Curve Fitting

It is easier to understand the graphical view rather than mathematical view, that’s why Curve Fitting Statistics is useful in statistics, curve fitting statistics is a way to represent large number of data in the form of curve. Proper definition of curve fitting statistics states that it is a way to represent best fit curve of large amount of data. With the help of these fitting curves, we can easily visualize the whole data and we can easily summarize the data means we can easily judge that where data is placed and we can easily judge relationship between data. Now we will discuss how we fit curve into statistics.
We use following steps, which shows whole procedure of statistics curve fitting-
Step1: First of all, we judge what type of equation we have, means its first order, second order or n- order like
First order equation f(x) = ax + b
Second order equation f(x) = ax² + bx + c
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 n- order equation f(x) = axⁿ + bx^n-1 + cx^n-2 +................+ d.
Step 2: After evaluating the curve equation, we will analyze how many constraints are there, constraints can be any points, angle or curvature like
If f(x) = ax + b, it is like a Slope equation where this type of curve contains exactly two fit points,
And angle of this curve equation is tan θ = a.
If f(x) = ax² + bx + c, then this type of curve contains exactly three fit points, where angle of this curve equation is tan θ = (dy / dx).
If f(x) = ax³ + bx² + cx + d, then this type of curve contains exactly four fit points, where angle of this curve equation is tan θ = (dy / dx).
Step 3: After evaluation of these constraints, we make suitable curves like ellipse, circle, rectangle, sphere etc.
For making this kind of curve, first we find out what kind of error is there in given curve equation by following method -
Step 1: If we have a function f(x) = a + bx or y – (a + bx) = 0, then we calculate sum of Square -
Where sum of square is R² (a, b) = ∑_i=1ⁿ [y – (a + bx)]².
Step 2: After evaluation of sum of square, we can calculate the condition where R² (a, b) is minimum,
dR² / da_i (a, b) = -2 . ∑_i=1ⁿ [y – (a + bx)],
For critical points, above condition should be 0
dR² / da_i (a, b) = 0,
dR² / da_i (a, b) = -2 . ∑_i=1ⁿ [y – (a + bx)] = 0,
After evaluating above sum equation, it produces value of ‘a’ and ‘b’-
Step 3: After evaluation of ‘a’ and ‘b’, we calculate standard errors for ‘a’ and ‘b’ by following formula -
Standard error of a = SE(a) = s . √(1 / n + x² / ss_x)
Standard error of b = SE(a) = s / √ ss_x
Where ss_x is a regression and ‘s’ is variance.
This method is used for finding errors in curve equation.