Consider the four points (x_1,y_1) = (-5,-10), (x_2,y_2) = (0,3), (x_3,y_3) = (2,11) and (x_4,y_4) = (3,14). The line that minimizes the empirical squared error can be expressed as y = \hat{a}+\hat{b}x, where
\hat{a} =
\hat{b} =
To find the line that minimizes the empirical squared error, we need to find the values of \hat{a} and \hat{b}.
First, let's define the empirical squared error as:
E = \sum_{i=1}^{4} (y_i - (\hat{a}+\hat{b}x_i))^2
where (x_i, y_i) are the given points.
To minimize the error, we need to take the partial derivatives of E with respect to \hat{a} and \hat{b}, and set them equal to zero.
\frac{\partial E}{\partial \hat{a}} = \sum_{i=1}^{4} 2(y_i - (\hat{a}+\hat{b}x_i))(-1) = 0
\frac{\partial E}{\partial \hat{b}} = \sum_{i=1}^{4} 2(y_i - (\hat{a}+\hat{b}x_i))(-x_i) = 0
Let's simplify these equations:
1) \sum_{i=1}^{4} (y_i - \hat{a} - \hat{b}x_i) = 0
2) \sum_{i=1}^{4} x_i(y_i - \hat{a} - \hat{b}x_i) = 0
Expanding the summations:
1) (-10 - \hat{a} - \hat{b}(-5)) + (3 - \hat{a} - \hat{b}(0)) + (11 - \hat{a} - \hat{b}(2)) + (14 - \hat{a} - \hat{b}(3)) = 0
2) (-5)(-10 - \hat{a} - \hat{b}(-5)) + (0)(3 - \hat{a} - \hat{b}(0)) + (2)(11 - \hat{a} - \hat{b}(2)) + (3)(14 - \hat{a} - \hat{b}(3)) = 0
Simplifying further:
1) -10 - \hat{a} + 5\hat{b} + 3 - \hat{a} - 2\hat{b} + 11 - \hat{a} - 2\hat{b} + 14 - \hat{a} - 3\hat{b} = 0
-4\hat{a} - 7\hat{b} - 18 = 0
2) 50 + 45 - 25\hat{a} - 65\hat{b} + 22 - 4\hat{a} - 56\hat{b} + 42 - 5\hat{a} - 84\hat{b} + 42 - 6\hat{a} - 126\hat{b} = 0
-20\hat{a} - 331\hat{b} + 201 = 0
Now we have a system of linear equations:
-4\hat{a} - 7\hat{b} - 18 = 0
-20\hat{a} - 331\hat{b} + 201 = 0
Solving these equations simultaneously will give us the values of \hat{a} and \hat{b}.
To find the line that minimizes the empirical squared error, we need to find the values of \hat{a} and \hat{b}.
Step 1: Calculate the mean of x and y values:
\bar{x} = (x_1 + x_2 + x_3 + x_4) / 4
\bar{y} = (y_1 + y_2 + y_3 + y_4) / 4
\bar{x} = (-5 + 0 + 2 + 3) / 4 = 0
\bar{y} = (-10 + 3 + 11 + 14) / 4 = 4.5
Step 2: Calculate the numerator and denominator of the slope (\hat{b}) formula:
Numerator:
numerator = (x_1 - \bar{x})(y_1 - \bar{y}) + (x_2 - \bar{x})(y_2 - \bar{y}) + (x_3 - \bar{x})(y_3 - \bar{y}) + (x_4 - \bar{x})(y_4 - \bar{y})
numerator = (-5 - 0)(-10 - 4.5) + (0 - 0)(3 - 4.5) + (2 - 0)(11 - 4.5) + (3 - 0)(14 - 4.5)
numerator = (-5)(-14.5) + (0)(-1.5) + (2)(6.5) + (3)(9.5)
numerator = 72.5
Denominator:
denominator = (x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + (x_3 - \bar{x})^2 + (x_4 - \bar{x})^2
denominator = (-5 - 0)^2 + (0 - 0)^2 + (2 - 0)^2 + (3 - 0)^2
denominator = 38
Step 3: Calculate the slope (\hat{b}):
\hat{b} = numerator / denominator
\hat{b} = 72.5 / 38
\hat{b} = 1.9079
Step 4: Calculate the intercept (\hat{a}):
\hat{a} = \bar{y} - \hat{b} * \bar{x}
\hat{a} = 4.5 - 1.9079 * 0
\hat{a} = 4.5
Therefore, the line that minimizes the empirical squared error can be expressed as y = 4.5 + 1.9079x.