Analysis of Time Series components.

Time series:
A time series is a set of observations measured at time or space intervals arranged in chronological order. For instance, the yearly demand of a commodity, weekly prices of an item, food production in India from year to year, etc. Many economists and statisticians have defined time series in different words.
Wessel and Wellet: When quantitative data  are arranged in the order of their occurrence,the resulting statistical series is called a time series.

Moris Hamburg: A time series is a set of statistical observations arranged in chronological order.

Patterson: A time series consists of statistical data
which are collected, recorded or observed over successive increments.

Ya-lun-Chou: A time series may be defined as a collection of magnitudes belonging to different time
periods, of some variable or composite of variables such as production of steel, per capita income, gross national product, price of tobacco or index of industrial production.

Cecil H. Meyers: A time series may be defined as a sequence of repeated measurements of a variable
made periodically through time.

Werner Z. Hissch: A time series is a sequence of values of the same variate corresponding to successive points of time.

What purpose is served by time series analysis?
The analysis of time series has seen found useful to economists and business persons, in particular, and also to scientists, sociologist, etc. It has also found its utility in meteorology, seismology,
Oceanography, geomorphology. etc., in earth sciences;, electrocardiograms, electroencephotograms in
medical sciences and problem of estimating missile trajectories.
Time series analysis helps in understanding the following phenomena.

(i) It helps in knowing the reał behaviour of the past.

(ii) It helps in predicting the future behaviour like demand, production. weather conditions, prices, etc.

(iii) It helps in płanning the future operations.

(iv) Analysis of time series helps to compare the present accomplishments with the past performances.

(v) Two or more times series can be compared belonging to the same reference period.

The main drawbacks of the time series analysıs.
The drawbacks of the time series analysis can be summarised as follows:

(i) The conclusions drawn on the basis of time series analysis are not cent per cent true.

(ii) Time series analysis is unable to fully adjust the influences affecting a time series like customs,  climate, policy changes, etc.

(iii) The complex forces affecting a time series existing at certain period may not be having
the same complex forces in future. Hence,the forecasts may not hold true.

Need of editing of data before time series analysis:
Ans. The data have to be critically examined and adjusted for various factors before the analysis of
time series, otherwise many discrepancies are likely to arise leading to wrong conclusions.
 For example, the production for January could be more than February. In reality, it may be due to more number of days in January than in February.

Components of a time series Analysis:
There are four components or elements of a time series, namely:
(1) Secular Trend-T                              (2) Seasonal Variation-S

(3) Cyclical Variation-C                       (4) Irregular Variation-I
 
1.secular trend: 
Term trend implies secular trend. It measures long-term changes occurring in a time series without bothering about short-term fluctuations occuring in between. In short, secular trend measures smooth and regular tong-term movements at a time series delineating the increasıng, decreasing or stagnant trend over a long span of time. The graph showing trend is a straight line running from left bottom to right top, left top to right bottom or parallel to abscissa  depicting growth, decline or stagnation respectively.In some situations curvilinear trend is also studied.

(2). seasonal variation: 
Short-term fluctuations observed in a time series data, particularly in a specified period usually within a year, are called seasonal variations. For instance, certain items have more sale in a particular season like ice cream in summer, rain coats in rainy season and woollens in winter season. Similarly, first week of a month records greater sale of grocery than the last week of a month. Certain items have tremendous sale on festivals only in a particular month. All such varitation in a time series come under seasonal variation. Seasonal variations are more akin to climatic and weather conditions.

(3). Cyclic variation:
 Cyclic variation relates to periodic changes, particularly in business. A cycle consists of more than a year period. The cycles in a time series depict the prosperity and recession, ups and downs, booms and slumps of a business. A complete cycle usually has four constituents namely, prosperity, recession, depression and recovery. Cycles related to business are termed as business cycles or trade cycles. The length of business cycles varies from one business to the other. The graph of cyclic variation is a curve having alternately convexity and concavity.

Cycles are never regular in periodicity and amplitude. Hardly any time series has strict cycles. Hence, in practice statisticians and economists often use the term undulations or oscillations instead of cycles.
Since a cycle covers a long span of time, the data required for the depiction of cycles should be recorded for a large number of years (periods).

 What factors are generally responsible for the occurrence of cycles? 
 A large number of factors are responsible for the occurrence of cycles.
 But a few important ones are given below:

(i) Likes and dislikes of the people change after a certain period and they cause cycles in business phenomenon.

(ii) Production of certain items is stopped and new items are produced. Again old items are adopted. Such changes form cycles.

(iii) Social customs change from time to time resulting into business cycles.

(iv) New scientific and technological developments affect the production and consumption of items which create cycles.


(4). Irregular variations: 
 Irregular variations or the so-called random variations are irregular in the sense that it is not possible to think of their time of occurrence, direction and magnitude. These variation usually occur due to epidemics, earthquakes, floods, wars, accidents, etc. Another name given to irregular variations is residual variations. This name is derived in the sense that all those variations which cannot be subsumed in trend, seasonal and cyclic variations, are assigned to irregular variations.


 Mathematical models for a time series analysis.
 In a traditional or classical time series analysis, the most commonly assumed mathematical mode!
is the multiplicative model. Here it is assumed that any particular observation Y at time t is as a  result of the product of the effect  of the four components of a time series namely. Trend (7), seasonal variation (S), cyclic variation (C) and the irregular variation (I), i.e.,

                                                    Y= T*S* C*I

Further the multiplicative model does the independence not of assume the four components time series.It is appropriate for the projections.
         Some people believe that result the of observation Y ia as a result of additive effect of the components T. S. C and I, i.e.,

                                                    Y= T + S + C + I

The additive model is  based on the assumption that the four components  are independent of each other,Additive model is  rarely  used as it is not appropriate for future events.
Some people have also advocated the use of  mixed models.A mixed models is mathematical relation
which is expressed as combination  of multiplicative and additive components of a time series.

They may be combined in a number of ways.Such types of models examples of are hardly used. Some of the mixed models are given below:
             
                                            Y =  T + S × C + I
                                            Y =  T + S × C × l
                                            Y =  T + S + C × I
                                            Y =  T × C + S × I



The essential requirements for proper analysis of a time series:
 The essential requirements for proper analysis of a time series are:

(i) Data should be available for a long period.

(ii) The value should have been available as far as possible at equal intervał of time. If not, they have to be adjusted.

(i) The time periods should be definite according to calendar.

(iv) The data should consist of a homogeneous set of values in respect of units of measurements and time scale.


 Names of different methods of measuring trend:

  1. Free-hand or graphic method
  2.  Semi-average method
  3.  Moving average method
  4.  Least square method





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