Method of least square
Download as PPTX, PDF10 likes14,616 views
The document discusses curve fitting and the principle of least squares. It explains that curve fitting involves finding the curve of best fit that passes through most data points. The principle of least squares states that the curve of best fit is the curve for which the sum of the squares of the errors between the data points and fitted curve is minimized. It provides examples of using the method of least squares to fit linear, quadratic, and exponential curves to data.
Method of least square
- 2. A LAW THAT CONNECTSTHETWOVARIABLE 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 PASSTHROUGH MOST OFTHE 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.
- 3. 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
- 4. 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
- 5. 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.
- 6. METHOD OF LEAST SQUARE bxayLET 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 bxiayiS 1 2)^(
- 7. For S to be minimum (2) (3) On simplification of above 2 equations (4) (5) 0)(0)1)((2 1 bxayorbxiayi a S n i 0)2^(0))((2 1 bxaxyorxibxiayi b S n i xbnay 2^xbxaxy EQUATION (4) &(5) ARE NORMAL EQUATIONS SOLVINGTHEMWILL GIVE USVALUE OF a,b
- 8. 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.
- 9. 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
- 10. 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)
- 11. 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.
- 12. 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.