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Time-Series Forecast ModelsTime-Series Forecast Models A time series is based on a sequence of evenly time-spaced data points, such as daily shipments, weekly sales, or quarter- ly earnings.
Forecasting time-series data implies that forecasts are predicted only from the past values of that variable, and that other varia- bles, no matter how potentially valuable, are ignored.
Mo
nth
ly S
ale
s (
in
un
its
)
Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec
Data Pointor
(observation)
MGMT E-5070
Decomposition of a Time SeriesDecomposition of a Time Series
Analyzing time series means breaking down past data into components and then project- ing them into the future
A time series typically has four components: trend, seasonality, cycles, and random variation
TIME SERIES MODELS ATTEMPT TO PREDICT TIME SERIES MODELS ATTEMPT TO PREDICT THE FUTURE BY USING HISTORICAL DATATHE FUTURE BY USING HISTORICAL DATA
Decomposition of a Time SeriesDecomposition of a Time Series
Trend Trend ( ( TT ) ) is the gradual upward or downward movement of the data over time.
SeasonalitySeasonality ( S ) ( S ) is a pattern of the demand fluc- tuation above or below the trend line that repeats at regular intervals.
CyclesCycles ( C ) ( C ) are patterns in annual data that occur every several years. They are usually tied into the business cycle.
Random variationsRandom variations ( R ) ( R ) are blips in the data that are caused by chance and unusual situations. They follow no discernible pattern.
Time Series & ComponentsTime Series & Components
TREND COMPONENTTREND COMPONENTSEASONAL PEAKSSEASONAL PEAKS
ACTUAL DEMAND LINEACTUAL DEMAND LINE
YEAR 1 YEAR 2 YEAR 3 YEAR 4
TIME
AVERAGE DEMAND OVER 4 YEARSAVERAGE DEMAND OVER 4 YEARS
PR
OD
UC
T O
R S
ER
VIC
E D
EM
AN
D
Time Series & ComponentsTime Series & ComponentsRANDOM VARIATIONSRANDOM VARIATIONS
Forecasters usually assume that the random variations are averaged out over time.
These random errors are often assumed to be normally distributed with a mean of zero.
IT IS ALSO ASSUMED THAT RANDOM VARIATIONSIT IS ALSO ASSUMED THAT RANDOM VARIATIONS DO NOT HEAVILY INFLUENCE DEMANDDO NOT HEAVILY INFLUENCE DEMAND
TheThe Moving Average Moving Average ModelModel
Assumes demand will stay fairly steady over time.
A two-month moving average forecast is found by summing the demand during the past two periods and dividing by “ 2 ” .
With each passing period, the most recent demand is added to the sum; the earliest demand is dropped. This smooths out short-term irregularities in the data series.
It has no trend, seasonal, or cyclical components.
TheThe Moving Average Moving Average ModelModel
( demands in previous n periods )
nn IS THE NUMBER OF PERIODS IN THE MOVING AVERAGE
Forecast = Σ
TheThe Mo Moving Average ving Average ModelModel
Year Demand Forecast
1 110 -
2 100 -
3 120 105105
4 140 110110
5 170 130130
TWO - PERIOD EXAMPLETWO - PERIOD EXAMPLE
110 + 100 / 2 = 105105100 + 120 / 2 = 110110120 + 140 / 2 = 130130
TheThe Mo Moving Average ving Average ModelModel
Year Demand Forecast
1 110 -
2 100 -
3 120 --
4 140 --
5 170 117.5117.5
6 150 132.5132.5
FOUR - PERIOD EXAMPLEFOUR - PERIOD EXAMPLE
110 + 100 + 120 + 140 / 4 = 117.5117.5100 + 120 + 140 + 170 / 4 = 132.5132.5
Weighted Moving Average ModelWeighted Moving Average Model
Makes the forecast more responsive to changes.
Used when there is a trend or pattern. Weights place more emphasis on recent values.
Deciding the weights requires some experience and good luck!
SEVERAL WEIGHTS SHOULD BE TRIED, AND THE ONES WITHSEVERAL WEIGHTS SHOULD BE TRIED, AND THE ONES WITH THE LOWEST FORECAST ERROR SHOULD BE SELECTEDTHE LOWEST FORECAST ERROR SHOULD BE SELECTED
Weighted MovinWeighted Moving Average Modelg Average Model
∑ ( weight in period i )( actual value in period)
∑ ( weights )
Weighted Moving Average ModelWeighted Moving Average ModelTHREE - PERIOD
88 (120) + 11 (100) + 11 (110)
1010= =
Period Weight Demand
Most recent 8 120
2nd Most recent 1 100
3rd Most recent 1 110
4th
PeriodForecast
117 units
‘10’representsthe sum ofthe weights
Weighted Moving Average ModelWeighted Moving Average ModelTHREE - PERIOD
88 (140) + 11 (120) + 11 (100)
1010= =
Period Weight Demand
Most recent 8 140
2nd Most recent 1 120
3rd Most recent 1 100
5th Period
Forecast134 units
Exponential Smoothing ModelExponential Smoothing Model
THENEW
FORECAST
LAST FORECASTED
DEMANDα 1 - α++==
The new forecast is equal to the old forecast adjusted by a fraction of the error( last period actual demand – last period forecast ) . The smoothing coefficient
( α ) is a weight for the last actual demand.
LASTACTUALDEMAND
First Order or Primary VersionFirst Order or Primary Version
A moving average technique that only requires thelast period actual demand and the last period
forecasted demand for input.
Exponential Smoothing ExampleExponential Smoothing Example
ASSUMING THAT α = .7 , THE NEXT FORECAST IS:
.7 ( 100 units ) + ( 1 - .7 )( 110 units )
70 + 33 = 103 units
LastForecast
Last ActualDemand
Exponential Smoothing ExampleExponential Smoothing Example
ASSUMING THAT α = .7 , THE NEXT NEW FORECAST IS:
.7 ( 120 units ) + ( 1 - .7 )( 103 units )
84 + 30.9 = 114.9 units
LastForecast
Last ActualDemand
The Smoothing CoefficientThe Smoothing Coefficient
The symbol is alpha ( α )
It can assume any value between 0 and 1 inclusive
It places a weight on the last actual period demand The value of alpha resulting in the lowest forecast error is selected for the model.
Smoothing Coefficient SelectionSmoothing Coefficient Selection
This range ( .0 – .3 ) places the heaviest weight on the historical demand periods.
The intent is to make the forecast reflect the long - term stability of the product’s demand, as well as to minimize short-term fluctuations that could distort future forecasts.
It is appropriate for products whose demand patterns are extremely stable over time and expected to remain so.
LOW - RANGELOW - RANGE
Smoothing Coefficient SelectionSmoothing Coefficient Selection
This range ( .4 – .6 ) splits weights between historical and most recent demand periods.
The intent is to make the forecast reflect the importance of each.
It is appropriate for products whose demand patterns are only slightly unstable.
MEDIUM - RANGEMEDIUM - RANGE
Smoothing Coefficient SelectionSmoothing Coefficient Selection
This range ( .7 – 1.0 ) places the heaviest weight on the most recent demand periods.
The intent is to make the forecast largely reflect the most recent demand experience.
It is appropriate for products that are entirely new, and for products whose demand patterns are unstable.
HIGH - RANGEHIGH - RANGE
Trend Projection ModelTrend Projection ModelA REGRESSION MODEL OVER TIMEA REGRESSION MODEL OVER TIME
This technique fits a trend line through a series of historical data points and then projects that trend line into the future forboth medium and long-range forecasting.
WE WILL FOCUS ON STRAIGHT-LINE TRENDS FOR NOWWE WILL FOCUS ON STRAIGHT-LINE TRENDS FOR NOW
Trend Projection ModelTrend Projection ModelA REGRESSION MODEL OVER TIMEA REGRESSION MODEL OVER TIME
TIME ( X )TIME ( X )
DE
MA
ND
( Y
)D
EM
AN
D (
Y )
THIS ALSO IMPLIES THAT THE MEAN
SQUARED ERROR (MSE) IS MINIMIZED
MSE IS AMEASURE OF
FORECASTERROR
We identify a straight line that minimizes the sumof the squares of the vertical distances from theregression line to each of the actual observations.
THE THE TREND TREND
LINELINE
Trend Projection ModelTrend Projection Model
Y = a + b X^
Y-AXIS INTERCEPT : THE POINT ON THE VERTICAL
AXIS THAT THE REGRESSION LINE CROSSES
THE SLOPE OF THE LEAST-SQUARES LINE: THE RATE OF CHANGE
IN ‘Y’ GIVEN CHANGE INTIME ‘X’
X AXIS
Y A
XIS
ORIGIN
THE SPECIFIED VALUE OF ‘X’( TIME )
THE PREDICTED VALUE ( FORECAST )
Trend Projection ModelTrend Projection ModelEXAMPLEEXAMPLE
Y = a + b ( X )
Y = 92.6667 + 10.9697 ( 11 ) 213.3333 units = 92.6667 + 120.6667
^
11th YEAR FORECAST Y - INTERCEPT SLOPE 11th YEAR
^