Decomposition of Time Series Lecture 03

Decomposition of Time Series 

Lecture 03 

Introduction

The observed time series for a phenomenon or activity “Yt” is based on four basic components known as secular trend (T), cyclical fluctuations (C), seasonal variation (S), and random variation (I). The primary objective is to break down the observed series into its various components and estimate the individual effect of each component. To accomplish this, we must assume certain things about how the various components are related. These components are therefore expected to have either an additive or a multiplicative model.

Additive Model

In an additive time series model, the observed series “Yt” is considered as the sum of four basic components.

The mathematical form of additive model is given by

Yt = Tt + Ct + St + It

Each of these components has the same units as the original series. In the additive model it is assumed that the four components are independent of each other.

The additive model is used when the magnitude of seasonal or accidental variations remains constant or approximately constant over time regardless of the overall level of time series.

Multiplicative Model

In the multiplicative time series model, the observed series “Yt” is considered as the product of four basic components.

The mathematical form of multiplicative model is given by

Yt = Tt x Ct x St x It

The multiplicative model is based on the assumption that the four components of a time series are not necessarily independent and they can affect one another. Under this model, the trend has the same units as the original series. The other factors are unitless factors such as percentage.

 A multiplicative model is appropriate when the magnitude of seasonal or random variation increases as the overall level of time series increases. By visualisation, if the time series is exponentially increases or decreases.

HISTORIGRAM

(Graph of Time Series)

In order to create a time series graph, we must consider both of the paired pieces of data. A typical Cartesian coordinate system serves as our foundation. The date or time intervals are plotted on the horizontal axis, and the values of the variable that we are measuring is plotted on the vertical axis. By doing this, each point on the graph may be connected to a measurement of a quantity and a date. Normally, straight lines connect the graph's points in the order in which they appear.

Practice Question

The average weekly temperature of a town is given below:

Week	1	2	3	4	5	6	7	8
Temperature	7	12	22	16	10	15	25	19

Construct a historigram, and determine which kind of time series model would be used for the phenomenon and why.

Solution: Week is taken along the x-axis and temperature along the y-axis.

The seasonal variation or accidental variation is not constant over time; therefore, a multiplicative model would be used.

Measurement of Secular Trend

Consider a classical time series model

Yt = Tt x Ct x St x It

Here we measure Tt or (Tt Ct) by the following methods:

1. The method of freehand curve.

2. The semi-average method.

3. The method of moving average.

4. The method of least squares.

Later on, remove its effect from observed time series data. This phenomenon is called detrending.

The Freehand Curve Method

In the freehand curve method, plot the observed data of an activity on graph paper and join the plotted points by segments of straight lines. Observe up and down movements on the graph and draw a smooth curve or a straight line freehand passing through the plotted points in such a way that the general direction of change in values is indicated. The trend values can be read from the graph.

Pros of Freehand Curve Method

i. It is an easy and adaptable method.

ii. There is no mathematical computation required.

iii. It's a time-saving technique.

Cons of the Freehand Curve Method

i. The freehand curve method is a subjective method, and there is a risk of personal bias.

ii. The application of this method required knowledge in the methodology and activity under study.

Practice Question

Find the trend line for the annual profit of a firm by the freehand curve method for the following data.

Years	2000	2001	2002	2003	2004	2005	2006	2007
Value	12	20	18	24	40	36	45	52

Solution:

The Semi-Average Method

The semi-average method is the simplest method to measure the secular trend. The data points are divided into two halves, and the average of each half is found. placed the average of each against the middle value and plotted by graph.

The step-by-step procedure is given below:

i. Split the data into two equal halves; ignore the middle value in case of an odd number of observations.

ii. Determine the arithmetic mean of each half.

iii. Placed the mean against the middle value of each half.

iv. Plot the data on graph paper.

Pros of Semi-Average Method

i. The semi-average method is easy to understand and useful when a rough sketch of the data is required.

ii. The data is linear, and no outliers exist.

iii. The semi-average method can be used for pilot survey data.

Cons of Semi-Average Method

i. The method of semi-average is not appropriate for nonlinear data.

ii. The accuracy is limited for an odd number of observations.

iii. The outliers can mislead the results of analysis.

Practice Question

Find the trend line for the annual profit of a firm bysemi-average method for the following data.

Years	2000	2001	2002	2003	2004	2005	2006	2007
Value	12	20	18	24	40	36	45	52

Solution:

Year	Values	Semi- Total	Semi- Average
2000	12
2001	20
		74	18.5
2002	18
2003	24
2004	40
2005	36
		173	43.25
2006	45
2007	52

• Read More: Moving Average Method