Introduction to least squares line:
There are many methods avilable for curve fitting. The most popular method of curve fitting is the principle of least squares line method. Curve fitting is a process of finding a functional relationship betweent the variables. It is useful in the study of correlation and regression.
Definition of least Squares Line :
Let (xi, yi) be the observed set of values of the variables (x, y), where i = 1, 2, 3,…,n. Let y = f(x) be a functional relationship between x and y. Then di = yi - f(xi) which is the difference between the observed value of y and the value of y is determined by the functional relation is called the residuals. The priniciple of least squares states that the parameters involved in f(x) should be chosen in such a way that `sum` di2 is minimum.
Fitting a Straight Line Using least Squares Method
Consider the fitting of the straight line y = ax + b to the data (xi, yi), i = 1, 2, 3, …, n. The residual
di is given by di = yi - (axi + b).
Therefore, `sum` di2 =`sum` (yi - axi - b)2 = R (say).
Since we are using the principle of least squares, we have to determine the value of a and b so that R is minimum.
Determine the Parameters of a and B Using Leats Squares Line Method:
Since R is minimum, `(del R)/(dela)` = 0 `=>` - 2 `sum` (yi - axi - b)xi = 0
`=>` `sum` (xiyi - axi2 - bxi) = 0.
Therefore, a`sum`xi2 + b `sum`xi = `sum`xiyi ————(1)
`(del R)/(del b)` = 0 `=>` - 2 `sum` (yi - axi - b) = 0
Therefore, a`sum`xi + nb = `sum`yi ————(2)
Equations (1) and (2) are called normal equations from which a and b can be found.
Note: If the given data is not in linear form it can be brought to linear form by some suitable transformation of variables. Then using the priniciple of least squares the curve of best fit can be achieved.
There are many methods avilable for curve fitting. The most popular method of curve fitting is the principle of least squares line method. Curve fitting is a process of finding a functional relationship betweent the variables. It is useful in the study of correlation and regression.
Definition of least Squares Line :
Let (xi, yi) be the observed set of values of the variables (x, y), where i = 1, 2, 3,…,n. Let y = f(x) be a functional relationship between x and y. Then di = yi - f(xi) which is the difference between the observed value of y and the value of y is determined by the functional relation is called the residuals. The priniciple of least squares states that the parameters involved in f(x) should be chosen in such a way that `sum` di2 is minimum.
Fitting a Straight Line Using least Squares Method
Consider the fitting of the straight line y = ax + b to the data (xi, yi), i = 1, 2, 3, …, n. The residual
di is given by di = yi - (axi + b).
Therefore, `sum` di2 =`sum` (yi - axi - b)2 = R (say).
Since we are using the principle of least squares, we have to determine the value of a and b so that R is minimum.
Determine the Parameters of a and B Using Leats Squares Line Method:
Since R is minimum, `(del R)/(dela)` = 0 `=>` - 2 `sum` (yi - axi - b)xi = 0
`=>` `sum` (xiyi - axi2 - bxi) = 0.
Therefore, a`sum`xi2 + b `sum`xi = `sum`xiyi ————(1)
`(del R)/(del b)` = 0 `=>` - 2 `sum` (yi - axi - b) = 0
Therefore, a`sum`xi + nb = `sum`yi ————(2)
Equations (1) and (2) are called normal equations from which a and b can be found.
Note: If the given data is not in linear form it can be brought to linear form by some suitable transformation of variables. Then using the priniciple of least squares the curve of best fit can be achieved.
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