“ TIME SERIES ANALYSIS- Measurement of Trend” A Lecture on “ TIME SERIES ANALYSIS- Measurement of Trend” for B.Sc(Statistics) 3rd year VI th semester, Paper-VII Title of the paper : Applied Statistics By Dr.G.Varalakshmi M.Sc.,Ph.D. Lecturerin Statistics D. K. Govt. College for women (A) Nellore 7/3/2019

“ TIME SERIES ANALYSIS- Measurement of Trend Objectives: To know the Definition of Time Series Analysis with examples To study the Components of Time Series To understand the Models of Time Series To know the Uses of Time Series Analysis To study the “Measurement of Trend-Methods” To solve the problems using the above said Methods 7/3/2019

DEFINITION OF TIMESERIES DATA There are three types of data CROSS SECTION DATA: Data collected on a variable at same point of time Ex: State-Population, Height- Weight TIME SERIES DATA: Data collected on a variable at different time periods Here time may be year or month or week or hour etc., Ex: Year-Population, Month-Sales, Week-Profits, Hour-Temparature PANEL DATA: A Combination of Cross Section Data and Time Series Data 7/3/2019

Example 7/3/2019

A Set of data depending on the time is called Time series - Kenny& Keeping A Time Series consists of data arranged in chronoligical - Croxton & Cowden t yt t1 Y1 t2 Y2 t3 Y3 . tn Yn 7/3/2019

ANALYSIS OF TIME SERIES When data relating to a variable are segregated by time there will be variations in the values of the variable from time to time The decomposition or seperation of time series value into its components is called Analysis of Time Series 7/3/2019

COMPONENTS OF TIME SERIES The various factors affecting a time series variable can be broadly classified into the following four categories Trend (Long Term Fluctuations) (T) Seasonal variations(Short Term Fluctuations) (S) Cyclical Variations (C) Irregular Variations (I ) 7/3/2019

TREND (LONG TERM FLUCTUATIONS) It is the general nature of the variable to show an increasing or decreasing tendency over a long period Trend is the slowly changing component of a time series Sudden and short term changes can’t be considered as due to trend Ex: Year-Population Year-Production Year- Profits 7/3/2019

SEASONAL VARIATIONS Seasonal variations are periodic variations which tend to repeat themselves at regular intervals of time(Season) The season may be a Quarter or Month or Day Ex:1. The Demand for A.Cs, Cool drinks etc increases during summer and will be bleak during rest of the year 2.Prices of certain seasonal crops will be low during that season and high during the rest of the year 3.Deposits in Banks during Croping Season 7/3/2019

CYCLICAL VARIATIONS Cyclical Variations are the oscillatory moments in a time series where the period of oscillation is more than a year Every kind of Business and Economic activity is susceptible to business cycles A Business Cycle consists of Four Phases or Periods 1. Prosperity : Material Cheap, Wages low, Prices High, So Profits High 2. Decline : Material high, wages increase, Price discounts offered , Purchasers wait for further reduction of prices-so profits reduced gradually 3.Depression: Increasing Pessimism in trade leads to closure of factory , since no profits leads to unemployment, low wages, low profits 4.Recovery : A period of improvement or recovery- Material low, Demand High, Production High 7/3/2019

IRREGULAR VARIATIONS (RANDOM VARIATIONS) Variations due to unforeseen events such as floods, droughts, earthquakes, wars, Lockouts, Strikes, Migrations These variations are purely random and beyond human control They do not repeat in a definite pattern Even then we need to study them 7/3/2019

Example 7/3/2019

MODELS OF TIME SERIES Additive Model : Yt =Tt +St + Ct + It Multiplicative Model : Yt =Tt St Ct It Mixed Model : Yt =Tt St + Ct + It 7/3/2019

USES OF TIME SERIES ANALYSIS This Analysis is useful for the Governments, Business Persons,NGOs, Banks, Private Organizations as follows Helps in understanding the PAST SITUATIONS Helps in finding the FUTURE PREDICTIONS Comparison of Actual data with the Predicted data and find the reasons for the difference Comparison of different TIMESERIES DATA 7/3/2019

METHODS OF MEASURING TREND Graphical Method Semi Averages Method Moving Averages Method Curve Fitting Method METHODS OF MEASURING SEASONAL VARIATIONS Simple Averages Method Ratio to Trend Method Ratio to Moving Averages Method Link Relatives Method 7/3/2019

GRAPHICAL METHOD YEAR SALES (in thousands) 2001 38 2002 40 2003 49 2004 45 2005 47 2006 52 2007 55 2008 2009 50 2010 53 2011 56 7/3/2019

SEMI AVERAGES METHOD YEAR SALES (in thousands) SEMI AVERAGES 2001 38 2002 40 2003 49 43.8 2004 45 2005 47 2006 52 2007 55 2008 2009 50 53.8 2010 53 2011 56 7/3/2019

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MOVING AVERAGE METHOD ODD MOVING PERIOD 7/3/2019

MOVING AVERAGE METHOD EVEN MOVING PERIOD 7/3/2019

MOVING AVERAGE METHOD EVEN MOVING PERIOD 7/3/2019

MOVING AVERAGE METHOD EVEN MOVING PERIOD 7/3/2019

CURVE FITTING Fitting of a straight line Fitting of a Parabola Fitting of Exponential curve Fitting of Power curve Fitting of growth curves 7/3/2019

FITTING OF STRIGHT LINE (ODD NUMBER OF OBSERVATIONS) 7/3/2019

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FITTING OF STRIGHT LINE (EVEN NUMBER OF OBSERVATIONS) 7/3/2019

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Normal Equations: y = n a+b ∑ x + c ∑ x2 y = a ∑ x+b ∑ x2+ c ∑ x3 FITTING OF PARABOLA y =a +b x + c x2 Normal Equations: y = n a+b ∑ x + c ∑ x2 y = a ∑ x+b ∑ x2+ c ∑ x3 y = a ∑ x2+b ∑ x3+ c ∑ x4 7/3/2019

QUESTIONS ???? 7/3/2019

THANK YOU 7/3/2019