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TIME SERIES ANALYSIS PRESENTED BY:- CHANDRA PRAKASH GOYAL RAVINDRA KUMAR KUMAWAT PRESENTATION ON -
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DEFINITION AND MEANING <ul><li>A time series is a sequence of values of the same variate corresponding to successive points in time.’ -Werner Z. Hirsch , An Introduction To Modern Statistics </li></ul><ul><li>HISTORICAL VARIABLES </li></ul><ul><li>When we observe numerical data at different points of time, the set of observations is known as time series. </li></ul><ul><li>Time is the most important factor because the variable is related to time which may be either year, month , week, day, hour or even minute or second. </li></ul><ul><li>Independent variable = Measurement of time </li></ul><ul><li>dependent variable = Effects of change in data with time </li></ul><ul><li>Example :- Yearly production of IRON during five year plan, Temperature/Hour of a patient in the Hospital, Speed of missile per second </li></ul>
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ROLE OF TIME SERIES ANALYSIS <ul><li>Time series analysis is of great significance in business decision making for the following reasons: </li></ul><ul><li>It helps in the understanding of past behavior. </li></ul>
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<ul><li>2) It helps in planning future operations . </li></ul>
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<ul><li>3) It helps in evaluating current accomplishments. </li></ul><ul><li>4) It facilitates comparison. </li></ul>
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COMPONENTS OF TIME SERIES <ul><li>Four types of patterns, variations, movements, fluctuations of time series called components of time series:- </li></ul><ul><li>Secular Trend, </li></ul><ul><li>Seasonal Variations, </li></ul><ul><li>Cyclical Variations, </li></ul><ul><li>Irregular Variations. </li></ul>Seasonal Variation Secular trend
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<ul><li>Classical time series:- </li></ul><ul><li>Here, Y=Trend, </li></ul><ul><li>S=Seasonal Variation, </li></ul><ul><li>C=Cyclical Variation, </li></ul><ul><li>I=Irregular Variation. </li></ul><ul><li>Another approach is to treat each observation of a time series as the sum of these four components. Symbolically, </li></ul>
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Preliminary Adjustments before Analyzing Time Series <ul><li>Before beginning the actual work of analyzing a time series it is necessary to make certain adjustments in the raw data .The adjustments are: </li></ul><ul><li>Adjustments for Calendar Variation, </li></ul><ul><li>Adjustments for Population changes, </li></ul><ul><li>Adjustments for Price changes, </li></ul><ul><li>adjustments for comparability. </li></ul>
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STRAIGHT LINE TREND-METHODS OF MEASUREMENT <ul><li>The following methods are used for measuring trend: </li></ul><ul><li>The Freehand or Graphic Method, </li></ul><ul><li>The Semi-average Method, </li></ul><ul><li>The Method of Least Squares. </li></ul><ul><li>The following are the methods of measuring non-linear trends: </li></ul><ul><li>Freehand or Graphic Method, </li></ul><ul><li>Moving Average Method, </li></ul><ul><li>Second Degree Parabola. </li></ul>NON LINEAR TREND
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CONVERSION OF ANNUAL TREND VALUES TO MONTHLY TREND VALUES <ul><li>In converting straight line trends from an annual to a monthly basis, two situations must be clearly distinguished. For series such as sales, production or earnings, the annual figure is the total of monthly figures. Here it is necessary to divide both a and b by 12 to reduce them to monthly level. The b values must then be divided 12 once again in order to convert from annual to monthly increments. </li></ul><ul><li>Example: Convert the following annual trend equation for Tea production to a monthly trend equation: </li></ul><ul><li>Y= 108+1.58X </li></ul><ul><li>(Origin 2003 , time unit I year, Y =tea production in million Kg.) </li></ul><ul><li>Solution: Monthly trend equation will be obtained by dividing a by 12 and b by 144. </li></ul><ul><li>thus the monthly trend equation will be: </li></ul><ul><li>Y=108/12+1.58/144 </li></ul><ul><li>(Origin July 1,2003, time unit 1 month, Y monthly production in million Kg.) </li></ul>
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<ul><li>In computing trends, the middle of the time series is often used as the origin in order to cut short the computations. But very often we need to change the origin of the trend equation to some other point in the series his is either to facilitate the comparison of trend values among neighboring years or to convert a trend equation from any annual to a monthly basis. For example , consider the trend equation: </li></ul><ul><li>Y= 110-2X </li></ul><ul><li>(Origin 2005, time unit 1year) </li></ul><ul><li>If we wish to shift the trend equation to 2008, We note that precedes the stated origin of 2005 by 7 time units. Consequently we must deduct 7 times annual increment that is b(-7), from the trend value of 2005 as below: </li></ul><ul><li>Y= 110-2(-7)= 110 + 14 = 124 </li></ul><ul><li>The value 124 becomes the trend value at the new origin 2008 and the trend equation may now be written as: </li></ul><ul><li>Y = 124 – 2 X </li></ul>SHIFTING THE TREND ORIGIN
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Measurement of Seasonal Variations <ul><li>When data are expressed annually, there is no seasonal variation. How ever, monthly or quarterly data frequently exhibit strong seasonal movement and considerable interests attaches to devising a pattern of average seasonal variation. </li></ul><ul><li>Example: If we observe the sales of a book seller, we find hat of the quarter July – September sales are maximum. If we know by how much the sales of this quarter are usually above or below the previous quarter for seasonal reasons, we shall be able to answer that there is upward tendency or downward tendency? </li></ul><ul><li>Before attempting to measure seasonal variation certain preliminary decisions must be made . </li></ul><ul><li>To obtain a statistical description of a pattern of seasonal variation it will be desirable to first free the data from the effect of trend , cycles and irregular variation. Once these other components have been eliminated we can calculate , I index form, a measure of seasonal variations which is usually referred to s seasonal index. Thus the measures of seasonal variation are called seasonal indexes(%). </li></ul><ul><li>There are four methods of measuring seasonal variations. </li></ul><ul><li>Methods of simple averages (weekly, Monthly or quarterly). </li></ul><ul><li>Ratio to trend method. </li></ul><ul><li>Ratio to moving average method . </li></ul><ul><li>Link relatives method. </li></ul>
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Example:-consumption of monthly electric power in million of Kw hours for street lighting in one of the states in India during 2004-2008 is given below: Find out seasonal variation by the method of monthly averages. Year Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 2004 318 281 278 250 231 216 223 245 269 302 325 347 2005 342 309 299 268 249 236 242 262 288 321 342 364 2006 367 328 320 287 269 251 259 284 309 345 367 394 2007 392 349 342 311 290 273 282 305 328 364 389 417 2008 420 378 370 334 314 296 305 330 356 396 422 452
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Solution: COMPUTATION OF SEASONAL INDICES BY THE METHOD OF MONTHLY AVERAGES Month Consumption of monthly electric power Monthly total for 5 years Five yearly Average Percent- age 2004 2005 2006 2007 2008 Jan. 318 342 367 392 420 1839 367.8 116.1 Feb. 281 309 328 349 378 1645 329 103.9 March 278 299 320 342 370 1609 321.8 101.6 April 250 268 287 311 334 1450 290 91.6 May 231 249 269 290 314 1353 270.6 85.4 June 216 236 251 273 296 1272 254.4 80.3 July 323 242 259 282 305 1311 262.2 82.8 Aug. 245 262 284 305 330 1426 285.2 90.1 Sept. 269 288 309 328 356 1550 310 97.9 Oct. 302 321 345 364 396 1728 345.6 109.1 Nov. 325 342 367 389 422 1845 369 116.5 Dec. 347 364 394 417 452 1974 394.8 124.7 Total 19002 3800.4 1200 Average 1.5835 316.7 100
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<ul><li>The above calculations are explained below: </li></ul><ul><li>Column No 7 gives the total for each month for five years. </li></ul><ul><li>In column No 8 each total of column no 7 has been divided by five to obtain an average to each month. </li></ul><ul><li>The average of monthly averages is obtained by dividing the total of monthly averages by 12. </li></ul><ul><li>In column No 9 each monthly average has been expressed as a percentage of the average of monthly averages. Thus, the percentage for January </li></ul><ul><li>If instead of monthly data we are given weekly or quarterly data, we shall compute weekly or quarterly averages by following the same procedure as explained above. </li></ul>
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Measurements of Cyclical Variations <ul><li>Despite the importance of business cycles, they are most difficult type of economic fluctuations to measure. This is because successive cycles vary widely in timing, amplitude and pattern, and because the cyclical rhythm is inextricably mixed with irregular factors. Because of these reasons it impossible to construct meaningful typical cycle indexes of curves similar to those that have been developed for trends and seasonal. The various methods used for measuring cyclical variations are : </li></ul><ul><li>Residual method </li></ul><ul><li>Reference Cycle analysis method </li></ul><ul><li>Direct method </li></ul><ul><li>Harmonic analysis method </li></ul>
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Measurement of Irregular Variations <ul><li>The irregular components in a time series represents the residue of fluctuations after trend cyclical and seasonal movements have been accounted for. Thus, if original divided by T, S and C; we get I, i.e. </li></ul><ul><li>(TSCI/TSC = I). In practice the cycle itself is so erratic and is so interwovenleft with irregular movements that it is impossible to separate them. In the analysis of a time series into its components fluctuations, therefore, trend and seasonal movements are usually measured directly, while cyclical and irregular fluctuations are left altogether after the other elements have been removed. </li></ul>
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<ul><li>REFERENCES </li></ul><ul><li>BUSINESS STATISTICS BY S.P .GUPTA AND M.P. GUPTA </li></ul><ul><li>KAILASH NATH NAGAR </li></ul><ul><li>WWW.WIKIPEDIA.COM </li></ul>
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<ul><li>Job Execution </li></ul><ul><li>Equal contribution given by both </li></ul>
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<ul><li>LIFE IS JUST A MIRROR WHAT YOU SEE OUT THERE, YOU MUST FIRST SEE INSIDE OF YOU. </li></ul>
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