Statistical Forecasting Models

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Statistical Forecasting Models

• (Lesson - 07)

Best Bet to See the Future
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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

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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.

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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

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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.

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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.

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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.

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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.

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Moving Average Models

• Simple Moving Average Forecast

• Weighted Moving Average Forecast

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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
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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

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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.

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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

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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.

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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)

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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)

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Exponential Smoothing Model

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

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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.

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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

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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

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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)

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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.

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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

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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.

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Forecasting with Crystal Ball
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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
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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.

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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.

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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)!

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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

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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

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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.

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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, . . . )

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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.

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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, . . . )
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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
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Next Lesson

• (Lesson - 09)
• Introduction to Optimization