SciPy Optimize and Statistics

SciPy provides scientific computing tools including optimization and statistics.

Linear Regression with scipy.stats

from scipy import stats
import numpy as np

x = np.array([1, 2, 3, 4, 5])
y = np.array([2.1, 4.0, 5.9, 8.1, 10.0])

# Linear regression
result = stats.linregress(x, y)

slope = result.slope
intercept = result.intercept
r_squared = result.rvalue ** 2
std_err = result.stderr
p_value = result.pvalue

print(f"Slope: {slope:.4f}")
print(f"Intercept: {intercept:.4f}")
print(f"R-squared: {r_squared:.4f}")

Arrhenius Analysis with Linear Regression

from scipy import stats
import numpy as np

# Temperature and conductivity data
temp_c = np.array([25, 40, 55, 70, 85])
conductivity = np.array([0.008, 0.015, 0.028, 0.048, 0.082])

# Convert to Arrhenius form
temp_k = temp_c + 273.15
inv_t = 1 / temp_k  # x-axis: 1/T
ln_sigma = np.log(conductivity)  # y-axis: ln(sigma)

# Linear regression: ln(sigma) = ln(A) - Ea/(R*T)
result = stats.linregress(inv_t, ln_sigma)

# Extract activation energy
R = 8.314  # J/(mol*K)
Ea_jmol = -result.slope * R  # J/mol
Ea_kjmol = Ea_jmol / 1000  # kJ/mol

# Pre-exponential factor
A = np.exp(result.intercept)

print(f"Activation Energy: {Ea_kjmol:.2f} kJ/mol")
print(f"Pre-exponential Factor: {A:.2e}")
print(f"R-squared: {result.rvalue**2:.4f}")

Curve Fitting with scipy.optimize

from scipy.optimize import curve_fit
import numpy as np

# Define Arrhenius function
def arrhenius(T, A, Ea):
    R = 8.314
    return A * np.exp(-Ea / (R * T))

# Fit to data
temp_k = np.array([298, 313, 328, 343, 358])
conductivity = np.array([0.01, 0.02, 0.035, 0.06, 0.1])

# Initial guesses
p0 = [1e5, 30000]  # A, Ea

# Fit curve
popt, pcov = curve_fit(arrhenius, temp_k, conductivity, p0=p0)
A_fit, Ea_fit = popt

print(f"Fitted A: {A_fit:.2e}")
print(f"Fitted Ea: {Ea_fit/1000:.2f} kJ/mol")

Calculate R-squared Manually

def calculate_r_squared(y_actual, y_predicted):
    ss_res = np.sum((y_actual - y_predicted) ** 2)
    ss_tot = np.sum((y_actual - np.mean(y_actual)) ** 2)
    return 1 - (ss_res / ss_tot)