FORECASTING
I. Introduction
-- What is forecasting?
· Scientific (educated) guess
· Based on past data or experience
· Rarely perfect
· Group forecast more accurate
· Shorter time horizon, more accurate
-- Why forecasting?
-- Forecasting time horizon
· Short range – usually less than 3 months
· Medium range – 3 months to 3 years
· Long range – over 3 years
-- Which forecasting method to use?
· Time horizon
· Costs vs. accuracy
· Easy to understand?
-- Pattern of Data
· Trend – gradual upward or downward movement
· Seasonality – repeated pattern
· Cycle – background pattern over long period of time
· Randomness – unexplainable, unpredictable, unknown fluctuations
II. Qualitative Forecasting Methods
- no past data is available, for managerial decision, long term forecast
· Jury of executive opinion
· The Delphi technique – systematic survey of experts
· Sales force composite
· Consumer market survey
III. Quantitative Forecasting Methods
1) Smoothing Techniques:
- for data with no clear pattern, short term forecast
· Naive: Ft+1 = Yt
· n Period Moving Average (MA-n): equal weight for n data
Forecast = S(most recent n data)/n
· n Period Weighted Moving Average (WMA-n): flexible weight assignment
Forecast = S(weight * most recent n data)/ S (weight)
· Exponential Smoothing with smoothing constant a: very little data record required, more weight on recent data, weight decreases exponentially
Ft+1 = Ft + a(Yt - Ft) = aYt + (1-a)Ft 0 a < 1,
-- Example: Forecast the score for test 5 by the above smoothing techniques with a = 0.1 and F1 = 77.
Test / 1 / 2 / 3 / 4 / 5Avg.Score / 72 / 82 / 85 / 90 / ?
-- Relationship between smoothing techniques
· average of past data, but differ in weight distribution
· all methods lag behind actual data change
· more weight on most recent data, more responsive, less smooth
-- Thinking challenge:
1. Which method requires the least amount of data?
2. What happens if the a in the exponential smoothing method increases?
2) Trend Projection by Linear Regression
- for data with linear trend pattern, medium term
Step 1. Find a trend line y = a + bx to fit the past data the best (minimizing mean squared errors)
· best slope:
· best intercept:
where and are averages of x’s and y’s
-- Example continues:
X / y / xy / x2 / y2* Excel commands:
· calculating b:“=slope(range of y’s, range of x’s)”
· calculating a:“=intercept(range of y’s, range of x’s)”
Step 2. Trend projection
· Trend projection forecast: Fnext = a + b*xnext
-- Example continues:
Step 3. Find forecast interval, if necessary
· Forecast interval = Fnext +- Z(1+ a)/2 *Sy,x
· standard error: Sy,x =
-- Example continues:
* Excel commands:
· calculating Sy,x:”=steyx(range of y’s, range of x’s)”
3) Causal Model by Linear Regression
- same set up as trend projection, except that x is an independent variable (other than time), y is a dependent variable, medium term
-- Example: The sales manager of a large apartment rental complex feels the demand for apartments may be related to the number of newspaper ads placed during the previous month. She has collected the data shown below.
Ads / 15 / 9 / 40 / 20 / 25 / 25 / 15 / 35Rental / 6 / 4 / 16 / 6 / 13 / 9 / 10 / 16
If the number of ads placed in this month is 30, what would be her estimate of rentals in the coming month?
-- Correlation coefficient r: measures the strength of linear relationship between x and y
· -1 £ r £ +1
· r =
* Excel command:
· calculating r:”=correl(range of y’s, range of x’s)”
-- Interpretation:
· r > 0:
· r = 0:
· r < 0:
· r2 = coefficient of determination: % of variation in the dependent variable (y) is explained by regression equation (linear relationship).
-- Example continued: How strong is the relationship between the ads placed and the rentals?
4) Decomposition of Time Series
- for data with both trend and seasonality patterns, short to medium term
-- Idea:
- Decompose the past data (filter out the seasonal influence from original data)
- Forecast trend pattern and seasonality pattern separately
- Combine the forecasts using the multiplicative model: Yt = Tt * St
-- Example: Data of a popular brand of sweater sale (by quarters) over the past three years:
2000 / Sale / 2001 / Sale / 2002 / Sale(t) / (Yt) / (t) / (Yt) / (t) / (Yt)
1 / 12 / 5 / 16 / 9 / 18
2 / 25 / 6 / 32 / 10 / 45
3 / 76 / 7 / 80 / 11 / 84
4 / 52 / 8 / 62 / 12 / 60
Forecast the sale of each season in 2003.
Step 1. Draw the historical data diagram to check if there is an obvious seasonal pattern
Step 2. Calculate the "seasonal indexes" (S) for each season.
· Seasonal Index = Avg. Seasonal Sale / Avg Sale
(spring) = ===> S(spring) =
(summer) = ===> S(summer) =
(fall) = ===> S(fall) =
(winter) = ===> S(winter) =
=
Step 3. Deseasonalize the original data: Tt = Yt / S
2000 / Sale / 2001 / Sale / 2002 / sale(t) / (Tt) / (t) / (Tt) / (t) / (Tt)
1 / 5 / 9
2 / 6 / 10
3 / 7 / 11
4 / 8 / 12
Step 4. Calculate the trend based on the deseasonalized demand by linear regression Tt = a + bt
a = b =
Step 5. Use trend projection to forecast the demand with trend only.
T13 = T14 =
T15 = T16 =
Step 6. Use seasonal indexes to modify the forecasts to reflect the seasonality patterns: F = T * S
F13 = F14 =
F15 = F16 =
IV. Choosing a Forecasting Method
- all criteria are a function of the forecast error
· forecast error = actual realization (Y)– forecast (F)
1. Bias – measures the direction of forecast
Bias = (Sforecast error)/n
2. Mean Absolute Deviation (MAD)
- same penalty for small and large errors
MAD = (S|forecast error|)/n
3. Mean Squared Error (MSE)
– more penalty on large errors
MSE = (Sforecast error2)/n
4. Mean Absolute Percent Error (MAPE)
- relative error, scale independent
MAPE = (S|forecast error|/Actual Data)/n
-- Comparing different forecasting methods
· Based on past performance
· Measure dependent
· Choose a (for exp. Smoothing) and n (for MA) to minimize MAD, MSE, or MAPE
-- Example: Same data as earlier
Test / Avg Score / Forecast / Err / |Err| / Err Square / |Err|/Actual
1 / 72 / 77 / -5 / 5 / 25 / 0.069444
2 / 82 / 76.5 / 5.5 / 5.5 / 30.25 / 0.067073
3 / 85 / 77.05 / 7.95 / 7.95 / 63.2025 / 0.093529
4 / 90 / 77.845 / 12.155 / 12.155 / 147.744 / 0.135056
Bias / MAD / MSE / MAPE
5.15125 / 7.65125 / 66.54913 / 0.091276
Exp Smoothing with alpha = / 0.2
Test / Avg Score / Forecast / Err / |Err| / Err Square / |Err|/Actual
1 / 72 / 77 / -5 / 5 / 25 / 0.069444
2 / 82 / 76 / 6 / 6 / 36 / 0.073171
3 / 85 / 77.2 / 7.8 / 7.8 / 60.84 / 0.091765
4 / 90 / 78.76 / 11.24 / 11.24 / 126.3376 / 0.124889
Bias / MAD / MSE / MAPE
5.01 / 7.51 / 62.0444 / 0.089817
Trend Projection with a = / 68 / b= / 5.7
Test / Avg Score / Forecast / Err / |Err| / Err Square / |Err|/Actual
1 / 72 / 73.7 / -1.7 / 1.7 / 2.89 / 0.023611
2 / 82 / 79.4 / 2.6 / 2.6 / 6.76 / 0.031707
3 / 85 / 85.1 / -0.1 / 0.1 / 0.01 / 0.001176
4 / 90 / 90.8 / -0.8 / 0.8 / 0.64 / 0.008889
Bias / MAD / MSE / MAPE
0 / 1.3 / 2.575 / 0.016346
V. Control of the Forecasting Process
- to check if the performance of a forecasting method changes
-- Tracking Signal control chart:
· plotting statistic TS(t) = RSFE(t)/MAD(t)
· RSFE(t) = running sum of the forecast error through time t
· MAD(t) = MAD through time t
· 3 sigma control chart limits: UCL = 3, LCL = -3, CL = 0
-- Interpretation?
-- Example: A forecasting method provides the following forecasts in the past 6 periods. Does its performance change over time?
T / Y / F / Error / RFSE(t) / MAD(t) / TS(t)1 / 8 / 6
2 / 7 / 9
3 / 10 / 6
4 / 2 / 6
5 / 12 / 8
6 / 11 / 7