Method of least square

METHOD OF LEAST SQUARE

BY: SOMYA BAGAI 11CSU148

EMPIRICAL LAW & CURVE FITTING

A LAW THAT CONNECTS THE TWO VARIABLE OF A GIVEN DATA IS CALLED EMPIRICAL LAW. SEVERAL EQUATIONS OF DIFFERENT TYPE CAN BE OBTAINED TO EXPRESS GIVEN DATA APPROX.

A CURVE OF “BEST FIT “WHICH CAN PASS THROUGH MOST OF THE POINTS OF GIVEN DATA (OR NEAREST) IS DRAWN .PROCESS OF FINDING SUCH EQUATION IS CALLED AS CURVE FITTING .

THE EQ’N OF THE CURVE IS USED TO FIND UNKNOWN VALUE.

SCATTER DIAGRAM

To find a relationship b/w the set of paired observations x and y(say), we plot their corresponding values on the graph , taking one of the variables along the x-axis and other along the y-axis i.e. (x1,y1) (X2,Y2)…..,(xn,yn).

The resulting diagram showing a collection of dots is called a scatter diagram. A smooth curve that approximate the above set of points is known as the approximate curve

PRINCIPLE OF LEAST SQUARE LET y=f(x) be equation of curve to be fitted to

given data points at the experimental value of PM is and the corresponding value of fitting curve is NM i.e .

),)....(2,2(),1,1( ynxnyxyx xix yi )1(xf

THIS DIFFERENCE IS CALLED ERROR.Similarly we say:

To make all errors positive ,we square each of them .

eMINMIPPN 11

)(

)2(22

)1(11

xnfynen

xfye

xfye

2^.........2^32^22^1 eneeeS THE CURVE OF BEST FIT IS THAT FOR WHICH THE SUM OF SQUARE OF ERRORS IS MINIMUM .THIS IS CALLED THE PRINCIPLE OF LEAST SQUARES.

METHOD OF LEAST SQUARE

bxay LET be the straight line to given data inputs. (1)

2^.......2^22^1

2)^1(2^1

)(1

111

eneeS

bxaye

bxaye

ytye

2^ei

n

i

bxiayiS1

2)^(

For S to be minimum

(2)

(3)

On simplification of above 2 equations

(4)

(5)

0)(0)1)((21

bxayorbxiayiaS

n

i

0)2^(0))((21

bxaxyorxibxiayibS

n

i

xbnay

2^xbxaxy

EQUATION (4) &(5) ARE NORMAL EQUATIONSSOLVING THEM WILL GIVE US VALUE OF a,b

To fit the parabola: y=a+bx+cx2 :

Form the normal equations ∑y=na+b∑x+c∑x2 , ∑xy=a∑x+b∑x2 +c∑x3 and ∑x2y=a∑x2 + b∑x3 + c∑x4 .

Solve these as simultaneous equations for a,b,c.

Substitute the values of a,b,c in y=a+bx+cx2 , which is the required parabola of the best fit.

In general, the curve y=a+bx+cx2 +……+kxm-1 can be fitted to a given data by writing m normal equations.

FITTING OF OTHER CURVES

x y

2 144

3 172.8

4 207.4

5 248.8

6 298.5

Q.FIT A RELATION OF THE FORM ab^x

xBnAY

xBAY

bxay

xaby

lnlnln

^

2^xBxAxY

x y Y xY X^2

2 144 4.969 9.939 4

3 172.8 5.152 15.45 9

4 207.4 5.3346 21.3385 16

5 248.8 5.5166 27.5832 25

6 298.5 5.698 34.192 36

20 26.6669 108.504 90

26.67=5A +B20, 108.504=20A+90B

A=4.6044 & B=0.1824

Y=4.605 + x(0.182)

ln y =4.605 +x(0.182)

y= e^(4.605).e^(0.182)

APPLICATIONS & ADVANTAGES

The most important application is in data fitting. The best fit in the least-squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model.

The graphical method has its drawbacks of being unable to give a unique curve of fit .It fails to give us values of unknown constants .principle of least square provides us with a elegant procedure to do so.

When the problem has substantial uncertainties in the independent variable (the 'x' variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares.

THANK YOU HAVE A NICE

DAY !