Find the sum of the squared errors for the linear model f(x) and the quadratic model g(x) using the given points.
f(x) = 2.0x − 3, g(x) = 0.09x^2 + 1.6x − 3
(−1, −4), (1, −2), (2, 0), (4, 6), (6, 9)
linear model-?
quadratic model- ?
Let me get this right - you want me to square all these differences to save you the trouble of doing it. I already took the course. Let us see what you did.
To find the sum of the squared errors for the linear model f(x) and the quadratic model g(x), we need to calculate the squared error for each data point and then sum up those squared errors.
Let's begin with the linear model f(x) = 2.0x - 3. We'll calculate the squared error for each data point and then sum them up.
For the given points (−1, −4), (1, −2), (2, 0), (4, 6), (6, 9), let's substitute the x-values into the linear model equation and compare the results with the corresponding y-values.
For the point (-1, -4):
Substituting x = -1 into f(x) = 2.0x - 3:
f(-1) = 2.0(-1) - 3 = -2 - 3 = -5
The squared error for this point is the difference between the calculated y-value and the given y-value, squared:
(-5 - (-4))^2 = (-1)^2 = 1
Repeat these steps for the remaining data points and sum up the squared errors:
For the point (1, -2):
f(1) = 2.0(1) - 3 = 2 - 3 = -1
(-1 - (-2))^2 = (1)^2 = 1
For the point (2, 0):
f(2) = 2.0(2) - 3 = 4 - 3 = 1
(1 - 0)^2 = (1)^2 = 1
For the point (4, 6):
f(4) = 2.0(4) - 3 = 8 - 3 = 5
(5 - 6)^2 = (-1)^2 = 1
For the point (6, 9):
f(6) = 2.0(6) - 3 = 12 - 3 = 9
(9 - 9)^2 = (0)^2 = 0
Summing up the squared errors for the linear model f(x):
1 + 1 + 1 + 1 + 0 = 4
Next, let's calculate the squared errors for the quadratic model g(x) = 0.09x^2 + 1.6x − 3. We'll follow the same process as we did for the linear model.
For each point, substitute the x-values into the quadratic model equation and compare the results with the corresponding y-values.
For the point (-1, -4):
Substituting x = -1 into g(x) = 0.09x^2 + 1.6x - 3:
g(-1) = 0.09(-1)^2 + 1.6(-1) - 3 = 0.09 - 1.6 - 3 = -3.51
The squared error for this point is calculated as the difference between the calculated y-value and the given y-value, squared:
(-3.51 - (-4))^2 = (0.49)^2 = 0.2401
Repeat these steps for the remaining data points and sum up the squared errors:
For the point (1, -2):
g(1) = 0.09(1)^2 + 1.6(1) - 3 = 0.09 + 1.6 - 3 = -1.31
(-1.31 - (-2))^2 = (0.69)^2 = 0.4761
For the point (2, 0):
g(2) = 0.09(2)^2 + 1.6(2) - 3 = 0.09(4) + 1.6(2) - 3 = 0.36 + 3.2 - 3 = 0.56
(0.56 - 0)^2 = (0.56)^2 = 0.3136
For the point (4, 6):
g(4) = 0.09(4)^2 + 1.6(4) - 3 = 0.09(16) + 1.6(4) - 3 = 1.44 + 6.4 - 3 = 4.84
(4.84 - 6)^2 = (-1.16)^2 = 1.3456
For the point (6, 9):
g(6) = 0.09(6)^2 + 1.6(6) - 3 = 0.09(36) + 1.6(6) - 3 = 3.24 + 9.6 - 3 = 9.84
(9.84 - 9)^2 = (0.84)^2 = 0.7056
Summing up the squared errors for the quadratic model g(x):
0.2401 + 0.4761 + 0.3136 + 1.3456 + 0.7056 = 3.081