Monday, 15 July 2013

time series

Time Series

A time series is a sequence of observations which are ordered in time (or space). If observations are made on some phenomenon throughout time, it is most sensible to display the data in the order in which they arose, particularly since successive observations will probably be dependent. Time series are best displayed in a scatter plot. The series value X is plotted on the vertical axis and time t on the horizontal axis. Time is called the independent variable (in this case however, something over which you have little control). Time series is set of data collected and arranged in accordance of time.
 According to Croxton and Cowdon
,”A Time series consists of data arranged chronologically ” Such data may be series of temperature of patients, series showing number of suicides in different months of year etc. The analysis of time series means separating out different components which influences values of series. The variations in the time series can be divided into two parts: long term variations and short term variations.Long term variations can be divided into two parts: Trend or Secular Trend and Cyclical variations.Short term variations can be divided into two parts:Seasonal variations and Irregular Variations. There are two kinds of time series data:
  1. Continuous, where we have an observation at every instant of time, e.g. lie detectors, electrocardiograms. We denote this using observation X at time t, X(t). 

  1. Discrete, where we have an observation at (usually regularly) spaced intervals. We denote this as Xt.
Economics - weekly share prices, monthly profits
Meteorology - daily rainfall, wind speed, temperature
Sociology - crime figures (number of arrests, etc), employment figures
Example of a time series plot


The four components of time series are:

1.Secular trend
2.Seasonal variation
3.Cyclical variation
4.Irregular variation

Secular trend:

                               A time series data may show upward trend or downward trend for a period of years and this may be due to factors like increase in population,change in technological progress ,large scale shift in consumers demands,etc.For example,population increases over a period of time,price increases over a period of years,production of goods on the capital market of the country increases over a period of years.These are the examples of upward trend.The sales of a commodity may decrease over a period of time because of better products coming to the market.This is an example of declining trend or downward trend.The increase or decrease in the movements of a time series is called Secular trend. 


The study of trend helps a businessman in forecasting and planning his future business activities. It helps an economist in formulating economic policies. It is used in making comparison between two or more time series. By isolating trend from the given time series we can study the short-term and irregular variations.

Seasonal variation: 

Seasonal variation are short-term fluctuation in a time series which occur periodically in a year.This continues to repeat  year after year.The major factors that are responsible for the repetitive pattern of seasonal variations are weather conditions and customs of people.More woollen clothes are sold in winter than in the season of summer .Regardless of the trend we can observe that in each year more ice creams are sold in summer and very little in Winter season.The sales in the departmental stores are more during festive seasons that in the normal days.They are due to the following two reasons.

(i)    Natural Cause
Changes in the climate and weather conditions have a profound effect on seasonal variations. For example, the sales of umbrella pick up very fast in rainy season, the demand for cold drinks goes up during summer etc.
(ii)    Man-made Cause
The social customs, traditions and conventions play an important role in fixing seasonal variations. For example, during festivals like Bihu, Puja, Id etc. the sales of many commodities go high, the price of ornaments go high during marriage seasons etc.

An accurate assessment of seasonal variations is an aid in business planning and scheduling such as in the area of production, inventory control, personnel, advertising and so on. A knowledge of seasonal variation helps a consumer in purchasing the articles at least prices during the slack seasons.

Cyclical variations:

                                   Cyclical variations are recurrent upward or downward movements in a time series but the period of cycle is greater than a year.Also these variations are not regular as seasonal variation.There are different types of cycles of varying in length and size.The ups and downs in business activities are the effects of cyclical variation.A business cycle showing these oscillatory movements has to pass through four phases-prosperity,recession,depression and recovery.In a business,these four phases are completed by passing one to another  in this order.

The knowledge of cyclical variations is of great importance to businessmen in the formulation of policies for stabilizing the business activity. With a knowledge of cyclical variation a businessman can face the challenges of recession and depression by taking the appropriate' decisions in advance.

Irregular variation:

 Irregular variations are fluctuations in time series that are short in duration,erratic in nature and follow no regularity in the occurrence pattern.These variations are also referred to as residual variations since by definition they represent what is left out in a time series after trend ,cyclical and seasonal variations.Irregular fluctuations results due to the occurrence of unforeseen events like floods,earthquakes,wars,famines,etc. These variations may be due to such isolated incidents as floods, famines, 'wars, earthquakes etc. which are completely irregular in nature. Thus, they are wholly unpredictable. This includes all types of variations in a time series which are not attributed to trend, seasonal or cyclical variations,


In business forecasting, it is important to analyze the characteristic movements of variations in the given time series  The following methods serve as a tool for this analysis:

1.Methods for Measurement of Secular T rend

i.Freehand curve Method (Graphical Method) 

ii.Method of selected points
iii.Method of semi-averages
iv.Method of moving averagesv.Method of Least Squares

2.Methods for Measurement of Seasonal Variations

i.Method of Simple Average
ii.Ratio to Trend Method
iii.Ratio to Moving Average Method
iv.Method of Link Relatives

3.Methods for Measurement for Cyclical Variations4. Methods for Measurement for IrregularVariations


 Measurement of Trend
To measure the secular trend, the short-term variations should be removed and irregularities should be smoothed out. The following are the methods of measuring trend.
1.     Graphic (or free hand curve) method
2.     Semi-average method
3.     Moving average method
4.     Least squares method 
The values of the time series are plotted on a graph paper with the time (t) along x-axis and the values of the variable (y) along y-axis. A freehand curve is drawn through these points in such a manner that it may show a general trend. A free hand curve removes the short-term variations and irregular movements.
It is the simplest method, Time and labour is saved. It is very flexibe method as it represents both linear and non-linear trends.
The main drawback of this method is that it is highly subjective as different persons will draw different free hand curves. Because of its subjective nature it is useless in forecasting.
Semi-Average Method
This method is sometimes used when a straight line appears to be an adequate expression of trend. In this method, the original data are divided into two equal parts. The averages of each part are then calculated. The average of each part is centred in the period of the time of the part from which it has been computed and then plotted on the graph paper. In this way, a line may be drawn to pass through the plotted points which gives the trend line. In case of odd number of years, the mid-year is eliminated while dividing the data into two equal parts.
This method is not subjective and ·everyone gets the same trend line. It is possible to extend the trend line both the ways to estimate future or past values. But the method assumes the presence of linear trend which may not exist.

Moving Average Method
Method of moving average consists in calculating a series of arithmetic means of some overlapping groups of the time series. There is no hard and fast rule for decision regarding period of moving average and the selection of period depends upon the objective that has to be attained from the figures of trend. The averaging process removes ups and downs in the data. If y1, y2, y3, ........ are the values of the time series for time periods t1, t2, t3........... then.
1st moving average of period In  1/m(y1+y2+-----+ym  )     
2nd moving average of period m   1/m(y2+y3+----+ym+1
3rd moving average of period m    1/m(y3+----+ym+2

Now, if the period m is odd, then the moving average values are placed against the middle values of the time periods. Again if the period m is even, then the moving average values are placed in between two middle time periods. After that, method of centering is applied to adjust the data with the original time periods.
The moving average values plotted against time give the trend line.

Merits and Demerits of Moving Average Method
        (i)     This method is easy to understand and calculate.
        (ii)    This method is not subjective.
(iii)    This method is flexible in the sense that a few more observations may be included of moving average is equal to or multiple of the cycle of a cyclical..
        (i)     Moving average can not be obtained for all the years.
(ii)    There is no definite rule for fixing the period of moving average.
(iii)    This method is suitable only when the trend is liner.

(iv)   This method is not suitable for forecasting.


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