426 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 400 350 300 250 200 150 100 50 0 1234 5 6 7 8 9 10 11 12 13 14 15 16 FIG. 10.1 10.3 COMPONENT OF TIME SERIES The analysis of Time Series consists of the description and measurement of various changes or movements as they appear in the series during a period of time. These changes or movements are called the components of elements of time series. Fluctuations in a time series are mainly due to four basic types of variations (or movements). These four types of component are: 1. Secular Trend or Long Term Movement (T) 2. Seasonal Variation or Seasonal Movement (S) 3. Cyclical Fluctuation or Cyclic Variation (C) 4. Residual, Irregular of Random Movement (I) (1) Secular Trend: In Business, Economics and in our daily conversation the term Secular Trend or simply trend is popularly used. Where we speak of rising trend of population or prices, we mean the gradual increase in population or prices over a period of Time. Similarly, by declining trend of production or sales, we mean gradual decrease in production or sales over a period of time. The concept of trend does not include short range Oscillations, but refers to the steady movement over a long period time. “Secular trend is the smooth, regular and long term movement of a series showing continuous growth stagnation or decline over a long period of time. Graphically it exhibits general direction and shape of time series”. The trend movement of an economic time series may be upward or downward. The upward trend may be due to population growth, technological advances, improved methods of Business Organization and Management, etc. Similarly, the downward trend may be due to lack of demand for the product, storage of raw materials to be used in production, decline in death rate due to advance in medical sciences, etc. (2) Seasonal Variation: Seasonal variation is a short-term periodic movement, which occurs more or less regularly within a stipulated period of one year or shorter. The major factors that cause seasonal variations are climate and weather conditions, customs and habits of people, religious festivals, etc. For instance, the demand for electric fans goes up in summer season, the sale of Ice-cream increases very much in summer and the sale of woolen cloths goes up in winter. Also the sales of jewelleries and ornaments go up in TIME SERIES AND FORECASTING 427 marriage seasons, the sales and profits of departmental stores go up considerably during festivals like Id, Christmas, etc. Although the period of seasonal variations refers to a year in business and economics, it can also be taken as a month, week, day, hour, etc. depending on the type of data available. Seasonal variation gives a clear idea about the relative position of each season and on this basis, it is possibe to plan for the season. (3) Cyclical Fluctuations: These refer to the long term oscillations, or swings about a trend line or curve. These cycles, as they are some times called, may or may not be periodic that is they may or may not follow exactly similar patterns after equal intervals of time. In business and economic activities, moments are considered cyclic only if they recur after intervals of more than one year. The ups and downs in business, recurring at intervals of times are the effects of cyclical variations. A business cycle showing the swing from prosperity through recession, depression, recovery and back again to prosperity. This movement varies in time, length and intensity. (4) Residual Irregular or Random Movement: Random movements are the variations in a time series which are caused by chance factors or unforeseen factors which cannot be predicted in advance. For example, natural calamities like flood, earthquake etc, may occur at any movement and at any time. They can be neither predicted nor controlled. But the occurrence of these events influences business activities to a great extent and causes irregular or random variations in time series data. 10.4 ANALYSIS OF TIME SERIES Time series analysis consists of a description (generally mathematical) of the component movements present. To understand the procedures involved in such a description consider graph (A), which shows Ideal Time Series, Graph (A1) shows the graph of long-term, or secular, trend line. Graph (A 2 ) shows this long-term trend line with a superimposed cyclic movement (assumed to be periodic) and graph (A 3 ) shows a seasonal movement superimposed of graph (A2). The concept in graph (A) suggests a technique for analyzing time series. Y t Long Term Trend FIG. 10.2 (Graph A1) In Traditional or classical time series analysis, it is normally assumed that there is multiplicative relationship between the four components. Symoblically. =yT I× × S × C (1) y = Result of the Four Components (or original data) 428 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES It is assumed that trend has no effect on seasonal component. Also it is assumed that the business cycle has no effect on the seasonal component. Instead of Multiplicative model (1) some statisticians may prefer an additive model. ++=yT+ I S C (2) where y is the sum of the four components. Y t Long Term Trend and Cyclical Movement Y t Long Term Trend, Cyclical and Seasonal Movement FIG. 10.3 10.4.1 Analysis of Trend or Secular Trend In time series analysis the analysis of secular trend is very important. It helps us to predict or forecast future results. There are four methods used in analyzing trend in time series analysis. They are: (a) Method of Free Hand Curve (or graphic) (b) Method of Semi Averages (c) Method of Moving Averages (d) Method of Least Square (=) Free Hand Method: It is simplest method for studying trend. In this graphic method, the time series data are first plotted on the graph paper taking time on the x-axis and observed values of the other variable on y-axis. Then points obtained are joined by a free hand smooth curve of first degree. The line so obtained is called the trend curve and it shows the direction of the trend. The vertical distical of this line from x-axis gives the trend value for each time period. This method should be used only when a quick approximate idea of the trend is required. Example 1. Fit a trend line to the following data by the free hand grpahical method. YY YY Y earear earear ear 2000 2001 2002 2003 2004 2005 2006 Sales 52 54 56 53.5 57 54.5 59 TIME SERIES AND FORECASTING 429 Sol. 60 59 58 57 56 55 54 53 52 51 Sales 1998 2000 2002 2004 2006 2008 Yea rs FIG. 10.4 (>) Method of Semi Averages: This method is very simple and gives greater accuracy than the method of free hand or graphical. In this method, the given data is first divided into two parts and an average for each part is found. Then these two averages are plotted on a graph paper with respect to the midpoint of the two respective time intervals. The line obtained on joining these two points is the required trend line and may be extend both ways to estimate intermediate values. Remark: If given data is in odd number, then divide the whole series into two equal parts ignoring the middle period. Example 2. Fit a trend line to the following data by the method of semi averages. Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Bank 53 79 76 66 69 94 105 87 79 104 97 92 101 Clearance Sol. Here 13n = i.e., odd no. of data Now divide the given data into two equal parts (by omitting 1997) Year Clearenc e Semi Total Semi Avarage 1991 53 1992 79 1993 76 53+79+76+66+69+94 437/6=72.8333 1994 66 1995 69 1996 94 1997 105 1998 87 1999 79 2000 104 87+79+104+97+92+101 560/6=93.333 2001 97 2002 92 2003 101 430 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 130 120 110 100 90 80 70 60 50 Clearance 1 2 3 4 5 6 7 8 9 10 11 12 13 Yea r FIG. 10.5 Now the two semi averages 72.833 and 93.333 are plotted against the middle of their respective periods. Example 3. Draw the trend line by semi-average method using the given data Year 1998 1999 2000 2001 2002 2003 Production 253 260 255 266 259 264 (In Tons) Sol. Here n = 6 Pr 1998 253 253 260 255 768 3 256 1999 260 2000 255 2001 266 2002 259 266 259 264 789 3 263 2003 264 Year oduction Semi Total Semi Average ++ = ++ = TIME SERIES AND FORECASTING 431 270 265 260 255 250 245 Production 12 34 5 6 Yea r FIG. 10.6 (?) Method of Moving Averages: In the moving average method, the trend is described by smoothing out the fluctuations of the data by means of a moving average. Let ()() () 11 22 , , , , , , nn ty ty ty be the given time series, where 123 , , , , n ttt t denote the time periods and 123 , , , , n yyy y denote the corresponding values of the variable. The p-period moving totals (or sums) are defined as 123 , , , , , p yyy y 23 1. , , , p yy y + 345 2 , , , , p yyy y + , and so on. The p-period moving averages are defined as 12 23 134 2 ,, pp p yy yyy y yy y pp p ++ ++ + ++ + ++ + etc. The p period moving totals (or sums) and moving averages be also called moving totals (or sums) of order p and moving averages of order p respectively. These moving averages are also called the trend values. When we estimate the trend, we should select the order or period of the moving average (such as 3 yearly moving average, 5 yearly moving average, 4 yearly moving average, 8 yearly moving average, etc.). This order should be equal to the length of cycles in the time series. The method of moving averages is the most frequently used approach for determining the trend because it is definitely simpler process of fitting a polynomial. (1) Calculation of Moving Averages when the Period is Odd: In the case of odd period we would obtain the trend values and trend line as follows: (=) In the case of 3–yearly period, first of all calculate the following moving totals (or sums) y 1 + y 2 + y 3 , y 2 + y 3 + y 4 , y 3 + y 4 + y 5 , etc. In the case of 5 yearly period, calculate the following moving total (or sums) y 1 + y 2 + y 3 + y 4 + y 5 , y 2 + y 3 + y 4 + y 5 + y 6 , y 3 + y 4 + y 5 + y 6 + y 7 , etc. A period may be a year, a week, a day, etc. (>) Place the moving totals at the centres of three respective time. 432 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES (? ) Calculte the corresponding moving averages for 3 yearly or 5 yearly periods by dividing the moving totals by 3 or 5 respectively. Place these at the centre of the respective time. (@) If required we can plot these moving averages or trend values against the periods and obtain the trend line (or curve) from which we can determine the increasing or decreasing trend of the data. (A) It is more convenient to calculate the moving averages when the period is odd than when it is even, because there is only one middle period when the peirod is odd so and the moving average can be easily centred. (2) Calculation of Moving Averages when the Period is Even: When an even number of data is included in the moving averages (as 4 years), the centre point of the group will be between two years. It is therefore necessary to adjust or shift (known technically as centre) these averages so that they concide with the years. The 4 yearly moving total and the 4 yearly moving average may be obtained by the methods already outlined for the odd period average. To centre the values, a 2 yearly moving average is taken of the even period moving average. A 2 yearly moving average is taken of the 4 yearly moving average. The resulting average is located between the two 4 yearly moving average values and, therefore, coincides with the years. The end results (i.e., a 2 yearly moving average of 4 yearly moving average) are known as the 4 yearly moving average centred. We shall follow the steps given below in calculating the moving average when the order is even, say 4. (i) We calculate the following moving totals: +++ 1234 yyyy , 2345 , yyyy+++ 3456 , yyyy+++ etc. (ii) Place these moving totals at the centres of the respective time spans. In the case or 4 yearly time period, there are two middle terms viz. 2nd and 3rd. Hence, place this moving total against the centre of these two middle terms. Similarly, place other moving totals at the centres of 3rd and 4th periods, 4th and 5th periods and so on. (iii) Calculate the corresponding moving averages for 4 yearly periods by dividing the moving totals by 4. Place these at the centres of the time spans. i.e., against the corresponding moving totals. (Note that the moving averages so placed do not coincide with the original time period.) (iv) We take the total of 4 yearly moving averages taking two terms at a time starting from the first and place the sum at the middle of these two terms. The same procedure is repeated for other averages. (v) Finally, we take the two-period averages of the above moving averages by dividing each by 2. These are the required trend values. This process is called centreing of moving averages. If required, we can plot these moving averages or trend values and obtain trend line or curve. The major disadvantage of this method is that some trend values at the beginning and end of the series cannot be determined. Example 4. Calculate 3 yearly moving averages or trend values for the following data. Year (t) 1998 1999 2000 2001 2002 2003 2004 2005 Value(y) 3 5 7 10 12 14 15 16 TIME SERIES AND FORECASTING 433 Sol. 3 yearly moving averages means that there are three values induced in a group. Calculation of 3 yearly moving averages Year Value 3 Yearly 3 Yearly Moving Total ()t ()y Moving Total (Trend Valuea) 1998 3 1999 5 15 5.00 2000 7 22 7.33 2001 10 29 9.67 2002 12 36 12.00 2033 14 41 13.67 2004 15 45 15.00 2005 16 Hence the trend values are 5.00, 7.33, 9.67, 12.00, 13.67, 15.00 Example 5. Compute the 4 yearly moving averages from the following data: Year 1991 1992 1993 1994 1995 1996 1997 1998 Annual sales 36 43 43 34 44 54 34 24 (Rs. In crores) Sol. Calculation of 4 yearly moving averages Year 2 Yearly 4 Yearly Annual Sales 4 Yearly 4 Yearly Total of Centred Moving Average (Rs in Crores) Moving Totals Moving Average col. 4 (Centred) (Trend values) 1991 36 1992 43 1993 43 156 39 1994 34 164 41 80 40 1995 44 175 43.75 84.75 42.375 1996 54 166 41.50 85.25 42.625 1997 34 156 39 80.50 40.25 1998 24 Hence the trend values are 40, 42.375, 42.625, 40.25. Example 6. Assuming 5 yearly moving averages, calculate trend value from the data given below and plot the results on a graph paper. Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 Production 105 107 109 112 114 116 118 121 123 124 125 127 129 (Thousand) 434 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Sol. Calculation of 5 yearly moving averages Year Value 5 Yearly 5 Yearly Moving Moving Total Averages (trend value) 1971 105 – – 1972 107 – – 1973 109 547 109.4 1974 112 558 111.6 1975 114 569 113.8 1976 116 581 116.2 1977 118 592 118.4 1978 121 602 120.4 1979 123 611 122.2 1980 124 620 124.0 1981 125 628 125.6 1982 127 – – 1983 129 – – Hence the trend values are 109.4, 111.6, 113.8, 116.2, 118.4, 120.4, 122.2, 124.0, and 125.6. 1 3 5 7 9 11 13 Yea rs 135 130 125 120 115 110 105 100 Values FIG. 10.7 (@) Method of Least Square: This method is widely used for the measurement of trend. In method of least square we minimize the sum of the squares of the deviation of observed values from their expected values with respect to the constants. Let Tabx=+ be the required trend line By the principal of least square, the line of the best fit is obtained when the sum of the squares of the differences, i S is minimum i.e. () 2 iii STabx =−− ∑ is minimum TIME SERIES AND FORECASTING 435 When i S is minimum, we obtain normal equations as Tnab x=+ ∑∑ 2 Tx a x b x=+ ∑∑∑ On solving these two equations, we get a = T n ∑ , b = Tx x ∑ ∑ 2 Remark: If we take the midpoint in time as the origin, the negative values in the first half of the series balance out the positive values in the second half so 0x = ∑ Example 7. Determine the equation of a straight-line which best fits the following data Year 1974 1975 1976 1977 1978 Sales 35 56 79 80 40 (in Rs. 000) Compute the trend values for all the years from 1974 to 1978. Sol. Let the equation of the straight-line of best fit, with the origin at the middle year 1976 and unit of x as 1 year, be yabx=+ By the method of least squares, the values of a and b given by y a N = ∑ and 2 xy b x = ∑ Here N = number of years = 5 Calculations for the line of best fit Year Sale (Rs ’000) y x x 2 xy 1974 35 –2 4 –70 1975 56 –1 1 –56 1976 79 0 0 0 1977 80 1 1 80 1978 40 2 4 80 1979 y ∑ = 290 0 2 x ∑ = 10 xy ∑ = 34 . 129 (Thousand) 434 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Sol. Calculation of 5 yearly moving averages Year Value 5 Yearly 5 Yearly Moving Moving Total Averages (trend value) 1971. refers to a year in business and economics, it can also be taken as a month, week, day, hour, etc. depending on the type of data available. Seasonal variation gives a clear idea about the relative. SERIES AND FORECASTING 427 marriage seasons, the sales and profits of departmental stores go up considerably during festivals like Id, Christmas, etc. Although the period of seasonal variations