Types of Correlation

Least Square Method Graph
In linear regression, the line of best fit is a straight line as shown in the following diagram:

The given data points are to be minimized by the method of reducing residuals or offsets of each point from the line. The vertical offsets are generally used in surface, polynomial and hyperplane problems, while perpendicular offsets are utilized in common practice.

Least Square Method Formula
The least-square method states that the curve that best fits a given set of observations, is said to be a curve having a minimum sum of the squared residuals (or deviations or errors) from the given data points. Let us assume that the given points of data are (x1, y1), (x2, y2), (x3, y3), …, (xn, yn) in which all x’s are independent variables, while all y’s are dependent ones. Also, suppose that f(x) is the fitting curve and d represents error or deviation from each given point.
Now, we can write:
d1 = y1 − f(x1)
d2 = y2 − f(x2)
d3 = y3 − f(x3)
…..
dn = yn – f(xn)

The least-squares explain that the curve that best fits is represented by the property that the sum of squares of all the deviations from given values must be minimum, i.e:

Sum = Minimum Quantity
Suppose when we have to determine the equation of line of best fit for the given data, then we first use the following formula.
The equation of least square line is given by Y = a + bX
Normal equation for ‘a’:
∑Y = na + b∑X
Normal equation for ‘b’:
∑XY = a∑X + b∑X2
Solving these two normal equations we can get the required trend line equation.
Thus, we can get the line of best fit with formula y = ax + b.

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