index for that season
Steps in the process:
Seasonal Index Example
Jan 80 85 105 90 94
Feb 70 85 85 80 94
Mar 80 93 82 85 94
Apr 90 95 115 100 94
May 113 125 131 123 94
Jun 110 115 120 115 94
Jul 100 102 113 105 94
Aug 88 102 110 100 94
Sept 85 90 95 90 94
Oct 77 78 85 80 94
Nov 75 72 83 80 94
Dec 82 78 80 80 94
Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index
Seasonal Index Example
Jan 80 85 105 90 94
Feb 70 85 85 80 94
Mar 80 93 82 85 94
Apr 90 95 115 100 94
May 113 125 131 123 94
Jun 110 115 120 115 94
Jul 100 102 113 105 94
Aug 88 102 110 100 94
Sept 85 90 95 90 94
Oct 77 78 85 80 94
Nov 75 72 83 80 94
Dec 82 78 80 80 94
Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index
0.957 Seasonal index = average 2005-2007 monthly demand
average monthly demand
= 90/94 = .957
Seasonal Index Example
Jan 80 85 105 90 94 0.957
Feb 70 85 85 80 94 0.851
Mar 80 93 82 85 94 0.904
Apr 90 95 115 100 94 1.064
May 113 125 131 123 94 1.309
Jun 110 115 120 115 94 1.223
Jul 100 102 113 105 94 1.117
Aug 88 102 110 100 94 1.064
Sept 85 90 95 90 94 0.957
Oct 77 78 85 80 94 0.851
Nov 75 72 83 80 94 0.851
Dec 82 78 80 80 94 0.851
Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index
Seasonal Index Example
Jan 80 85 105 90 94 0.957
Feb 70 85 85 80 94 0.851
Mar 80 93 82 85 94 0.904
Apr 90 95 115 100 94 1.064
May 113 125 131 123 94 1.309
Jun 110 115 120 115 94 1.223
Jul 100 102 113 105 94 1.117
Aug 88 102 110 100 94 1.064
Sept 85 90 95 90 94 0.957
Oct 77 78 85 80 94 0.851
Nov 75 72 83 80 94 0.851
Dec 82 78 80 80 94 0.851
Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index
Expected annual demand = 1,200
Jan 1,20012 x .957 = 96
Feb 1,20012 x .851 = 85
Forecast for 2008
Seasonal Index Example
140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 –
| | | | | | | | | | | |
J F M A M J J A S O N D
Time
Demand
2008 Forecast 2007 Demand 2006 Demand 2005 Demand
San Diego Hospital
10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 –
9,000 –| | | | | | | | | | | |
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
67 68 69 70 71 72 73 74 75 76 77 78
Month
Inpatient Days
9530
9551
9573
9594
9616
9637
9659
9680
9702
9724
9745
9766 Trend Data
San Diego Hospital
1.06 – 1.04 – 1.02 – 1.00 – 0.98 – 0.96 – 0.94 –
0.92 –| | | | | | | | | | | |
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
67 68 69 70 71 72 73 74 75 76 77 78
Month
Index for Inpatient Day
s1.04
1.02 1.01
0.99
1.03 1.04
1.00
0.98 0.97
0.99
0.97 0.96
Seasonal Indices
San Diego Hospital
10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 –
9,000 –| | | | | | | | | | | |
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
67 68 69 70 71 72 73 74 75 76 77 78
Month
Inpatient Days 9911
9265 9764
9520 9691
9411 9949
9724
9542
9355 10068
9572 Combined Trend and Seasonal Forecast
Associative Forecasting
Used when changes in one or more independent variables can be used to predict the changes in the
dependent variable
Most common technique is linear regression analysis
We apply this technique just as we did in the time series example
Associative Forecasting
Forecasting an outcome based on predictor variables using the least squares technique
y = a + bx^
where y = computed value of the variable to be predicted (dependent variable)
a = y-axis intercept
b = slope of the regression line x = the independent variable though to predict the value of the
dependent variable
^
Associative Forecasting Example
Sales Local Payroll ($ millions), y ($ billions), x
2.0 1
3.0 3
2.5 4
2.0 2
2.0 1
3.5 7
4.0 – 3.0 – 2.0 – 1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll
Associative Forecasting Example
Sales, y Payroll, x x2 xy
2.0 1 1 2.0
3.0 3 9 9.0
2.5 4 16 10.0
2.0 2 4 4.0
2.0 1 1 2.0
3.5 7 49 24.5
∑y = 15.0 ∑x = 18 ∑x2 = 80 ∑xy = 51.5
x = ∑x/6 = 18/6 = 3 y = ∑y/6 = 15/6 = 2.5
b = = = .25∑xy - nxy
∑x2 - nx2
51.5 - (6)(3)(2.5) 80 - (6)(32)
a = y - bx = 2.5 - (.25)(3) = 1.75
Associative Forecasting Example
4.0 – 3.0 – 2.0 – 1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll
y = 1.75 + .25x^ Sales = 1.75 + .25(payroll)
If payroll next year is estimated to be $6 billion, then:
Sales = 1.75 + .25(6) Sales = $3,250,000
3.25
Standard Error of the Estimate
þ A forecast is just a point estimate of a future value
þ This point is actually the mean of a probability distribution
4.0 – 3.0 – 2.0 – 1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll 3.25
Standard Error of the Estimate
where y = y-value of each data point
yc = computed value of the dependent variable, from the
regression equation
n = number of data points
Sy,x = ∑(y - yc)2 n - 2
Standard Error of the Estimate
Computationally, this equation is considerably easier to use
We use the standard error to set up prediction intervals around the point
estimate
Sy,x = ∑y2 - a∑y - b∑xy n - 2
Standard Error of the Estimate
4.0 – 3.0 – 2.0 – 1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll 3.25
Sy,x = =∑y2 - a∑y - b∑xy n - 2
39.5 - 1.75(15) - .25(51.5) 6 - 2
Sy,x = .306
The standard error of the estimate is $306,000 in sales
þ How strong is the linear relationship between the variables?
þ Correlation does not necessarily imply causality!
þ Coefficient of correlation, r, measures degree of association
þ Values range from 1 to +1
Correlation
Correlation Coefficient
r = n Σξψ − ΣξΣψ
[νΣξ2 − (Σξ)2][νΣψ2 − (Σψ)2]
Correlation Coefficient
r = n Σξψ − ΣξΣψ
[νΣξ2 − (Σξ)2][νΣψ2 − (Σψ)2]
y
(a) Perfect x
positive correlation:
r = +1
y
(b) Positive x
correlation:
0 < r < 1
y
(c) No x
correlation:
r = 0
y
(d) Perfect x
negative correlation:
r = -1
þ Coefficient of Determination, r2, measures the
percent of change in y predicted by the change in x
þ Values range from 0 to 1
þ Easy to interpret
Correlation
For the Nodel Construction example:
r = .901 r2 = .81
Multiple Regression Analysis
If more than one independent variable is to be used in the model, linear regression can be extended to
multiple regression to accommodate several independent variables
y = a + b1x1 + b2x2 … ^
Computationally, this is quite complex and generally done on the computer
Multiple Regression Analysis
y = 1.80 + .30x1 - 5.0x2^
In the Nodel example, including interest rates in the model gives the new equation:
An improved correlation coefficient of r = .96 means this model does a better job of predicting the change in construction sales
Sales = 1.80 + .30(6) 5.0(.12) = 3.00 Sales = $3,000,000
þ Measures how well the forecast is predicting actual values
þ Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)
þ Good tracking signal has low values
þ If forecasts are continually high or low, the forecast has a bias error
Monitoring and Controlling Forecasts
Tracking Signal
Monitoring and Controlling Forecasts
Tracking
signal RSFE
= MAD
Tracking
signal =
∑(Actual demand in period i -
Forecast demand in period i)
( |Αχτυαλ − Φορεχαστ|/ν) ∑
Tracking Signal
Tracking signal +
0 MADs –
Upper control limit
Lower control limit
Time
Signal exceeding limit
Acceptable range
Tracking Signal Example
Cumulative Absolute Absolute
Actual Forecast Forecast Forecast
Qtr Demand Demand Error RSFE Error Error MAD
1 90 100 -10 -10 10 10 10.0
2 95 100 -5 -15 5 15 7.5
3 115 100 +15 0 15 30 10.0
4 100 110 -10 -10 10 40 10.0
5 125 110 +15 +5 15 55 11.0
6 140 110 +30 +35 30 85 14.2
Cumulative Absolute Absolute
Actual Forecast Forecast Forecast
Qtr Demand Demand Error RSFE Error Error MAD
1 90 100 -10 -10 10 10 10.0
2 95 100 -5 -15 5 15 7.5
3 115 100 +15 0 15 30 10.0
4 100 110 -10 -10 10 40 10.0
5 125 110 +15 +5 15 55 11.0
6 140 110 +30 +35 30 85 14.2
Tracking Signal Example
Tracking Signal (RSFE/MAD)
-10/10 = -1 -15/7.5 = -2
0/10 = 0 -10/10 = -1 +5/11 = +0.5 +35/14.2 = +2.5
The variation of the tracking signal between 2.0 and +2.5 is within acceptable limits
Adaptive Forecasting
It’s possible to use the computer to
continually monitor forecast error and adjust the values of the α and β coefficients used in exponential smoothing to continually
minimize forecast error
This technique is called adaptive smoothing
Focus Forecasting
Developed at American Hardware Supply, focus forecasting is based on two principles: