1. Statistical Forecasting Models
(Lesson - 07)
Best Bet to See the Future
Dr. C. Ertuna

2. Statistical Forecasting Models
Time Series Models: independent variable is time.
Moving Average
Exponential Smoothening
Holt-Winters Model
Explanatory Methods: independent variable is one or more factor(s).
Regression
Dr. C. Ertuna

3. Time Series Models
Statistical Time Series Models are very useful for short range forecasting problems such as weekly sales.
Time series models assume that whatever forces have influenced the variables in question (sales) in the recent past will continue into the near future.
Dr. C. Ertuna

4. Time Series Components
A time series can be described by models based on the following components
Tt Trend Component
St Seasonal Component
Ct Cyclical Component
It Irregular Component
Using these components we can define a time series as the sum of its components or an additive model
Alternatively, in other circumstances we might define a time series as the product of its components or a multiplicative model – often represented as a logarithmic model
Dr. C. Ertuna

5. Components of Time Series Data
A linear trend is any long-term increase or decrease in a time series in which the rate of change is relatively constant.
A seasonal component is a pattern that is repeated throughout a time series and has a recurrence period of at most one year.
A cyclical component is a pattern within the time series that repeats itself throughout the time series and has a recurrence period of more than one year.
Dr. C. Ertuna

6. Components of Time Series Data
The irregular (or random) component refers to changes in the time-series data that are unpredictable and cannot be associated with the trend, seasonal, or cyclical components.
Dr. C. Ertuna

7. Stationary Time Series Models
Time series with constant mean and variance are called stationary time series.
When Trend, Seasonal, or Cyclical effects are not significant then
Moving Average Models and
Exponential Smoothing Models
are useful over short time periods.
Dr. C. Ertuna

8. Moving Average Models
Simple Moving Average forecast is computed as the average of the most recent k-observations.
Weighted Moving Average forecast is computed as the weighted average of the most recent k-observations where the most recent observation has the highest weight.
Dr. C. Ertuna

9. Moving Average Models Simple Moving Average Forecast
Weighted Moving Average Forecast
Dr. C. Ertuna

10. Weighted Moving Average
To determine best weights and period (k) we can use forecast accuracy.
MSE = Mean Square Error is a good measure for forecast accuracy.
RMSE = is the square root of the MSE.
Data: Evens - Burglaries
Dr. C. Ertuna

11. Weighted Moving Average
Tools / Solver
Set Target Cell: Cell containing RMSE value
Equal to: Min
By Changing Cells: Cells containing weights
Subject to constraints: Cell containing sum of the weight = 1
Options / (check) Assume Non-Negativity
Solve Keep Solver Solution OK
Dr. C. Ertuna

12. Weighted Moving Average
Best weights for a given “k” (in this case “3”) is determined by solver trough minimizing RMSE.
Same procedure could be applied to models with different k’s and the one with lowest RMSE could be considered as the model with best forecasting period.
Dr. C. Ertuna

13. Moving Average Models Tools/ Data Analysis / Moving Average
Input Range: Observations with title (No time)
Output Range: Select next column to the input range and 1-Row below of the first observation
Chart misaligns the forecasted values! Forecasted 59th month is aligned with 58th month
Dr. C. Ertuna

14. Exponential Smoothing
Exponential smoothing is a time-series smoothing and forecasting technique that produces an exponentially weighted moving average in which each smoothing calculation or forecast is dependent upon all previously observed values.
The smoothing factor “α” is a value between 0 and 1, where α closer to 1 means more weigh to the recent observations and hence more rapidly changing forecast.
Dr. C. Ertuna

15. Exponential Smoothing Modelor
where:
Ft= Forecast value for period t
Yt-1 = Actual value for period t-1
Ft-1 = Forecast value for period t-1
 = Alpha (smoothing constant)
Dr. C. Ertuna

16. Exponential Smoothing Model
Tools/ Data Analysis / Exponential Smoothing.
Input Range: Observations with title (No time)
Output Range: Select next column to the input range and first Row of the first observation
Damping Factor: 1-α (not α)
Dr. C. Ertuna

17. Exponential Smoothing Model
To determine best “α” we can use forecast accuracy.
MSE = Mean Square Error is a good measure for forecast accuracy.
Dr. C. Ertuna

18. Holt-Winters Model
The Holt-Winters forecasting model could be used in forecasting trends. Holt-Winters model consists of both an exponentially smoothing component (E, w) and a trend component (T, v) with two different smoothing factors.
Dr. C. Ertuna

19. Holt-Winters Model where: Ft+k= Forecast value k periods from t
Yt-1 = Actual value for period t-1
Et-1 = Estimated value for period t-1
Tt = Trend for period t
w = Smoothing constant for estimates
v = Smoothing factor for trend
k = number of periods
E1 and T1 are not defined.
E2 = Y2
T2 = Y2 – Y1
Dr. C. Ertuna

20. Holt-Winters Model E_2 = Y_2 and T_2 = (Y_2-Y_1)
E_12 = $D$1*C14+(1-$D$1)*(D13+E13)
T_12 = $E$1*(D14-D13)+(1-$E$1)*E13
F_13 = D14+E14
Dr. C. Ertuna

21. Holt-Winters Model
Set E (smoothing component), T (trend component), and F (forecasted values) columns next to Y (actual observations) in the same sequence
Determine initial “w” and “v” values
Leave E,T &F blanc for the base period (t=1)
Set E2 = Y2
Set T2 = Y2-Y1 Note: (F2 is blanc)
Dr. C. Ertuna

22. Holt-Winters Model Formulate E3 = w*Y3 + (1-w)*(E2+T2)
Formulate T3 = v*(E3-E2) + (1-v)*T2
Formulate F3 = E2 + T2
Copy the formulas down until reaching one cell further than the last observation (Yn).
Compute MSE using Y’s and F’s
Use solver to determine optimal “w” and “v”.
Dr. C. Ertuna

23. Holt-Winters Model Solver set up for Holt Winters:
Target Cell: MSE (min)
Changing Cells: w and v
Constrains: w <= 1
w >= 0
v <= 1
v >= 0
Dr. C. Ertuna

24. Forecasting with Crystal Ball
CBTools / CB Predictor
[Input Data] Select
Range, First Raw, First Column Next
[Data Attribute] Data is in Next
[Method Gallery] Select All Next
[Results] Number of periods to forecast [1]
Select Past Forecasts at cell Run
periods, etc.
Dr. C. Ertuna

25. Forecasting with Crystal Ball
Dr. C. Ertuna

26. Forecasting with Crystal Ball
Method Parameters:
Method
Parameter
Value
Best:
Double Exponential Smoothing
Alpha
0.999
Beta
0.051
2nd:
Single Exponential Smoothing
3rd:
Single Moving Average
Periods 1
4th:
Double Moving Average 2
Method Errors:
Method
RMSE
MAD
MAPE
Best:
Double Exponential Smoothing
1.5043
0.9871
7.68%
2nd:
Single Exponential Smoothing
1.5147
1.1566
9.03%
3rd:
Single Moving Average
1.5453
1.2042
9.40%
4th:
Double Moving Average
2.0855
1.592
11.16%
Forecast:
Date
Lower: 5%
Forecast
Upper: 95%
2000
11.9
14.4
17.0
Dr. C. Ertuna

27. Performance of a Model
Performance of a model is measured by Theil’s U.
The Theil's U statistic falls between 0 and 1.
When U = 0, that means that the predictive performance of the model is excellant and when U = 1 then it means that the forecasting performance is not better than just using the last actual observation as a forecast.
Dr. C. Ertuna

28. Theil’s U versus RMSE
The difference between RMSE (or MAD or MAPE) and Theil’s U is that the formars are measure of ‘fit’; measuring how well model fits to the historical data.
The Theil's U on the other hand measures how well the model predicts against a ‘naive’ model. A forecast in a naive model is done by repeating the most recent value of the variable as the next forecasted value.
Dr. C. Ertuna

29. Choosing Forecasting Model
The forecasting model should be the one with lowest Theil’s U.
If the best Theil’s U model is not the same as the best RMSE model then you need to run CB again by checking only the best Theil’s U model to obtain forecasted value.
P.S. CB uses forecasting value of the lowest RMSE model (best model according CB)!
Dr. C. Ertuna

30. Determining Performance
Theil’s U determins the forecasting performance of the model.
The interpretation in daily language is as follows:
Interpret (1- Thei’l U)
1.00 – High (strong) forecasting power
0.80 – Moderately high forecasting power
0.60 – Moderate forecasting power
0.40 – Weak forecasting power
0.20 – Very weak forecasting power
Dr. C. Ertuna

31. Regression or Time Series Forecast
Here is the guiding principle when to apply Regression and when to apply Time Series Forecast.
As some thing changes (one or more independent variables) how does another thing (dependent variable) change is an issue of directional relationship For directional relationships we can use regression.
 If the independent variable is TIME (as time changes how does a variable change) Then we can use either regression or time series forecasting models
Dr. C. Ertuna

32. Explanatory Methods
Simple Linear Regression Model: The simplest inferential forecasting model is the simple linear regression model, where time (t) is the independent variable and the least square line is used to forecast the future values of Yt.
Dr. C. Ertuna

33. Regression in Forecasting Trends
where:
Yt = Value of trend at time t
0 = Intercept of the trend line
1 = Slope of the trend line
t = Time (t = 1, 2, )
Dr. C. Ertuna

34. Regression in Forecasting Seasonality
Many time series have distinct seasonal pattern. (For example room sales are usually highest around summer periods.)
Multiple regression models can be used to forecast a time series with seasonal components.
The use of dummy variables for seasonality is common.
Dummy variables needed = total number of seasonality –1
For example: Quarterly Seasonal: 3 Dummies are needed, Monthly Seasonal: 11 Dummies needed, etc.
The load of each seasonal variable (dummy) is compared to the one which is hidden in intercept.
Dr. C. Ertuna

35. Regression in Forecasting Seasonality
where:
Q1 = 1 , if quarter is 1, = 0 otherwise
Q2 = 1 , if quarter is 2, = 0 otherwise
Q3 = 1 , if quarter is 3, = 0 otherwise
2 = the load of Q1 above Q4
0 = the overall intercept + the load of Q4
t = Time (t = 1, 2, )
Dr. C. Ertuna

36. Seasonal Regression E(Y_Q1) = -10801.6 + 5.52 * Year.1 + 8.06
Dr. C. Ertuna

37. Introduction to Optimization
Next Lesson
(Lesson - 09)
Introduction to Optimization
Dr. C. Ertuna