Statistical Forecasting Models - PowerPoint PPT Presentation

Loading...

PPT – Statistical Forecasting Models PowerPoint presentation | free to download - id: 469ae9-MTRiN



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

Statistical Forecasting Models

Description:

Statistical Forecasting Models (Lesson - 07) Best Bet to See the Future Statistical Forecasting Models Time Series Models: independent variable is time. – PowerPoint PPT presentation

Number of Views:238
Avg rating:3.0/5.0
Slides: 38
Provided by: zeli1
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: Statistical Forecasting Models


1
Statistical Forecasting Models
  • (Lesson - 07)

Best Bet to See the Future
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

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.

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

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.

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.

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.

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.

9
Moving Average Models
  • Simple Moving Average Forecast
  • Weighted Moving Average Forecast

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
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

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 ks and the one with lowest RMSE could
    be considered as the model with best forecasting
    period.

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

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 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.

15
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)

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-a (not a)

17
Exponential Smoothing Model
  • To determine best a we can use forecast
    accuracy.
  • MSE Mean Square Error is a good measure for
    forecast accuracy.

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.

19
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
  1. E1 and T1 are not defined.
  2. E2 Y2
  3. T2 Y2 Y1

20
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

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

22
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.

23
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

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.

25
Forecasting with Crystal Ball
26
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
27
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.

28
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.

29
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)!

30
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

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

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.

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, . . . )

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.

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, . . . )
36
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
37
Next Lesson
  • (Lesson - 09)
  • Introduction to Optimization
About PowerShow.com