GATE Electrical 2023 | Question: 25

Question

Three points in the $x-y$ plane are $(-1,0.8),(0,2.2)$ and $(1,2.8).$ The value of the slope of the best fit straight line in the least square sense is___________(Round off to $2$ decimal places)

Answer 1

Ans :- 1.00

For NNN points (xi,yi)(x_i,y_i)(xi​,yi​), the least-squares slope mmm is

m=N∑xiyi−(∑xi)(∑yi)/ N∑xi^2−(∑xi)^2.

Answer 2

Formula for slope of the best-fit line (least squares) for the points: 

​

Answer is 1

Answer 3

By matrix method optimal vector $\text{w}^* = (XX^\top)^{-1}XY$  where $X$ and $Y$ are vectors containing values values of x and y respectively.

$$X=\begin{bmatrix}
1& 1 & 1 \\
-1 & 0  & 1  \\
\end{bmatrix}, \quad Y= \begin{bmatrix}
 0.8\\
 2.2\\
 2.8\\
\end{bmatrix}$$

Note first row of $X$ is for the intercept term of the line.

$$(XX^\top)^{-1} = \frac{1}{6}\begin{bmatrix}
2 & 0\\
0 & 3 \\
\end{bmatrix} \quad XY=\begin{bmatrix}
5.8 \\
2 \\
\end{bmatrix}$$

$$\boxed{\text{w}^* =(XX^\top)^{-1}XY =\begin{bmatrix}
\frac{11.6}{6} \\
1 \\
\end{bmatrix}} $$

so best fit line : $y=\vec{x} \cdot \text{w}^*$ or $y=\text{w}_0 +x_1\text{w}_1$

We have slope = $\boxed{\text{w}_1 = 1}$