Least squares curve fitting is one of the basic and commonly used method for curve fitting .

suppose we have a data set of points

the first step in curve fitting is to choose a promising or intended function form , essentially a parameterised function ,to fit the data points ,Polynomial function form is one of the form that is used widely

here we have to determine the coefficients from the above equation , so that we can find the set of these coefficients ( data points ) such that the function best fits the given data .

by using the least square method we can minimise the root-mean-square error

here is the error at the point

so , is the distance between the data value and the fitted value on the curve

the least square error method finds the set of coordinates for and minimises the root mean square of error . As a result , the Fitted curve will be

x | 0 | 1 | 2 | 3 | 4 | 5 |

y | 0 | 1 | 1.414 | 1.73 | 2 | 2.24 |

Now after selecting the data set ,we need to decide the function that we will use the function to fit the data .

Now let us analyse the data by a plotting them as an intuitive view of the relationship between x and y

(IN general curve-fitting of n data points , an (n-1) degree of polynomial fits best for all the data points )

Since we have taken 6 data points ,the polynomial of degree 5 ,can pass exactly through all the data points

Now lets us try to find the values of for a polynomial of degree 2

from the above table

The equation is and the normal equations are

,

,

,

Now Substituting these values in the normal equations

On solving the above 3 equations ,we get

now on substitution the above values in in the equation

we get

` ````
```%use s2
var data_ls = SortedOrderedPairs(
doubleArrayOf(0.0, 1.0, 2.0, 3.0, 4.0, 5.0),
doubleArrayOf(0.0, 1.0, 1.414, 1.74, 2.0, 2.24))
var ls : LeastSquares = LeastSquares(2)
// here the 2 represents the highest degree of polynomial we are using
var fls : UnivariateRealFunction = ls.fit(data_ls)
var f_0 = fls.evaluate(0.0)
var f_1 = fls.evaluate(1.0)
var f_2 = fls.evaluate(2.0)
var f_3 = fls.evaluate(3.0)
var f_4 = fls.evaluate(4.0)
var f_5 = fls.evaluate(5.0)
//the error value can be obtained by
println(String.format("f(%f) = %f and the error value is %f", 0.0, f_0, 0.0-f_0))
println(String.format("f(%f) = %f and the error value is %f", 1.0, f_1, 1.0-f_1))
println(String.format("f(%f) = %f and the error value is %f", 2.0, f_2, 1.414-f_2))
println(String.format("f(%f) = %f and the error value is %f", 3.0, f_3, 1.732-f_3))
println(String.format("f(%f) = %f and the error value is %f", 4.0, f_4, 2.0-f_4))
println(String.format("f(%f) = %f and the error value is %f", 5.0, f_5, 2.24-f_5))

Output :

f(0.000000) = 0.098571 and the error value is -0.098571 f(1.000000) = 0.829029 and the error value is 0.170971 f(2.000000) = 1.401771 and the error value is 0.012229 f(3.000000) = 1.816800 and the error value is -0.084800 f(4.000000) = 2.074114 and the error value is -0.074114 f(5.000000) = 2.173714 and the error value is 0.066286

Now let us calculate the same for the above data set for a linear equation

from above table we get

Now let us the least squares for the linear equation

Now the normal equations for the linear equation are

now let us substitute the value in the equation

On solving equations (1) and (2) we get

here ,now the equation becomes

` ````
```%use s2
var data_ls = SortedOrderedPairs(
doubleArrayOf(0.0, 1.0, 2.0, 3.0, 4.0, 5.0),
doubleArrayOf(0.0, 1.0, 1.414, 1.74, 2.0, 2.24))
var ls : LeastSquares = LeastSquares(1)
// here the 2 represents the highest degree of polynomial we are using
var fls : UnivariateRealFunction = ls.fit(data_ls)
var f_0 = fls.evaluate(0.0)
var f_1 = fls.evaluate(1.0)
var f_2 = fls.evaluate(2.0)
var f_3 = fls.evaluate(3.0)
var f_4 = fls.evaluate(4.0)
var f_5 = fls.evaluate(5.0)
//the error value can be obtained by
println(String.format("f(%f) = %f and the error value is %f", 0.0, f_0, 0.0-f_0))
println(String.format("f(%f) = %f and the error value is %f", 1.0, f_1, 1.0-f_1))
println(String.format("f(%f) = %f and the error value is %f", 2.0, f_2, 1.414-f_2))
println(String.format("f(%f) = %f and the error value is %f", 3.0, f_3, 1.732-f_3))
println(String.format("f(%f) = %f and the error value is %f", 4.0, f_4, 2.0-f_4))
println(String.format("f(%f) = %f and the error value is %f", 5.0, f_5, 2.24-f_5))

Output :

f(0.000000) = 0.361429 and the error value is -0.361429 f(1.000000) = 0.776457 and the error value is 0.223543 f(2.000000) = 1.191486 and the error value is 0.222514 f(3.000000) = 1.606514 and the error value is 0.125486 f(4.000000) = 2.021543 and the error value is -0.021543 f(5.000000) = 2.436571 and the error value is -0.196571

Now let us compare the plots for the above 2 examples

` ````
```%use s2
// plotting the above function using JGnuplot
val p = JGnuplot(false)
p.addPlot("0.36 +0.41*x")
p.addPlot("0.1 +0.81*x-0.08*x*x")
p.getXAxis().setBoundaries(-20.0, 20.0)
p.getYAxis().setBoundaries(-20.0, 20.0)
p.plot()

Output :

when compared the results with the linear equation and the quadratic equation for the above data set ,the data seems to be fitted in curve of higher order polynomial when compared to the linear equation..

However, for analyzing the trend of the given data the linear equations seems to be more accurate , in the quadratic equation for the data the predicted values have a sudden fall . Thus we can say the the data can be fitted more precisely in the higher degree polynomial with less noise and linear equations are used in finding the trend of the data set given.

**Example : 3**

now let us try for an other example with a different data set.

x | 2 | 4 | 5 | 8 |

y | 5 | 6 | 5 | 7 |

from the above table

The general equations are

for getting a equation of degree 3 which is

Now let us calculate the necessary terms for calculating the general form equations

from the above table we get that

on solving the above equation we get

the final equation is

` ````
```%use s2
var data_ls = SortedOrderedPairs(
doubleArrayOf(2.0, 4.0, 5.0, 8.0),
doubleArrayOf(5.0, 6.0, 5.0, 7.0))
var ls : LeastSquares = LeastSquares(3)
// here the 3 represents the highest degree of polynomial we are using
var fls : UnivariateRealFunction = ls.fit(data_ls)
var f_0 = fls.evaluate(2.0)
var f_1 = fls.evaluate(4.0)
var f_2 = fls.evaluate(5.0)
var f_3 = fls.evaluate(8.0)
//the error value can be obtained by
println(String.format("f(%f) = %f and the error value is %f", 2.0, f_0, 5.0-f_0))
println(String.format("f(%f) = %f and the error value is %f", 4.0, f_1, 6.0-f_1))
println(String.format("f(%f) = %f and the error value is %f", 5.0, f_2, 5.0-f_2))
println(String.format("f(%f) = %f and the error value is %f", 8.0, f_3, 7.0-f_3))

Output :

f(2.000000) = 5.000000 and the error value is 0.000000 f(4.000000) = 6.000000 and the error value is 0.000000 f(5.000000) = 5.000000 and the error value is -0.000000 f(8.000000) = 7.000000 and the error value is 0.000000

` ````
```%use s2
// plotting the above function using JGnuplot
val p = JGnuplot(false)
p.addPlot("-13.22 +15.53*x-3.74*x*x+0.26*x*x*x")
p.getXAxis().setBoundaries(0.0, 11.0)
p.getYAxis().setBoundaries(-10.0, 11.0)
p.plot()

Output :

Here we can observe that the error values for the above data when calculated the (n-1)th degree equations for n data points ,the curve fits perfectly with out any noises . the corresponding values for (n-1) and (n) are given below

(n-1) = 3

n = 4