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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.

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.

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

III

1998

III

1999

III

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

15.000

1.1765

12.3333

0.9673

III

9.0000

0.7059

14.6667

1.1503

Average

12.7500

Seasonal Model - Hospital Example

Year

Quarter

eriod

# of

Nurses

Trend

Forecast

Seasonal

Index

Seasonal

Forecast

Error

1997

11.2

1.1765

13.1

0.8

11.4

0.9671

11.1

1.1

III

11.7

0.7059

8.3

5.2

12.0

1.1505

13.8

0.0

1998

12.3

1.1765

14.5

2.3

12.6

0.9671

12.2

3.3

III

12.9

0.7059

9.1

3.6

13.2

1.1505

15.2

10.0

1999

13.5

1.1765

15.9

0.7

13.8

0.9671

13.3

0.1

III

14.1

0.7059

9.9

0.0

14.3

1.1505

16.5

2.2

Total

29.4

MSE

2.5

2000

14.6

1.1765

17.2

14.9

0.9671

14.4

III

15.2

0.7059

10.7

15.5

1.1505

17.8

Month

Sales

MA (2)

MA(3)

February

March

April

May

14.5

16.7

June

13.5

14.0

July

14.5

14.3

August

15.7

September

18.0

MAD

(

)

MSE

MAPE

Month

Period

Sales

MA (2)

Error2

Abs.

Deviation

MA(3)

Error2

Abs.

Deviation

February

March

April

May

14.5

2.25

1.5

16.7

13.4

3.7

June

13.5

6.25

2.5

14.0

4.0

2.0

July

14.5

12.25

3.5

14.3

13.4

3.7

August

15.7

18.8

4.3

September

18.0

MAD

2.9

3.42

MSE

9.15

12.42



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(

)

Forecast

Error

Errors

Absolute

deviation

1997

III

1998

III

14.7

3.7

13.44

3.7

13.7

1.7

2.78

1.7

1999

12.3

2.7

7.11

2.7

12.7

0.3

0.11

0.3

III

13.3

3.3

11.11

3.3

12.7

5.3

28.44

5.3

Total

119

MSE

13.22

MAD

3.22

2000

13.7

III

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

(

=0.2)

Forecast

Error2

Abs

deviation

1997

14.00

16.00

4.00

III

13.20

51.84

7.20

11.76

5.02

2.24

1998

12.21

14.38

3.79

12.97

1.07

1.03

III

13.17

4.72

2.17

12.74

0.55

0.74

1999

12.59

5.80

2.41

13.07

0.01

0.07

III

13.06

9.35

3.06

12.45

30.84

5.55

Total

MSE

12.69

MAD

2.93

2000

13.56

III

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

11.15

8.10

2.85

11.44

2.09

1.44

III

11.73

32.88

5.73

12.02

3.90

1.98

1998

12.31

13.58

3.69

12.60

1.95

1.40

III

12.90

3.59

1.90

13.19

1.40

1.19

1999

13.48

2.32

1.52

13.77

0.59

0.77

III

14.06

16.45

4.06

14.35

13.35

3.65

MSE/MAD

8.35

2.51

2000

14.64

14.93

III

15.22

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