Slides 13b: Time-Series Models; Measuring Forecast Error

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Slides 13b Time-Series ModelsMeasuring
Forecast Error
MGS3100 Chapter 13
Forecasting
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Forecasting Models
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Time Series Models

• General Form Y T C S e, where
• T Trend - long term movement of mean
• C (Business) Cycle - an upturn or downturn not
  caused by seasonal variation effect of the
  economy
• S Seasonal Variation - repetitive pattern
  observed over a specific time period
• e Error (random variation)
• Practical Forecast Form Y T S
• C is important, but difficult to forecast
• Dont forecast an error!

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Components of a Time Series
Time series value
Linear trend and seasonality time series
Future
Linear trend time series
A stationary time series
Time
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Time Series Stationary Models

• Stationary Model Assumptions
• Assumes item forecasted will stay steady over
  time (constant mean random variation only)
• Techniques will smooth out short-term
  irregularities
• Forecast for period t1 is equal to forecast for
  period tk the forecast is revised only when new
  data becomes available.
• Stationary Model Types
• Naïve Forecast
• Moving Average
• Weighted Moving Average
• Exponential Smoothing

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Stationary Time Series ModelsThe Naïve Model

• Whatever happened last period will happen again
  this time
• The model is simple and flexible
• Provides a baseline to measure other models
• Attempts to capture seasonal factors at the
  expense of ignoring trend

or
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Measures of Forecast Error

• Bias - The arithmetic sum of the errors
• MAD - Mean Absolute Deviation
• MAPE Mean Absolute Percentage Error
• Mean Square Error (MSE) - Similar to simple
  sample variance
• Standard Error - Standard deviation of the
  sampling distribution (the square
• root of the MSE)
• Bias, MAD, and MAPE - typically
• used for time series

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Naïve Forecast
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Naïve Forecast Graph
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Stationary Time Series ModelsMoving Averages

• The Moving Average Method
• The forecast is the average of the last n
  observations of the time series.

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Moving Averages
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Moving Averages Forecast
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Moving Averages Graph
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Stability vs. Responsiveness

• Should I use a 2-period moving average or a
  3-period moving average?
• The larger the n the more stable the forecast.
• A 2-period model will be more responsive to
  change.
• We dont want to chase outliers.
• But we dont want to take forever to correct for
  a real change.
• We must balance stability with responsiveness.

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Stationary Time Series ModelsWeighted Moving
Averages

• The Weighted Moving Average Method
• Historical values of the time series are assigned
  different weights when performing the forecast

w1Yt w2Yt-1 w3Yt-2 wnYt-n1
Swi 1
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Weighted Moving Average
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Weighted Moving Average
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Stationary Time Series ModelsExponential
Smoothing

• Exponential Smoothing
• Moving average technique that requires a minimum
  amount of past data
• Uses a smoothing constant a with a value between
  0 and 1 (Usual range 0.1 to 0.3)
• Forecast for period t Forecast for period t-1
  plus a times the difference between the actual
  value and forecast in period t-1 Yt Yt-1
  a(Yt-1 - Yt-1), or
• Can also be expressed as Yt a(Yt-1) (1-
  a)(Yt-1)
• a(Actual value in period t-1) (1- a)(Forecast
  in period t-1)

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Exponential Smoothing Data
Class Exercise What is the forecast for January
of the following year? How about March? Find
the Bias, Mad MAPE. (Note a equals 0.1.)
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Exponential Smoothing (Alpha .419)
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Exponential Smoothing
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Evaluating the Performance of Forecasting
Techniques

• Several forecasting methods have been presented.
• Which one of these forecasting methods gives the
  best forecast?

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Performance Measures Sample Example

• Find the forecasts and the errors for each
  forecasting technique applied to the following
  stationary time series.

• Time 1 2 3 4 5 6 Time series 100 110
  90 80 105 115
• 3-Period Moving average 100 93.33 91.6
• Error for the 3-Period MA - 20 11.67 23.4
• 3-Period Weighted MA(.5, .3, .2) 98 89 85.5
• Error for the 3-Period WMA - 18 16 29.5

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Performance Measures MAD for the Sample
Example
18.35
21.17
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Performance Measures MAPE for the Sample
Example
.188
.211
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Performance Measures Selecting Model Parameters

• Use the performance measures to select a good set
  of values for each model parameter.
• For the moving average
• the number of periods (n).
• For the weighted moving average
• The number of periods (n),
• The weights (wi).
• For the exponential smoothing
• The exponential smoothing factor (a).
• Excel Solver can be used to determine the values
  of the model parameters.

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Trend Seasonality

• Trend analysis
• Technique that fits a trend equation (or curve)
  to a series of historical data points
• Projects the equation into the future for medium
  and long term forecasts. Typically do not want to
  forecast into the future more than half the
  number of time periods used to generate the
  forecast
• Seasonality analysis
• Adjustment to time series data due to variations
  at certain periods.
• Adjust with seasonal index - ratio of average
  value of the item in a season to the overall
  annual average value.
• Examples demand for coal in winter months
  demand for soft drinks in the summer and over
  major holidays

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Linear Trend AnalysisMidwestern Manufacturing
Sales
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Least Squares for Linear Regression Midwestern
Manufacturing
Objective Minimize the squared deviations!
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Least Squares Method
Where
predicted value of the dependent variable
(demand)
X value of the independent variable (time)
a Y-axis intercept - b b Slope of
the regression line
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Linear Trend Data Error Analysis
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Least Squares Graph
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Another way to Determine TrendUse the Excel
Regression Function

• Run linear regression to test b1 in the model
  Ytb0b1tet
• Excel results

0.71601
This large P-value indicates that there is
little evidence that trend exists

• Conclusion A stationary model is appropriate.

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Forecasting Seasonal Data Quick Method
Ratio Demand / Average Demand
Seasonal Index ratio of the average value of
the item in a season to the overall average
annual value. Example average of year 1
January ratio to year 2 January ratio. (0.851
1.064)/2 0.957
If Year 3 average monthly demand is expected to
be 100 units. Forecast demand Year 3 January
100 X 0.957 96 units Forecast demand Year 3
May 100 X 1.309 131 units
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Forecasting Seasonal Data With Trend

• Calculate the seasonal indices (as shown on the
  previous slide)
• Calculate deseasonalized treand by dividing the
  actual value (Y) by the seasonal index for that
  period
• Deseasonalized Trend Y / Seasonal index
  (e.g., 80
  units/ 0.957 83.595)
• Find the trend line, and extend the trend line
  into the desired forecast period.

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Forecasting Seasonal Data With Trend Calculating
the Seasonal Forecast

• 4. Now that we have the Seasonal Indices and
  Trend line, we can reseasonalize the data and
  generate the seasonalized forecast by
  multiplying the trend line values in the forecast
  period by the appropriate seasonal indices for
  each time period as follows

Y Trend x Seasonal Index