If E= 0 then all the n points will lie on y = f(x). If E ≠ 0, f(x) is chosen such that E is minimum, i.e., the best fitting curve to the set of points is that for which E is minimum. 

This method is known as the least square method. 

This method does not attempt to determine the form of the curve y = f(x) but it determines the values of the parameters of the equation of the curve.

FITTING OF LINEAR CURVES

The normal equations for the straight line y = a + bx are

∑y = na + b∑x

∑xy = a∑x + b∑x²

These equations can be solved simultaneously to give the best values of a and b. 

The best fitting straight line is obtained by substituting the values of a and b in the equation y = a + bx.

Question:

Fit a straight line to the following data:

Solution:

Let the straight line be fitted to the data be

                y = a + bx

The normal equations are

∑y = na + b∑x

∑xy = a∑x + b∑x²

Substituting these values in normal equations

           3.95 = 6a + 900b                        …(i)

           621 = 900a + 14200b                …(ii)

Solving equations (i) and (ii),

a = 0.0476   and   b = 0.0041

Hence, the required equation of the straight line is

                        y = 0.0476 + 0.0041x

HOW TO FIND THE REQUIRED EQUATION OF STRAIGHT LINE USING fx82MS CALCULATOR:

FITTING OF QUADRATIC CURVES

The normal equations for the curve y = a+bx+cx² are

∑y = na + b∑x + c∑x²

∑xy = a∑x + b∑ x² + c∑x³

∑x²y = a∑x²+ b∑x³ + c∑x⁴

These equations can be solved simultaneously to give the best values of a, b, and c. 

The best fitting parabola is obtained by substituting the values of a, b, and c in the equation y = a+bx+cx².

Question:

Fit a parabola to the following data:

Solution:

Let the equation of the parabola be a+bx+cx²

The normal equations are

∑y = na + b∑x + c∑x²

∑xy = a∑x + b∑ x² + c∑x³

∑x²y = a∑x²+ b∑x³ + c∑x⁴

Here n = 5,

Substituting these values in the normal equations,

      200 = 5a + 15b + 55c                       …(i)

      832 = 15a + 55b + 225c                  …(ii)

    3672 = 55a + 225b + 979c               …(iii)

Solving equations (i), (ii) and (iii) we get

a = 10.4,  b = -11.0857  and  c = 5.7143

Hence, the required equation of the parabola is

           y = 10.4 – 11.0857x + 5.7143x² 

HOW TO FIND THE REQUIRED EQUATION OF PARABOLA USING fx82MS CALCULATOR:

FITTING OF EXPONENTIAL AND LOGARITHMIC CURVES

For the curve y = ab^x

Taking Logarithm on both sides of the equation y = ab^x,

log_ey = log_ea + xlog_eb

Putting log_ey = Y, log_ea = A, log_eb = B

Y = A + BX

The normal equations are

∑Y = nA + B∑X

∑XY = a∑X + b∑X²

Solving these equations, A and B, and, hence, a and b can be found. 

The best fitting exponential curve is obtained by substituting the values of a and b in the equation y = ab^x.

Question:

Fit a curve of the form y = ab^x to the following data by the method of least squares:

Solution:

                                     y = ab^x

Taking Logarithm on both sides of the equation y = ab^x,

log_ey = log_ea + xlog_eb

Putting log_ey = Y, log_ea = A, log_eb = B

Y = A + BX

The normal equations are

∑Y = nA + B∑X

∑XY = a∑X + b∑X²

Here n = 7,

          34.4718   = 7A + 28B                            … (i)

          142.2551 = 28A + 140B                       … (ii)

Solving equations (i) and (ii),

           A = 4.3006   and   B = 0.156

                   log_ea = A

                   log_ea = 4.3006

                     a = 73.744

log_eb = B

log_eb = 0.156

 b = 1.1688

^NOTES