Forecasting  Suppose your fraternity/sorority house consumed the following number of cases of beer for the last 6 weekends: 8, 5, 7, 3, 6, 9  How many cases do you think your fraternity/sorority will consume this weekend?

Forecasting  We could use a Moving Average forecasting method  Using a three period moving average, we would get the following forecast: = 6 3

Forecasting  What if we used a two period moving average? = 7.5 2

Forecasting  The number of periods used in the moving average forecast affects the “responsiveness” of the forecasting method: 1 Period 2 Period 3 Period

Forecasting  We can look at the Moving Average method as using a weighted average:  Rather than equal weights, it might make sense to use weights which favor more recent consumption values = (3) + (6) + (9) = 6

Forecasting  With the Weighted Moving Average, we have to select weights that are individually greater than zero and less than 1, and as a group sum to 1:  Valid Weights:.5,.3,.2.6,.3,.1 1, 1,  Invalid weights:.5,.2,.1.6, -0.1, 0.5.5,.4,.3,.2

Forecasting  A Weighted Moving Average forecast with weights of: 1, 1, is as performed as follows:  How do you make the Weighted Moving Average forecast more responsive? (3) + (6) + (9) = 7

Forecasting Terminology Initialization ExPost Forecast Historical Data No Historical Data

Forecasting Terminology  Applying this terminology to our problem using the Moving Average forecast: Initialization ExPost Forecast Model Evaluation

“We are now looking at a future from here, and the future we were looking at in February now includes some of our past, and we can incorporate the past into our forecast. 1993, the first half, which is now the past and was the future when we issued our first forecast, is now over” Laura D’Andrea Tyson, Head of the President’s Council of Economic Advisors, quoted in November of 1993 in the Chicago Tribune, explaining why the Administration reduced its projections of economic growth to 2 percent from the 3.1percent it predicted in February. Forecasting Terminology

Exponential Smoothing  Exponential Smoothing is designed to give the benefits of the Weighted Moving Average forecast with out the cumbersome problem of specifying weights. In Exponential Smoothing, there is only one parameter:   = smoothing constant (between 0 and 1) F(t+1) =  A(t) + (1-  ) F(t) F(initial) = F(2)=[A(1) +A(2)] / 2

tA(t)F(t) Exponential Smoothing  Using  = 0.4, we get Initialization ExPost Forecast

Practice Problem tA(t)MA (n=3) Weighted MA (0.6, 0.3, 0.1) Exponential (  = 0.4)

Practice Problem tA(t)MA (n=3) Weighted MA (0.6, 0.3, 0.1) Exponential (  = 0.4)

Expanding the Exponential Smoothing Formula F(t+1) =  A(t) + (1 –  ) F(t) =  A(t) + (1 –  ) [  A(t-1) + (1 –  ) F(t-1)] =  A(t) + (1 –  )  A(t-1) + (1 –  ) 2 F(t-1) =  A(t) + (1 –  )  A(t-1) + (1 –  ) 2 [  A(t-2) + (1 –  ) F(t-2) ]  A(t) + (1 –  )  A(t-1) + (1 –  ) 2  A(t-2) + (1 –  ) 3 F(t-2) and so on... Thus, the exponential smoothing formula considers all previous actual data

Expanding the Exponential Smoothing Formula

Outliers (eloping point) Outlier

Data with Trends

Simple Linear Regression Model Y = mx + b Simple linear regression can be used to forecast data with trends Y is the regressed forecast value or dependent variable in the model, b is the intercept value of the regression line, and m is the slope of the regression line x (Time) Y b m

Simple Linear Regression Model In linear regression, the squared errors are minimized Error

Formulas for Calculating “m” and “b”

Simple Linear Regression Problem Applying the model to the following data:

Calculate “m” and “b”

Evaluate Results

Practice Problem Question: Given the data below, what is the simple linear regression model that can be used to predict sales?

Calculate “m” and “b”

Evaluate Results F(t) = (t)

Simple Linear Regression in Excel  If the Analysis ToolPak is loaded, extensive regression analysis can be performed using the regression function

Simple Linear Regression in Excel  To get the slope and intercept easily, use the slope and intercept functions: = slope(y-range, x-range) = intercept(y-range, x-range )

Limitations in Linear Regression As with the moving average model, all data points count equally with simple linear regression

Holt’s Trend Model  To forecast data with trends, we can use an exponential smoothing model with trend, frequently known as Holt’s model:  We will use linear regression to initialize the model L(t) =  A(t) + (1-  ) F(t) T(t) =  L(t) - L(t-1) ] + (1-  ) T(t-1) F(t+1) = L(t) + T(t)

Holt’s Trend Model First, we’ll initialize the model: L(4) = (9.9)=60.1 T(4) = 9.9

Updating in Holt’s Trend Model 52 L(t) =  A(t) + (1-  ) F(t)   L(5) = 0.3 (52) (70)=64.6 T(t) =  [L(t) - L(t-1) ] + (1-  ) T(t-1) T(5) = 0.4 [64.6 – 60.1] (9.9) = 7.74 F(t+1) = L(t) + T(t) F(6) = =

Updating in Holt’s Trend Model 63   L(6) = 0.3 (63) (72.34)=69.54 T(6) = 0.4 [69.54 – 64.60] (7.74) = 6.62 F(7) = =

Holt’s Model Results Initialization ExPost Forecast

Initialization ExPost Forecast Holt’s Model Results Regression

Practice Problem

Seasonal Model (No Trend)

L(t) =  A(t) / S(t-p) + (1-  ) L(t-1) S(t) =  A(t) / L(t)] + (1-  ) S(t-p) Seasonal Model Exponential Formulas p is the number of periods in a season Quarterly data: p = 4 Monthly data: p = 12 F(t+1) = L(t) * S(t+1-p)

Seasonal Model Initialization S(5) = 0.60 S(6) = 1.00 S(7) = 1.55 S(8) = 0.85 L(8) = 26.5

Seasonal Model Forecasting  

Seasonal Model Forecasting

Practice Problem  

Practice Problem Initialization S(5) = 1.51 S(6) = 0.55 S(7) = 0.72 S(8) = 1.22 L(8) =

Practice Problem Forecasting  

Winter’s Model for Data with Trend and Seasonal Components L(t) =  A(t) / S(t-p) + (1-  )[L(t-1)+T(t-1)] T(t) =  L(t) - L(t-1)] + (1-  ) T(t-1) S(t) =  A(t) / L(t)] + (1-  ) S(t-p) F(t+1) =  L(t) + T(t)] S(t+1-p)

Seasonal-Trend Model Decomposition  To initialize Winter’s Model, we will use Decomposition Forecasting, which itself can be used to make forecasts.

52 Decomposition Forecasting  There are two ways to decompose forecast data with trend and seasonal components: – Use regression to get the trend, use the trend line to get seasonal factors – Use averaging to get seasonal factors, “deseasonalize” the data, then use regression to get the trend. We will use the second method

Decomposition Forecasting  The following data contains trend and seasonal components:

Decomposition Forecasting  The seasonal factors are obtained by the same method used for the Seasonal Model forecast:

Decomposition Forecasting  With the seasonal factors, the data can be deseasonalized by dividing the data by the seasonal factors: Regression on the Deseasonalizeddata will give the trend

Regression Results

Decomposition Forecast  Regression on the deseasonalized data produces the following results: Slope (m) = 7.71 Intercept (b) =  Forecasts can be performed using the following equation [mx + b](seasonal factor) Note that the parentheses do matter!

Decomposition Forecasting  The decomposition data can be used to make forecasts

Winter’s Model Initialization  We can use the decomposition forecast to define the following Smoothing Model parameters:  So from our previous model, we have L(n) = b + m (n) T(n) = m S(j) = S(j-p) L(n) = (7.71) = T(n) = 7.71 S(5) = 0.80 S(6) = 1.35 S(7) = 1.05 S(8) = 0.79

Winter’s Model Example

Practice Problem    Forecast Sales for 2006 using Winter’s Model Use 2003 and 2004 to initialize the model Use:

Practice Problem Initialization

Practice Problem Forecasting

Evaluating Forecasts “Trust, but Verify” Ronald W. Reagan  Computer software gives us the ability to mess up more data on a greater scale more efficiently  While software like SAP can automatically select models and model parameters for a set of data, and usually does so correctly, when the data is important, a human should review the model results  One of the best tools is the human eye

Visual Review  How would you evaluate this forecast?

Forecast Evaluation Initialization ExPost Forecast Where Forecast is Evaluated Do not include initialization data in evaluation

Errors All error measures compare the forecast model to the actual data for the ExPost Forecast region

Errors Measure All error measures are based on the comparison of forecast values to actual values in the ExPost Forecast region—do not include data from initialization.

Bias and MAD

 Bias tells us whether we have a tendency to over- or under- forecast. If our forecasts are “in the middle” of the data, then the errors should be equally positive and negative, and should sum to zero  MAD (Mean Absolute Deviation) is the average error, ignoring whether the error is positive or negative.  Errors are bad, and the closer to zero an error is, the better the forecast is likely to be.  Error measures tell how well the method worked in the ExPost forecast region. How well the forecast will work in the future is uncertain Bias and MAD

Practice

Exp.  = 0.3 Exp.  = 0.8

Absolute vs. Relative Measures  Forecasts were made for two sets of data. Which forecast was better? Data Set 1 Bias = MAD = Data Set 2 Bias = 182 MAD =  What would you say now? Data Set 1Data Set 2

MPE and MAPE  When the numbers in a data set are larger in magnitude, then the error measures are likely to be large as well, even though the fit might not be as “good”  Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE) are relative forms of the Bias and MAD, respectively.  MPE and MAPE can be used to compare forecasts for different sets of data

MPE and MAPE  Mean Percentage Error (MPE)  Mean Absolute Percentage Error (MAPE)

MPE and MAPE Data Set 1

MPE and MAPE Data Set 2

MPE and MAPE Data Set 2Data Set 1

Practice

Tracking Signal  What’s happened in this situation? How could we detect this in an automatic forecasting environment?

Tracking Signal  The tracking signal can be calculated after each actual sales value is recorded. The tracking signal is calculated as:  The tracking signal is a relative measure, like MPE and MAPE, so it can be compared to a set value (typically 4 or 5) to identify when forecasting parameters and/or models need to be changed.

Tracking Signal

TS = -5.78

Practice