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Statistical Forecasting Models

- (Lesson - 07)

Best Bet to See the Future

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

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.

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

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.

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.

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.

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.

Moving Average Models

- Simple Moving Average Forecast

- Weighted Moving Average Forecast

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

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

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 ks and the one with lowest RMSE could

be considered as the model with best forecasting

period.

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

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 a is a value between 0 and

1, where a closer to 1 means more weigh to the

recent observations and hence more rapidly

changing forecast.

Exponential Smoothing Model

or

- 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)

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-a (not a)

Exponential Smoothing Model

- To determine best a we can use forecast

accuracy. - MSE Mean Square Error is a good measure for

forecast accuracy.

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.

Holt-Winters Model

- where
- Ftk 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

Holt-Winters Model

- E_2 Y_2 and T_2 (Y_2-Y_1)
- E_12 D1C14(1-D1)(D13E13)
- T_12 E1(D14-D13)(1-E1)E13
- F_13 D14E14

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 (t1)
- Set E2 Y2
- Set T2 Y2-Y1 Note (F2 is blanc)

Holt-Winters Model

- Formulate E3 wY3 (1-w)(E2T2)
- 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 Ys and Fs
- Use solver to determine optimal w and v.

Holt-Winters Model

- Solver set up for Holt Winters
- Target Cell MSE (min)
- Changing Cells w and v
- Constrains w lt 1
- w gt 0
- v lt 1
- v gt 0

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.

Forecasting with Crystal Ball

Forecasting with Crystal Ball

Method Errors Method Errors

Method Method RMSE MAD MAPE

Best Double Exponential Smoothing Double Exponential Smoothing 1.5043 0.9871 7.68

2nd Single Exponential Smoothing Single Exponential Smoothing 1.5147 1.1566 9.03

3rd Single Moving Average Single Moving Average 1.5453 1.2042 9.40

4th Double Moving Average Double Moving Average 2.0855 1.592 11.16

Method Parameters Method Parameters Method Parameters

Method Method Parameter Value

Best Double Exponential Smoothing Double Exponential Smoothing Double Exponential Smoothing Alpha 0.999

Beta 0.051

2nd Single Exponential Smoothing Single Exponential Smoothing Single Exponential Smoothing Alpha 0.999

3rd Single Moving Average Single Moving Average Single Moving Average Periods 1

4th Double Moving Average Double Moving Average Double Moving Average Periods 2

Forecast Forecast

Date Lower 5 Forecast Upper 95

2000 11.9 14.4 17.0

Performance of a Model

- Performance of a model is measured by Theils 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.

Theils U versus RMSE

- The difference between RMSE (or MAD or MAPE) and

Theils 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.

Choosing Forecasting Model

- The forecasting model should be the one with

lowest Theils U. - If the best Theils U model is not the same as

the best RMSE model then you need to run CB again

by checking only the best Theils U model to

obtain forecasted value. - P.S. CB uses forecasting value of the lowest RMSE

model (best model according CB)!

Determining Performance

- Theils U determins the forecasting performance

of the model. - The interpretation in daily language is as

follows - Interpret (1- Theil U)
- 1.00 0.80 High (strong) forecasting power
- 0.80 0.60 Moderately high forecasting power
- 0.60 0.40 Moderate forecasting power
- 0.40 0.20 Weak forecasting power
- 0.20 0.00 Very weak forecasting power

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

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.

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, . . . )

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.

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, . . . )

Seasonal Regression

E(Y_Q1) -10801.6 5.52 Year.1 8.06 E(Y_Q2)

-10801.6 5.52 Year.2 -3.50 E(Y_Q3)

-10801.6 5.52 Year.3 5.51 E(Y_Q4)

-10801.6 5.52 Year.4

Next Lesson

- (Lesson - 09)
- Introduction to Optimization