# Forecasting by gjjur4356

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```									          LESSON 5: FORECASTING
STATIONARY TIME SERIES METHODS

Outline

•   Simple Moving Average
•   Weighted Moving Average
•   Exponential Smoothing
•   Comparison of Simple Moving Average and
Exponential Smoothing
Time Series Methods

• In this lesson we shall discuss some time series
forecasting methods. All methods discussed in this lesson
are designed for stationary series. Recall from the
previous lesson that a stationary series contains only the
average and no trend, seasonality, cyclicity, etc.
• No method is superior to any other method in every
context. In a particular context, various methods can be
used and evaluated using a suitable measure (e.g., MAD,
MSE, MAPE etc.) discussed in the previous lesson. Then,
it is possible to use the method that works best in that
context. See the Taco Bell example.
• A comparison among the methods is done near the end of
the lesson.
Time Series Methods

• All these methods will be illustrated with the following
example: Suppose that a hospital would like to forecast
the number of patients arrival from the following historical
data:
Week               Patients Arrival
1                       400
2                       380
3                       411
4                       415
• Note: Although week 4 data is given, some methods
require that forecast for period 4 is first computed before
computing forecast for period 5.
Time Series Methods
Simple Moving Average
A moving average of order N is simply the
arithmetic average of the most recent N
450 —   observations. For 3-week moving averages N=3;
for 6-week moving averages N=6; etc.
430 —
Patient arrivals

410 —

390 —

370 —             Actual patient
arrivals

|       |         |     |     |      |
0         5      10        15    20    25     30
Week
Time Series Methods
Simple Moving Average

450 —                           Patient
Week    Arrivals
430 —                     1         400
2         380
Patient arrivals

410 —                     3         411

390 —       Given 3-week data, one-step-ahead forecast
370 —
for week 4 or two-step-ahead forecast for
week 5 is simply the arithmetic average of
the first 3-week data
|        |      |       |          |    |
0     5       10     15      20         25   30
Week
Time Series Methods
Simple Moving Average

450 —                        Patient
Week     Arrivals
430 —                 1         400
2         380
Patient arrivals

410 —                 3         411

390 —              One - step - ahead forecast
370 —              for week4
F4 
|     |     |       |          |          |
0     5    10    15      20         25         30
Week
Time Series Methods
Simple Moving Average

450 —                        Patient
Week     Arrivals
430 —                 1         400
2         380
Patient arrivals

410 —                 3         411

390 —              Two- step - ahead forecast
370 —              for week5
F5 
|     |     |       |          |         |
0     5    10    15      20         25        30
Week
Time Series Methods
Simple Moving Average
One-step-ahead forecast for week 5 is computed
from the arithmetic average of weeks 2, 3 and 4
450 —   data                          Patient
Week     Arrivals
430 —                         2         380
3         411
Patient arrivals

410 —                         4         415

390 —                      One - step - ahead forecast
370 —                      for week5
F5 
|       |       |       |          |          |
0         5      10      15      20         25         30
Week
Time Series Methods
Simple Moving Average

450 —   3-week MA
forecast
430 —
Patient arrivals

410 —

390 —

370 —         Actual patient
arrivals

|       |         |     |    |    |
0     5      10        15    20   25   30
Week
Time Series Methods
Simple Moving Average

450 —   3-week MA              6-week MA
forecast               forecast
430 —
Patient arrivals

410 —

390 —

370 —         Actual patient
arrivals

|       |         |          |      |    |
0     5      10        15         20     25   30
Week
Taco Bell determined
that the demand for
each 15-minute interval
can be estimated from a
6-week simple moving
average of sales.

The forecast was used
to determine the
number of employees
needed.
Time Series Methods
Weighted Moving Average

In the simple moving average method each of the N
periods is equally important for the purpose of
forecasting.
Weighted moving average is more general than the
simple moving average and assigns different weights to
different periods. Let,
wt i  weight assigned to period t  i 
Dt i  actual data for period t  i 
i  1,2,, N

Then, the one-step ahead forecast for period t
Ft  wt 1 Dt 1  wt 2 Dt 2    wt  N Dt  N
Time Series Methods
Weighted Moving Average

450 —    3-week MA
forecast    Weighted Moving Average
430 —                       Assigned weights
t-1    0.70
Patient arrivals

410 —                         t-2    0.20
t-3    0.10
390 —
F4 
370 —

|       |     |       |        |       |
0      5      10    15      20       25      30
Week
Time Series Methods
Weighted Moving Average

450 —    3-week MA
forecast     Weighted Moving Average
430 —                       Assigned weights
t-1    0.70
Patient arrivals

410 —                         t-2    0.20
t-3    0.10
390 —
F5 
370 —

|       |     |       |        |       |
0      5      10    15      20       25      30
Week
Time Series Methods
Exponential Smoothing

• Exponential smoothing method computes a forecast
value which is the weighted average of the most
recent data and forecast values.
• The weight assigned to the most recent data is called
the smoothing constant,  and the weight assigned to
the most recent forecast is (1- ).
• The method requires an initial forecast value. The
initial forecast value may be obtained by some other
forecasting technique.
• If the smoothing constant,  is large, the forecast
values fluctuate with the actual data. If  is small, the
fluctuation is less.
Time Series Methods
Exponential Smoothing
• The one-step-ahead forecast for period t
Ft  Dt 1  1   Ft 1
• Notice that therefore,
Ft  Dt 1  1   Dt  2  1   Ft  2 
 Dt 1  1   Dt  2  1    Ft  2
2

 Dt 1  1   Dt  2  1    Dt 3  1   Ft 3 
2

 Dt 1  1   Dt  2  1    Dt 3  1    Ft 3
2             3


• With further expansion of the expression for forecast
for period t it can be seen that the forecast for period t
depends on all previous data!!
Time Series Methods
Exponential Smoothing

450 —
Exponential Smoothing
430 —                       = 0.10

Ft = Dt-1 + (1 - )Ft - 1
Patient arrivals

410 —

390 —

370 —

|     |     |          |           |            |
0     5    10    15         20          25           30
Week
Time Series Methods
Exponential Smoothing

450 —
Exponential Smoothing
430 —                       = 0.10

Ft = Dt-1 + (1 - )Ft - 1
Patient arrivals

410 —

390 —                Initial forecast value
F3 = (400 +
370 —                380)/2=390
D3 = 411
|     |     |          |           |            |
0     5    10    15         20          25           30
Week
Time Series Methods
Exponential Smoothing

450 —
Exponential Smoothing
430 —                             = 0.10

Ft = Dt-1 + (1 - )Ft - 1
Patient arrivals

410 —

390 —                       Initial forecast value
F3 = (400 +
370 —                       380)/2=390
D3 = 411
F4 
|     |            |          |           |            |
0     5    10           15         20          25           30
Week
Time Series Methods
Exponential Smoothing

450 —
Exponential Smoothing
430 —                           = 0.10

Ft = Dt + (1 - )Ft - 1
Patient arrivals

410 —

F4 = 392.1
390 —
D4 = 415
370 —
F5 
|     |        |          |          |           |
0     5    10       15         20         25          30
Week
Time Series Methods
Exponential Smoothing

450 —

430 —
Patient arrivals

410 —

390 —

370 —

|     |     |     |      |    |
0     5    10    15    20     25   30
Week
Comparison of Exponential Smoothing and
Simple Moving Average

• Both Methods
– Are designed for stationary demand
– Require a single parameter
– Lag behind a trend, if one exists
– Have the same distribution of forecast error if
  2 /( N  1)
Comparison of Exponential Smoothing and
Simple Moving Average

• Moving average uses only the last N periods data,
exponential smoothing uses all data
• Exponential smoothing uses less memory and requires
fewer steps of computation; store only the most recent
forecast!