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CURVE FITTING BY LEAST SQARES

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                CURVE FITTING BY LEAST SQUARES

    

     INTRODUCTION

           Let us suppose that we wish to approximate (describe) a certain type of function that best expresses that association that exists between variables. A scatter plot of the set of values of the variables makes it possible to visualize a smooth cure that effectively approximates the given data set. A more useful way to represent this sort of approximating curve is by means of an equation or a formula. A term applied to the process of determining the equation and/or estimating the parameters appearing in the equation of approximating curve, is commonly called curve fitting
.

           It is relevant to point out that the relationship between the variables may be functional or regression. In functional relationship, a variable Y, has a true corresponding to each possible value or of another variable X, i.e. there is no question of random variation in the values of Y, and we make no relationship, i.e. problems of approximation and not of regression (already discussed earlier). Such relationships which are common in the natural sciences may be linear or non-linear.

  APPROXIMATING CURES AND THE PRINCIPLE OF LEAST SQUARES

     The data sets encountered in practice greatly vary in nature. It is therefore necessary to decide which type of approximating curve and their equation should be used. For this purpose, some of many common type of approximating curves and their equation are given below:

  1. A straight line or linear curve,                                                 Y= a + bX
  2. A parabola of second degree or quadratic curve,                Y= a + bX + cX2
  3.  A parabola of third degree or cubic curve,                                      Y = a + bX + cX2 + dX3 

    and so on.
        
          In all these equation Y is the dependent and X, the independent variable. In some situation, however, the variable X and Y can be reversed.

         We may approximate a given set of data by drawing a free hand curve, covering most of the points plotted. But it is clear that different individual would draw different curve according to their personal judgment. Therefore this procedure of fitting a curve is not satisfactory.

            The principle of least squares is applicable to curve fitting where the purpose is simply one of smoothing ( or approximation ) of a set of observations. Accordingly, we choose to determine the value of the parameters in the equations of approximating curves so as to make the sum of squares of residuals a minimum. A residual has been defined as the difference between the observed value and the corresponding value of the approximating curve.
         
           

     
    

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