[2022 Fall] Assignment7-2¶

Course: AP3021

Q7-2¶

Fit an exponential model to the data. Plot the data and the equation on standard/semi-log graph. (Python+繪圖)¶

In [1]:

import numpy as np
import matplotlib.pyplot as plt

x = np.array([0.4, 0.8,  1.2,  1.6,    2,  2.3])
y = np.array([800, 980, 1500, 1945, 2900, 3600])
lnx = np.log(x)
lny = np.log(y)
a0 = 1
a1 = 1

space=np.linspace(0, 2.5) # line space

A = np.zeros((2, 2))
B = np.zeros(2)

# Construct Matrix for solution
n = np.size(lnx)
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(lny)
B[1] = np.sum(x * lny)

# 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]
exp_regress = np.exp(regress)

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

# R = (A[0, 0] * B[1] - A[0,1] * B[0]) / 
# (np.sqrt(A[0, 0] * A[1, 1] - A[0, 1] ** 2) *
# np.sqrt(A[0, 0] * np.sum(y ** 2) - B[0] ** 2))

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

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

plt.show()

import numpy as np import matplotlib.pyplot as plt x = np.array([0.4, 0.8, 1.2, 1.6, 2, 2.3]) y = np.array([800, 980, 1500, 1945, 2900, 3600]) lnx = np.log(x) lny = np.log(y) a0 = 1 a1 = 1 space=np.linspace(0, 2.5) # line space A = np.zeros((2, 2)) B = np.zeros(2) # Construct Matrix for solution n = np.size(lnx) 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(lny) B[1] = np.sum(x * lny) # 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] exp_regress = np.exp(regress) # calculate St, Sr, Syx(STD), R^2, R St = np.sum((lny - np.mean(lny)) ** 2) Sr = np.sum((lny - X[0] - X[1] * x) ** 2) Syx = np.sqrt(Sr / (n - (m + 1))) r2 = (St - Sr) / St r = np.sqrt(r2) # R = (A[0, 0] * B[1] - A[0,1] * B[0]) / # (np.sqrt(A[0, 0] * A[1, 1] - A[0, 1] ** 2) * # np.sqrt(A[0, 0] * np.sum(y ** 2) - B[0] ** 2)) print('Standard Deviation:', Syx) print('Coefficient of determination:', r2) print('Correlation Coefficient:', r) plt.scatter(x, y, s=15) plt.plot(space, exp_regress, 'r') plt.title("Exponential Regression") plt.legend(['Regression Line', 'Data Point']) plt.grid() plt.show()

Condition number:  23.747545336711372
Standard Deviation: 0.053091820126552336
Coefficient of determination: 0.9935702501670921
Correlation Coefficient: 0.9967799406925744

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Last update: 2024-04-27