Analysis of Time Series and Forecasting

Chapter 15

Learning Objectives

Objectives of Time series

Components of Time series

Additive & multiplicative model

Moving Average method

Least Square Method

Method of simple averages (Seasonal Variation)

De-seasonalisation of data

Exponential Smoothing Method

Types of Forecasting Models

Qualitative methods – judgmental methods– Forecasts generated subjectively by the

forecaster– Educated guesses

Quantitative methods – based on mathematical modeling:– Forecasts generated through mathematical

modeling

Quantitative Approaches to Forecasting

Quantitative methods are based on an analysis of historical data concerning one or more time series.A time series is a set of observations measured at successive points in time or over successive periods of time.If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method.If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method. Quantitative approaches are generally preferred. In this chapter we will focus on quantitative approaches to forecasting.

Time Series and Forecasting

Time series refers to any group of statistical information accumulated at regular intervals. It is a quantitative method used to determine patterns in data collected over time.

Future prediction is always a matter of great concern in any organization. It is of great interest in any field such as economics, business, whether, stock market, etc.

In forecasting, these patterns are projected to arrive at an estimate for the future.

Objectives of Time Series Analysis

Understanding the past behaviour: If we look at the times series of sales of readymade garment of a company then we can get an idea about the sales trend of that company in the past and on the basis of this trend, we can predict future sales.

Estimation and Forecasting: A forecast does not test what will happen but indicates what would happen if the past behaviour continues.

Components of Time series

Secular trend: Overall/general tendency if the series is observed over a long period of time.

Cyclical variation: Cyclical movements. It do not follow any regular pattern but move in a somewhat unpredictable manner and period is more than one year.

Seasonal variation: Periodic patterns of change within a year that tend to be repeated from year to year.

Irregular variation: Describing the variation which is completely unpredictable, changing in a random manner.

Components of Time Series Data

1 2 3 4 5 6 7 8 9 10 11 12 13

Year

Seasonal

Cyclical

Trend

Irregularfluctuations

Trend Analysis

Secular trend represents the long-term direction of the series.

Study of secular trends allows us to describe the historical pattern.

Studying secular trends permits us to project past patterns, or trends, into the future.

Studying the secular trend of a time series allows us to eliminate the trend component from the series.

Trends can be linear or curvilinear.

Fitting the linear trend: Least square method is used to find the best fitting line.

When time series are best described by curves, then method of fitting the second-degree equation (parabolic curve) is used.

Cyclical Variation

Cyclical variation is the component of a time series that tend to oscillate above and below the secular trend line for period more than one year.

The cycles may sometimes be peculiar to a particular industry but more generally they relate to a the whole economy and are called business cycles. In business, cyclic variations are commonly associated with economic cycles, successful booms and slumps in economy. It has got 4 phases i.e prosperity, recession, depression and recovery.

Contd…

Seasonal Variation

Seasonal variation is a component of a time series which is defined as the repetitive and predictable movement around the trend line in one year or less.

In order to detect seasonal variation, time intervals must be measured in small units (days, weeks, months or quarters).

Main reasons for seasonal variations are the different climatic conditions in the year, customs, traditions, festivals, etc. Industries dealing with agricultural products, soft drinks, woolen garments, fashion tourism etc. are noticeably affected by seasonal variations.

Contd…

Irregular Variation

The final component of a time series is irregular variation. Typically, irregular variation occurs over short intervals and follows a random pattern.

Because of the unpredictability of irregular variation, we do not attempt to explain it mathematically. However, we can often isolate its causes, but not all causes of irregular variation can be identified so easily.

Additive Model

Assumptions: The four components are independent and additive.

Y=T+S+C+I

The components C,S & I are measured as deviations from the secular trend and are all expressed in the same units as the original time series data.

Multiplicative ModelAssumptions: The four components are not independent and are multiplicative of each other.

Y=T*S*C*I

Only trend is expressed in the same units as the given data. Others are expressed as proportions.

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Examples

Decline in sales of cold drink in December

An Era of Prosperity

A fire in factory delaying production for three weeks

A need for increase in wheat production due to consistent increase in population

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Excel Instructions for Drawing a Scatter Plot

1. Enter data in the Excel spreadsheet.2. Click on Insert on the toolbar and then click on the Chart tab. The

Chart Wizard will appear. In step 1 on select the XY (scatter) chart type and then click next.

3. In step 2 specify the cells where your data is located in the data range box.

4. In step 3 you can give your chart a title and label your axes. In step 4 specify where you want the chart to be placed.

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During the past ten weeks, sales of cases of Comfort brand headache medicine at Robert's Drugs have been as follows:

Week Sales Week Sales 1 110 6 120 2 115 7 130 3 125 8 115 4 120 9 110 5 125 10 130

Plot this data.

Example: Robert’s Drugs

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Plot Robert’s Drugs Example

Excel Spreadsheet Showing Input Data. Specify cells A4:B13 as the Data Range. A B

1 Robert's Drugs2

3 Week (t ) Salest

4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 13014 11

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Plot Robert’s Drugs Example

Robert's Drug Example

105

110

115

120

125

130

135

0 5 10 15

Week, t

Sa

les

I labeled Robert’s DrugExample as The Chart title

I labeled Week, t as My Value (x)axis

I labeled Sales as My Value (y)axis

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Smoothing Methods

In cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular components of the time series.

Three common smoothing methods are:– Moving average– Weighted moving average– Exponential smoothing

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Smoothing Methods: Moving Average

Moving Average Method

The moving average method consists of computing an average of the most recent n data values for the series and using this average for forecasting the value of the time series for the next period.

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Robert Drug’s Example: Moving Average

Our scatter plot for Robert’s Drug Sales has no significant trend, seasonal, or cyclical effects. Thus we should employ a smoothing technique for forecasting sales.

Forecast the sales for period 11 using a three period moving average (MA3).

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Example: Robert’s Drugs: Moving Average

Steps to Moving Average Using Excel

Step 1: Select the Tools pull-down menu.

Step 2: Select the Data Analysis option.

Step 3: When the Data Analysis Tools dialog appears, choose Moving Average.

Step 4: When the Moving Average dialog box appears:

Enter B4:B13 in the Input Range box.

Enter 3 in the Interval box.

Enter C5 in the Output Range box.

Select OK.

This specifies the value of n

This is the column following our data,and one row below whereour data begins.

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Robert’s Drugs: Moving Average

MA3 (Three period Moving average) for Robert’s Drug Example

Ft is the forecast for week t.

F4 (forecast for week 4)=116.7

F11 (forecast for week 11)=118.3

Thus we would forecast the sales for Week 11 to be 118.3

Robert's Drugn=3

Week (t ) Yt Ft

1 1102 115 #N/A3 125 #N/A4 120 116.66675 125 1206 120 123.33337 130 121.66678 115 1259 110 121.666710 130 118.333311 118.3333

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Smoothing Methods: Weighted Moving Average

Weighted Moving Average Method

The weighted moving average method consists of computing a weighted average of the most recent n data values for the series and using this weighted average for forecasting the value of the time series for the next period. The more recent observations are typically given more weight than older observations. For convenience, the weights usually sum to 1.

The regular moving average gives equal weight to past data values when computing a forecast for the next period. The weighted moving average allows different weights to be allocated to past data values.

There is no Excel command for computing this so you must do this manually. You can either manually enter the formulas into excel and apply to all periods or compute value by hand.

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Smoothing Methods: Weighted Moving Average

Use a 3 period weighted moving average to forecast the sales for week 11 giving a weight of 0.6 to the most recent period, 0.3 to the second most recent period, and 0.1 to the third most recent period.

F11 = (0.6)*130 + (0.3)*110 + (0.1)* 115= 122.5

Thus we would forecast the sales for week 11 to be 122.5.

Sales for themost recentperiod

Sales for 2nd most recentperiod

Sales for 3rd most recentperiod

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Smoothing Methods: Exponential Smoothing

Exponential Smoothing– Using exponential smoothing, the forecast for the next

period is equal to the forecast for the current period plus a

proportion () of the forecast error in the current period.

– Using exponential smoothing, the forecast is calculated by:

Ft+1=Yt + (1- )Ft

where: is the smoothing constant (a number between 0 and

1)Ft is the forecast for period t

Ft +1 is the forecast for period t+1

Yt is the actual data value for period t

This is the same as Ft+1 = Ft + α (Yt – Ft)

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Robert’s Drugs: Exponential Smoothing

Forecast the sales for period 11 using Exponential Smoothing α= 0.1.

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Robert’s Drugs: Exponential Smoothing

Steps to Exponential Smoothing Using ExcelStep 1: Select the Tools pull-down menu.

Step 2: Select the Data Analysis option.

Step 3: When the Data Analysis Tools dialog appears, choose Exponential Smoothing.

Step 4: When the Exponential Smoothing dialog box appears:

Enter B4:B13 in the Input Range box.

Enter 0.9 (for = 0.1) in Damping Factor box.

Enter C4 in the Output Range box.

Select OK.

Damping factoris always 1-α

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Robert’s Drugs: Exponential Smoothing

F11 = 0.1 * Y10 + .9 F10

= .1 *130 + .9 * 115.4099 = 116.87

Robert's Drugsα=0.1

Week (t ) Salest Ft

1 110 #N/A2 115 1103 125 110.54 120 111.955 125 112.7556 120 113.97957 130 114.58168 115 116.12349 110 116.0111

10 130 115.409911

Thus we would forecast sales for week 11 to be 116.87

Thank U

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