[2022 Fall] Assignment7-1¶

Course: AP3021

Q7-1¶

a. Use least-squares regression to fit a straight line. Compute the standard error [Eq.17.9] of the estimate and the correlation coefficient. Assess the fit. (Python+繪圖)¶

In [1]:

import matplotlib.pyplot as plt
import numpy as np

x = np.array([1,   2, 3, 4, 5, 6, 7,  8,  9])
y = np.array([1, 1.5, 2, 3, 4, 5, 8, 10, 13])

space = np.linspace(1, 10) # line space
A = np.zeros((2, 2))
B = np.zeros(2)

# Construct Matrix for solution
n = np.size(x)
m = 1
A[0, 0] = np.size(x)
A[0, 1] = np.sum(x)
A[1, 0] = np.sum(x)
A[1, 1] = np.sum(x ** 2)
B[0] = np.sum(y)
B[1] = np.sum(x * y)

# Solve AX=B to know a0(X[0]), a1(X[1])
X = np.inner(np.linalg.inv(A), B)
print("Condition number: ", np.linalg.cond(A))

# construct regression line
regress = X[1] * space + X[0]

St = np.sum((y - np.mean(y)) ** 2)
Sr = np.sum((y - X[0] - X[1] * x) ** 2)
Syx = np.sqrt(Sr / (n-(m+1)))
r2 = (St - Sr) / St
r = np.sqrt(r2)

print('Standard Deviation:', Syx)
print('Coefficient of determination:', r2)
print('Correlation Coefficient:', r)

plt.scatter(x, y, s=10)
plt.plot(space, regress, 'r')
plt.title("Linear Regression")
plt.legend(['Regression Line', 'Data Point'])
plt.grid()

plt.show()

import matplotlib.pyplot as plt import numpy as np x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9]) y = np.array([1, 1.5, 2, 3, 4, 5, 8, 10, 13]) space = np.linspace(1, 10) # line space A = np.zeros((2, 2)) B = np.zeros(2) # Construct Matrix for solution n = np.size(x) m = 1 A[0, 0] = np.size(x) A[0, 1] = np.sum(x) A[1, 0] = np.sum(x) A[1, 1] = np.sum(x ** 2) B[0] = np.sum(y) B[1] = np.sum(x * y) # Solve AX=B to know a0(X[0]), a1(X[1]) X = np.inner(np.linalg.inv(A), B) print("Condition number: ", np.linalg.cond(A)) # construct regression line regress = X[1] * space + X[0] St = np.sum((y - np.mean(y)) ** 2) Sr = np.sum((y - X[0] - X[1] * x) ** 2) Syx = np.sqrt(Sr / (n-(m+1))) r2 = (St - Sr) / St r = np.sqrt(r2) print('Standard Deviation:', Syx) print('Coefficient of determination:', r2) print('Correlation Coefficient:', r) plt.scatter(x, y, s=10) plt.plot(space, regress, 'r') plt.title("Linear Regression") plt.legend(['Regression Line', 'Data Point']) plt.grid() plt.show()

Condition number:  158.0603399688975
Standard Deviation: 1.3066526967183245
Coefficient of determination: 0.9143610668789809
Correlation Coefficient: 0.9562222894698601

b. Use polynomial regression to fit a parabola to the data. Compute the standard error and correlation coefficient. Compare with (a). (Python+繪圖)¶

In [2]:

import matplotlib.pyplot as plt
import numpy as np

x = np.array([1,   2, 3, 4, 5, 6, 7,  8,  9])
y = np.array([1, 1.5, 2, 3, 4, 5, 8, 10, 13])

space = np.linspace(1, 10)

A = np.zeros((3, 3))
B = np.zeros(3)

# Construct Matrix for solution
n = np.size(x)
m = 2
A[0, 0] = np.size(x)
A[0, 1] = np.sum(x)
A[1, 0] = np.sum(x)
A[1, 1] = np.sum(x ** 2)
A[0, 2] = np.sum(x ** 2)
A[2, 0] = np.sum(x ** 2) 
A[1, 2] = np.sum(x ** 3)
A[2, 1] = np.sum(x ** 3)
A[2, 2] = np.sum(x ** 4)
B[0] = np.sum(y)
B[1] = np.sum(x * y)
B[2] = np.sum((x ** 2) * y)

# Solve AX=B to know a0(X[0]), a1(X[1])
X = np.inner(np.linalg.inv(A), B)
print("Condition number:", np.linalg.cond(A)) # Knowing the condition number

# construct regression line
regress = X[2] * (space ** 2) + X[1] * space + X[0]

# calculate St, Sr, Syx(STD), R^2, R
St = np.sum((y - np.mean(y)) ** 2)
Sr = np.sum((y - X[0] - X[1] * x - X[2] * (x ** 2)) ** 2)
Syx = np.sqrt(Sr / (n - (m + 1)))
r2 = (St - Sr) / St
r = np.sqrt(r2)

#print(A, B, X)
print('Standard Deviation:', Syx)
print('Coefficient of determination:', r2)
print('Correlation Coefficient:', r)

plt.scatter(x, y, s=15)
plt.plot(space, regress, 'r')
plt.title("Polynomial Regression")
plt.legend(['Regression Line', 'Data Point'])
plt.grid()

plt.show()

import matplotlib.pyplot as plt import numpy as np x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9]) y = np.array([1, 1.5, 2, 3, 4, 5, 8, 10, 13]) space = np.linspace(1, 10) A = np.zeros((3, 3)) B = np.zeros(3) # Construct Matrix for solution n = np.size(x) m = 2 A[0, 0] = np.size(x) A[0, 1] = np.sum(x) A[1, 0] = np.sum(x) A[1, 1] = np.sum(x ** 2) A[0, 2] = np.sum(x ** 2) A[2, 0] = np.sum(x ** 2) A[1, 2] = np.sum(x ** 3) A[2, 1] = np.sum(x ** 3) A[2, 2] = np.sum(x ** 4) B[0] = np.sum(y) B[1] = np.sum(x * y) B[2] = np.sum((x ** 2) * y) # Solve AX=B to know a0(X[0]), a1(X[1]) X = np.inner(np.linalg.inv(A), B) print("Condition number:", np.linalg.cond(A)) # Knowing the condition number # construct regression line regress = X[2] * (space ** 2) + X[1] * space + X[0] # calculate St, Sr, Syx(STD), R^2, R St = np.sum((y - np.mean(y)) ** 2) Sr = np.sum((y - X[0] - X[1] * x - X[2] * (x ** 2)) ** 2) Syx = np.sqrt(Sr / (n - (m + 1))) r2 = (St - Sr) / St r = np.sqrt(r2) #print(A, B, X) print('Standard Deviation:', Syx) print('Coefficient of determination:', r2) print('Correlation Coefficient:', r) plt.scatter(x, y, s=15) plt.plot(space, regress, 'r') plt.title("Polynomial Regression") plt.legend(['Regression Line', 'Data Point']) plt.grid() plt.show()

Condition number: 29878.808747043466
Standard Deviation: 0.34477129211586627
Coefficient of determination: 0.9948894656299115
Correlation Coefficient: 0.9974414597508525

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Last update: 2024-04-27