29 Introduction to Time Series Analysis and its Importance

Dr. Harmanpreet Singh Kapoor

Module 35: Introduction to Time Series Analysis and its Importance

 

  • Learning Objectives
  • Introduction.
  • Functional relationship
  • Classification of time series
  • Components of time series
  • Mathematical model of time series
  • Utility of time series
  • Requirement of time series
  • Applications of time series
  • Summary
  • Suggested Readings

 

  1. Learning Objectives

 

The objective of this module is to give basic introduction of Time series analysis and explain its meaning and concepts to understand its vast application areas. Classification of time series and components of time series will be discussed in detail. Mathematical model of time series. Utility, requirement of time series and its application will also be discussed for in depth knowledge of this topic.

 

 

  1. Introduction

 

In general, Series is a sequence of observations in a specific order. Similarly, Time series is a sequence of observations in specific order but these observations are time dependent. Actual time series is realization of stochastic process. In different sectors like Economics, Commerce and Environment events happen over a period of time. For example- temperature of a day at different time interval and price of commodity in one month to other month or it may be day or year, production of crop varies over year, production of an item in an industry increase or decrease depending on the demand of the product, sale of goods depend upon the price as well as need of the product over the time like in change of technology, consumption of items increase with increase in earning over time or decrease due to lack of money like visit to hotels depends upon person’s earning that varies over time. These examples show a series of data which are dependent on time, this measurement of variable at different time point is known as “Time interval” in literature terms. Time interval may be hours, days, quarters, months and years. In modern literature time series is also referred as both data and process. Hence time series has a great importance in Business, Economics and other areas. This is the main reason for the development of many statistics tools specially for these areas exist in literature. Even a particular subject named as Econometrics originate to cater the need of this topic. Econometrics is now among the most widely taught subject in the colleges and other institutions due to importance of happening of events over a period of time.

 

Definition:

 

“A Time Series is a set of statistical observations arranged in chronological order” by Morris Hamburg.

“A time series may be defined as a collection of reading belonging to different time periods, of some economic variables or composite of variables” by Ya-lun Chou.

“Time series is a chronological sequence of observations of quantitative variables that are recorded successively at regular and specified time interval”.

 

  1. Functional relationship:

 

Time series is a sequence of observations with respect to study variable and time interval. It means there is a relationship between study variable and time interval.

 

For example: Suppose a researcher records temperature of a place on a particular day in any weather condition then recorded observation would be different in different time interval like hours, days and months. There is a relationship between temperature and time interval.

 

This relationship can be defined in a mathematical functional relationship. The functional relationship of time series can be described as:

=  (  )

 

where                                                                                                                                          is the value of variable under considered time t.

 

According to this functional relationship temperature of a place is a variable whose values are considered at time t and t represents different time interval of equal length.

 

  1. Classification of the Time series:

 

Time series can be classified in two types, first is according to dependency on time and second is according to variable numeracy.

 

Dependency on time can be further categorized into:

  1. Discrete time series
  2. Continuous time series

 

Variable numeracy has further two categories

 

  1. Univariate time series
  2. Multivariate time series

 

We will now discuss these classification in detail:

 

  1. Discrete time series: A Time series is discrete time series when variable dependency on time is of discrete form. A variable is said to be discrete if it takes only integer values like days, months etc. For example- closing price of daily stock market.
  2. Continuous time series: A Time series is continuous time series when variable depends on time is of continuous nature. A variable is said to be continuous if it takes values in continuous form like in second, minutes etc. For example- hourly reading of temperature.
  3. Univariate time series: A time series is univariate time series when there is only one type of study variable or one characteristics under consideration. For example: share price of one company.
  4. Multivariate time series: A Time series is multivariate time series when there is more than one type of study variable that is of interest in time series. For example- share price of one company, share price of other company.
  5. Components of time series:

Study variable take different values in a particular time series. This difference in value is not due to only one component like time but affected by more than one component or multiple components. These multiple components pull up and down the values of the characteristics under study. For example: the crop yield depends on weather condition, seed quality, land quality, fertilizer etc. Here crop yield is study variable and weather condition, seed quality, land quality are other multiple components that are also responsible for the yield of the crop.

 

These multiple components are classified into four categories. Such as:

1) Secular Trend (long term movement)

2) Seasonal variation (short term movement)

3) Cyclic variation

4) Random or Irregular Movements

 

These four categories are known as components of time series. Now let’s discuss components of time series one by one with examples.

 

  • 1) Secular trend:

 

Secular trend or generally trend is a long term movement. In general, Trend shows tendency of the time series data in long term period. This tendency may be downward and upward. It should be kept in mind that this tendency of upward and downward movement should not remain same throughout the time span. It is also possible that it can be observed in different time section or time span. However one can say that the overall tendency of the series seem like moving upward or downward but in actual there are many factor that persist for a long period. Tendency depends on the effects of the factor. If effect of factors on time series data is of increasing then it shows upward tendency, if is on decreasing then it shows downward tendency.

 

Trend consists of Long term effects. Long term cannot be defined exactly because in some cases 2 year time may be enough and for some cases it is not enough.

 

For some cases 5 minutes is a very long period, for other case it may not be appropriate. The term of long or short period of time depends on the nature or objective of the data. In some cases, a period of hours can be considered as very large while in other cases even the period of years is not sufficient to be named as long time. For example if data of agriculture production

 

of 24 month shows increment in the growth of the crop then it is not considered as secular trend for 2 years, the count of bacterial population of a culture in every five minutes, if this tendency of increment persist for a week then it is a secular trend. One important thing is that the values for short period (may be 2-3 years in some cases) are mainly affected by cyclic variations. Hence the values are not able to reveal true trend. In order to solve this problem, one should consider the multiple cycle periods to see the impact of cyclic variations on the values. For some cases, due to nature of  the data like increment of bacterial population persists due to strong germicide in an hour then reading in every five minutes or in seconds would lead to a general pattern that can be termed as a secular trend.

 

This tendency of the data may be linear and nonlinear trend. It depends on the nature of the data. If the time series values are plotted on the graph with more concentration around a straight line then trend pattern of time series values are said to be linear on the other hand if the time series values are not close to straight line then it is called non-linear trend like increase in the price of gold, crude oil in the international market is not constant and varies with different phases of time. Hence the trend of time series will help you to get a general idea about the pattern of the behavior of the data under consideration. This will help in forecasting and future planning of different sectors like business, weather and economy based on previous values. With the use of trend analysis one can compare two or more time series over different time periods and draw important conclusions based on them.

 

  • 2) Seasonal variation:

 

Seasonal variation is a short term movement in a time series. Short term movement means observational data collects in less than one year of time period i.e. data collected monthly, quarterly, weekly, daily, hourly etc. If collected data are annual then there is no seasonal variation. As the name suggests, seasonal variation means variation that occur due to change in season and natural forces in any study variable. For example prices, production and consumption of any commodities, sales and profit in any departmental store etc.

 

Some variations in data are due to natural forces like seasons. For example: sale of umbrella increases in rainy season, consumption of ice-cream increases in summer, price of woolen increases in winter. Some variation are due to man-made convention such as sale of jewellery and price of clothes go up due to marriage season and in festival like Diwali, Christmas, and Durga Pooja.

 

The following image shows sales of clothes changing in different seasons

 

  • 3) Cyclic variation:

Cyclic variation is a repeated pattern in a time period of a time series, this time period will be more than one year. The data of sale of ice cream for few years shows that the sale of ice-cream is high in summer and low in winter. Therefore this data shows some repeated pattern in time series data, this pattern is known as cyclic variation .

 

Generally cyclic variation shows in business cycle. Business cycle have four phases. Hence this is referred as four phase cycle such as prosperity, recession, depression and recovery. Each phase in business cycle persists for a very long term variation approximately 5 to 7 years. On the other side, cyclic variation may be short term and long term movement that depend on time period but time period must be greater than one year.

  • 4) Irregular component:

 

Irregular component is also known as random or residual component. Every time series must contain this fluctuation and this fluctuation cannot be removed from the data. Irregular component can be found by removing all three components, so that irregular component is not affected by other three components of time series.

 

Irregular component is beyond the control of human being and considered as unpredictable. This type of fluctuation may be seen due to effect of factors like earthquake, wars, floods, famines, political unrest revolution, epidemics etc.

 

Irregular variation are unpredictable due to its accidental variation.

For example- rise in prices of LPG gas cylinder due to strike of workers etc.

 

 

 

  1. Mathematical model of time series:

 

In analysis of time series, there are types of problems like to identify the components and their effect on the data sets, to isolate, analyze and measure them independently. In literature, there are some mathematical model to decompose the time series into its components.

 

Additive model: Additive model of a time series can be expressed as:

=                +         +          +

where        is time series value at time t.

is trend value at time t.

is seasonality value at time t.

is cyclic value at time t.

is random or irregular value at time t.

All four component of time series in additive model operate independently to each other. Thus, in additive model , there is no effect of one component to other component. Additive model can be used to measure one or more component by elimination i.e. by subtraction.

For example- time series decomposition of additive model with visualization as in the following image:

Multiplicative model:

 

If all component of time series operate proportionally to the general level of the series, multiplicative model is appropriate. Multiplicative model can be expressed as:

=

where

is the value of time series at time t.

, , and is the value of trend, seasonal ,cyclic and irregular component respectively. Multiplicative decomposition of time series is same as the additive decomposition of logarithmic values of the original time series that is:

For example: time series transform Multiplicative to Additive decomposition by using logarithmic with visualization is given in Figure 7.

 

It means that when the observations are related with each other in proportionate manner then by taking log of the observation we can convert the multiplicative model into additive model as it is easier to detect the components in additive models than multiplicative model. By taking log of observations one can smooth the curve. In practical cases, it is most widely used concept to take log of observations to determine the model that fits in an appropriate manner on the observations.

 

Most of time series related to business and economic confirms to multiplicative model. Multiplicative model are used in measure of one or more components i.e. by division. For example if trend value is known then isolate them:

=

 

If data is annual then seasonal components is not there i.e.

=

For example-assume birth time series decomposition of Multiplicative model with visualization as in the following image:

Hence from Figures 6-8, one can understand the mathematical model of time series components. One can choose an appropriate model based on the relationship among components in the data values.

 

Mixed model:

 

Some time series data shows different type of models under different type of assumption. This different type of model is known as combination of additive and multiplicative model or mixed model of time series. Some types of different model are:

=                           +

=                                     +

=               +         +

 

where

 

is the value of time series at time t.

and      is the value of trend, seasonal, cyclic and irregular component.

 

  1. Utility of time series:

 

Utility of time series is given as:

 

  • 1) It helps in understanding past behavior of the time series: to understand the past behavior of any time series, to analyze variation between data sets over a period of time.
  • 2) It helps in future prediction: by observing past behavior of time series data we may see how observation are varying. According to behavior of data one can use this information for forecasting. We know that time series depend on past data. For better prediction of values one require long period data.
  • 3) It helps in understand current situation of the problem: the actual performance can be analyzed by taking difference between expected performance and cause of variation of the data.
  • 4) It facilitates comparisons between different time series:

 

  1. Requirements of time series:

 

Time series have following requirements:

 

1) Data must consist of a homogenous set of values.

2) The data should be available for sufficiently long period.

3) Time gap between various values must, as far as possible be equal.

4) The gaps, if any, in data should be made up by interpolation.

 

  1. Application of Time series:

 

Time series analysis has wide application area. Most important area of time series analysis is economic and business. Some other application areas are: weather forecasting, pattern recognition, earthquake prediction, mathematical science and mathematical finance. Time series analysis is mostly used in forecasting or predication.

 

  1. Summary

 

This module is an introduction to time series. Here we give a basic introduction of Time series analysis and attempt to explain its meaning, concepts that help in understanding its vast application areas. Classification of time series like univariate, multivariate, discrete and continuous are discussed and their differences analyzed from one another. After that components of time series like secular trend, seasonal, cyclic and irregular are discussed in detail with graphical understanding so that they are easy to learn and differentiate from one another. Mathematical model of time series like additive model, multiplicative model and mixed models are shown with their mathematical formulation, to know about the importance of time series and its applications. Utility, requirement of time series and its application are discussed for in depth knowledge of this topic.

 

  1. Suggested Readings

 

  • Gupta, S. P., Statistical Methods, Sultan Chand & Sons, New Delhi, 2012.
  • Gupta, S. C. and Kapoor, V. K., Fundamentals of Applied Statistics, Sultan Chand & Sons, New Delhi, 2009.
  • Sharma, J. K., Business Statistics, Vikas Publishing House, 2014.
  • Tsay, R. S., Time Series and Forecasting: Brief History and Future Research, Journal of the American
  • Statistical Association, Vol.95, pp. 638-643, 2000.