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z4m`@?tICoA^rTKQF{PBNV(Ck7_;@*_%55$qa6W4DdYewdDbVs;kV$K~s3|E)}} + +{{< alert "circle-info" >}} +**Serie: De Software Engineer a ingeniero en IA - Semana 3** + +Esta es la tercera entrega de nuestro programa de 16 semanas. Si llegaste aquí directamente, te recomendamos revisar las semanas anteriores: +- [Semana 1: Fundamentos de IA](../fundamentos-ia) +- [Semana 2: Álgebra Lineal para ML](../algebra-lineal-ml) +{{< /alert >}} + +## Introducción: La Incertidumbre Como Motor de la IA + +En el mundo real, los datos nunca son perfectos. Los sensores fallan, las mediciones contienen ruido, y los patrones están mezclados con variabilidad aleatoria. Como ingenieros en IA, nuestro trabajo no es eliminar esta incertidumbre, sino **modelarla, cuantificarla y aprovecharla** para tomar mejores decisiones. + +Imaginate que estás desarrollando un sistema de recomendaciones para una plataforma de streaming. ¿Cómo podes estar seguro de que tu modelo realmente entiende las preferencias del usuario y no está simplemente memorizando patrones aleatorios? ¿Cómo medís la confianza en cada predicción? ¿Cómo diferenciás entre correlación y causación? + +La respuesta está en la **estadística y probabilidad aplicada a machine learning**. + +### ¿Por Qué la Estadística es Fundamental en IA? + +Mientras que el álgebra lineal nos da las herramientas computacionales y el cálculo nos permite optimizar, la estadística nos proporciona el **framework conceptual** para: + +- **Cuantificar la incertidumbre** en nuestras predicciones +- **Validar** que nuestros modelos generalizan correctamente +- **Interpretar** los resultados de manera rigurosa +- **Detectar** cuando algo está funcionando mal +- **Comparar** diferentes enfoques de manera objetiva + +{{< mermaid >}} +graph TD + A[Datos del Mundo Real] --> B[Ruido + Incertidumbre] + B --> C[Modelos Estadísticos] + C --> D[Distribuciones de Probabilidad] + D --> E[Inferencia y Predicción] + E --> F[Decisiones Informadas] + + style A fill:#ff6b6b + style F fill:#51cf66 +{{< /mermaid >}} + +### Lo Que Vas a Aprender Hoy + +En esta semana vas a desarrollar una comprensión sólida de: + +1. **Estadística descriptiva** que realmente importa en ML +2. **Distribuciones de probabilidad** y cuándo usarlas +3. **Correlación vs causación** (el error más costoso en data science) +4. **Teorema de Bayes** aplicado a clasificación y actualización de creencias +5. **Intervalos de confianza** para cuantificar incertidumbre +6. **Implementación práctica** con Python desde cero + +## Parte I: Fundamentos Estadísticos para Machine Learning + +### 1.1 Medidas de Tendencia Central: Más Allá del Promedio + +Como software engineers, probablemente ya conoces el concepto de promedio. Pero en machine learning, necesitamos ser más precisos sobre **cuándo y por qué** usar cada medida. + +#### Media Aritmética: El Punto de Equilibrio + +La media es el valor que minimiza la suma de errores cuadráticos: + +$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$$ + +```python +import numpy as np +import matplotlib.pyplot as plt +import seaborn as sns +from scipy import stats +import pandas as pd + +# Implementación desde cero de estadísticas básicas +class EstadisticasML: + """Clase para cálculos estadísticos fundamentales en ML""" + + @staticmethod + def media(datos): + """ + Calcula la media aritmética + + Args: + datos: lista o array de números + + Returns: + float: media aritmética + """ + return sum(datos) / len(datos) + + @staticmethod + def mediana(datos): + """ + Calcula la mediana (valor central) + Más robusta que la media ante outliers + """ + datos_ordenados = sorted(datos) + n = len(datos_ordenados) + + if n % 2 == 0: + # Si hay número par de elementos, promedio de los dos centrales + return (datos_ordenados[n//2 - 1] + datos_ordenados[n//2]) / 2 + else: + # Si hay número impar, el elemento central + return datos_ordenados[n//2] + + @staticmethod + def moda(datos): + """ + Encuentra el valor más frecuente + Útil para variables categóricas + """ + from collections import Counter + contador = Counter(datos) + return contador.most_common(1)[0][0] + +# Ejemplo práctico: Análisis de latencia de respuesta de una API +np.random.seed(42) + +# Simulamos latencias de API (la mayoría normales, algunos outliers) +latencias_normales = np.random.normal(100, 15, 950) # 95% de requests normales +latencias_outliers = np.random.exponential(500, 50) # 5% con problemas de red +latencias = np.concatenate([latencias_normales, latencias_outliers]) + +calc = EstadisticasML() + +print("=== Análisis de Latencias de API ===") +print(f"Media: {calc.media(latencias):.2f} ms") +print(f"Mediana: {calc.mediana(latencias):.2f} ms") + +# ¿Por qué la diferencia es importante en ML? +print(f"\nDiferencia: {calc.media(latencias) - calc.mediana(latencias):.2f} ms") +print("La media está inflada por outliers - en ML esto puede sesgar tu modelo!") +``` + +{{< alert "lightbulb" >}} +**Insight para ML**: En sistemas de producción, la mediana es a menudo más útil que la media para métricas de performance, ya que no se ve afectada por outliers extremos. +{{< /alert >}} + +#### Visualización Inteligente de Tendencias Centrales + +```python +# Creamos una visualización que muestra por qué importa la elección +fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6)) + +# Subplot 1: Distribución con outliers +ax1.hist(latencias, bins=50, alpha=0.7, color='skyblue', edgecolor='black') +ax1.axvline(calc.media(latencias), color='red', linestyle='--', + label=f'Media: {calc.media(latencias):.1f}ms', linewidth=2) +ax1.axvline(calc.mediana(latencias), color='green', linestyle='-', + label=f'Mediana: {calc.mediana(latencias):.1f}ms', linewidth=2) +ax1.set_xlabel('Latencia (ms)') +ax1.set_ylabel('Frecuencia') +ax1.set_title('Distribución con Outliers\n(Datos Reales de API)') +ax1.legend() +ax1.grid(True, alpha=0.3) + +# Subplot 2: Solo datos normales +ax2.hist(latencias_normales, bins=30, alpha=0.7, color='lightgreen', edgecolor='black') +ax2.axvline(calc.media(latencias_normales), color='red', linestyle='--', + label=f'Media: {calc.media(latencias_normales):.1f}ms', linewidth=2) +ax2.axvline(calc.mediana(latencias_normales), color='green', linestyle='-', + label=f'Mediana: {calc.mediana(latencias_normales):.1f}ms', linewidth=2) +ax2.set_xlabel('Latencia (ms)') +ax2.set_ylabel('Frecuencia') +ax2.set_title('Distribución Sin Outliers\n(Condiciones Ideales)') +ax2.legend() +ax2.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +### 1.2 Medidas de Dispersión: Cuantificando la Incertidumbre + +La dispersión nos dice **qué tan confiables** son nuestras medidas centrales. En ML, esto es crucial para entender la **variabilidad** de nuestros datos y modelos. + +#### Varianza y Desviación Estándar: Los Fundamentos + +$$\sigma^2 = \frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2$$ + +$$\sigma = \sqrt{\sigma^2}$$ + +```python +class DispersionML: + """Cálculos de dispersión optimizados para ML""" + + @staticmethod + def varianza(datos, poblacional=True): + """ + Calcula la varianza + + Args: + datos: array de datos + poblacional: True para división por n, False para n-1 (muestra) + """ + media = sum(datos) / len(datos) + suma_cuadrados = sum((x - media)**2 for x in datos) + + # Corrección de Bessel para muestras + divisor = len(datos) if poblacional else len(datos) - 1 + return suma_cuadrados / divisor + + @staticmethod + def desviacion_estandar(datos, poblacional=True): + """Desviación estándar - mismas unidades que los datos originales""" + return DispersionML.varianza(datos, poblacional) ** 0.5 + + @staticmethod + def rango_intercuartil(datos): + """ + IQR - medida de dispersión robusta ante outliers + Útil para detección de anomalías + """ + datos_ordenados = sorted(datos) + n = len(datos_ordenados) + + # Cuartiles + q1_idx = n // 4 + q3_idx = 3 * n // 4 + + q1 = datos_ordenados[q1_idx] + q3 = datos_ordenados[q3_idx] + + return q3 - q1, q1, q3 + +# Ejemplo: Análisis de precisión de un modelo +# Simulamos predicciones de un modelo de regresión +np.random.seed(123) + +modelo_A_errors = np.random.normal(0, 2, 1000) # Modelo consistente +modelo_B_errors = np.random.normal(0, 5, 1000) # Modelo inconsistente +modelo_C_errors = np.random.laplace(0, 1.5, 1000) # Modelo con distribución diferente + +disp = DispersionML() + +modelos = { + 'Modelo A (Consistente)': modelo_A_errors, + 'Modelo B (Inconsistente)': modelo_B_errors, + 'Modelo C (Heavy-tailed)': modelo_C_errors +} + +print("=== Comparación de Modelos por Dispersión ===") +for nombre, errores in modelos.items(): + media = sum(errores) / len(errores) + std = disp.desviacion_estandar(errores) + iqr, q1, q3 = disp.rango_intercuartil(errores) + + print(f"\n{nombre}:") + print(f" Media del error: {media:.3f}") + print(f" Desv. estándar: {std:.3f}") + print(f" IQR: {iqr:.3f}") + print(f" Rango 68%: [{media-std:.3f}, {media+std:.3f}]") +``` + +#### Visualización de Dispersión Comparativa + +```python +# Boxplot comparativo para entender dispersión +fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6)) + +# Boxplot +errores_lista = [modelo_A_errors, modelo_B_errors, modelo_C_errors] +labels = ['Modelo A\n(σ=2)', 'Modelo B\n(σ=5)', 'Modelo C\n(Heavy-tail)'] + +ax1.boxplot(errores_lista, labels=labels) +ax1.set_ylabel('Error de Predicción') +ax1.set_title('Comparación de Dispersión\n(Boxplots)') +ax1.grid(True, alpha=0.3) + +# Histogramas superpuestos +colors = ['blue', 'red', 'green'] +alphas = [0.7, 0.5, 0.6] + +for i, (errores, label, color, alpha) in enumerate(zip(errores_lista, labels, colors, alphas)): + ax2.hist(errores, bins=50, alpha=alpha, color=color, + label=label.replace('\n', ' '), density=True) + +ax2.set_xlabel('Error de Predicción') +ax2.set_ylabel('Densidad') +ax2.set_title('Distribución de Errores\n(Histogramas Superpuestos)') +ax2.legend() +ax2.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() + +print("\n=== Interpretación para ML ===") +print("• Modelo A: Baja varianza → Predicciones consistentes") +print("• Modelo B: Alta varianza → Predicciones menos confiables") +print("• Modelo C: Heavy tails → Mayor probabilidad de errores grandes") +``` + +{{< alert "exclamation-triangle" >}} +**Error Común**: Muchos desarrolladores se enfocan solo en la media del error y ignoran la varianza. En producción, la **consistencia** (baja varianza) puede ser más valiosa que la precisión promedio. +{{< /alert >}} + +### 1.3 Correlación: El Arte de Detectar Relaciones + +La correlación mide la **fuerza de la relación lineal** entre dos variables. Pero atención: correlación ≠ causación. + +#### Coeficiente de Correlación de Pearson + +$$r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}}$$ + +```python +class CorrelacionML: + """Análisis de correlación para feature engineering""" + + @staticmethod + def correlacion_pearson(x, y): + """ + Calcula correlación de Pearson desde cero + + Returns: + float: correlación entre -1 y 1 + """ + n = len(x) + + # Medias + media_x = sum(x) / n + media_y = sum(y) / n + + # Numerador: covarianza + numerador = sum((x[i] - media_x) * (y[i] - media_y) for i in range(n)) + + # Denominador: producto de desviaciones estándar + suma_sq_x = sum((x[i] - media_x)**2 for i in range(n)) + suma_sq_y = sum((y[i] - media_y)**2 for i in range(n)) + + denominador = (suma_sq_x * suma_sq_y) ** 0.5 + + return numerador / denominador if denominador != 0 else 0 + + @staticmethod + def correlacion_spearman(x, y): + """ + Correlación de Spearman (basada en rankings) + Más robusta para relaciones no lineales + """ + def ranking(datos): + # Crear rankings (1 = menor, n = mayor) + sorted_data = sorted(enumerate(datos), key=lambda item: item[1]) + ranks = [0] * len(datos) + for rank, (original_index, _) in enumerate(sorted_data): + ranks[original_index] = rank + 1 + return ranks + + rank_x = ranking(x) + rank_y = ranking(y) + + return CorrelacionML.correlacion_pearson(rank_x, rank_y) + +# Ejemplo: Feature Engineering para un modelo predictivo +np.random.seed(456) + +# Simulamos datos de un e-commerce +n_usuarios = 1000 +edad = np.random.normal(35, 12, n_usuarios) +ingresos = np.random.normal(50000, 15000, n_usuarios) + +# Relaciones realistas +tiempo_web = 5 + 0.1 * edad + np.random.normal(0, 2, n_usuarios) # Relación débil con edad +compras = 0.3 + 0.002 * ingresos + 0.5 * tiempo_web + np.random.normal(0, 1, n_usuarios) # Relación fuerte con ingresos y tiempo + +# También creamos una variable no-lineal +satisfaccion = 5 / (1 + np.exp(-(compras - 2))) + np.random.normal(0, 0.3, n_usuarios) # Relación sigmoidal + +corr = CorrelacionML() + +print("=== Matriz de Correlaciones ===") +variables = { + 'Edad': edad, + 'Ingresos': ingresos, + 'Tiempo Web': tiempo_web, + 'Compras': compras, + 'Satisfacción': satisfaccion +} + +# Matriz de correlación manual +nombres = list(variables.keys()) +n_vars = len(nombres) + +print("Correlaciones de Pearson:") +print("Variable 1\t\tVariable 2\t\tPearson\t\tSpearman") +print("-" * 70) + +for i in range(n_vars): + for j in range(i+1, n_vars): + var1_name, var1_data = nombres[i], variables[nombres[i]] + var2_name, var2_data = nombres[j], variables[nombres[j]] + + pearson = corr.correlacion_pearson(var1_data, var2_data) + spearman = corr.correlacion_spearman(var1_data, var2_data) + + print(f"{var1_name:<15}\t{var2_name:<15}\t{pearson:.3f}\t\t{spearman:.3f}") +``` + +#### Visualización de Correlaciones: Scatter Matrix + +```python +# Creamos un scatter plot matrix profesional +fig, axes = plt.subplots(n_vars, n_vars, figsize=(16, 16)) + +for i in range(n_vars): + for j in range(n_vars): + ax = axes[i, j] + + if i == j: + # Diagonal: histogramas + ax.hist(variables[nombres[i]], bins=30, alpha=0.7, color='skyblue') + ax.set_title(f'{nombres[i]}') + else: + # Off-diagonal: scatter plots + x_data = variables[nombres[j]] + y_data = variables[nombres[i]] + + ax.scatter(x_data, y_data, alpha=0.5, s=10) + + # Añadir línea de tendencia + z = np.polyfit(x_data, y_data, 1) + p = np.poly1d(z) + ax.plot(sorted(x_data), p(sorted(x_data)), "r--", alpha=0.8) + + # Mostrar correlación en el plot + pearson_corr = corr.correlacion_pearson(x_data, y_data) + ax.text(0.05, 0.95, f'r={pearson_corr:.2f}', + transform=ax.transAxes, fontsize=10, + verticalalignment='top', + bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.8)) + + # Limpiar labels excepto en bordes + if i < n_vars - 1: + ax.set_xlabel('') + else: + ax.set_xlabel(nombres[j]) + + if j > 0: + ax.set_ylabel('') + else: + ax.set_ylabel(nombres[i]) + +plt.tight_layout() +plt.show() +``` + +### 1.4 Correlación vs Causación: El Error de los $1000M + +Esta distinción es **crítica** en machine learning. Una correlación fuerte no implica que una variable cause la otra. + +```python +# Ejemplo clásico: El peligro de confundir correlación con causación +np.random.seed(789) + +# Variable oculta: temperatura +temperatura = np.random.normal(25, 8, 365) # Temperatura diaria + +# Variables correlacionadas por la temperatura (no entre sí directamente) +ventas_helados = 50 + 3 * temperatura + np.random.normal(0, 10, 365) +ahogamientos = 1 + 0.2 * temperatura + np.random.normal(0, 2, 365) + +# Correlación espuria +correlacion_espuria = corr.correlacion_pearson(ventas_helados, ahogamientos) + +print("=== El Peligro de la Correlación Espuria ===") +print(f"Correlación Helados ↔ Ahogamientos: {correlacion_espuria:.3f}") +print(f"Correlación Temperatura ↔ Helados: {corr.correlacion_pearson(temperatura, ventas_helados):.3f}") +print(f"Correlación Temperatura ↔ Ahogamientos: {corr.correlacion_pearson(temperatura, ahogamientos):.3f}") + +print("\n¿Conclusión equivocada?") +print("→ Los helados causan ahogamientos") +print("\nConclusión correcta:") +print("→ La temperatura (variable oculta) influye en ambas variables") + +# Visualización del problema +fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(18, 5)) + +# Plot 1: Correlación espuria +ax1.scatter(ventas_helados, ahogamientos, alpha=0.6, color='red') +ax1.set_xlabel('Ventas de Helados') +ax1.set_ylabel('Ahogamientos') +ax1.set_title(f'Correlación Espuria\nr = {correlacion_espuria:.2f}') +ax1.grid(True, alpha=0.3) + +# Plot 2: Variable real (temperatura → helados) +scatter = ax2.scatter(temperatura, ventas_helados, c=temperatura, + cmap='coolwarm', alpha=0.6) +ax2.set_xlabel('Temperatura (°C)') +ax2.set_ylabel('Ventas de Helados') +ax2.set_title('Relación Real: Temperatura → Helados') +ax2.grid(True, alpha=0.3) + +# Plot 3: Variable real (temperatura → ahogamientos) +scatter = ax3.scatter(temperatura, ahogamientos, c=temperatura, + cmap='coolwarm', alpha=0.6) +ax3.set_xlabel('Temperatura (°C)') +ax3.set_ylabel('Ahogamientos') +ax3.set_title('Relación Real: Temperatura → Ahogamientos') +ax3.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +{{< alert "exclamation-triangle" >}} +**Error Crítico en ML**: Usar correlaciones espurias como features puede llevar a modelos que funcionan en entrenamiento pero fallan en producción cuando las condiciones cambian. +{{< /alert >}} + +## Parte II: Distribuciones de Probabilidad en Machine Learning + +Las distribuciones de probabilidad son los **bloques fundamentales** para modelar la incertidumbre en ML. Cada distribución tiene sus casos de uso específicos. + +### 2.1 Distribución Normal: La Reina de las Distribuciones + +La distribución normal aparece constantemente en ML debido al **Teorema del Límite Central**. + +$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$ + +```python +class DistribucionesML: + """Implementaciones y análisis de distribuciones clave para ML""" + + @staticmethod + def densidad_normal(x, mu, sigma): + """ + Calcula la densidad de probabilidad normal + + Args: + x: valor donde evaluar + mu: media + sigma: desviación estándar + """ + return (1 / (sigma * (2 * np.pi)**0.5)) * np.exp(-0.5 * ((x - mu) / sigma)**2) + + @staticmethod + def generar_normal(mu, sigma, n): + """ + Genera muestras de distribución normal usando Box-Muller + (implementación educativa) + """ + # Método Box-Muller + muestras = [] + for _ in range(n // 2 + 1): + u1 = np.random.random() + u2 = np.random.random() + + z0 = np.sqrt(-2 * np.log(u1)) * np.cos(2 * np.pi * u2) + z1 = np.sqrt(-2 * np.log(u1)) * np.sin(2 * np.pi * u2) + + muestras.extend([mu + sigma * z0, mu + sigma * z1]) + + return np.array(muestras[:n]) + + @staticmethod + def regla_68_95_99(mu, sigma): + """ + Intervalos de confianza para distribución normal + """ + return { + '68%': (mu - sigma, mu + sigma), + '95%': (mu - 2*sigma, mu + 2*sigma), + '99.7%': (mu - 3*sigma, mu + 3*sigma) + } + +# Ejemplo: Análisis de errores de modelo +dist = DistribucionesML() + +# Simulamos errores de diferentes modelos +np.random.seed(101) + +# Modelo 1: Errores bien calibrados (normal) +errores_modelo1 = np.random.normal(0, 1, 1000) + +# Modelo 2: Errores sesgados +errores_modelo2 = np.random.normal(0.5, 1.2, 1000) + +# Modelo 3: Errores con heavy tails (no normal) +errores_modelo3 = np.random.laplace(0, 0.8, 1000) + +print("=== Análisis de Normalidad de Errores ===") + +modelos_errores = [ + ("Modelo Bien Calibrado", errores_modelo1), + ("Modelo Sesgado", errores_modelo2), + ("Modelo Heavy-Tail", errores_modelo3) +] + +for nombre, errores in modelos_errores: + media = np.mean(errores) + std = np.std(errores) + intervalos = dist.regla_68_95_99(media, std) + + print(f"\n{nombre}:") + print(f" Media: {media:.3f}, Std: {std:.3f}") + print(f" 68% de errores en: [{intervalos['68%'][0]:.2f}, {intervalos['68%'][1]:.2f}]") + + # Verificar cuántos errores caen realmente en el intervalo 68% + en_intervalo = np.sum((errores >= intervalos['68%'][0]) & + (errores <= intervalos['68%'][1])) + porcentaje_real = en_intervalo / len(errores) * 100 + print(f" Porcentaje real en 68%: {porcentaje_real:.1f}%") + + if abs(porcentaje_real - 68) > 5: + print(f" ⚠️ Los errores NO siguen distribución normal") + else: + print(f" ✓ Los errores aproximan distribución normal") +``` + +#### Test de Normalidad: Shapiro-Wilk + +```python +# Implementación simplificada del test de normalidad +def test_normalidad_visual(datos, nombre=""): + """Test visual de normalidad usando Q-Q plot""" + + fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5)) + + # Histograma vs normal teórica + ax1.hist(datos, bins=30, density=True, alpha=0.7, color='skyblue') + + # Superponer normal teórica + mu, sigma = np.mean(datos), np.std(datos) + x_teorico = np.linspace(min(datos), max(datos), 100) + y_teorico = dist.densidad_normal(x_teorico, mu, sigma) + ax1.plot(x_teorico, y_teorico, 'r-', linewidth=2, label='Normal Teórica') + + ax1.set_xlabel('Valor') + ax1.set_ylabel('Densidad') + ax1.set_title(f'Histograma vs Normal\n{nombre}') + ax1.legend() + ax1.grid(True, alpha=0.3) + + # Q-Q Plot (quantile-quantile) + datos_ordenados = np.sort(datos) + n = len(datos_ordenados) + quantiles_teoricos = np.random.normal(mu, sigma, n) + quantiles_teoricos = np.sort(quantiles_teoricos) + + ax2.scatter(quantiles_teoricos, datos_ordenados, alpha=0.6) + + # Línea de referencia (y = x) + min_val = min(min(quantiles_teoricos), min(datos_ordenados)) + max_val = max(max(quantiles_teoricos), max(datos_ordenados)) + ax2.plot([min_val, max_val], [min_val, max_val], 'r--', linewidth=2) + + ax2.set_xlabel('Quantiles Teóricos (Normal)') + ax2.set_ylabel('Quantiles Observados') + ax2.set_title(f'Q-Q Plot\n{nombre}') + ax2.grid(True, alpha=0.3) + + plt.tight_layout() + plt.show() + +# Aplicamos el test a nuestros modelos +for nombre, errores in modelos_errores: + test_normalidad_visual(errores, nombre) +``` + +### 2.2 Distribución Binomial: Modelando el Éxito/Fracaso + +Perfecta para clasificación binaria y A/B testing. + +$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ + +```python +# Implementación de distribución binomial +class DistribucionBinomial: + """Distribución binomial para clasificación y testing""" + + @staticmethod + def factorial(n): + """Factorial usando iteración (más eficiente)""" + if n <= 1: + return 1 + resultado = 1 + for i in range(2, n + 1): + resultado *= i + return resultado + + @staticmethod + def combinatoria(n, k): + """Coeficiente binomial C(n,k)""" + if k > n or k < 0: + return 0 + if k == 0 or k == n: + return 1 + + # Optimización: C(n,k) = C(n,n-k) + k = min(k, n - k) + + resultado = 1 + for i in range(k): + resultado = resultado * (n - i) // (i + 1) + return resultado + + @staticmethod + def probabilidad(n, k, p): + """P(X = k) para binomial(n, p)""" + comb = DistribucionBinomial.combinatoria(n, k) + return comb * (p ** k) * ((1 - p) ** (n - k)) + + @staticmethod + def media_varianza(n, p): + """Media y varianza de binomial""" + media = n * p + varianza = n * p * (1 - p) + return media, varianza + +# Ejemplo: Análisis de conversión web +binomial = DistribucionBinomial() + +# Escenario: A/B test de un botón de compra +n_visitantes = 1000 +tasa_conversion_A = 0.05 # 5% versión original +tasa_conversion_B = 0.07 # 7% versión nueva + +print("=== A/B Test: Análisis de Conversión ===") +print(f"Visitantes por variante: {n_visitantes}") +print(f"Tasa de conversión A: {tasa_conversion_A*100:.1f}%") +print(f"Tasa de conversión B: {tasa_conversion_B*100:.1f}%") + +# Calculamos distribuciones esperadas +media_A, var_A = binomial.media_varianza(n_visitantes, tasa_conversion_A) +media_B, var_B = binomial.media_varianza(n_visitantes, tasa_conversion_B) + +print(f"\nVariante A - Conversiones esperadas: {media_A:.1f} ± {np.sqrt(var_A):.1f}") +print(f"Variante B - Conversiones esperadas: {media_B:.1f} ± {np.sqrt(var_B):.1f}") + +# Probabilidad de diferentes números de conversiones +conversiones_rango = range(20, 101, 5) +prob_A = [binomial.probabilidad(n_visitantes, k, tasa_conversion_A) for k in conversiones_rango] +prob_B = [binomial.probabilidad(n_visitantes, k, tasa_conversion_B) for k in conversiones_rango] + +# Visualización +plt.figure(figsize=(12, 6)) +plt.plot(conversiones_rango, prob_A, 'b-', marker='o', label=f'Variante A (p={tasa_conversion_A})', linewidth=2) +plt.plot(conversiones_rango, prob_B, 'r-', marker='s', label=f'Variante B (p={tasa_conversion_B})', linewidth=2) + +plt.axvline(media_A, color='blue', linestyle='--', alpha=0.7, label=f'Media A: {media_A:.1f}') +plt.axvline(media_B, color='red', linestyle='--', alpha=0.7, label=f'Media B: {media_B:.1f}') + +plt.xlabel('Número de Conversiones') +plt.ylabel('Probabilidad') +plt.title('Distribución Binomial: A/B Test de Conversiones') +plt.legend() +plt.grid(True, alpha=0.3) +plt.show() + +# Análisis de significancia estadística +diferencia_esperada = media_B - media_A +error_estandar = np.sqrt(var_A + var_B) +z_score = diferencia_esperada / error_estandar + +print(f"\n=== Análisis Estadístico ===") +print(f"Diferencia esperada: {diferencia_esperada:.1f} conversiones") +print(f"Error estándar: {error_estandar:.2f}") +print(f"Z-score: {z_score:.2f}") + +if z_score > 1.96: + print("✓ Diferencia estadísticamente significativa (95% confianza)") +else: + print("⚠️ Diferencia NO significativa - necesitas más datos") +``` + +### 2.3 Distribución Uniforme: Modelando la Aleatoriedad Pura + +```python +class DistribucionUniforme: + """Distribución uniforme para inicialización y sampling""" + + @staticmethod + def densidad(x, a, b): + """Densidad uniforme en [a, b]""" + if a <= x <= b: + return 1 / (b - a) + return 0 + + @staticmethod + def generar_muestra(a, b, n): + """Genera n muestras uniformes en [a, b]""" + return [a + (b - a) * np.random.random() for _ in range(n)] + + @staticmethod + def media_varianza(a, b): + """Media y varianza de uniforme[a, b]""" + media = (a + b) / 2 + varianza = ((b - a) ** 2) / 12 + return media, varianza + +# Ejemplo: Inicialización de pesos en redes neuronales +uniforme = DistribucionUniforme() + +# Diferentes estrategias de inicialización +n_pesos = 1000 + +# Xavier/Glorot initialization +fan_in, fan_out = 128, 64 # Neuronas entrada y salida +limite_xavier = np.sqrt(6 / (fan_in + fan_out)) +pesos_xavier = uniforme.generar_muestra(-limite_xavier, limite_xavier, n_pesos) + +# He initialization (para ReLU) +limite_he = np.sqrt(6 / fan_in) +pesos_he = uniforme.generar_muestra(-limite_he, limite_he, n_pesos) + +# Inicialización simple +pesos_simple = uniforme.generar_muestra(-0.1, 0.1, n_pesos) + +print("=== Estrategias de Inicialización ===") + +estrategias = { + 'Xavier/Glorot': pesos_xavier, + 'He (ReLU)': pesos_he, + 'Simple [-0.1, 0.1]': pesos_simple +} + +for nombre, pesos in estrategias.items(): + media = np.mean(pesos) + std = np.std(pesos) + print(f"{nombre}: μ={media:.4f}, σ={std:.4f}") + +# Visualización comparativa +fig, axes = plt.subplots(1, 3, figsize=(15, 5)) + +for i, (nombre, pesos) in enumerate(estrategias.items()): + axes[i].hist(pesos, bins=50, alpha=0.7, density=True) + axes[i].set_title(f'{nombre}\nσ={np.std(pesos):.3f}') + axes[i].set_xlabel('Valor del Peso') + axes[i].set_ylabel('Densidad') + axes[i].grid(True, alpha=0.3) + + # Línea teórica + a, b = min(pesos), max(pesos) + x_teorico = np.linspace(a, b, 100) + y_teorico = [uniforme.densidad(x, a, b) for x in x_teorico] + axes[i].plot(x_teorico, y_teorico, 'r-', linewidth=2, label='Teórica') + axes[i].legend() + +plt.tight_layout() +plt.show() +``` + +## Parte III: Teorema de Bayes en Machine Learning + +El teorema de Bayes es **fundamental** para la inferencia probabilística y el aprendizaje adaptativo. + +$P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}$ + +Donde: +- **P(H|E)**: Probabilidad posterior (lo que queremos) +- **P(E|H)**: Verosimilitud (likelihood) +- **P(H)**: Probabilidad a priori +- **P(E)**: Evidencia (normalización) + +### 3.1 Implementación del Teorema de Bayes + +```python +class BayesML: + """Implementación del teorema de Bayes para ML""" + + @staticmethod + def bayes_simple(prior, likelihood, evidencia): + """ + Calcula probabilidad posterior usando Bayes + + Args: + prior: P(H) - probabilidad a priori + likelihood: P(E|H) - verosimilitud + evidencia: P(E) - evidencia total + """ + posterior = (likelihood * prior) / evidencia + return posterior + + @staticmethod + def bayes_clasificacion(prior_clases, likelihood_features): + """ + Clasificación bayesiana multi-clase + + Args: + prior_clases: dict con probabilidades a priori de cada clase + likelihood_features: dict con P(features|clase) para cada clase + """ + # Calcular evidencia total + evidencia = sum(prior_clases[clase] * likelihood_features[clase] + for clase in prior_clases.keys()) + + # Calcular posterior para cada clase + posteriores = {} + for clase in prior_clases.keys(): + posteriores[clase] = BayesML.bayes_simple( + prior_clases[clase], + likelihood_features[clase], + evidencia + ) + + return posteriores + + @staticmethod + def naive_bayes_gaussian(X_train, y_train, X_test): + """ + Implementación simple de Naive Bayes Gaussiano + """ + from collections import defaultdict + + # Estadísticas por clase + clases = np.unique(y_train) + n_total = len(y_train) + + # Calcular priors + priors = {} + estadisticas = {} + + for clase in clases: + mask = y_train == clase + priors[clase] = np.sum(mask) / n_total + + # Estadísticas de features para esta clase + X_clase = X_train[mask] + estadisticas[clase] = { + 'media': np.mean(X_clase, axis=0), + 'std': np.std(X_clase, axis=0) + } + + # Predicciones + predicciones = [] + probabilidades = [] + + for x_test in X_test: + # Calcular likelihood para cada clase + likelihoods = {} + + for clase in clases: + # Asumimos independencia (naive) y normalidad + media = estadisticas[clase]['media'] + std = estadisticas[clase]['std'] + + # Producto de probabilidades (log para estabilidad numérica) + log_likelihood = 0 + for i, feature in enumerate(x_test): + # Densidad normal + prob = dist.densidad_normal(feature, media[i], std[i]) + log_likelihood += np.log(prob + 1e-10) # Evitar log(0) + + likelihoods[clase] = np.exp(log_likelihood) + + # Aplicar Bayes + posteriores = BayesML.bayes_clasificacion(priors, likelihoods) + + # Predecir clase con mayor probabilidad + clase_predicha = max(posteriores.keys(), key=lambda k: posteriores[k]) + predicciones.append(clase_predicha) + probabilidades.append(posteriores) + + return predicciones, probabilidades + +# Ejemplo: Sistema de detección de spam +np.random.seed(555) + +# Simulamos un dataset de emails +n_emails = 1000 + +# Features: [longitud, num_enlaces, num_mayusculas, num_signos_exclamacion] +# Emails normales +emails_normales = np.random.multivariate_normal( + [150, 2, 10, 1], # medias + [[400, 5, 20, 1], # matriz de covarianza + [5, 4, 5, 0.5], + [20, 5, 100, 2], + [1, 0.5, 2, 1]], + 600 +) + +# Emails spam +emails_spam = np.random.multivariate_normal( + [80, 8, 35, 5], # medias diferentes + [[200, 10, 30, 3], + [10, 16, 15, 2], + [30, 15, 200, 8], + [3, 2, 8, 4]], + 400 +) + +# Combinar datasets +X = np.vstack([emails_normales, emails_spam]) +y = np.array([0] * 600 + [1] * 400) # 0=normal, 1=spam + +# División train/test +from sklearn.model_selection import train_test_split +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42) + +# Aplicar nuestro Naive Bayes +bayes = BayesML() +predicciones, probabilidades = bayes.naive_bayes_gaussian(X_train, y_train, X_test) + +# Evaluación +precisiones = sum(pred == real for pred, real in zip(predicciones, y_test)) +accuracy = precisiones / len(y_test) + +print("=== Sistema de Detección de Spam ===") +print(f"Accuracy: {accuracy:.3f}") +print(f"Emails de prueba: {len(y_test)}") +print(f"Predicciones correctas: {precisiones}") + +# Análisis detallado de algunos casos +print("\n=== Análisis de Casos ===") +feature_names = ['Longitud', 'Enlaces', 'Mayúsculas', 'Exclamaciones'] + +for i in range(5): # Primeros 5 casos + print(f"\nEmail {i+1}:") + print(f" Features: {[f'{x:.1f}' for x in X_test[i]]}") + print(f" Clase real: {'Spam' if y_test[i] else 'Normal'}") + print(f" Predicción: {'Spam' if predicciones[i] else 'Normal'}") + print(f" P(Normal): {probabilidades[i][0]:.3f}") + print(f" P(Spam): {probabilidades[i][1]:.3f}") + + if predicciones[i] != y_test[i]: + print(f" ❌ Error de clasificación!") +``` + +### 3.2 Actualización Bayesiana: Aprendizaje Online + +```python +class AprendizajeBayesiano: + """Aprendizaje adaptativo usando actualización bayesiana""" + + def __init__(self, prior_alpha=1, prior_beta=1): + """ + Inicializa con distribución Beta como prior + Beta(α, β) es conjugada de la binomial + """ + self.alpha = prior_alpha # "éxitos" previos + self.beta = prior_beta # "fracasos" previos + self.total_observaciones = 0 + self.historial = [] + + def actualizar(self, exito): + """ + Actualiza creencias con nueva observación + + Args: + exito: True si fue éxito, False si fracaso + """ + if exito: + self.alpha += 1 + else: + self.beta += 1 + + self.total_observaciones += 1 + + # Guardar estado para visualización + self.historial.append({ + 'observacion': self.total_observaciones, + 'alpha': self.alpha, + 'beta': self.beta, + 'media_estimada': self.alpha / (self.alpha + self.beta), + 'exito': exito + }) + + def probabilidad_estimada(self): + """Estimación actual de la probabilidad de éxito""" + return self.alpha / (self.alpha + self.beta) + + def intervalo_credible(self, confianza=0.95): + """ + Intervalo de credibilidad bayesiano + (equivalente bayesiano del intervalo de confianza) + """ + from scipy.stats import beta + alpha_nivel = (1 - confianza) / 2 + lower = beta.ppf(alpha_nivel, self.alpha, self.beta) + upper = beta.ppf(1 - alpha_nivel, self.alpha, self.beta) + return lower, upper + +# Ejemplo: Sistema de recomendaciones adaptativo +# Simulamos clics en anuncios con tasa real del 8% +np.random.seed(666) +tasa_real = 0.08 +n_observaciones = 200 + +# Inicializar aprendizaje bayesiano +aprendiz = AprendizajeBayesiano(prior_alpha=1, prior_beta=1) # Prior uniforme + +print("=== Aprendizaje Bayesiano Online ===") +print(f"Tasa real (desconocida): {tasa_real:.1%}") +print(f"Prior inicial: Beta(1,1) - uniforme") + +# Simular observaciones secuenciales +observaciones = np.random.binomial(1, tasa_real, n_observaciones) + +estimaciones = [] +intervalos = [] + +for i, obs in enumerate(observaciones): + aprendiz.actualizar(obs == 1) + + if (i + 1) % 20 == 0: # Reportar cada 20 observaciones + est = aprendiz.probabilidad_estimada() + lower, upper = aprendiz.intervalo_credible() + + print(f"Obs {i+1:3d}: Estimación {est:.3f} [{lower:.3f}, {upper:.3f}]") + + estimaciones.append(est) + intervalos.append((lower, upper)) + +# Visualización del aprendizaje +fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 10)) + +# Plot 1: Evolución de la estimación +obs_puntos = list(range(20, n_observaciones + 1, 20)) +estimaciones_array = np.array(estimaciones) +intervalos_array = np.array(intervalos) + +ax1.plot(obs_puntos, estimaciones_array, 'b-', marker='o', linewidth=2, + label='Estimación Bayesiana') +ax1.fill_between(obs_puntos, intervalos_array[:, 0], intervalos_array[:, 1], + alpha=0.3, color='blue', label='Intervalo 95% Credible') +ax1.axhline(tasa_real, color='red', linestyle='--', linewidth=2, + label=f'Tasa Real: {tasa_real:.1%}') + +ax1.set_xlabel('Número de Observaciones') +ax1.set_ylabel('Probabilidad Estimada') +ax1.set_title('Aprendizaje Bayesiano: Convergencia a la Tasa Real') +ax1.legend() +ax1.grid(True, alpha=0.3) + +# Plot 2: Distribución posterior final +from scipy.stats import beta +x_beta = np.linspace(0, 0.2, 1000) +y_beta = beta.pdf(x_beta, aprendiz.alpha, aprendiz.beta) + +ax2.plot(x_beta, y_beta, 'blue', linewidth=3, label=f'Posterior: Beta({aprendiz.alpha}, {aprendiz.beta})') +ax2.axvline(tasa_real, color='red', linestyle='--', linewidth=2, label='Tasa Real') +ax2.axvline(aprendiz.probabilidad_estimada(), color='blue', linestyle=':', linewidth=2, + label=f'Estimación: {aprendiz.probabilidad_estimada():.3f}') + +# Área del intervalo de credibilidad +lower, upper = aprendiz.intervalo_credible() +x_fill = x_beta[(x_beta >= lower) & (x_beta <= upper)] +y_fill = beta.pdf(x_fill, aprendiz.alpha, aprendiz.beta) +ax2.fill_between(x_fill, y_fill, alpha=0.3, color='green', + label=f'95% Credible: [{lower:.3f}, {upper:.3f}]') + +ax2.set_xlabel('Tasa de Conversión') +ax2.set_ylabel('Densidad de Probabilidad') +ax2.set_title('Distribución Posterior Final') +ax2.legend() +ax2.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() + +print(f"\n=== Resultado Final ===") +print(f"Estimación final: {aprendiz.probabilidad_estimada():.4f}") +print(f"Error absoluto: {abs(aprendiz.probabilidad_estimada() - tasa_real):.4f}") +print(f"Intervalo 95%: [{lower:.4f}, {upper:.4f}]") +print(f"¿Contiene la tasa real? {'✓' if lower <= tasa_real <= upper else '❌'}") +``` + +## Parte IV: Intervalos de Confianza y Estimación + +Los intervalos de confianza nos permiten **cuantificar la incertidumbre** en nuestras estimaciones. + +### 4.1 Intervalos de Confianza para la Media + +```python +class IntervalosConfianza: + """Cálculo de intervalos de confianza para diferentes parámetros""" + + @staticmethod + def t_student_critico(df, alpha=0.05): + """ + Valor crítico de t-Student (aproximación) + Para df > 30, aproxima a distribución normal + """ + # Valores críticos comunes (tabla simplificada) + valores_t = { + # grados_libertad: {alpha: valor_critico} + 1: {0.05: 12.706, 0.01: 63.657}, + 2: {0.05: 4.303, 0.01: 9.925}, + 5: {0.05: 2.571, 0.01: 4.032}, + 10: {0.05: 2.228, 0.01: 3.169}, + 20: {0.05: 2.086, 0.01: 2.845}, + 30: {0.05: 2.042, 0.01: 2.750}, + } + + # Para df > 30, usar aproximación normal + if df > 30: + if alpha == 0.05: + return 1.96 # Z crítico para 95% + elif alpha == 0.01: + return 2.576 # Z crítico para 99% + + # Buscar el df más cercano en la tabla + df_disponibles = sorted(valores_t.keys()) + df_usar = min(df_disponibles, key=lambda x: abs(x - df)) + + return valores_t[df_usar].get(alpha, 1.96) + + @staticmethod + def intervalo_media(muestra, confianza=0.95): + """ + Intervalo de confianza para la media poblacional + Usa t-Student para muestras pequeñas, normal para grandes + """ + n = len(muestra) + media_muestral = np.mean(muestra) + std_muestral = np.std(muestra, ddof=1) # Corrección de Bessel + + alpha = 1 - confianza + + if n >= 30: + # Muestra grande: usar distribución normal + z_critico = IntervalosConfianza.t_student_critico(31, alpha) + error_estandar = std_muestral / np.sqrt(n) + margen_error = z_critico * error_estandar + else: + # Muestra pequeña: usar t-Student + df = n - 1 + t_critico = IntervalosConfianza.t_student_critico(df, alpha) + error_estandar = std_muestral / np.sqrt(n) + margen_error = t_critico * error_estandar + + return (media_muestral - margen_error, media_muestral + margen_error) + + @staticmethod + def intervalo_proporcion(exitos, total, confianza=0.95): + """ + Intervalo de confianza para una proporción + Usa aproximación normal con corrección de continuidad + """ + p_hat = exitos / total + alpha = 1 - confianza + z_critico = IntervalosConfianza.t_student_critico(31, alpha) + + # Error estándar de la proporción + error_estandar = np.sqrt(p_hat * (1 - p_hat) / total) + margen_error = z_critico * error_estandar + + # Corrección de continuidad para muestras pequeñas + if total < 100: + margen_error += 0.5 / total + + lower = max(0, p_hat - margen_error) + upper = min(1, p_hat + margen_error) + + return lower, upper + +# Ejemplo: Evaluación de rendimiento de modelo con intervalos +np.random.seed(777) + +# Simulamos accuracy de un modelo en diferentes muestras de test +modelo_accuracy_real = 0.85 # Accuracy real (desconocida) + +print("=== Intervalos de Confianza para Accuracy ===") + +tamaños_muestra = [30, 100, 500, 1000] + +for n in tamaños_muestra: + # Simular resultados de test + predicciones_correctas = np.random.binomial(n, modelo_accuracy_real) + accuracy_observada = predicciones_correctas / n + + # Calcular intervalo de confianza + ic = IntervalosConfianza() + lower, upper = ic.intervalo_proporcion(predicciones_correctas, n, 0.95) + + # También calculamos con fórmula alternativa para comparar + intervalo_media = ic.intervalo_media([1]*predicciones_correctas + [0]*(n-predicciones_correctas), 0.95) + + print(f"\nMuestra de {n:4d} casos:") + print(f" Accuracy observada: {accuracy_observada:.3f}") + print(f" IC 95% (proporción): [{lower:.3f}, {upper:.3f}]") + print(f" Ancho del intervalo: {upper - lower:.3f}") + print(f" ¿Contiene el real? {'✓' if lower <= modelo_accuracy_real <= upper else '❌'}") +``` + +### 4.2 Bootstrap: Intervalos de Confianza Sin Asumir Distribuciones + +```python +class Bootstrap: + """Método de bootstrap para estimación de intervalos de confianza""" + + @staticmethod + def bootstrap_muestra(datos, n_bootstrap=1000): + """ + Genera muestras bootstrap + + Args: + datos: muestra original + n_bootstrap: número de remuestras + """ + n = len(datos) + muestras_boot = [] + + for _ in range(n_bootstrap): + # Muestreo con reemplazo + muestra_boot = [datos[np.random.randint(0, n)] for _ in range(n)] + muestras_boot.append(muestra_boot) + + return muestras_boot + + @staticmethod + def intervalo_percentil(estadisticas_boot, confianza=0.95): + """ + Intervalo de confianza usando método de percentiles + """ + alpha = 1 - confianza + lower_percentil = (alpha / 2) * 100 + upper_percentil = (1 - alpha / 2) * 100 + + lower = np.percentile(estadisticas_boot, lower_percentil) + upper = np.percentile(estadisticas_boot, upper_percentil) + + return lower, upper + + @staticmethod + def bootstrap_estadistica(datos, estadistica_func, n_bootstrap=1000, confianza=0.95): + """ + Bootstrap completo para cualquier estadística + + Args: + datos: muestra original + estadistica_func: función que calcula la estadística de interés + n_bootstrap: número de remuestras + confianza: nivel de confianza + """ + # Generar muestras bootstrap + muestras_boot = Bootstrap.bootstrap_muestra(datos, n_bootstrap) + + # Calcular estadística para cada muestra + estadisticas_boot = [estadistica_func(muestra) for muestra in muestras_boot] + + # Calcular intervalo + lower, upper = Bootstrap.intervalo_percentil(estadisticas_boot, confianza) + + return { + 'estadisticas_bootstrap': estadisticas_boot, + 'intervalo': (lower, upper), + 'estadistica_original': estadistica_func(datos) + } + +# Ejemplo: Bootstrap para métricas de ML complejas +np.random.seed(888) + +# Simulamos predicciones y valores reales de un modelo de regresión +n_test = 200 +y_true = np.random.normal(100, 15, n_test) +# Modelo con sesgo y ruido +y_pred = 0.9 * y_true + 5 + np.random.normal(0, 8, n_test) + +# Combinamos en un dataset para bootstrap +datos_test = list(zip(y_true, y_pred)) + +print("=== Bootstrap para Métricas de ML ===") + +# Definimos diferentes métricas +def mse(datos): + """Mean Squared Error""" + return np.mean([(true - pred)**2 for true, pred in datos]) + +def mae(datos): + """Mean Absolute Error""" + return np.mean([abs(true - pred) for true, pred in datos]) + +def r_squared(datos): + """Coeficiente de determinación""" + y_true_vals = [true for true, pred in datos] + y_pred_vals = [pred for true, pred in datos] + + y_mean = np.mean(y_true_vals) + ss_res = sum((true - pred)**2 for true, pred in datos) + ss_tot = sum((true - y_mean)**2 for true in y_true_vals) + + return 1 - (ss_res / ss_tot) if ss_tot != 0 else 0 + +def correlacion_personalizada(datos): + """Correlación entre predicciones y valores reales""" + y_true_vals = [true for true, pred in datos] + y_pred_vals = [pred for true, pred in datos] + return CorrelacionML.correlacion_pearson(y_true_vals, y_pred_vals) + +# Aplicar bootstrap a cada métrica +bootstrap = Bootstrap() +metricas = { + 'MSE': mse, + 'MAE': mae, + 'R²': r_squared, + 'Correlación': correlacion_personalizada +} + +resultados_bootstrap = {} + +for nombre, func in metricas.items(): + resultado = bootstrap.bootstrap_estadistica(datos_test, func, n_bootstrap=2000) + resultados_bootstrap[nombre] = resultado + + lower, upper = resultado['intervalo'] + original = resultado['estadistica_original'] + + print(f"\n{nombre}:") + print(f" Valor observado: {original:.4f}") + print(f" IC 95%: [{lower:.4f}, {upper:.4f}]") + print(f" Ancho del IC: {upper - lower:.4f}") + +# Visualización de las distribuciones bootstrap +fig, axes = plt.subplots(2, 2, figsize=(15, 12)) +axes = axes.ravel() + +for i, (nombre, resultado) in enumerate(resultados_bootstrap.items()): + ax = axes[i] + estadisticas = resultado['estadisticas_bootstrap'] + original = resultado['estadistica_original'] + lower, upper = resultado['intervalo'] + + # Histograma de la distribución bootstrap + ax.hist(estadisticas, bins=50, alpha=0.7, density=True, color='skyblue') + + # Líneas importantes + ax.axvline(original, color='red', linestyle='-', linewidth=2, + label=f'Observado: {original:.3f}') + ax.axvline(lower, color='green', linestyle='--', linewidth=2, + label=f'IC 95%: [{lower:.3f}, {upper:.3f}]') + ax.axvline(upper, color='green', linestyle='--', linewidth=2) + + # Área del intervalo de confianza + x_fill = np.linspace(lower, upper, 100) + # Aproximar densidad para el área + hist_vals, bin_edges = np.histogram(estadisticas, bins=50, density=True) + bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2 + from scipy.interpolate import interp1d + f_interp = interp1d(bin_centers, hist_vals, kind='linear', + bounds_error=False, fill_value=0) + y_fill = f_interp(x_fill) + ax.fill_between(x_fill, y_fill, alpha=0.3, color='green') + + ax.set_xlabel(f'{nombre}') + ax.set_ylabel('Densidad') + ax.set_title(f'Distribución Bootstrap: {nombre}') + ax.legend() + ax.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() + +print(f"\n=== Interpretación de Intervalos ===") +print("• Intervalos estrechos → Estimaciones más precisas") +print("• Intervalos amplios → Mayor incertidumbre") +print("• Bootstrap NO asume distribución específica") +print("• Útil para métricas complejas sin fórmula teórica") +``` + +## Parte V: Proyecto Práctico - Análisis Estadístico Completo + +Ahora vamos a aplicar todos los conceptos en un proyecto real de análisis estadístico para machine learning. + +```python +# Proyecto: Análisis estadístico de dataset de precios de casas +# Vamos a generar un dataset sintético realista + +class ProyectoEstadisticoML: + """Proyecto completo de análisis estadístico para ML""" + + def __init__(self): + self.datos = None + self.resultados = {} + + def generar_dataset(self, n_casas=1000): + """ + Genera dataset sintético de precios de casas + Con relaciones realistas entre variables + """ + np.random.seed(999) + + # Variables base + tamaño = np.random.gamma(2, 50, n_casas) # m², distribución gamma + antiguedad = np.random.exponential(15, n_casas) # años, exponencial + habitaciones = np.random.poisson(3, n_casas) + 1 # count, poisson + + # Variables derivadas con ruido + ubicacion_score = np.random.beta(2, 2, n_casas) # 0-1, beta + + # Precio con relaciones realistas + ruido + precio_base = ( + tamaño * 2000 + # $2000 por m² + (10 - antiguedad) * 1000 + # depreciación + habitaciones * 5000 + # valor por habitación + ubicacion_score * 50000 # premium por ubicación + ) + + # Añadir ruido realista (heteroscedástico) + ruido_relativo = 0.1 + 0.05 * ubicacion_score # más ruido en ubicaciones premium + ruido = np.random.normal(0, precio_base * ruido_relativo) + precio = precio_base + ruido + + # Asegurar valores positivos y realistas + precio = np.maximum(precio, 50000) + antiguedad = np.minimum(antiguedad, 50) + + self.datos = pd.DataFrame({ + 'precio': precio, + 'tamaño': tamaño, + 'antiguedad': antiguedad, + 'habitaciones': habitaciones, + 'ubicacion_score': ubicacion_score + }) + + print(f"Dataset generado: {n_casas} casas con 5 variables") + return self.datos + + def analisis_descriptivo(self): + """Análisis estadístico descriptivo completo""" + print("\n" + "="*50) + print("ANÁLISIS ESTADÍSTICO DESCRIPTIVO") + print("="*50) + + calc = EstadisticasML() + disp = DispersionML() + + resultados = {} + + for columna in self.datos.columns: + datos = self.datos[columna].values + + # Estadísticas básicas + media = calc.media(datos) + mediana = calc.mediana(datos) + std = disp.desviacion_estandar(datos, poblacional=False) + iqr, q1, q3 = disp.rango_intercuartil(datos) + + # Coeficiente de variación + cv = std / media if media != 0 else 0 + + # Asimetría (skewness) aproximada + skewness = (media - mediana) / std if std != 0 else 0 + + resultados[columna] = { + 'media': media, + 'mediana': mediana, + 'std': std, + 'cv': cv, + 'iqr': iqr, + 'q1': q1, + 'q3': q3, + 'skewness': skewness, + 'min': min(datos), + 'max': max(datos) + } + + print(f"\n{columna.upper()}:") + print(f" Media: {media:10.2f} | Mediana: {mediana:10.2f}") + print(f" Std: {std:12.2f} | CV: {cv:10.2%}") + print(f" IQR: {iqr:12.2f} | Asimetría: {skewness:8.2f}") + print(f" Rango: [{min(datos):8.1f}, {max(datos):8.1f}]") + + self.resultados['descriptivo'] = resultados + return resultados + + def analisis_correlaciones(self): + """Análisis de correlaciones y relaciones""" + print("\n" + "="*50) + print("ANÁLISIS DE CORRELACIONES") + print("="*50) + + corr = CorrelacionML() + variables = list(self.datos.columns) + n_vars = len(variables) + + # Matriz de correlación + matriz_corr = np.zeros((n_vars, n_vars)) + + print("\nMatriz de Correlación de Pearson:") + print(" " * 15, end="") + for var in variables: + print(f"{var[:8]:>8}", end="") + print() + + for i in range(n_vars): + print(f"{variables[i][:14]:<14}", end=" ") + for j in range(n_vars): + if i == j: + correlacion = 1.0 + else: + datos_i = self.datos[variables[i]].values + datos_j = self.datos[variables[j]].values + correlacion = corr.correlacion_pearson(datos_i, datos_j) + + matriz_corr[i, j] = correlacion + print(f"{correlacion:8.3f}", end="") + print() + + # Identificar correlaciones fuertes + print("\n=== Correlaciones Significativas (|r| > 0.5) ===") + for i in range(n_vars): + for j in range(i+1, n_vars): + if abs(matriz_corr[i, j]) > 0.5: + print(f"{variables[i]} ↔ {variables[j]}: r = {matriz_corr[i, j]:.3f}") + + self.resultados['correlaciones'] = matriz_corr + return matriz_corr + + def test_normalidad(self): + """Test de normalidad para cada variable""" + print("\n" + "="*50) + print("ANÁLISIS DE NORMALIDAD") + print("="*50) + + resultados_normalidad = {} + + for columna in self.datos.columns: + datos = self.datos[columna].values + + # Test visual usando regla 68-95-99.7 + media = np.mean(datos) + std = np.std(datos) + + # Contar datos en intervalos + en_1std = np.sum((datos >= media - std) & (datos <= media + std)) + en_2std = np.sum((datos >= media - 2*std) & (datos <= media + 2*std)) + en_3std = np.sum((datos >= media - 3*std) & (datos <= media + 3*std)) + + pct_1std = en_1std / len(datos) * 100 + pct_2std = en_2std / len(datos) * 100 + pct_3std = en_3std / len(datos) * 100 + + # Determinar normalidad + normal_68 = abs(pct_1std - 68) < 5 + normal_95 = abs(pct_2std - 95) < 5 + es_normal = normal_68 and normal_95 + + print(f"\n{columna.upper()}:") + print(f" 68% esperado vs real: 68.0% vs {pct_1std:.1f}% {'✓' if normal_68 else '❌'}") + print(f" 95% esperado vs real: 95.0% vs {pct_2std:.1f}% {'✓' if normal_95 else '❌'}") + print(f" 99.7% esperado vs real: 99.7% vs {pct_3std:.1f}%") + print(f" Conclusión: {'Normal' if es_normal else 'No Normal'}") + + resultados_normalidad[columna] = { + 'es_normal': es_normal, + 'pct_1std': pct_1std, + 'pct_2std': pct_2std, + 'pct_3std': pct_3std + } + + self.resultados['normalidad'] = resultados_normalidad + return resultados_normalidad + + def intervalos_confianza_bootstrap(self): + """Intervalos de confianza usando bootstrap""" + print("\n" + "="*50) + print("INTERVALOS DE CONFIANZA (BOOTSTRAP)") + print("="*50) + + bootstrap = Bootstrap() + + # Definir estadísticas de interés + def precio_medio(datos_df): + return datos_df['precio'].mean() + + def precio_por_m2(datos_df): + return datos_df['precio'].mean() / datos_df['tamaño'].mean() + + def correlacion_precio_tamaño(datos_df): + return CorrelacionML.correlacion_pearson( + datos_df['precio'].values, + datos_df['tamaño'].values + ) + + estadisticas = { + 'Precio Medio': precio_medio, + 'Precio por m²': precio_por_m2, + 'Corr(precio, tamaño)': correlacion_precio_tamaño + } + + # Bootstrap para cada estadística + print("Estadística" + " "*15 + "Valor" + " "*8 + "IC 95%") + print("-" * 60) + + for nombre, func in estadisticas.items(): + # Adaptar bootstrap para DataFrames + def bootstrap_df_func(datos_list): + # datos_list es una lista de índices + sample_df = self.datos.iloc[datos_list].reset_index(drop=True) + return func(sample_df) + + # Generar índices bootstrap + n = len(self.datos) + muestras_indices = [] + for _ in range(1000): + indices_boot = [np.random.randint(0, n) for _ in range(n)] + muestras_indices.append(indices_boot) + + # Calcular estadística para cada muestra + estadisticas_boot = [bootstrap_df_func(indices) for indices in muestras_indices] + + # Calcular intervalo + lower, upper = Bootstrap.intervalo_percentil(estadisticas_boot) + original = func(self.datos) + + print(f"{nombre:<25} {original:10.2f} [{lower:8.2f}, {upper:8.2f}]") + + def visualizaciones_profesionales(self): + """Crear visualizaciones profesionales del análisis""" + print("\n" + "="*50) + print("GENERANDO VISUALIZACIONES...") + print("="*50) + + # Configuración de estilo + plt.style.use('default') + + # 1. Histogramas y distribuciones + fig, axes = plt.subplots(2, 3, figsize=(18, 12)) + axes = axes.ravel() + + for i, columna in enumerate(self.datos.columns): + if i < 5: # Solo las primeras 5 variables + ax = axes[i] + datos = self.datos[columna].values + + # Histograma + ax.hist(datos, bins=50, alpha=0.7, density=True, color='skyblue') + + # Estadísticas + media = np.mean(datos) + std = np.std(datos) + ax.axvline(media, color='red', linestyle='-', linewidth=2, label=f'Media: {media:.1f}') + ax.axvline(np.median(datos), color='green', linestyle='--', linewidth=2, + label=f'Mediana: {np.median(datos):.1f}') + + ax.set_xlabel(columna.replace('_', ' ').title()) + ax.set_ylabel('Densidad') + ax.set_title(f'Distribución: {columna.replace("_", " ").title()}') + ax.legend() + ax.grid(True, alpha=0.3) + + # Ocultar el último subplot si no se usa + if len(self.datos.columns) == 5: + axes[5].axis('off') + + plt.suptitle('Distribuciones de Variables', fontsize=16, y=0.98) + plt.tight_layout() + plt.show() + + # 2. Matriz de correlación como heatmap + fig, ax = plt.subplots(figsize=(10, 8)) + + matriz = self.resultados.get('correlaciones') + variables = list(self.datos.columns) + + # Crear heatmap manual + im = ax.imshow(matriz, cmap='RdBu_r', vmin=-1, vmax=1) + + # Configurar ticks + ax.set_xticks(range(len(variables))) + ax.set_yticks(range(len(variables))) + ax.set_xticklabels([v.replace('_', ' ').title() for v in variables]) + ax.set_yticklabels([v.replace('_', ' ').title() for v in variables]) + + # Rotar labels del eje x + plt.setp(ax.get_xticklabels(), rotation=45, ha="right", rotation_mode="anchor") + + # Añadir valores en las celdas + for i in range(len(variables)): + for j in range(len(variables)): + text = ax.text(j, i, f'{matriz[i, j]:.2f}', + ha="center", va="center", color="black" if abs(matriz[i, j]) < 0.5 else "white") + + ax.set_title("Matriz de Correlación") + fig.colorbar(im, ax=ax) + plt.tight_layout() + plt.show() + + # 3. Scatter plots con líneas de tendencia + fig, axes = plt.subplots(2, 2, figsize=(16, 12)) + axes = axes.ravel() + + # Relaciones importantes con precio + relaciones = [ + ('tamaño', 'precio'), + ('antiguedad', 'precio'), + ('habitaciones', 'precio'), + ('ubicacion_score', 'precio') + ] + + for i, (var_x, var_y) in enumerate(relaciones): + ax = axes[i] + + x_data = self.datos[var_x].values + y_data = self.datos[var_y].values + + # Scatter plot + ax.scatter(x_data, y_data, alpha=0.6, s=20) + + # Línea de tendencia + z = np.polyfit(x_data, y_data, 1) + p = np.poly1d(z) + ax.plot(sorted(x_data), p(sorted(x_data)), "r--", alpha=0.8, linewidth=2) + + # Correlación + corr_val = CorrelacionML.correlacion_pearson(x_data, y_data) + ax.text(0.05, 0.95, f'r = {corr_val:.3f}', transform=ax.transAxes, + bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.8)) + + ax.set_xlabel(var_x.replace('_', ' ').title()) + ax.set_ylabel(var_y.replace('_', ' ').title()) + ax.set_title(f'{var_x.replace("_", " ").title()} vs {var_y.replace("_", " ").title()}') + ax.grid(True, alpha=0.3) + + plt.suptitle('Relaciones entre Variables', fontsize=16) + plt.tight_layout() + plt.show() + + print("✓ Visualizaciones completadas") + + def resumen_ejecutivo(self): + """Resumen ejecutivo con insights para ML""" + print("\n" + "="*60) + print("RESUMEN EJECUTIVO PARA MACHINE LEARNING") + print("="*60) + + print("\n🎯 INSIGHTS CLAVE:") + + # 1. Variables más importantes + matriz_corr = self.resultados.get('correlaciones') + if matriz_corr is not None: + variables = list(self.datos.columns) + precio_idx = variables.index('precio') + + correlaciones_precio = [(variables[i], abs(matriz_corr[precio_idx, i])) + for i in range(len(variables)) if i != precio_idx] + correlaciones_precio.sort(key=lambda x: x[1], reverse=True) + + print(f"\n1. VARIABLES MÁS PREDICTIVAS DEL PRECIO:") + for i, (var, corr) in enumerate(correlaciones_precio[:3], 1): + print(f" {i}. {var.replace('_', ' ').title()}: |r| = {corr:.3f}") + + # 2. Calidad de los datos + normalidad = self.resultados.get('normalidad', {}) + vars_normales = sum(1 for v in normalidad.values() if v['es_normal']) + total_vars = len(normalidad) + + print(f"\n2. CALIDAD DE LOS DATOS:") + print(f" • Variables con distribución normal: {vars_normales}/{total_vars}") + print(f" • Outliers detectados: Revisar variables con CV > 50%") + + # 3. Recomendaciones para ML + print(f"\n3. RECOMENDACIONES PARA ML:") + + descriptivo = self.resultados.get('descriptivo', {}) + for var, stats in descriptivo.items(): + cv = stats.get('cv', 0) + if cv > 0.5: + print(f" ⚠️ {var}: Alta variabilidad (CV={cv:.1%}) - Considerar transformación") + elif abs(stats.get('skewness', 0)) > 1: + print(f" ⚠️ {var}: Distribución asimétrica - Considerar log-transform") + + print(f"\n4. PRÓXIMOS PASOS:") + print(f" 1. Feature engineering basado en correlaciones fuertes") + print(f" 2. Transformaciones para variables asimétricas") + print(f" 3. Detección y tratamiento de outliers") + print(f" 4. Validación cruzada con estratificación") + + print(f"\n✅ Análisis estadístico completo - Listo para modelado ML") + +# Ejecutar proyecto completo +print("INICIANDO PROYECTO DE ANÁLISIS ESTADÍSTICO") +print("="*60) + +proyecto = ProyectoEstadisticoML() + +# 1. Generar dataset +datos = proyecto.generar_dataset(1000) + +# 2. Análisis descriptivo +proyecto.analisis_descriptivo() + +# 3. Análisis de correlaciones +proyecto.analisis_correlaciones() + +# 4. Test de normalidad +proyecto.test_normalidad() + +# 5. Intervalos de confianza +proyecto.intervalos_confianza_bootstrap() + +# 6. Visualizaciones +proyecto.visualizaciones_profesionales() + +# 7. Resumen ejecutivo +proyecto.resumen_ejecutivo() +``` + +{{< alert "lightbulb" >}} +**Insight del Proyecto**: Este análisis estadístico completo te proporciona las bases para tomar decisiones informadas sobre preprocesamiento, selección de features y elección de algoritmos de ML. +{{< /alert >}} + +## Parte VI: Tests de Hipótesis para Machine Learning + +Los tests de hipótesis son fundamentales para validar si las diferencias que observamos en nuestros modelos son estadísticamente significativas. + +### 6.1 Test t de Student para Comparación de Modelos + +```python +class TestsHipotesisML: + """Tests de hipótesis aplicados a machine learning""" + + @staticmethod + def test_t_independiente(muestra1, muestra2, alpha=0.05): + """ + Test t para muestras independientes + H0: μ1 = μ2 (no hay diferencia entre medias) + H1: μ1 ≠ μ2 (hay diferencia significativa) + """ + n1, n2 = len(muestra1), len(muestra2) + media1, media2 = np.mean(muestra1), np.mean(muestra2) + var1, var2 = np.var(muestra1, ddof=1), np.var(muestra2, ddof=1) + + # Pooled standard error + pooled_se = np.sqrt(var1/n1 + var2/n2) + + # t-statistic + t_stat = (media1 - media2) / pooled_se + + # Grados de libertad (aproximación de Welch) + df = ((var1/n1 + var2/n2)**2) / ((var1/n1)**2/(n1-1) + (var2/n2)**2/(n2-1)) + + # Valor crítico (aproximado) + t_critico = IntervalosConfianza.t_student_critico(int(df), alpha) + + # p-value aproximado (bilateral) + p_value = 2 * (1 - 0.5 * (1 + np.sign(t_stat) * np.sqrt(t_stat**2 / (t_stat**2 + df)))) + + return { + 'estadistico': t_stat, + 'p_value': p_value, + 'grados_libertad': df, + 'valor_critico': t_critico, + 'significativo': abs(t_stat) > t_critico, + 'diferencia_medias': media1 - media2, + 'intervalo_diferencia': ((media1 - media2) - t_critico * pooled_se, + (media1 - media2) + t_critico * pooled_se) + } + + @staticmethod + def test_paired_t(antes, despues, alpha=0.05): + """ + Test t pareado para comparar rendimiento antes/después + H0: diferencia_media = 0 + H1: diferencia_media ≠ 0 + """ + diferencias = np.array(despues) - np.array(antes) + n = len(diferencias) + media_diff = np.mean(diferencias) + std_diff = np.std(diferencias, ddof=1) + se_diff = std_diff / np.sqrt(n) + + t_stat = media_diff / se_diff + df = n - 1 + t_critico = IntervalosConfianza.t_student_critico(df, alpha) + + return { + 'estadistico': t_stat, + 'grados_libertad': df, + 'valor_critico': t_critico, + 'significativo': abs(t_stat) > t_critico, + 'mejora_media': media_diff, + 'intervalo_mejora': (media_diff - t_critico * se_diff, + media_diff + t_critico * se_diff) + } + +# Ejemplo: Comparación de dos algoritmos de ML +np.random.seed(123) + +# Simulamos accuracy de dos modelos en diferentes folds de validación cruzada +modelo_A_accuracy = np.random.normal(0.82, 0.03, 20) # Modelo base +modelo_B_accuracy = np.random.normal(0.85, 0.035, 20) # Modelo mejorado + +# Limitamos accuracies a [0, 1] +modelo_A_accuracy = np.clip(modelo_A_accuracy, 0, 1) +modelo_B_accuracy = np.clip(modelo_B_accuracy, 0, 1) + +print("=== COMPARACIÓN DE MODELOS CON TEST T ===") +print(f"Modelo A - Media: {np.mean(modelo_A_accuracy):.4f}, Std: {np.std(modelo_A_accuracy, ddof=1):.4f}") +print(f"Modelo B - Media: {np.mean(modelo_B_accuracy):.4f}, Std: {np.std(modelo_B_accuracy, ddof=1):.4f}") + +# Test t independiente +tests = TestsHipotesisML() +resultado = tests.test_t_independiente(modelo_A_accuracy, modelo_B_accuracy) + +print(f"\n=== RESULTADOS DEL TEST T ===") +print(f"Estadístico t: {resultado['estadistico']:.3f}") +print(f"Grados de libertad: {resultado['grados_libertad']:.1f}") +print(f"Valor crítico (α=0.05): ±{resultado['valor_critico']:.3f}") +print(f"p-value: {resultado['p_value']:.4f}") +print(f"Diferencia de medias: {resultado['diferencia_medias']:.4f}") +print(f"IC 95% diferencia: [{resultado['intervalo_diferencia'][0]:.4f}, {resultado['intervalo_diferencia'][1]:.4f}]") + +if resultado['significativo']: + print(f"\n✅ CONCLUSIÓN: Diferencia estadísticamente significativa") + print(f" El Modelo B es significativamente mejor que el Modelo A") +else: + print(f"\n❌ CONCLUSIÓN: No hay diferencia estadísticamente significativa") + +# Visualización de la comparación +fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(18, 6)) + +# Plot 1: Boxplot comparativo +ax1.boxplot([modelo_A_accuracy, modelo_B_accuracy], labels=['Modelo A', 'Modelo B']) +ax1.set_ylabel('Accuracy') +ax1.set_title('Comparación de Modelos\n(Boxplot)') +ax1.grid(True, alpha=0.3) + +# Plot 2: Histogramas superpuestos +ax2.hist(modelo_A_accuracy, bins=15, alpha=0.7, label='Modelo A', color='blue', density=True) +ax2.hist(modelo_B_accuracy, bins=15, alpha=0.7, label='Modelo B', color='red', density=True) +ax2.axvline(np.mean(modelo_A_accuracy), color='blue', linestyle='--', linewidth=2) +ax2.axvline(np.mean(modelo_B_accuracy), color='red', linestyle='--', linewidth=2) +ax2.set_xlabel('Accuracy') +ax2.set_ylabel('Densidad') +ax2.set_title('Distribución de Accuracy') +ax2.legend() +ax2.grid(True, alpha=0.3) + +# Plot 3: Diferencias pareadas (si fuera paired test) +diferencias = modelo_B_accuracy - modelo_A_accuracy +ax3.hist(diferencias, bins=15, alpha=0.7, color='green') +ax3.axvline(np.mean(diferencias), color='darkgreen', linestyle='-', linewidth=2, + label=f'Diferencia media: {np.mean(diferencias):.4f}') +ax3.axvline(0, color='black', linestyle='--', linewidth=1, label='Sin diferencia') +ax3.set_xlabel('Diferencia de Accuracy (B - A)') +ax3.set_ylabel('Frecuencia') +ax3.set_title('Distribución de Diferencias') +ax3.legend() +ax3.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() + +# Ejemplo de test pareado: antes/después de feature engineering +print("\n" + "="*60) +print("TEST PAREADO: ANTES/DESPUÉS DE FEATURE ENGINEERING") +print("="*60) + +# Simulamos el mismo modelo evaluado antes y después de feature engineering +np.random.seed(456) +accuracy_antes = np.random.normal(0.78, 0.04, 15) +# Después del feature engineering: mejora correlacionada +mejora = np.random.normal(0.04, 0.015, 15) # Mejora promedio de 4% +accuracy_despues = accuracy_antes + mejora + +# Clip to valid range +accuracy_antes = np.clip(accuracy_antes, 0, 1) +accuracy_despues = np.clip(accuracy_despues, 0, 1) + +print(f"Antes FE - Media: {np.mean(accuracy_antes):.4f}") +print(f"Después FE - Media: {np.mean(accuracy_despues):.4f}") +print(f"Mejora promedio: {np.mean(accuracy_despues - accuracy_antes):.4f}") + +resultado_paired = tests.test_paired_t(accuracy_antes, accuracy_despues) + +print(f"\n=== RESULTADOS DEL TEST T PAREADO ===") +print(f"Estadístico t: {resultado_paired['estadistico']:.3f}") +print(f"Valor crítico: ±{resultado_paired['valor_critico']:.3f}") +print(f"Mejora media: {resultado_paired['mejora_media']:.4f}") +print(f"IC 95% mejora: [{resultado_paired['intervalo_mejora'][0]:.4f}, {resultado_paired['intervalo_mejora'][1]:.4f}]") + +if resultado_paired['significativo']: + print(f"\n✅ El feature engineering produjo una mejora significativa") +else: + print(f"\n❌ La mejora no es estadísticamente significativa") +``` + +### 6.2 ANOVA para Comparación Múltiple de Modelos + +```python +class ANOVAMultiple: + """ANOVA para comparar múltiples modelos simultáneamente""" + + @staticmethod + def anova_one_way(grupos): + """ + ANOVA de un factor para comparar múltiples grupos + H0: μ1 = μ2 = ... = μk (todas las medias son iguales) + H1: Al menos una media es diferente + """ + # Calcular estadísticas por grupo + k = len(grupos) # número de grupos + n_total = sum(len(grupo) for grupo in grupos) + + # Media global + todos_valores = np.concatenate(grupos) + media_global = np.mean(todos_valores) + + # Suma de cuadrados entre grupos (SSB) + ssb = sum(len(grupo) * (np.mean(grupo) - media_global)**2 for grupo in grupos) + + # Suma de cuadrados dentro de grupos (SSW) + ssw = sum(np.sum((grupo - np.mean(grupo))**2) for grupo in grupos) + + # Grados de libertad + df_between = k - 1 + df_within = n_total - k + df_total = n_total - 1 + + # Cuadrados medios + msb = ssb / df_between + msw = ssw / df_within + + # Estadístico F + f_stat = msb / msw if msw > 0 else float('inf') + + # Valor crítico F (aproximación) + # Para α = 0.05, usamos aproximaciones comunes + if df_between <= 3 and df_within >= 10: + f_critico = 3.0 # Aproximación conservadora + else: + f_critico = 2.5 # Aproximación más liberal + + return { + 'f_estadistico': f_stat, + 'f_critico': f_critico, + 'df_between': df_between, + 'df_within': df_within, + 'ssb': ssb, + 'ssw': ssw, + 'msb': msb, + 'msw': msw, + 'significativo': f_stat > f_critico + } + + @staticmethod + def test_post_hoc_tukey(grupos, nombres_grupos): + """ + Test post-hoc de Tukey para comparaciones múltiples + (Versión simplificada) + """ + k = len(grupos) + comparaciones = [] + + # MSW del ANOVA + anova_result = ANOVAMultiple.anova_one_way(grupos) + msw = anova_result['msw'] + + # Comparaciones por pares + for i in range(k): + for j in range(i+1, k): + n1, n2 = len(grupos[i]), len(grupos[j]) + media1, media2 = np.mean(grupos[i]), np.mean(grupos[j]) + + # Error estándar para la diferencia + se_diff = np.sqrt(msw * (1/n1 + 1/n2)) + + # Estadístico Q (distribución estudentizada) + q_stat = abs(media1 - media2) / se_diff + + # Valor crítico Q (aproximación) + q_critico = 3.5 # Para k=3-5 grupos, α=0.05 + + comparaciones.append({ + 'grupos': f"{nombres_grupos[i]} vs {nombres_grupos[j]}", + 'diferencia': media1 - media2, + 'q_estadistico': q_stat, + 'q_critico': q_critico, + 'significativo': q_stat > q_critico + }) + + return comparaciones + +# Ejemplo: Comparación de múltiples algoritmos +np.random.seed(789) + +# Simulamos performance de 4 algoritmos diferentes +algoritmos = { + 'Random Forest': np.random.normal(0.85, 0.03, 20), + 'XGBoost': np.random.normal(0.87, 0.025, 20), + 'SVM': np.random.normal(0.82, 0.04, 20), + 'Logistic Regression': np.random.normal(0.79, 0.035, 20) +} + +# Clip to valid accuracy range +for nombre in algoritmos: + algoritmos[nombre] = np.clip(algoritmos[nombre], 0, 1) + +print("=== COMPARACIÓN MÚLTIPLE DE ALGORITMOS (ANOVA) ===") + +# Mostrar estadísticas descriptivas +for nombre, scores in algoritmos.items(): + print(f"{nombre:20}: Media = {np.mean(scores):.4f}, Std = {np.std(scores, ddof=1):.4f}") + +# ANOVA +grupos = list(algoritmos.values()) +nombres = list(algoritmos.keys()) + +anova = ANOVAMultiple() +resultado_anova = anova.anova_one_way(grupos) + +print(f"\n=== RESULTADOS ANOVA ===") +print(f"F-estadístico: {resultado_anova['f_estadistico']:.3f}") +print(f"F-crítico: {resultado_anova['f_critico']:.3f}") +print(f"Grados de libertad: between={resultado_anova['df_between']}, within={resultado_anova['df_within']}") + +if resultado_anova['significativo']: + print(f"✅ Hay diferencias significativas entre algoritmos") + + # Test post-hoc + print(f"\n=== TEST POST-HOC (Tukey) ===") + comparaciones = anova.test_post_hoc_tukey(grupos, nombres) + + for comp in comparaciones: + significativo_symbol = "✅" if comp['significativo'] else "❌" + print(f"{comp['grupos']:35}: Diff={comp['diferencia']:7.4f}, Q={comp['q_estadistico']:6.2f} {significativo_symbol}") + +else: + print(f"❌ No hay diferencias significativas entre algoritmos") + +# Visualización ANOVA +fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6)) + +# Boxplot de todos los algoritmos +ax1.boxplot(grupos, labels=[n[:8] for n in nombres]) # Truncar nombres largos +ax1.set_ylabel('Accuracy') +ax1.set_title('Comparación de Algoritmos\n(ANOVA)') +ax1.grid(True, alpha=0.3) +plt.setp(ax1.get_xticklabels(), rotation=45) + +# Gráfico de medias con intervalos de confianza +medias = [np.mean(grupo) for grupo in grupos] +stds = [np.std(grupo, ddof=1) for grupo in grupos] +errors = [std/np.sqrt(len(grupo)) for std, grupo in zip(stds, grupos)] + +x_pos = range(len(nombres)) +ax2.bar(x_pos, medias, yerr=errors, capsize=5, alpha=0.7, + color=['blue', 'red', 'green', 'orange']) +ax2.set_xticks(x_pos) +ax2.set_xticklabels([n[:8] for n in nombres], rotation=45) +ax2.set_ylabel('Accuracy Media') +ax2.set_title('Medias con Intervalos de Confianza') +ax2.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +## Parte VII: Detección de Outliers Estadística + +Los outliers pueden distorsionar nuestros modelos de ML. Aprendamos métodos estadísticos para detectarlos. + +```python +class DeteccionOutliers: + """Métodos estadísticos para detección de outliers en ML""" + + @staticmethod + def metodo_iqr(datos, factor=1.5): + """ + Método del rango intercuartil (IQR) + Outliers: < Q1 - factor*IQR o > Q3 + factor*IQR + """ + q1 = np.percentile(datos, 25) + q3 = np.percentile(datos, 75) + iqr = q3 - q1 + + limite_inferior = q1 - factor * iqr + limite_superior = q3 + factor * iqr + + outliers_mask = (datos < limite_inferior) | (datos > limite_superior) + outliers = datos[outliers_mask] + + return { + 'outliers': outliers, + 'indices_outliers': np.where(outliers_mask)[0], + 'limite_inferior': limite_inferior, + 'limite_superior': limite_superior, + 'n_outliers': len(outliers), + 'porcentaje_outliers': len(outliers) / len(datos) * 100 + } + + @staticmethod + def metodo_zscore(datos, umbral=3): + """ + Método del Z-score + Outliers: |z| > umbral (típicamente 3) + """ + media = np.mean(datos) + std = np.std(datos) + + z_scores = np.abs((datos - media) / std) + outliers_mask = z_scores > umbral + + return { + 'z_scores': z_scores, + 'outliers': datos[outliers_mask], + 'indices_outliers': np.where(outliers_mask)[0], + 'umbral': umbral, + 'n_outliers': np.sum(outliers_mask), + 'porcentaje_outliers': np.sum(outliers_mask) / len(datos) * 100 + } + + @staticmethod + def metodo_zscore_modificado(datos, umbral=3.5): + """ + Z-score modificado usando mediana (más robusto) + Usa MAD (Median Absolute Deviation) en lugar de std + """ + mediana = np.median(datos) + mad = np.median(np.abs(datos - mediana)) + + # Factor de escala para aproximar desviación estándar + mad_scaled = mad * 1.4826 + + z_scores_mod = np.abs((datos - mediana) / mad_scaled) + outliers_mask = z_scores_mod > umbral + + return { + 'z_scores_modificados': z_scores_mod, + 'outliers': datos[outliers_mask], + 'indices_outliers': np.where(outliers_mask)[0], + 'mediana': mediana, + 'mad': mad, + 'n_outliers': np.sum(outliers_mask), + 'porcentaje_outliers': np.sum(outliers_mask) / len(datos) * 100 + } + + @staticmethod + def metodo_isolation_forest_simple(datos, contaminacion=0.1): + """ + Versión simplificada de Isolation Forest + Basada en profundidad promedio de aislamiento + """ + n = len(datos) + profundidades = [] + + # Simulamos múltiples "árboles" de aislamiento + n_trees = 100 + for _ in range(n_trees): + # Para cada punto, simulamos profundidad de aislamiento + profundidades_arbol = [] + datos_copia = datos.copy() + + for punto in datos: + profundidad = 0 + datos_temp = datos_copia.copy() + + # Simular divisiones hasta aislar el punto + while len(datos_temp) > 1 and profundidad < 20: + # División aleatoria + min_val, max_val = np.min(datos_temp), np.max(datos_temp) + if max_val == min_val: + break + + split_val = np.random.uniform(min_val, max_val) + + # Determinar en qué lado queda nuestro punto + if punto <= split_val: + datos_temp = datos_temp[datos_temp <= split_val] + else: + datos_temp = datos_temp[datos_temp > split_val] + + profundidad += 1 + + profundidades_arbol.append(profundidad) + + profundidades.append(profundidades_arbol) + + # Promedio de profundidades + prof_promedio = np.mean(profundidades, axis=0) + + # Puntos con menor profundidad son más anómalos + umbral_prof = np.percentile(prof_promedio, contaminacion * 100) + outliers_mask = prof_promedio <= umbral_prof + + return { + 'profundidades': prof_promedio, + 'outliers': datos[outliers_mask], + 'indices_outliers': np.where(outliers_mask)[0], + 'umbral_profundidad': umbral_prof, + 'n_outliers': np.sum(outliers_mask), + 'porcentaje_outliers': np.sum(outliers_mask) / len(datos) * 100 + } + +# Ejemplo: Detección de outliers en datos de latencia de API +np.random.seed(999) + +# Generamos datos de latencia con outliers realistas +latencia_normal = np.random.gamma(2, 50, 900) # Latencias normales +latencia_outliers_red = np.random.exponential(500, 80) # Problemas de red +latencia_outliers_server = np.random.uniform(1000, 2000, 20) # Problemas de servidor + +latencias = np.concatenate([latencia_normal, latencia_outliers_red, latencia_outliers_server]) +np.random.shuffle(latencias) # Mezclar para simular orden temporal + +print("=== DETECCIÓN DE OUTLIERS EN LATENCIAS DE API ===") +print(f"Total de muestras: {len(latencias)}") +print(f"Latencia media: {np.mean(latencias):.2f} ms") +print(f"Latencia mediana: {np.median(latencias):.2f} ms") + +detector = DeteccionOutliers() + +# Aplicar diferentes métodos +metodos = { + 'IQR (factor=1.5)': detector.metodo_iqr(latencias, 1.5), + 'Z-score (umbral=3)': detector.metodo_zscore(latencias, 3), + 'Z-score Modificado': detector.metodo_zscore_modificado(latencias, 3.5), + 'Isolation Forest': detector.metodo_isolation_forest_simple(latencias, 0.1) +} + +print(f"\n=== RESULTADOS POR MÉTODO ===") +for nombre, resultado in metodos.items(): + print(f"\n{nombre}:") + print(f" Outliers detectados: {resultado['n_outliers']} ({resultado['porcentaje_outliers']:.1f}%)") + if len(resultado['outliers']) > 0: + print(f" Rango outliers: [{np.min(resultado['outliers']):.1f}, {np.max(resultado['outliers']):.1f}] ms") + +# Visualización comparativa +fig, axes = plt.subplots(2, 2, figsize=(16, 12)) +axes = axes.ravel() + +metodos_lista = list(metodos.items()) +for i, (nombre, resultado) in enumerate(metodos_lista): + ax = axes[i] + + # Histograma con outliers marcados + ax.hist(latencias, bins=50, alpha=0.7, color='lightblue', label='Datos normales') + + if len(resultado['outliers']) > 0: + ax.hist(resultado['outliers'], bins=20, alpha=0.8, color='red', + label=f'Outliers ({resultado["n_outliers"]})') + + ax.set_xlabel('Latencia (ms)') + ax.set_ylabel('Frecuencia') + ax.set_title(f'{nombre}\n{resultado["n_outliers"]} outliers ({resultado["porcentaje_outliers"]:.1f}%)') + ax.legend() + ax.grid(True, alpha=0.3) + +plt.tight_layout() +plt.show() + +# Análisis de consenso entre métodos +print(f"\n=== ANÁLISIS DE CONSENSO ===") + +# Crear matriz de detección +indices_todos = set(range(len(latencias))) +detecciones = {} + +for nombre, resultado in metodos.items(): + detecciones[nombre] = set(resultado['indices_outliers']) + +# Intersección (outliers detectados por múltiples métodos) +interseccion_todos = set.intersection(*detecciones.values()) +union_todos = set.union(*detecciones.values()) + +print(f"Outliers detectados por TODOS los métodos: {len(interseccion_todos)}") +print(f"Outliers detectados por AL MENOS UN método: {len(union_todos)}") + +if len(interseccion_todos) > 0: + valores_consenso = latencias[list(interseccion_todos)] + print(f"Valores de consenso: {sorted(valores_consenso)[:5]}... (primeros 5)") + +# Matriz de consenso +print(f"\n=== MATRIZ DE CONSENSO ===") +nombres_metodos = list(metodos.keys()) +for i, metodo1 in enumerate(nombres_metodos): + for j, metodo2 in enumerate(nombres_metodos): + if i <= j: + if i == j: + overlap = len(detecciones[metodo1]) + else: + overlap = len(detecciones[metodo1].intersection(detecciones[metodo2])) + print(f"{metodo1[:10]} ∩ {metodo2[:10]}: {overlap:3d} outliers") + +print(f"\n💡 RECOMENDACIÓN:") +if len(interseccion_todos) > 0: + print(f" • Eliminar outliers detectados por múltiples métodos ({len(interseccion_todos)} casos)") + print(f" • Investigar outliers detectados por un solo método") +else: + print(f" • No hay consenso fuerte - revisar umbrales") + print(f" • Considerar contexto del dominio para decisión final") +``` + +## Resumen y Conexión con la Semana 4 + +### Conceptos Clave Dominados + +Has completado un recorrido exhaustivo por los fundamentos estadísticos de machine learning: + +{{< mermaid >}} +graph LR + A[Estadística Descriptiva] --> B[Distribuciones] + B --> C[Correlación vs Causación] + C --> D[Teorema de Bayes] + D --> E[Intervalos de Confianza] + E --> F[Tests de Hipótesis] + F --> G[Detección de Outliers] + G --> H[Proyecto Completo] + + style A fill:#e1f5fe + style H fill:#c8e6c9 +{{< /mermaid >}} + +### Checklist de Competencias ✅ + +- **Estadística Descriptiva**: Calcular e interpretar medidas de tendencia central y dispersión +- **Distribuciones de Probabilidad**: Aplicar distribuciones normal, binomial y uniforme en contextos de ML +- **Correlación vs Causación**: Identificar y evitar interpretaciones erróneas +- **Teorema de Bayes**: Implementar clasificación bayesiana y actualización de creencias +- **Intervalos de Confianza**: Cuantificar incertidumbre con métodos paramétricos y bootstrap +- **Tests de Hipótesis**: Comparar modelos con rigor estadístico +- **Detección de Outliers**: Aplicar múltiples métodos para identificar anomalías + +### Preparación para la Semana 4: NumPy y Pandas + +Los conceptos estadísticos que dominaste hoy son los **cimientos** para la manipulación eficiente de datos con NumPy y Pandas: + +{{< alert "circle-info" >}} +**Próxima Semana**: Implementarás estos cálculos estadísticos de manera vectorizada y escalable usando las librerías fundamentales de data science en Python. +{{< /alert >}} + +**Lo que conectarás la próxima semana:** + +1. **Estadísticas con NumPy**: Vectorización de todos los cálculos que hiciste hoy +2. **Análisis exploratorio con Pandas**: Aplicar tests estadísticos a datasets reales +3. **Visualización estadística**: Crear gráficos profesionales para análisis +4. **Pipelines de preprocesamiento**: Integrar detección de outliers y transformaciones + +### Ejercicios para Consolidar + +{{< alert "lightbulb" >}} +**Desafío Semanal**: Antes de la próxima sesión, aplica el análisis estadístico completo que desarrollaste a un dataset público de tu elección. Algunos datasets recomendados: + +- [Kaggle: House Prices](https://www.kaggle.com/c/house-prices-advanced-regression-techniques) +- [UCI: Adult Income](https://archive.ics.uci.edu/ml/datasets/adult) +- [Seaborn: Tips Dataset](https://github.com/mwaskom/seaborn-data) + +**Objetivo**: Generar un reporte estadístico completo que incluya todos los elementos que vimos hoy. +{{< /alert >}} + +### Recursos Adicionales + +Para profundizar en los conceptos estadísticos aplicados a machine learning: + +{{< alert "book-open" >}} +**Lecturas Recomendadas**: + +- "The Elements of Statistical Learning" - Hastie, Tibshirani & Friedman +- "Pattern Recognition and Machine Learning" - Christopher Bishop +- "Think Stats" - Allen B. Downey (enfoque práctico con Python) + +**Documentación Técnica**: +- [SciPy Stats Documentation](https://docs.scipy.org/doc/scipy/reference/stats.html) +- [NumPy Statistical Functions](https://numpy.org/doc/stable/reference/routines.statistics.html) +{{< /alert >}} + +¡Felicitaciones! Has construido una base sólida en estadística aplicada a machine learning. En la próxima semana, estas herramientas conceptuales se transformarán en código eficiente y escalable con las librerías fundamentales del ecosistema científico de Python. + +--- + +*¿Preguntas sobre algún concepto estadístico? ¿Dudas sobre la implementación de algún test? Recordá que la estadística es la base de la toma de decisiones informadas en machine learning - cada concepto que dominaste hoy te ahorrará errores costosos en producción.*