TIME SERIES ANALYSIS

FOR

FORECASTING

Rupika Abeynayake

Professor in Applied Statistics

Frequently there is a time lag between awareness of an

impending event or need and occurrence of the event

This lead-time is the main reason for planning and forecasting. If

the lead-time is zero or very small, there is no need for planning

If the lead-time is long, and the outcome of the final event is

conditional on identifiable factors, planning can perform an

important role

In management and administrative situations the need for

planning is great because the lead-time for decision making

ranges from several years to a few days or hours to a few second.

Therefore, forecasting is an important aid in effective and

efficient planning

Introduction

Forecasting is a prediction of what will occur in the future, and it

is an uncertain process

One of the most powerful methodologies for generating

forecasts is time series analysis

A data set containing observations on a single phenomenon

observed over multiple time periods is called time-series. In time

series data, both values and the ordering of the data points have

meaning. For many agricultural products, data are usually

collected over time

Introduction…

Time series analysis and its applications have become

increasingly important in various fields of research, such as

business, economics, agriculture, engineering, medicine, social

sciences, politics etc.

Realization of the fact that “ Time is Money " in business

activities, the time series analysis techniques presented here,

would be a necessary tool for applying to a wide range of

managerial decisions successfully where time and money are

directly related

Introduction

On the time scale we are standing at a certain point called

the point of reference (Yt) and we look backward over past

observations (Yt-1, Yt-2,…,Yt-n+1) and foreword into the future

(Ft+1, Ft+2, …,Ft+m)

Once a forecasting model has been selected, we fit the

model to the known data and obtain the fitted values. For

the known observations this allows calculation fitted errors

(Yt-1- Ft-1)

A measure of goodness-of-fit of the model and as new

observations become available we can examine forecasting

errors (Yt+1- Ft+1)

Forecasting scenario

Measuring Forecast accuracy

n

1t

ten

1MEMean error :

n

1t

t |e|n

1MAEMean absolute error :

Mean squared error :

Percentage error (PE) :

n

1t

2

t |e|n

1MSE

PEt = 100*(Yt - Ft )/Yt

Mean percentage error (MPE) :

n

1t

tPEn

1MPE

Mean absolute percentage error (MAPE) :

n

1t

t |PE|n

1MAPE

Main components of time series data

There are four types of components in time series analysis

Seasonal component (S)

Trend component (T)

Cyclical component (C)

Irregular component ( I )

Yt = St , Tt , Ct , I

Time SeriesYt=S,T,C, I

Seasonal removing using smoothing

Trend removal using regression

Cyclical removing using % ratio I

Moving averages

Smoothing techniques are used to reduce irregularities (random

fluctuations) in time series data

Moving averages rank among the most popular techniques for the

preprocessing of time series. They are used to filter random "white

noise" from the data, to make the time series smoother

There are several methods of Moving averages

Simple Moving Averages

Double moving averages

Centered Moving Average

Weighted Moving Average

Simple Moving Averages

Moving Averages (MA) is effective and efficient approach provided

the time series is stationary in both mean and variance

The simple moving average required an odd number of

observations to be included in each average at the middle of the

data value being averaged

Takes a certain number of past periods and add them together;then divide by the number of periods gives the simple movingaverage

The following formula is used in finding the moving average oforder n, MA(n) for a period t+1

MAt+1 = [Yt + Yt-1 + ... +Yt-n+1] / n

Month Time period

Observed values

Three-month moving

average 3 MA

Five-month moving

average 5 MA

Jan 1 266.0 - -

Fab 2 145.9 --

-

-Mar 3 183.1

Apr 4 119.3 198.333

149.433

160.900

156.033

193.533

208.267

216.367

180.067

217.400

215.100

238.900

-

May 5 180.3 -

Jun 6 168.5

Jul 7 231.8

Aug 8 224.5

Sep 9 192.8

Oct 10 122.9

Nov 11 336.5

Dec 12 185.9

Jan 13 194.3 .

Feb 14 149.5 .

Example

178.92

159.42

176.60

184.88

199.58

188.10

221.70

212.52

206.48.

Example…

Actual

Smoothed

Forecast

Actual

Smoothed

Forecast

20100

350

250

150

50

Ob

s

Time

MSD:

MAD:

MAPE:

Length:

Moving Average

5165.47

57.39

32.03

3

Moving Average

Example…

Actual

Smoothed

Forecast

Actual

Smoothed

Forecast

20100

300

200

100

Ob

s

Time

MSD:

MAD:

MAPE:

Length:

Moving Average

4496.37

52.74

26.23

5

Moving Average

2.1Example

Month Time period

Observed values

Four-month moving average

4 MA

2 X 4 MA

Jan 1 266.0 -178.6

-

Fab 2 145.9 -

Mar 3 183.1 157.6 -

Apr 4 119.3 162.8 -

May 5 180.3 174.9 -

Jun 6 168.5 201.3

Jul 7 231.8 204.4

Aug 8 224.5 193.0

Sep 9 192.8 219.2

Oct 10 122.9 209.5

Nov 11 336.5 209.9

Dec 12 185.9 216.6

Jan 13 194.3 .

Feb 14 149.5 .

167.863

159.975

168.887

188.125

202.837

198.700

206.088

214.350

209.712

167.863

This method is very powerful with comparing simple moving

averages

Weighted MA(3) can be expressed as,

Weighted MA(3) = w1.Yt + w2.Yt-1 + w3.Yt-2

where w1, w2, & w3 are weights

There are many schemes selecting appropriate weights (Kendall,

Stuart, and Ord (1983)

Weights are any positive numbers such that,

w1 + w2 + w3 = 1

One of the methods of calculating weights is,

w1 = 3/(1 + 2 + 3) = 3/6, w2 = 2/6, and w3 = 1/6

Weighted Moving Average

2.Exponential Smoothing Techniques

One of the most successful forecasting methods is the exponential

smoothing (ES)

ES is an averaging technique that uses unequal weights and assigns

exponentially decreasing weights as the observations get older

There are several exponential smoothing techniques

Single Exponential Smoothing

Holt’s linear method

Holt-Winters’ trend and seasonality method

Single Exponential Smoothing

The method of single exponential forecasting takes the

forecast for the previous period and adjust it using the

forecast error. [(Forecast error = (Yt – Ft)]

Ft+1 = Ft + a (Yt - Ft)

Ft+1 = a Yt + (1 - a) Ft

where:

Yt is the actual value

Ft is the forecasted value

a is the weighting factor, which ranges from 0 to 1

t is the current time period

Choosing the Best Value for Parameter a (alpha)

In practice, the smoothing parameter is often chosen by a grid

search of the parameter space

That is, different solutions for = 0.1 to are tried starting with

0.9, with increments of 0.1.

Then is chosen so as to produce the smallest sums of squares

(or mean squares) for the residuals.

Month Timeperiod

Observedvalues

Exponentially Smoothed values

a = 0.1 a = 0.5 a = 0.9Jan 1 200.0

Feb 2 135.0 200.0 200.0 200.0

Mar 3 195.0 193.5 167.5 141.5

Apr 4 197.5 193.7 181.3 189.7

May 5 310.0 194.0 189.4 196.7

Jun 6 175.0 205.6 249.7 298.7

Jul 7 155.0 202.6 212.3 187.4

Aug 8 130.0 197.8 183.7 158.2

Sep 9 220.0 191.0 156.8 132.8

Oct 10 277.5 193.9 188.4 211.3

Nov 11 235.0 202.3 233.0 270.9

Dec 12 - 205.6 234.0 238.6

Analysis of Errors (Test period : 2 – 11)

a = 0.1 a = 0.5 a = 0.9

Mean Error 5.56 6.80 4.29

Mean Absolute Error 47.76 56.94 61.32

Mean Absolute percentage Error (MAPE)

24.58 29.20 30.81

Mean Square Error (MSE) 3438.33 4347.24 5039.37

Theil’s U-statistics 0.81 0.92 0.98

Time Series Plots

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12

Month

Sh

ipm

en

ts Observed values

SES = 0.1

SES = 0.5

SES = 0.9

Holt’s linear method

Holt (1957) extended single exponential smoothing to linear

exponential smoothing to allow forecasting of data with trends

The forecast for Hollt’s linear exponential smoothing is found using

two smoothing constants, a and (values between 0 & 1), and

three equations

))(1( 11 tttt bLYL aa

11 )1()( tttt bLLb

mbLF ttmt

Smoothing of data

Smoothing of trend

Forecast for m period a head

The initialization process : L1 = Y1 and b1 = Y2 – Y1 or b1 = (Y4-Y1) / 3

Holt-Winters’ trend and seasonality method

Holt’s method was extended by Winters (1960) to capture

seasonality

The Holt-Winter’s method is based on three smoothing

equations, one for level, one for trend, and one for seasonality

))(1( 11

tt

st

t

t bLS

YL aa

11 )1()( tttt bLLb

st

t

t

t SL

YS )1(

mstttmt SmbLF )(

Level

Trend

Seasonal

Forecast

Example…

Actual

Predicted

Forecast

Actual

Predicted

Forecast

150100500

700

600

500

400

300

200

100

Yt

Time

MSD:

MAD:

MAPE:

Delta (season):

Gamma (trend):

Alpha (level):

Smoothing Constants

356.695

12.557

5.163

0.200

0.200

0.200

Winters' Multiplicative Model for Yt

Season Sales Average Sales Seasonal

Factor

Spring 200 250 200/250 = 0.8

Summer 350 250 350/250 = 1.4

Fall 300 250 300/250 = 1.2

Winter 150 250 150/250 = 0.6

Total 1000 1000

Seasonal Factor

Ratio-to-moving-average

Season Average Sales

(1100/4)

Next Year

Forecast

FORECAST

Spring 275 275*0.8 220

Summer 275 275*1.4 385

Fall 275 275*1.2 330

Winter 275 275*0.6 165

Total 1100 1100

If Next year expected sale increment is 10%

The table below represent the Quarterly sales Figures

Year Q1 Q2 Q3 Q4

2008 20 30 39 60

2009 40 51 62 81

2010 50 64 74 85

Time

Period

Quarter Time

index

Sales Centered

MA (4)

Sales/MA*100

2008 Q1 1 20

2008 Q2 2 30

2008 Q3 3 39

2008 Q4 4 60 39.750 150.943

2009 Q1 5 40 44.875 89.136

2009 Q2 6 51 50.375 101.241

2009 Q3 7 62 55.875 110.962

2009 Q4 8 81 59.750 135.565

2010 Q1 9 50 62.625 79.840

2010 Q2 10 64 65.750 97.338

2010 Q3 11 74 67.750 109.225

2010 Q4 12 85

Year Q1 Q2 Q3 Q4

2008 150.94

2009 89.14 101.24 110.96 135.56

2010 79.84 97.33 109.22

Mean

84.49 99.29

110.09

5 143.25

437.125

AF 0.915 0.915 0.915 0.915

Seasonal

Index

77.3085 90.85 100.73

6

131.073

7

399.9693

Adjusted Factor (AF) = 400/437.125=0.915