Time series mnr
23
Quantitative Methods for Business PGDMA-624 Time Series Forecasting Note: Adapted from “Quantitative Methods for Business by Anderson et all.
Transcript of Time series mnr
Quantitative Methods for Business
PGDMA-624
Time Series Forecasting
Note: Adapted from “Quantitative Methods for Business by Anderson et all.
20-7-2010 2
Quantitative Approaches to Forecasting
Quantitative methods are based on an analysis of historical data concerning one or more time series.
A time series is a set of observations measured at successive points in time or over successive periods of time.
If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method.
If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method.
20-7-2010 3
Time Series Methods
Three time series methods are: smoothing trend projection trend projection adjusted for seasonal influence
Two primary methods: causal models and time series methods
Causal Models (Regression Models)Let Y be the quantity to be forecasted and (X1,
X2, . . . , Xn) are n variables that have predictive power for Y. A causal model is Y = f (X1, X2, . . . , Xn).
A typical relationship is a linear one:Y = a0 + a1X1 + . . . + an Xn
20-7-2010 4
Time Series Methods
A time series is just collection of past values of the variable being predicted. Also known as naïve methods. Goal is to isolate patterns in past data. (See Figures on following pages)
Components of Time series Trend Seasonality Cycles Irregular Component or Randomness
20-7-2010 5
Components of a Time Series The trend component accounts for the gradual shifting
of the time series over a long period of time. Any regular pattern of sequences of values above and
below the trend line is attributable to the cyclical component of the series.
The seasonal component of the series accounts for regular patterns of variability within certain time periods, such as over a year.
The irregular component of the series is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot attempt to predict its impact on the time series in advance.
20-7-2010 7
Notation Conventions
Let Y1, Y2, . . . Yn, . . . be the past values of the series to be predicted (demands?). If we are making a forecast during period t (for the future), assume we have observed Yt , Yt-1 etc. Let Ft = forecast made in period t
Models of Time Series
Additive Model: Y= S+T+C+I
Multiplicative Model : Y =S.T.C.I
20-7-2010 8
Evaluation of Forecasts
The forecast error in period t, et, is the difference between the forecast for demand in period t and the actual value of demand in t.
For one step ahead forecast: et = Yt – Ft
To evaluate Forecasting accuracy we develop a chart of Forecasting errors using: Mean Square error =MSE = (1/n) Σ ei 2
Root Mean Square error = RMSE = √MSE
Mean absolute Deviation: MAD = (1/n) Σ | e i |
20-7-2010 9
Forecasting for Stationary Series
A stationary time series has the form:Dt = + t where is a constant and t is a random variable with mean 0 and var
Stationary series indicate stable processes without observable trends
Two common methods for forecasting stationary series are moving averages and exponential smoothing.
Time Series with irregular (Random) component
20-7-2010 10
Week
Sales (1000 of
gallons)1 172 213 194 235 186 167 208 189 22
10 2011 1512 22
Example
Sales (1000 of gallons)
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12
Week
Sale
s(10
00s
of o
nes)
20-7-2010 11
Moving Averages
In words: the arithmetic average of the n most recent observations. For a one-step-ahead forecast:
Ft = (1/N) (Y t - 1 + Y t - 2 + . . . + Y t - n )
20-7-2010 12
Summary of Moving Averages
Advantages of Moving Average Method Easily understood Easily computed Provides stable forecasts
Disadvantages of Moving Average Method Requires saving lots of past data points: at least the N
periods used in the moving average computation Lags behind a trend Ignores complex relationships in data
20-7-2010 13
What about Weighted Moving Averages?
This method looks at past data and tries to logically attach importance to certain data over other data
Weighting factors must add to one
Can weight recent higher than older or specific data above others
Selecting length of moving averages
20-7-2010 14
Exponential Smoothing Method
A type of weighted moving average that applies declining weights to past data.
1. New Forecast = (most recent observation) + (1 - (last forecast)
where 0 < and generally is small for stability of forecasts ( around .1 to .2)
20-7-2010 15
Exponential Smoothing (cont.)
In symbols:Ft+1 = Yt + (1 - ) Ft
= Yt + (1 - ) ( Yt-1 + (1 - ) Yt-1)
= Yt + (1 - )( )Yt-1 + (1 - ( )Yt - 2 + . . .
Hence the method applies a set of exponentially declining weights to past data. It is easy to show that the sum of the weights is exactly one.
20-7-2010 16
Exponential Smoothing (cont.)
In symbols:Ft+1 = Yt + (1 - ) Ft
= Yt + (1 - ) ( Yt-1 + (1 - ) Yt-1)
= Yt + (1 - )( )Yt-1 + (1 - ( )Yt - 2 + . . .
Hence the method applies a set of exponentially declining weights to past data. It is easy to show that the sum of the weights is exactly one.
20-7-2010 18
Comparison of MA and ES
Similarities Both methods are appropriate for stationary
series Both methods depend on a single parameter Both methods lag behind a trend
20-7-2010 19
Comparison of MA and ES
Differences ES carries all past history (forever!) MA eliminates “bad” data after N periods MA requires all N past data points to compute
new forecast estimate while ES only requires last forecast and last observation of ‘demand’ to continue
20-7-2010 20
Forecasting with Trendand Seasonal Components
Steps of Multiplicative Time Series Model1. Calculate the centered moving averages (CMAs).2. Center the CMAs on integer-valued periods.3. Determine the seasonal and irregular factors (StIt ).
4. Determine the average seasonal factors.5. Scale the seasonal factors (St ).
6. Determine the deseasonalized data.7. Determine a trend line of the deseasonalized data.8. Determine the deseasonalized predictions.9. Take into account the seasonality.
20-7-2010 21
Year Quarter Sales(1000s)1 1 4.8 2 4.1 3 6 4 6.52 1 5.8 2 5.2 3 6.8 4 7.43 1 6 2 5.6 3 7.5 4 7.84 1 6.3 2 5.9 3 8 4 8.4
Trend and Seasonal Components
Sales(1000s)
0123456789
Y1Q1
Y1Q2
Y1Q3
y1Q4
Y2Q1
Y2Q2
Y2Q3
y2Q4
Y3Q1
Y3Q2
Y3Q3
y3Q4
Y4Q1
Y4Q2
Y4Q3
y4Q4
Year
Sale
s(00
0 s)
20-7-2010 22
Calculating Seasonal Indices De seasonalizing time series
Estimating Trend
Forecasting by adjusting seasonal variations
Fore casting
20-7-2010 23
Using Regression for Times Series ForecastingUsing Regression for Times Series Forecasting
Regression Methods Can be Used When a Trend Regression Methods Can be Used When a Trend is Present. is Present.
Model: Dt = a + bt + Model: Dt = a + bt + tt.. If t is scaled to 1, 2, 3, . . . , -- it becomes a If t is scaled to 1, 2, 3, . . . , -- it becomes a
number i -- then the least squares estimates for number i -- then the least squares estimates for aa and and bb can be computed as follows: (n is the can be computed as follows: (n is the number of observation we have)number of observation we have)
