Analysis of time series.| Components of time series.

 

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(vi) Time series,Analysis of Time Series components.

Components of time series

  Time  series  

      You may have heard people saying that the price of a particular commodity has increased or decreased with time. This commodity can be anything like gold, silver, any eatables, petrol, diesel etc. Also, you may have heard that the rate of interest has increased in banks. The rate of interest for home loans has decreased. What are all these? How are they useful to us? These types of data are the time series of data.

        A time series is a collection of observations of well-defined data items obtained through repeated measurements over time. For example, measuring the value of retail sales each month of the year would comprise a time series. Data collected irregularly or only once are not time series.

        In other words, the arrangement of data in accordance with their time of occurrence is a time series. It is the chronological arrangement of data. Here, time is just a way in which one can relate the entire phenomenon to suitable reference points. Time can be hours, days, months or years.

A time series depicts the relationship between two variables. Time is one of those variables and the second is any quantitative variable.It may be increasing for some and decreasing for some points in time.

Mathematically;

  =f(t),  

Where    is the value of  variables under consideration at time  t.

As;   t; ,  

 ; , , ---

‘U’  is the variable , and   ‘t’   is the time.

 

 

Uses of Time Series

·         

The most important use of studying time series is that it helps us to predict the future behaviour of the variable based on past experience

·         It is helpful for business planning as it helps in comparing the actual current performance with the expected one

·         From time series, we get to study the past behaviour of the phenomenon or the variable under consideration

·         We can compare the changes in the values of different variables at different times or places, etc.

 Components of time series

        The factors that are responsible for bringing about changes in a time series, also called the components of time series, are as follows:

• Secular Trends (or General Trends) or long term trend.
• Seasonal Movements/ variations,
• Cyclical Movements/ variations. ( periodic changes )
• Irregular Fluctuation/ random fluctuations or movements.

 Secular Trends (or General Trends) or long term trend.

The secular trend or simply trend is the main component of a time series which results from long term effects of socio-economic and political factors. This trend may show the growth or decline in a time series over a long period. This is the type of tendency which continues to persist for a very long period. Prices and export and import data, for example, reflect obviously increasing tendencies over time . The trend shows the general tendency of the data to increase or decrease during a long period of time. A trend is a smooth, general, long-term, average tendency. It is not always necessary that the increase or decrease is in the same direction throughout the given period of time.

   It is observable that the tendencies may increase, decrease or are stable in different sections of time.  But the overall trend must be upward, downward or stable. The population, agricultural production, items manufactured, number of births and deaths, number of industry or any factory, number of schools or colleges are some of its example showing some kind of tendencies of movement.

(a)         Linear and Non-Linear Trend

   If we plot the time series values on a graph in accordance with time t. The pattern of the data clustering shows the type of trend. If the set of data cluster more or less round a straight line, then the trend is linear otherwise it is non-linear (Curvilinear).

      Seasonal Trends/variations

          These are short term movements occurring in data due to seasonal factors. The short term is generally considered as a period in which changes occur in a time series with variations in weather or festivities. For example,  it is commonly observed that the consumption of ice-cream during summer is generally high and hence an ice-cream dealer’s sales would be higher in some months of the year while relatively lower during winter months. Employment, output, exports, etc., are subject to change due to variations in weather. Similarly, the sale of garments, umbrellas, greeting cards and fire-works are subject to large variations during festivals like Valentine’s Day, Eid, Christmas, New Year’s, etc. These types of variations in a time series are isolated only when the series is provided biannually, quarterly or monthly

These variations come into play either because of the natural forces or man-made conventions. The various seasons or climatic conditions play an important role in seasonal variations. Such as production of crops depends on seasons, the sale of umbrella and raincoats in the rainy season, and the sale of electric fans and A.C. shoots up in summer seasons.

Cyclic Movements/variations

   The variations in a time series which operate themselves over a span of more than one year are the cyclic variations. This oscillatory movement has a period of oscillation of more than a year. One complete period is a cycle. This cyclic movement is sometimes called the ‘Business Cycle’.     

 These are long term oscillations occurring in a time series. These oscillations are mostly observed in economics data and the periods of such oscillations are generally extended from five to twelve years or more. These oscillations are associated with the well known business cycles. These cyclic movements can be studied provided a long series of measurements, free from irregular fluctuations, .

It is a four-phase cycle comprising of the phases of prosperity, recession, depression, and recovery. The cyclic variation may be regular are not periodic. The upswings and the downswings in business depend upon the joint nature of the economic forces and the interaction between them.

Irregular movements/ Fluctuations

      These are sudden changes occurring in a time series which are unlikely to be repeated. They are components of a time series which cannot be explained by trends, seasonal or cyclic movements. These variations are sometimes called residual or random components. These variations, though accidental in nature, can cause a continual change in the trends, seasonal and cyclical oscillations during the forth coming period. Floods, fires, earthquakes, revolutions, epidemics, strikes etc., are the root causes of such irregularities

      They are not regular variations and are purely random or irregular. These fluctuations are unforeseen, uncontrollable, unpredictable, and are erratic. These forces are earthquakes, wars, flood, famines, and any other disasters.

Mathematical Model for Time Series Analysis

Mathematically, a time series is given as

y_t = f (t)

Here, y_t is the value of the variable under study at time t. If the population is the variable under study at the various time period t₁, t₂, t₃, … , t_n. Then the time series is

t: t₁, t₂, t₃, … , t_n

y_t: y_t1, y_t2, y_t3, …, y_tn

or, t: t₁, t₂, t₃, … , t_n

y_t: y₁, y₂, y₃, … , y_n

Additive Model for Time Series Analysis

If y_t is the time series value at time t. T_t, S_t, C_t, and R_t are the trend value, seasonal, cyclic and random fluctuations at time t respectively. According to the Additive Model, a time series can be expressed as

y_t = T_t + S_t + C_t + R_t.

This model assumes that all four components of the time series act independently of each other.

Multiplicative Model for Time Series Analysis

    The multiplicative model assumes that the various components in a time series operate proportionately to each other. According to this model

y_t = T_t × S_t × C_t × R_t

Mixed models

Different assumptions lead to different combinations of additive and multiplicative models as

y_t = T_t + S_t + C_t R_t.

The time series analysis can also be done using the model y_t = T_t + S_t × C_t × R_t or y_t = T_t × C_t + S_t × R_t etc.

Measure of trend etc  in next  video

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