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Acounting Assignment For JOHN MUREITHI Only
Chapter 7: Forecasting Time Series Models
Lan Wang
CSU East Bay
Some Time Series Terms
Stationary Data - a time series variable exhibiting no significant upward or downward trend over time.
Moving average
Exponential smoothing
Some Time Series Terms
Nonstationary Data - a time series variable exhibiting a significant upward or downward trend over time.
Regression analysis
Some Time Series Terms
Seasonal Data - a time series variable exhibiting a repeating patterns at regular intervals over time.
Seasonal index
Simple Moving Average
Average random fluctuations in a time series to infer short-term changes in direction
Assumption: future observations will be similar to recent past
Moving average for next period = average of most recent k observations
Moving Average Example
The monthly sales for Telco Batteries, Inc. were as follows:
MONTH SALES
February 21
March 15
April 14
May 13
June 16
July 18
August 20
a. Calculate a 3 month moving average forecast for September
b. Calculate a 2 month moving average forecast for September
c. Which moving average forecast is more accurate?
Moving Average Example
Error Metrics and Forecast Accuracy
Mean absolute deviation (MAD)
Mean square error (MSE)
Mean absolute percentage error (MAPE)
The quality of a forecast depends on how accurate it is in predicting future values of a time series.
8
Telco Batteries Example - continued
Exponential Smoothing
Exponential smoothing model:
Ft+1 = (1 – a )Ft + aAt
= Ft + a (At – Ft )
Ft+1 is the forecast for time period t+1,
Ft is the forecast for period t,
At is the observed value in period t, and
a is a constant between 0 and 1, called the smoothing constant.
Highly effective approach.
10
Exponential Smoothing
The monthly sales for Telco Batteries, Inc. were as follows:
MONTH SALES
February 21
March 15
April 14
May 13
June 16
July 18
August 20
a. Calculate an Exponential Smoothing forecast with alpha = 0.2, for September
b. Calculate an Exponential Smoothing forecast with alpha = 0.3, for September
c. Which Exponential Smoothing forecast is more accurate?
Exponential Smoothing Example - Continued
alpha
Month Sales 0.2 0.3 AD(0.2) AD(0.3) SE(0.2) SE(0.3) APE(0.2) APE(0.3)
February 21 21 21
March 15 21 21 6.00 6.00 36.00 36.00 0.40 0.40
April 14 19.80 19.20 5.80 5.20 33.64 27.04 0.41 0.37
May 13 18.64 17.64 5.64 4.64 31.81 21.53 0.43 0.36
June 16 17.51 16.25 1.51 0.25 2.29 0.06 0.09 0.02
July 18 17.21 16.17 0.79 1.83 0.62 3.34 0.04 0.10
August 20 17.37 16.72 2.63 3.28 6.93 10.75 0.13 0.16
September 17.89 17.71
MAD 3.73 3.53
MSE 18.55 16.45
MAPE 0.25 0.23
AD - Absolute Deviation SE Squared error
APE - Absolute Percentage Error
Practice
Attendance in each time period. Please forecast the attendance using exponential smoothing (alpha=0.4 and 0.6).
Use MAD, MSE as guidance, find the better alpha setting for each forecasting model.
Trend Models
Trend is the long-term sweep or general direction of movement in a time series.
We’ll now consider some nonstationary time series techniques that are appropriate for data exhibiting upward or downward trends.
An Example
WaterCraft Inc. is a manufacturer of personal water crafts (also known as jet skis).
The company has enjoyed a fairly steady growth in sales of its products.
The officers of the company are preparing sales and manufacturing plans for the coming year.
Forecasts are needed of the level of sales that the company expects to achieve each quarter.
Forecasting Models With Linear Trends
Double Moving Average
Double Exponential Smoothing
Based on the linear trend equation (simple linear regression with time as the independent variable.
Autoregressive models
Linear Trend Model Example
Quarter Guests (in thousands)
Winter 2003 73
Spring 2003 104
Summer 2003 168
Fall 2003 74
Winter 2004 65
Spring 2004 82
Summer 2004 124
Fall 2004 52
Winter 2005 89
Spring 2005 146
Summer 2005 205
Fall 2005 98
Attendance at Orlando’s newest Disneylike attraction, Vacation World, are as shown in the table
Develop a regression equation that models the trend in the data.
Calculate the attendance forecast for year 2006 using the regression equation developed.
Model with Trend & Seasonality
Seasonality is a regular, repeating pattern in time series data.
May be additive or multiplicative in nature…
Multiplicative time series model is commonly used as shown below:
Y = T*S
where Y = actual value of time series
T = trend component
S = seasonal component
The goal of the time series decomposition method is to identify the values of components of a time series (trend, cyclical, seasonal, irregular), and use these components for forecasting re-composition of the model.
Seasonal Index Example
Quarter Guests (in thousands)
Winter 2003 73
Spring 2003 104
Summer 2003 168
Fall 2003 74
Winter 2004 65
Spring 2004 82
Summer 2004 124
Fall 2004 52
Winter 2005 89
Spring 2005 146
Summer 2005 205
Fall 2005 98
Attendance at Orlando’s newest Disneylike attraction, Vacation World, are as shown in the table
Compute the seasonal index using the data
Calculate the seasonal forecast for the year 2006.
Get the linear trends in 2006
Methods available
1. Use excel function
=Tend(Ys, Xs, Period No.)
See Disney.xls
2. Run regression to read intercept and coefficient.
3. Fit a trend line while plotting out the data
Y=3.87X+81.49
Then, compute Seasonal Indices
2003 2004 2005 Quarter Average Seasonally Index
Winter 73 65 89 75.6667 0.7094
Spring 104 82 146 110.6667 1.0375
Summer 168 124 205 165.6667 1.5531
Fall 74 52 98 74.6667 0.7000
106.6667
Finally, get the Forecasts
Forecast = Linear Trend * Corresponding Seasonal Index
E. g., in fall 2003, or 4th period, the average index for fall is 0.7, S4=0.7;
the linear trend for 4th period is : L4=2.87*4+81.49=97
Therefore, F4=L4*S4=97*0.7=68
Final Case Study Forecasting in Hospital Example
Forecasting in Hospital
The number of nurses needed in Hayward Hospital’s surgical division varies from quarter to quarter. This variation causes the hospital difficulty in hiring and scheduling nurses in the surgical division. It seems to the operations manager at the hospital that there are always either too many nurses or not enough nurses scheduled to do the work in the surgical divan from quarter to quarter. Furthermore, nurses cannot be shifted to and from other departments due to the special surgical training required in the wing and because of an understanding with the nurses union. If too many nurses are scheduled, the salary expense and fringe benefits are too high and personnel problems seem to increase. On the other hand, if too few are scheduled overtime must be worked, increasing overhead costs and angering doctors.
The operations staff has been using a simple rule to schedule nurses. The average of the number of nurses needed in the past four quarters is the number scheduled to work next quarter. The operations manager wonders if there is a better way to forecast the number of nurses needed. She has had an operations analyst prepare historical data for the past three years with the number of nurses needed in the surgical division:
Forecasting in Hospital
Year
Quarter
Period
# of
nurses
1997
I
1
14
II
2
10
III
3
6
IV
4
14
1998
I
5
16
II
6
14
III
7
11
IV
8
12
1999
I
9
15
II
10
13
III
11
10
IV
12
18
Hospital Example (K=3)
Hospital Example (Alpha = 0.2 )
Hospital Example: Linear Trend Projection
Regression line (Obtained from Excel)
Demand(y) = 10.8636 + 0.2902 * Time(x)
Hospital Example: Linear Trend Projection
Computing Multiplicative Seasonal Indices
Hospital Example
Quarter
1997
1998
1999
Quarterly
average
Seasonal
Index
I
14
16
15
15.000
0
1.1765
II
10
14
13
12.3333
0.9673
III
6
11
10
9.0000
0.7059
IV
14
12
18
14.6667
1.1503
Average
12.7500
Seasonal Model - Hospital Example
Year
Quarter
P
eriod
# of
Nurses
Trend
Forecast
Seasonal
Index
Seasonal
Forecast
Error
2
1997
I
1
14
11.2
1.1765
13.1
0.8
II
2
10
11.4
0.9671
11.1
1.1
III
3
6
11.7
0.7059
8.3
5.2
IV
4
14
12.0
1.1505
13.8
0.0
1998
I
5
16
12.3
1.1765
14.5
2.3
II
6
14
12.6
0.9671
12.2
3.3
III
7
11
12.9
0.7059
9.1
3.6
IV
8
12
13.2
1.1505
15.2
10.0
1999
I
9
15
13.5
1.1765
15.9
0.7
II
10
13
13.8
0.9671
13.3
0.1
III
11
10
14.1
0.7059
9.9
0.0
IV
12
18
14.3
1.1505
16.5
2.2
Total
29.4
MSE
2.5
2000
I
13
14.6
1.1765
17.2
II
14
14.9
0.9671
14.4
III
15
15.2
0.7059
10.7
IV
16
15.5
1.1505
17.8
Month
Sales
MA (2)
MA(3)
February
21
March
15
April
14
18
May
13
14.5
16.7
June
16
13.5
14.0
July
18
14.5
14.3
August
20
17
15.7
September
19
18.0
n
F
A
=
MAD
n
1
=
i
t
t
å
-
(
)
n
F
A
=
MSE
n
1
=
i
2
t
t
å
-
n
=
MAPE
n
1
=
i
å
-
t
t
t
A
F
A
Month
Period
Sales
MA (2)
Error2
Abs.
Deviation
MA(3)
Error2
Abs.
Deviation
February
1
21
March
2
15
April
3
14
18
16
4
May
4
13
14.5
2.25
1.5
16.7
13.4
3.7
June
5
16
13.5
6.25
2.5
14.0
4.0
2.0
July
6
18
14.5
12.25
3.5
14.3
13.4
3.7
August
7
20
17
9
3
15.7
18.8
4.3
September
8
19
18.0
MAD
2.9
3.42
MSE
9.15
12.42
tF
10
bb
t
0
5
10
15
20
123456789101112
Time Period
Number of Nurses
Chart1
14
10
6
14
16
14
11
12
15
13
10
18
Number of Nurses
Time Period
Number of Nurses
Sheet1
Year Quarter Period Number of Nurses
1996 I 1 14
II 2 10
III 3 6
IV 4 14
1997 I 5 16
II 6 14
III 7 11
IV 8 12
1998 I 9 15
II 10 13
III 11 10
IV 12 18
Sheet1
0
0
0
0
0
0
0
0
0
0
0
0
Number of Nurses
Time Period
Number of Nurses
Sheet2
Sheet3
Year
Quarter
Period
# of
nurses
MA(
3
)
Forecast
Error
Errors
2
Absolute
deviation
1997
I
1
14
II
2
10
III
3
6
IV
4
14
10
4
16
4
1998
I
5
16
10
6
36
6
II
6
14
12
2
4
2
III
7
11
14.7
-
3.7
13.44
3.7
IV
8
12
13.7
-
1.7
2.78
1.7
1999
I
9
15
12.3
2.7
7.11
2.7
II
10
13
12.7
0.3
0.11
0.3
III
11
10
13.3
-
3.3
11.11
3.3
IV
12
18
12.7
5.3
28.44
5.3
Total
119
29
MSE
13.22
MAD
3.22
2000
I
13
13.7
II
14
III
15
IV
16
Year
Quarter
Period
# of nurses
MA(3)
Forecast
Error
Errors2
Absolute deviation
1997
I
1
14
II
2
10
III
3
6
IV
4
14
10
4
16
4
1998
I
5
16
10
6
36
6
II
6
14
12
2
4
2
III
7
11
14.7
-3.7
13.44
3.7
IV
8
12
13.7
-1.7
2.78
1.7
1999
I
9
15
12.3
2.7
7.11
2.7
II
10
13
12.7
0.3
0.11
0.3
III
11
10
13.3
-3.3
11.11
3.3
IV
12
18
12.7
5.3
28.44
5.3
Total
119
29
MSE
13.22
MAD
3.22
2000
I
13
13.7
II
14
III
15
IV
16
Year
Quarter
Period
# of
Nurses
ES
(
a
=0.2)
Forecast
Error2
Abs
deviation
1997
I
1
14
II
2
10
14.00
16.00
4.00
III
3
6
13.20
51.84
7.20
IV
4
14
11.76
5.02
2.24
1998
I
5
16
12.21
14.38
3.79
II
6
14
12.97
1.07
1.03
III
7
11
13.17
4.72
2.17
IV
8
12
12.74
0.55
0.74
1999
I
9
15
12.59
5.80
2.41
II
10
13
13.07
0.01
0.07
III
11
10
13.06
9.35
3.06
IV
12
18
12.45
30.84
5.55
Total
MSE
12.69
MAD
2.93
2000
I
13
13.56
II
1
4
III
15
IV
16
Year
Quarter
Period
# of Nurses
ES (=0.2)
Forecast
Error2
Abs deviation
1997
I
1
14
II
2
10
14.00
16.00
4.00
III
3
6
13.20
51.84
7.20
IV
4
14
11.76
5.02
2.24
1998
I
5
16
12.21
14.38
3.79
II
6
14
12.97
1.07
1.03
III
7
11
13.17
4.72
2.17
IV
8
12
12.74
0.55
0.74
1999
I
9
15
12.59
5.80
2.41
II
10
13
13.07
0.01
0.07
III
11
10
13.06
9.35
3.06
IV
12
18
12.45
30.84
5.55
Total
MSE
12.69
MAD
2.93
2000
I
13
13.56
II
14
III
15
IV
16
Year Quarter Period
(X)
# of Nurses
(Y)
1997 I 1 14
II 2 10
III 3 6
IV 4 14
1998 I 5 16
II 6 14
III 7 11
IV 8 12
1999 I 9 15
II 10 13
III 11 10
IV 12 18
Average 6.5 12.75
Total 78 153
Year
Quarter
Period (X)
# of Nurses (Y)
1997
I
1
14
II
2
10
III
3
6
IV
4
14
1998
I
5
16
II
6
14
III
7
11
IV
8
12
1999
I
9
15
II
10
13
III
11
10
IV
12
18
Average
6.5
12.75
Total
78
153
Year
Quarter
Period
# of Nurses
Forecast
Error2
Abs
deviation
1997
I
1
14
11.15
8.10
2.85
II
2
10
11.44
2.09
1.44
III
3
6
11.73
32.88
5.73
IV
4
14
12.02
3.90
1.98
1998
I
5
16
12.31
13.58
3.69
II
6
14
12.60
1.95
1.40
III
7
11
12.90
3.59
1.90
I
V
8
12
13.19
1.40
1.19
1999
I
9
15
13.48
2.32
1.52
II
10
13
13.77
0.59
0.77
III
11
10
14.06
16.45
4.06
IV
12
18
14.35
13.35
3.65
MSE/MAD
8.35
2.51
2000
I
13
14.64
II
14
14.93
III
15
15.22
IV
16
15.51
Year
Quarter
Period
# of Nurses
Forecast
Error2
Abs deviation
1997
I
1
14
11.15
8.10
2.85
II
2
10
11.44
2.09
1.44
III
3
6
11.73
32.88
5.73
IV
4
14
12.02
3.90
1.98
1998
I
5
16
12.31
13.58
3.69
II
6
14
12.60
1.95
1.40
III
7
11
12.90
3.59
1.90
IV
8
12
13.19
1.40
1.19
1999
I
9
15
13.48
2.32
1.52
II
10
13
13.77
0.59
0.77
III
11
10
14.06
16.45
4.06
IV
12
18
14.35
13.35
3.65
MSE/MAD
8.35
2.51
2000
I
13
14.64
II
14
14.93
III
15
15.22
IV
16
15.51
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