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import numpy as np
# Matrix operations
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
# Basic operationsprint("Matrix multiplication:", A @ B)
print("Element-wise:", A * B)
print("Transpose:", A.T)
print("Inverse:", np.linalg.inv(A))
print("Determinant:", np.linalg.det(A))
# Eigenvalues and eigenvectors (PCA foundation)
eigenvalues, eigenvectors = np.linalg.eig(A)
print(f"Eigenvalues: {eigenvalues}")
# Singular Value Decomposition (dimensionality reduction)
U, S, Vt = np.linalg.svd(A)
print(f"Singular values: {S}")
# Solving linear systems: Ax = b
b = np.array([5, 11])
x = np.linalg.solve(A, b)
print(f"Solution: {x}")
# Cosine similarity (NLP, recommendations)defcosine_similarity(v1, v2):
return np.dot(v1, v2) / (np.linalg.norm(v1) * np.linalg.norm(v2))
vec1 = np.array([1, 2, 3])
vec2 = np.array([4, 5, 6])
print(f"Cosine similarity: {cosine_similarity(vec1, vec2):.4f}")
4. Regression Analysis
import numpy as np
from sklearn.linear_model import LinearRegression, Ridge, Lasso
from sklearn.metrics import r2_score, mean_squared_error
import statsmodels.api as sm
# Multiple Linear Regression with statsmodels
X = np.random.randn(100, 3)
y = 2*X[:, 0] + 3*X[:, 1] - X[:, 2] + np.random.randn(100)*0.5
X_with_const = sm.add_constant(X)
model = sm.OLS(y, X_with_const).fit()
print(model.summary())
print(f"R-squared: {model.rsquared:.4f}")
print(f"Coefficients: {model.params}")
print(f"P-values: {model.pvalues}")
# Regularization comparison
X_train, y_train = X[:80], y[:80]
X_test, y_test = X[80:], y[80:]
models = {
"OLS": LinearRegression(),
"Ridge": Ridge(alpha=1.0),
"Lasso": Lasso(alpha=0.1)
}
for name, model in models.items():
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
print(f"{name}: R²={r2_score(y_test, y_pred):.4f}, RMSE={np.sqrt(mean_squared_error(y_test, y_pred)):.4f}")
Tools & Technologies
Tool
Purpose
Version (2025)
NumPy
Numerical computing
1.26+
SciPy
Scientific computing
1.12+
pandas
Data manipulation
2.2+
statsmodels
Statistical models
0.14+
scikit-learn
ML algorithms
1.4+
Troubleshooting Guide
Issue
Symptoms
Root Cause
Fix
Low p-value, small effect
Significant but meaningless
Large sample size
Check effect size
High variance
Unstable estimates
Small sample, outliers
More data, robust methods
Multicollinearity
Inflated coefficients
Correlated features
VIF check, remove features
Heteroscedasticity
Invalid inference
Non-constant variance
Weighted least squares
Best Practices
# ✅ DO: Check assumptions before testingfrom scipy.stats import shapiro
stat, p = shapiro(data)
if p > 0.05:
print("Data is approximately normal")
# ✅ DO: Use effect sizes, not just p-values# ✅ DO: Correct for multiple comparisons (Bonferroni)# ✅ DO: Report confidence intervals# ❌ DON'T: p-hack by trying many tests# ❌ DON'T: Confuse correlation with causation# ❌ DON'T: Ignore sample size requirements
