MSE) or the higher the R2, the better the forecasting model
n
(Error)
n
Forecast) - (Actual MSE
22
nActual
|Forecast - Actual|
MAPE
%100*
Evaluation of Forecasting Model
Forecasting Models 10
Time Series Model Building
Historical data collection
Data plotting (time series plot)
Forecasting model building
Evaluation and selection of model
Forecasting with the final selected model
Forecasting Models 11
Components of A Time Series
Trend: long term overall up or down
movement
Seasonality: periodic pattern repeating every
year
Cycles: up & down movement repeating over long time
frame
Random Variations: random movements follow no pattern
Forecasting Models 12
Components of A Time Series
Time
Trend
Randommovement
Time
Cycle
Time
Seasonalpattern
Dem
and
Time
Trend with seasonal pattern
Forecasting Models 13
Types of Time Series Models
Nonseasonal Model Trend Naïve Moving average Exponential
Seasonal Model Time Series Decomposition
Forecasting Models 14
Trend Model 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,…
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Forecasting Models 15
Trend Model (Cont.)Nonlinear Trend Models
Power: Yt = atb
Quadratic: Yt = a + bt + ct2
0
500
1000
1500
2000
2500
3000
1 2 3 4 5 6 7 8 9 10
Power
Quadratic
Forecasting Models 16
Naïve Model The simplest time series forecasting model
Idea: “what happened last time (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
Forecasting Models 17
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 observation are treated equally (equal
weights) Suitable for relatively stable time series with no trend or
seasonal pattern
nnt
Y2t
Y1t
Y=
n
periodsn previousin valuesactual of Sumt
F
Forecasting Models 18
Smoothing Effect of MA Model
Longer-period moving averages (larger n) react to actual changes more slowly
Forecasting Models 19
Moving Average Model Weighted n-Period Moving Average
Typically weights are decreasing:
w1>w2>…>wn
Sum of the weights = wi = 1
Flexible weights reflect relative importance of each previous observation in forecasting
Optimal weights can be found via Solver
ntY
nw
2tY
2w
1tY
1 w=
tF
Forecasting Models 20
Weighted MA: An Illustration
Month Weight Data
August 17% 130
September 33% 110
October 50% 90
November forecast:
FNov = (0.50)(90)+(0.33)(110)+(0.17)(130)
= 103.4
Forecasting Models 21
Simple Exponential SmoothingA special type of weighted moving average
Include all past observations Use a unique set of weights that weight recent observations
much more heavily than very old observations:
( )
( )
( )
1
1
1
2
3
weight
Decreasing weights given to older observations
0 1
Today
Forecasting Models 22
Simple ES: The Model
New forecast = weighted sum of last period actual value and last period
forecast : Smoothing constant Ft : Forecast for period t
Ft-1: Last period forecast
Yt-1: Last period actual value
321
32
21
)1()1(
)1()1(
tttt
tttt
YaYYF
YYYF
11 )1( ttt FYF
Forecasting Models 23
Simple Exponential Smoothing Properties of Simple Exponential
Smoothing
Widely used and successful model
Requires very little data
Larger , more responsive forecast; Smaller
, smoother forecast (See Table 13.2)
“best” can be found by Solver
Suitable for relatively stable time series
Forecasting Models 24
Holt’s Model– Exponential Smoothing with Trend
Ft Forecast for period t
Lt Level term (intercept)
Tt Trend term (slope)
Yt-1 Last period actual value Smoothing constant for Level L Smoothing constant for Trend T
1-t1-ttt
1-t1-t1-tt
ttt
)T -1 ( ) L - L ( T
)T )(L - 1 ( Y L
T L F
Forecasting Models 25
Basic Idea: a time series is composed of several basic components: Trend, Seasonality, Cycle, and Random Error
The multiplicative decomposition model:
These components contribute to time series value in a multiplicative way
Time Series Decomposition Model
ttt tt Error ySeasonalit Cycle Trend Y
Forecasting Models 26
Time Series Decomposition The basic model is:
Y = Trend Cyclical Seasonal Error
Since we cannot easily extract or predict cycles, we will assume that the trend component will capture cycles during the forecast period
Since we have to live with error (cannot predict it), our model is simplified to: Y = Trend Seasonal
Forecasting Models 27
I. Estimate Seasonal Component - Seasonal Index
Step 1: Calculate 1-Year Moving Averages For quarterly data, use 4-period MA For monthly data, use 12-period MA
Step 2: Calculate Centered Moving Averages Simple average of two adjacent MAs
Step 3: Calculate Seasonal Ratio (SR) SR = Y / CMA
Step 4: Calculate Seasonal Index (SI)
Forecasting Models 28
Period Quarter Y Moving Average Centered Moving Avg Seasonal Ratio1 1 Y1
2 2 Y2
MA1=(Y1 +Y2 +Y3+ Y4)/43 3 Y3 CMA1=(MA1+MA2)/2 SR3,1=Y3/CMA1
MA2=(Y2 +Y3 +Y4+ Y5)/44 4 Y4 CMA2=(MA2+MA3)/2 SR4,1=Y4/CMA2
MA3=(Y3+ Y4+Y5+Y6)/45 1 Y5 CMA3=(MA3+MA4)/2 SR1,2=Y5/CMA3
MA4=(Y4 +Y5 +Y6+ Y7)/46 2 Y6 CMA4=(MA4+MA5)/2 SR2,2=Y6/CMA4
MA5=(Y5+Y6 +Y7+ Y8)/47 3 Y7 CMA5=(MA5+MA6)/2 SR3,2=Y7/CMA5
MA6=(Y6 +Y7 +Y8+ Y9)/48 4 Y8 CMA6=(MA6+MA7)/2 SR4,2=Y8/CMA6
MA7=(Y7 +Y8 +Y9+ Y10)/4
Calculate Seasonal Index (Steps 1-3)
Forecasting Models 29
Illustration of Moving Average(MA) and Centered Moving Average (CMA)
Quarter 1 Quarter 2 Quarter 3 Quarter 4
Y1 Y2 Y3 Y4
1/1 6/303/312/15 5/15 8/15 11/159/30 12/31
MA1
3/32/15
Y5
Quarter 1
MA2
CMA
Forecasting Models 30
Calculate Seasonal Index (Step 4)
Year Quarter 1 Quarter 2 Quarter 3 Quarter 412345
-SR1,2
SR1,3
SR1,4
SR1,5
-SR2,2
SR2,3
SR2,4
SR2,5
SR3,1
SR3,2
SR3,3
SR3,4
-
SR4,1
SR4,2
SR4,3
SR4,4
-Average ASR1 ASR2 ASR3 ASR4
S.I. SI1 = ASR1/GA SI2 = ASR2/GA SI3 = ASR3/GA SI4 = ASR4/GA
ASR = Average Seasonal Ratio GA = Grand Average = average of all ASR’s The average of Seasonal Indices (SI) must be 1
Forecasting Models 31
II. Estimate Trend Component Step 1: Remove seasonal effect
Deseasonalized datat = Yt / SIt
Step 2: Fit a trend line to deseasonalized data using least squares method Step 3: Calculate the trend value for each
period Note: If the deseasonalized data look stable (no
apparent trend), simple exponential smoothing may be used in Steps 2 and 3 to calculate the forecast (rather than trend) for each period.
Forecasting Models 32
III. Forecast
Combine seasonal and trend components
Ft = Trend Valuet Seasonal Indext
This final step is also called reseasonalizing
Trend Valuet is the trend estimate for the period t,
based on the trend model fitted to the deseasonalized data
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