Learning Objectives:

• Find equation of the best fit line for a scatter plot using concept of least square regression.

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A straight line is described generically by f(x) = ax + b. In linear curve fitting, the goal is to identify the coefficients ‘a’ and ‘b’ such that f(x) ‘fits’ the given data well. The error between actual points and the ’best-fit’ line has to be minimized. that error such that the line will be the most accurate representation of the points. Here,

1. Positive or negative error have the same value (data point is above or below the line)

2. Weight greater errors more heavily.
   
   

Both things can be achieved by considering the squares of the errors rather than just plain values of errors. \[\mathrm{error} = \Sigma (d_i)^2\] \[\mathrm{error} = (y_1 - f(x_1))^2 + (y_2 - f(x_2))^2 + (y_3 - f(x_3))^2 + .....\] Our fit is a straight line, so now substitute f(x) = ax + b
\[\mathrm{error} = \sum_{i=1}^{n} (y_i - f(x_i))^2 = \sum_{i=1}^{n} (y_i - (ax_i + b))^2\] We have to minimize the error, so we use the concept of maxima/minima from Calculus, take derivative and equate it to zero. Since, there are two unknowns we will take partial derivative twice, once with respect to a and once with respect to b. \[\frac{\partial(\mathrm{error})}{\partial a} = -2 \sum_{i=1}^{n} x_i((y_i - (ax_i + b))) = 0\] \[\frac{\partial(\mathrm{error})}{\partial b} = -2 \sum_{i=1}^{n} ((y_i - (ax_i + b))) = 0\] \[a \sum x_i^2 + b\sum x_i = \sum x_i y_i\] \[a \sum x_i + b \times n = \sum y_i\] These equations can also be put in the matrix form, \[\begin{bmatrix} n & \sum x_i\\ \sum x_i & \sum x_i^2 \end{bmatrix}\begin{bmatrix} b\\a \end{bmatrix} = \begin{bmatrix} \sum y_i \\ \sum x_i y_i \end{bmatrix}\] We will first have to calculate all the \(\Sigma\) from the given data, and then using the above matrix form of equations, get the values of a and b.
Then, the best fit line will be \[y = ax + b\] This line later can be used for prediction of future values such as estimation of population, etc, or in case of missing scientific data, it gives reasonable value by interpolation.

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Learning Objectives:

• Calculate the value of correlation coefficient and interpret it.

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Pearson’s correlation coefficient between two variables is defined as the covariance of the two variables divided by the product of their standard deviations.
\[r = \dfrac{\Sigma X Y}{n\sigma_x \sigma_y}\] or, \[r = \dfrac{\Sigma X Y}{\sqrt{\Sigma X^2 \Sigma Y^2}}\] where,
X = deviation from mean, = x- \(\bar{x}\)
Y = deviation from mean, = y- \(\bar{y}\)
\(\sigma_x\), \(\sigma_y\) = standard deviation of x and y series
n = no. of values of the two variables.

The correlation coefficient ranges from -1 to 1.

\[\large r^{2}= b_{xy}* b_{yx}\]

• If the points are perfectly aligned in a straight line, then r = 1 (for positive slope)

• If the points are perfectly aligned in a straight line, then r = -1 (for negative slope)

• For random points which are scattered without any pattern, r = 0

where r = Karl Pearson’s coefficient of correlation

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