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Statistics & Mathematics

统计学与数学

Mathematical foundations for data science, machine learning, and statistical analysis.
数据科学、机器学习与统计分析的数学基础。

Quick Start

快速开始

python
import numpy as np
import scipy.stats as stats
from sklearn.linear_model import LinearRegression
python
import numpy as np
import scipy.stats as stats
from sklearn.linear_model import LinearRegression

Descriptive Statistics

Descriptive Statistics

data = np.array([23, 45, 67, 32, 45, 67, 89, 12, 34, 56]) print(f"Mean: {np.mean(data):.2f}") print(f"Median: {np.median(data):.2f}") print(f"Std Dev: {np.std(data, ddof=1):.2f}") print(f"IQR: {np.percentile(data, 75) - np.percentile(data, 25):.2f}")
data = np.array([23, 45, 67, 32, 45, 67, 89, 12, 34, 56]) print(f"Mean: {np.mean(data):.2f}") print(f"Median: {np.median(data):.2f}") print(f"Std Dev: {np.std(data, ddof=1):.2f}") print(f"IQR: {np.percentile(data, 75) - np.percentile(data, 25):.2f}")

Hypothesis Testing

Hypothesis Testing

sample_a = [23, 45, 67, 32, 45] sample_b = [56, 78, 45, 67, 89] t_stat, p_value = stats.ttest_ind(sample_a, sample_b) print(f"T-statistic: {t_stat:.4f}, p-value: {p_value:.4f}")
if p_value < 0.05: print("Reject null hypothesis: significant difference") else: print("Fail to reject null hypothesis")
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sample_a = [23, 45, 67, 32, 45] sample_b = [56, 78, 45, 67, 89] t_stat, p_value = stats.ttest_ind(sample_a, sample_b) print(f"T-statistic: {t_stat:.4f}, p-value: {p_value:.4f}")
if p_value < 0.05: print("Reject null hypothesis: significant difference") else: print("Fail to reject null hypothesis")
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Core Concepts

核心概念

1. Probability Distributions

1. 概率分布

python
import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
python
import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt

Normal Distribution

Normal Distribution

mu, sigma = 100, 15 normal_dist = stats.norm(loc=mu, scale=sigma) x = np.linspace(50, 150, 100)
mu, sigma = 100, 15 normal_dist = stats.norm(loc=mu, scale=sigma) x = np.linspace(50, 150, 100)

PDF, CDF calculations

PDF, CDF calculations

print(f"P(X < 85): {normal_dist.cdf(85):.4f}") print(f"P(X > 115): {1 - normal_dist.cdf(115):.4f}") print(f"95th percentile: {normal_dist.ppf(0.95):.2f}")
print(f"P(X < 85): {normal_dist.cdf(85):.4f}") print(f"P(X > 115): {1 - normal_dist.cdf(115):.4f}") print(f"95th percentile: {normal_dist.ppf(0.95):.2f}")

Binomial Distribution (discrete)

Binomial Distribution (discrete)

n, p = 100, 0.3 binom_dist = stats.binom(n=n, p=p) print(f"P(X = 30): {binom_dist.pmf(30):.4f}") print(f"P(X <= 30): {binom_dist.cdf(30):.4f}")
n, p = 100, 0.3 binom_dist = stats.binom(n=n, p=p) print(f"P(X = 30): {binom_dist.pmf(30):.4f}") print(f"P(X <= 30): {binom_dist.cdf(30):.4f}")

Poisson Distribution (events per time)

Poisson Distribution (events per time)

lambda_param = 5 poisson_dist = stats.poisson(mu=lambda_param) print(f"P(X = 3): {poisson_dist.pmf(3):.4f}")
lambda_param = 5 poisson_dist = stats.poisson(mu=lambda_param) print(f"P(X = 3): {poisson_dist.pmf(3):.4f}")

Central Limit Theorem demonstration

Central Limit Theorem demonstration

population = np.random.exponential(scale=10, size=100000) sample_means = [np.mean(np.random.choice(population, 30)) for _ in range(1000)] print(f"Sample means are approximately normal: mean={np.mean(sample_means):.2f}")
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population = np.random.exponential(scale=10, size=100000) sample_means = [np.mean(np.random.choice(population, 30)) for _ in range(1000)] print(f"Sample means are approximately normal: mean={np.mean(sample_means):.2f}")
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2. Hypothesis Testing Framework

2. 假设检验框架

python
from scipy import stats
import numpy as np

class HypothesisTest:
    """Framework for statistical hypothesis testing."""

    @staticmethod
    def two_sample_ttest(group_a, group_b, alpha=0.05):
        """Independent samples t-test."""
        t_stat, p_value = stats.ttest_ind(group_a, group_b)
        effect_size = (np.mean(group_a) - np.mean(group_b)) / np.sqrt(
            (np.var(group_a) + np.var(group_b)) / 2
        )
        return {
            "t_statistic": t_stat,
            "p_value": p_value,
            "significant": p_value < alpha,
            "effect_size_cohens_d": effect_size
        }

    @staticmethod
    def chi_square_test(observed, expected=None, alpha=0.05):
        """Chi-square test for categorical data."""
        if expected is None:
            chi2, p_value, dof, expected = stats.chi2_contingency(observed)
        else:
            chi2, p_value = stats.chisquare(observed, expected)
            dof = len(observed) - 1
        return {
            "chi2_statistic": chi2,
            "p_value": p_value,
            "degrees_of_freedom": dof,
            "significant": p_value < alpha
        }

    @staticmethod
    def ab_test_proportion(conversions_a, total_a, conversions_b, total_b, alpha=0.05):
        """Two-proportion z-test for A/B testing."""
        p_a = conversions_a / total_a
        p_b = conversions_b / total_b
        p_pooled = (conversions_a + conversions_b) / (total_a + total_b)

        se = np.sqrt(p_pooled * (1 - p_pooled) * (1/total_a + 1/total_b))
        z_stat = (p_a - p_b) / se
        p_value = 2 * (1 - stats.norm.cdf(abs(z_stat)))

        return {
            "conversion_a": p_a,
            "conversion_b": p_b,
            "lift": (p_b - p_a) / p_a * 100,
            "z_statistic": z_stat,
            "p_value": p_value,
            "significant": p_value < alpha
        }
python
from scipy import stats
import numpy as np

class HypothesisTest:
    """Framework for statistical hypothesis testing."""

    @staticmethod
    def two_sample_ttest(group_a, group_b, alpha=0.05):
        """Independent samples t-test."""
        t_stat, p_value = stats.ttest_ind(group_a, group_b)
        effect_size = (np.mean(group_a) - np.mean(group_b)) / np.sqrt(
            (np.var(group_a) + np.var(group_b)) / 2
        )
        return {
            "t_statistic": t_stat,
            "p_value": p_value,
            "significant": p_value < alpha,
            "effect_size_cohens_d": effect_size
        }

    @staticmethod
    def chi_square_test(observed, expected=None, alpha=0.05):
        """Chi-square test for categorical data."""
        if expected is None:
            chi2, p_value, dof, expected = stats.chi2_contingency(observed)
        else:
            chi2, p_value = stats.chisquare(observed, expected)
            dof = len(observed) - 1
        return {
            "chi2_statistic": chi2,
            "p_value": p_value,
            "degrees_of_freedom": dof,
            "significant": p_value < alpha
        }

    @staticmethod
    def ab_test_proportion(conversions_a, total_a, conversions_b, total_b, alpha=0.05):
        """Two-proportion z-test for A/B testing."""
        p_a = conversions_a / total_a
        p_b = conversions_b / total_b
        p_pooled = (conversions_a + conversions_b) / (total_a + total_b)

        se = np.sqrt(p_pooled * (1 - p_pooled) * (1/total_a + 1/total_b))
        z_stat = (p_a - p_b) / se
        p_value = 2 * (1 - stats.norm.cdf(abs(z_stat)))

        return {
            "conversion_a": p_a,
            "conversion_b": p_b,
            "lift": (p_b - p_a) / p_a * 100,
            "z_statistic": z_stat,
            "p_value": p_value,
            "significant": p_value < alpha
        }

Usage

Usage

result = HypothesisTest.ab_test_proportion( conversions_a=120, total_a=1000, conversions_b=150, total_b=1000 ) print(f"Lift: {result['lift']:.1f}%, p-value: {result['p_value']:.4f}")
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result = HypothesisTest.ab_test_proportion( conversions_a=120, total_a=1000, conversions_b=150, total_b=1000 ) print(f"Lift: {result['lift']:.1f}%, p-value: {result['p_value']:.4f}")
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3. Linear Algebra Essentials

3. 线性代数基础

python
import numpy as np
python
import numpy as np

Matrix operations

Matrix operations

A = np.array([[1, 2], [3, 4]]) B = np.array([[5, 6], [7, 8]])
A = np.array([[1, 2], [3, 4]]) B = np.array([[5, 6], [7, 8]])

Basic operations

Basic operations

print("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))
print("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 and eigenvectors (PCA foundation)

eigenvalues, eigenvectors = np.linalg.eig(A) print(f"Eigenvalues: {eigenvalues}")
eigenvalues, eigenvectors = np.linalg.eig(A) print(f"Eigenvalues: {eigenvalues}")

Singular Value Decomposition (dimensionality reduction)

Singular Value Decomposition (dimensionality reduction)

U, S, Vt = np.linalg.svd(A) print(f"Singular values: {S}")
U, S, Vt = np.linalg.svd(A) print(f"Singular values: {S}")

Solving linear systems: Ax = b

Solving linear systems: Ax = b

b = np.array([5, 11]) x = np.linalg.solve(A, b) print(f"Solution: {x}")
b = np.array([5, 11]) x = np.linalg.solve(A, b) print(f"Solution: {x}")

Cosine similarity (NLP, recommendations)

Cosine similarity (NLP, recommendations)

def cosine_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}")
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def cosine_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}")
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4. Regression Analysis

4. 回归分析

python
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
python
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

Multiple Linear Regression with statsmodels

X = np.random.randn(100, 3) y = 2X[:, 0] + 3X[:, 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}")
X = np.random.randn(100, 3) y = 2X[:, 0] + 3X[:, 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

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}")
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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}")
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Tools & Technologies

工具与技术

ToolPurposeVersion (2025)
NumPyNumerical computing1.26+
SciPyScientific computing1.12+
pandasData manipulation2.2+
statsmodelsStatistical models0.14+
scikit-learnML algorithms1.4+
工具用途版本（2025）
NumPy数值计算1.26+
SciPy科学计算1.12+
pandas数据处理2.2+
statsmodels统计模型0.14+
scikit-learn机器学习算法1.4+

Troubleshooting Guide

故障排查指南

IssueSymptomsRoot CauseFix
Low p-value, small effectSignificant but meaninglessLarge sample sizeCheck effect size
High varianceUnstable estimatesSmall sample, outliersMore data, robust methods
MulticollinearityInflated coefficientsCorrelated featuresVIF check, remove features
HeteroscedasticityInvalid inferenceNon-constant varianceWeighted least squares
问题症状根本原因解决方法
p值低但效应量小结果显著但无实际意义样本量过大检查效应量
高方差估计结果不稳定样本量小、存在异常值增加数据、使用稳健方法
多重共线性系数被放大特征之间存在相关性检查VIF、移除特征
异方差性推断结果无效方差非恒定使用加权最小二乘法

Best Practices

最佳实践

python
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python
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✅ DO: Check assumptions before testing

✅ DO: Check assumptions before testing

from scipy.stats import shapiro stat, p = shapiro(data) if p > 0.05: print("Data is approximately normal")
from 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: Use effect sizes, not just p-values

✅ DO: Correct for multiple comparisons (Bonferroni)

✅ DO: Correct for multiple comparisons (Bonferroni)

✅ DO: Report confidence intervals

✅ DO: Report confidence intervals

❌ DON'T: p-hack by trying many tests

❌ DON'T: p-hack by trying many tests

❌ DON'T: Confuse correlation with causation

❌ DON'T: Confuse correlation with causation

❌ DON'T: Ignore sample size requirements

❌ DON'T: Ignore sample size requirements

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Resources

参考资源

Skill Certification Checklist:
  • Can calculate descriptive statistics
  • Can perform hypothesis tests (t-test, chi-square)
  • Can implement A/B testing
  • Can perform regression analysis
  • Can use matrix operations for ML
技能认证检查清单:
  • 能够计算描述性统计量
  • 能够执行假设检验（t检验、卡方检验）
  • 能够实施A/B测试
  • 能够进行回归分析
  • 能够使用矩阵运算支持机器学习