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SmartCertify ML — Math Utilities
Linear algebra, statistics, and probability utilities.
"""
import numpy as np
from scipy import stats
from typing import List, Tuple, Optional
# ─── Linear Algebra Utilities ─────────────────────────────────
def cosine_similarity_vectors(a: np.ndarray, b: np.ndarray) -> float:
"""Compute cosine similarity between two vectors."""
norm_a = np.linalg.norm(a)
norm_b = np.linalg.norm(b)
if norm_a == 0 or norm_b == 0:
return 0.0
return float(np.dot(a, b) / (norm_a * norm_b))
def euclidean_distance(a: np.ndarray, b: np.ndarray) -> float:
"""Compute Euclidean distance between two vectors."""
return float(np.linalg.norm(a - b))
def matrix_rank(matrix: np.ndarray) -> int:
"""Compute rank of a matrix."""
return int(np.linalg.matrix_rank(matrix))
def compute_svd(matrix: np.ndarray) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
"""Compute Singular Value Decomposition."""
U, S, Vt = np.linalg.svd(matrix, full_matrices=False)
return U, S, Vt
def normalize_vector(v: np.ndarray) -> np.ndarray:
"""L2-normalize a vector."""
norm = np.linalg.norm(v)
if norm == 0:
return v
return v / norm
# ─── Statistics Utilities ─────────────────────────────────────
def compute_confidence_interval(
data: np.ndarray, confidence: float = 0.95
) -> Tuple[float, float]:
"""Compute confidence interval for the mean of data."""
n = len(data)
mean = np.mean(data)
se = stats.sem(data)
h = se * stats.t.ppf((1 + confidence) / 2, n - 1)
return (float(mean - h), float(mean + h))
def compute_z_score(value: float, mean: float, std: float) -> float:
"""Compute z-score for a value given mean and standard deviation."""
if std == 0:
return 0.0
return (value - mean) / std
def compute_p_value(z_score: float, two_tailed: bool = True) -> float:
"""Compute p-value from z-score."""
p = 2 * (1 - stats.norm.cdf(abs(z_score))) if two_tailed else (1 - stats.norm.cdf(z_score))
return float(p)
def ks_test(data: np.ndarray, distribution: str = "norm") -> Tuple[float, float]:
"""Kolmogorov-Smirnov test for distribution fit."""
statistic, p_value = stats.kstest(data, distribution)
return float(statistic), float(p_value)
def compute_entropy(probabilities: np.ndarray) -> float:
"""Compute Shannon entropy of a probability distribution."""
probabilities = probabilities[probabilities > 0]
return float(-np.sum(probabilities * np.log2(probabilities)))
def compute_kl_divergence(p: np.ndarray, q: np.ndarray) -> float:
"""Compute KL divergence D(P || Q)."""
p = np.asarray(p, dtype=np.float64)
q = np.asarray(q, dtype=np.float64)
# Avoid division by zero
mask = (p > 0) & (q > 0)
return float(np.sum(p[mask] * np.log(p[mask] / q[mask])))
# ─── Probability Utilities ────────────────────────────────────
def gaussian_probability(x: float, mean: float, std: float) -> float:
"""Compute probability density of x under Gaussian distribution."""
return float(stats.norm.pdf(x, loc=mean, scale=std))
def bayesian_update(
prior: float, likelihood: float, evidence: float
) -> float:
"""Apply Bayes' theorem: P(H|E) = P(E|H) * P(H) / P(E)."""
if evidence == 0:
return 0.0
return (likelihood * prior) / evidence
def softmax(x: np.ndarray) -> np.ndarray:
"""Compute softmax probabilities."""
e_x = np.exp(x - np.max(x))
return e_x / e_x.sum()
# ─── Feature Analysis ─────────────────────────────────────────
def compute_correlation_matrix(data: np.ndarray) -> np.ndarray:
"""Compute Pearson correlation matrix."""
return np.corrcoef(data, rowvar=False)
def compute_mutual_information(x: np.ndarray, y: np.ndarray, bins: int = 20) -> float:
"""Compute mutual information between two variables."""
hist_2d, _, _ = np.histogram2d(x, y, bins=bins)
pxy = hist_2d / hist_2d.sum()
px = pxy.sum(axis=1)
py = pxy.sum(axis=0)
mi = 0.0
for i in range(bins):
for j in range(bins):
if pxy[i, j] > 0 and px[i] > 0 and py[j] > 0:
mi += pxy[i, j] * np.log2(pxy[i, j] / (px[i] * py[j]))
return mi
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