pystatistics 1.6.0

GPU-accelerated statistical computing for Python

PyStatistics

GPU-accelerated statistical computing for Python.

What's New

Major expansion: 5 new modules, 2 new GLM families, ~650 new tests.

• GLM families: Gamma and negative binomial regression (fit(X, y, family='gamma'), fit(X, y, family='negative.binomial'))
• Ordinal regression: Proportional odds model (polr(y, X) matching R MASS::polr)
• Multinomial regression: Softmax regression (multinom(y, X) matching R nnet::multinom)
• PCA / Factor analysis: pca(X) and factor_analysis(X, n_factors) with varimax/promax rotation
• Time series: Complete framework — ACF/PACF, stationarity tests (ADF, KPSS), ETS, ARIMA/SARIMA, auto.arima, decomposition (classical + STL)
• GAMs: Penalized regression splines (gam(y, smooths=[s('x1')], smooth_data={...}) matching R mgcv::gam)

Previous Releases

1.2.1 — No silent model switches; backend='gpu' is honest; reproducible Monte Carlo via seed=; module structure refactoring.

1.1 — Named coefficients via names=; result.coef dict; OLS/Cox summary improvements matching R output.

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Design Philosophy

PyStatistics maintains two parallel computational paths with distinct goals:

• CPU implementations aim for R-level reproducibility. CPU backends are validated against R reference implementations to near machine precision (rtol = 1e-10). When a CPU result disagrees with R, PyStatistics has a bug.

• GPU implementations prioritize modern numerical performance and scalability. GPU backends use FP32 arithmetic and algorithms optimized for throughput. They are validated against CPU backends, not directly against R.

• Divergence between CPU and GPU outputs may occur due to floating-point precision, algorithmic differences, or both. This is by design, not a defect. The section below specifies exactly how much divergence is acceptable.

Operating Principles

1. Correctness > Fidelity > Performance > Convenience
2. Fail fast, fail loud — no silent fallbacks or "helpful" defaults
3. Explicit over implicit — require parameters, don't assume intent
4. Two-tier validation — CPU vs R, then GPU vs CPU

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Statistical Equivalence: GPU vs CPU

GPU backends produce results in FP32 (single precision) while CPU backends use FP64 (double precision). This section defines exactly what "statistically equivalent" means and when it breaks down.

All tolerances below are relative (rtol) unless stated otherwise. They apply to well-conditioned problems (condition number < 10^6) at moderate scale (n < 1M, p < 1000). Degradation at larger scale or worse conditioning is documented below.

Tier 1: Parameter Estimates

Quantity	Tolerance	Notes
Coefficients / means	rtol <= 1e-3	Tightest at ~1e-4 for simple LM
Fitted values	rtol <= 1e-3	Directly derived from coefficients
GPU-CPU correlation	> 0.9999	Binding constraint at all scales

Tier 2: Uncertainty Estimates

Quantity	Tolerance	Notes
Standard errors	rtol <= 1e-2	Computed from (X'WX)^-1 which amplifies FP32 rounding
Covariance matrices (MLE)	rtol <= 5e-2	Hessian inversion is sensitive to precision

Standard errors are the weakest link in the GPU pipeline. They depend on the inverse of X'WX (or X'X for LM), which squares the condition number. A well-conditioned problem at FP64 can become a poorly-conditioned inversion at FP32.

Tier 3: Model Fit Statistics

Quantity	Tolerance	Notes
Deviance	rtol <= 1e-4	Scalar reduction — tightest GPU metric
Log-likelihood	abs <= 1.0	Absolute, not relative (log scale)
AIC / BIC values	rtol <= 1e-3	Derived from log-likelihood + rank
R-squared (LM)	rtol <= 1e-3	Ratio of reductions

Tier 4: Inference Decisions

Quantity	Guarantee	Notes
Model ranking under AIC/BIC	Identical	For models with AIC/BIC gap > 2
Rejection at alpha = 0.05	Identical	For p-values outside [0.01, 0.10]
Rejection at alpha = 0.05	Not guaranteed	For p-values in [0.01, 0.10] ("boundary zone")

The boundary zone exists because a ~1% relative difference in a test statistic near the critical value can flip a rejection decision. This is inherent to FP32, not a software defect. If a p-value falls in the boundary zone, use the CPU backend for the definitive answer.

When Guarantees Degrade

Large scale (n > 1M): FP32 accumulation over millions of rows introduces drift. Element-wise tolerance relaxes to rtol = 1e-2, but correlation remains > 0.9999. This means GPU coefficients track CPU coefficients nearly perfectly in direction, with small magnitude drift from accumulated rounding.

Ill-conditioned problems (condition number > 10^6): The GPU backend refuses by default and raises NumericalError. Passing force=True overrides this, but no numerical guarantees apply. Use the CPU backend for ill-conditioned problems.

Pathological missing data patterns (MLE): FP32 L-BFGS-B optimization can stall in near-flat regions of the likelihood surface. Means may deviate by up to rtol = 0.5 in extreme cases. The GPU backend will issue a convergence warning. Use the CPU backend for complex missingness patterns.

Why FP32?

Consumer GPUs (NVIDIA RTX series) execute FP32 at 5-10x the throughput of FP64. Apple Silicon GPUs (MPS) do not support FP64 at all. FP32 is the only path to practical GPU acceleration on hardware that researchers actually have. The tolerances above are the honest cost of that acceleration.

CUDA vs MPS: Not All GPU Backends Are Equal

Certain operations (notably scatter_add_ with sparse targets) are 1000x slower on Apple MPS than on NVIDIA CUDA due to Metal's weaker atomic memory support. PyStatistics detects these cases and either fails fast or routes to CPU. See docs/GPU_BACKEND_NOTES.md for detailed benchmarks and guidance on when GPU helps vs hurts.

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Quick Start

import numpy as np

# --- Descriptive statistics ---
from pystatistics.descriptive import describe, cor, quantile

data = np.random.randn(1000, 5)
result = describe(data)
print(result.mean, result.sd, result.skewness, result.kurtosis)

# Correlation (Pearson, Spearman, Kendall)
r = cor(data, method='spearman')
print(r.correlation_matrix)

# Quantiles (all 9 R types supported)
q = quantile(data, type=7)
print(q.quantiles)

# --- Hypothesis testing ---
from pystatistics.hypothesis import t_test, chisq_test, p_adjust

result = t_test([1,2,3,4,5], [3,4,5,6,7])
print(result.statistic, result.p_value, result.conf_int)
print(result.summary())  # R-style print.htest output

# Multiple testing correction
p_adjusted = p_adjust([0.01, 0.04, 0.03, 0.005], method='BH')

# --- Linear regression ---
from pystatistics.regression import fit

X = np.random.randn(1000, 5)
y = X @ [1, 2, 3, -1, 0.5] + np.random.randn(1000) * 0.1
result = fit(X, y, names=['x1', 'x2', 'x3', 'x4', 'x5'])
print(result.summary())          # R-style output with variable names
print(result.coef)                # {'x1': 1.00, 'x2': 2.00, ...}
print(result.coef['x3'])          # 3.00

# Logistic regression
y_binary = (X @ [1, -1, 0.5, 0, 0] + np.random.randn(1000) > 0).astype(float)
result = fit(X, y_binary, family='binomial')
print(result.summary())

# GPU acceleration (any model)
result = fit(X, y, backend='gpu')

# --- Monte Carlo methods ---
from pystatistics.montecarlo import boot, boot_ci, permutation_test

# Bootstrap for the mean
data = np.random.randn(100)
def mean_stat(data, indices):
    return np.array([np.mean(data[indices])])

result = boot(data, mean_stat, R=2000, seed=42)
print(result.t0, result.bias, result.se)

# Bootstrap confidence intervals (all 5 types)
ci_result = boot_ci(result, type='all')
print(ci_result.ci['perc'])  # percentile CI
print(ci_result.ci['bca'])   # BCa CI

# Permutation test
x = np.random.randn(30)
y = np.random.randn(30) + 1.0
def mean_diff(x, y): return np.mean(x) - np.mean(y)
result = permutation_test(x, y, mean_diff, R=9999, seed=42)
print(result.p_value, result.summary())

# --- Survival analysis ---
from pystatistics.survival import kaplan_meier, survdiff, coxph, discrete_time

time = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
event = np.array([1, 0, 1, 1, 0, 1, 1, 0, 1, 1])

# Kaplan-Meier survival curve
km = kaplan_meier(time, event)
print(km.survival, km.se, km.ci_lower, km.ci_upper)

# Log-rank test (compare groups)
group = np.array([0, 0, 0, 0, 0, 1, 1, 1, 1, 1])
lr = survdiff(time, event, group)
print(lr.statistic, lr.p_value, lr.summary())

# Cox proportional hazards (CPU only)
X = np.column_stack([np.random.randn(10)])
cox = coxph(time, event, X)
print(cox.coefficients, cox.hazard_ratios, cox.summary())

# Discrete-time survival (GPU-accelerated)
dt = discrete_time(time, event, X, backend='auto')
print(dt.coefficients, dt.hazard_ratios, dt.baseline_hazard)

# --- ANOVA ---
from pystatistics.anova import anova_oneway, anova, anova_rm, anova_posthoc, levene_test

# One-way ANOVA
y = np.concatenate([np.random.randn(20) + mu for mu in [0, 1, 3]])
group = np.array(['A']*20 + ['B']*20 + ['C']*20)
result = anova_oneway(y, group)
print(result.summary())          # R-style ANOVA table
print(result.eta_squared)        # effect sizes

# Post-hoc: Tukey HSD
posthoc = anova_posthoc(result, method='tukey')
print(posthoc.summary())         # pairwise comparisons with adjusted p-values

# Factorial ANOVA (Type II SS, matches R's car::Anova)
result = anova(y, {'treatment': tx, 'dose': dose}, ss_type=2)

# ANCOVA (continuous covariate)
result = anova(y, {'group': group}, covariates={'age': age}, ss_type=2)

# Repeated measures with sphericity correction
result = anova_rm(y, subject=subj, within={'condition': cond}, correction='auto')
print(result.sphericity[0].gg_epsilon)  # Greenhouse-Geisser correction

# Levene's test for homogeneity of variances
lev = levene_test(y, group, center='median')  # Brown-Forsythe variant
print(lev.f_value, lev.p_value)

# --- Mixed models ---
from pystatistics.mixed import lmm, glmm

# Random intercept model (matches R lme4::lmer + lmerTest)
result = lmm(y, X, groups={'subject': subject_ids})
print(result.summary())         # lmerTest-style output with Satterthwaite df
print(result.icc)               # intraclass correlation coefficient
print(result.ranef['subject'])  # BLUPs (conditional modes) per subject

# Random intercept + slope
result = lmm(y, X, groups={'subject': subject_ids},
             random_effects={'subject': ['1', 'time']},
             random_data={'time': time_array})

# Crossed random effects (subjects x items)
result = lmm(y, X, groups={'subject': subj_ids, 'item': item_ids})

# Model comparison via LRT (requires ML, not REML)
m1 = lmm(y, X_reduced, groups={'subject': subj_ids}, reml=False)
m2 = lmm(y, X_full, groups={'subject': subj_ids}, reml=False)
print(m1.compare(m2))  # LRT chi-squared, df, p-value

# GLMM — logistic with random intercept
result = glmm(y_binary, X, groups={'subject': subject_ids},
              family='binomial')
print(result.summary())

# GLMM — Poisson with random intercept
result = glmm(y_count, X, groups={'subject': subject_ids},
              family='poisson')

# --- Gamma GLM ---
from pystatistics.regression import fit

y_positive = np.abs(np.random.randn(200)) + 0.1
X = np.random.randn(200, 3)
result = fit(X, y_positive, family='gamma')
print(result.summary())

# --- Ordinal regression ---
from pystatistics.ordinal import polr

y_ordinal = np.random.choice([1, 2, 3, 4, 5], size=200)
X = np.random.randn(200, 3)
result = polr(y_ordinal, X)
print(result.coefficients, result.thresholds)
print(result.summary())

# --- Time series (ARIMA) ---
from pystatistics.timeseries import arima, auto_arima, acf

ts = np.cumsum(np.random.randn(200))  # random walk
acf_result = acf(ts, nlags=20)
result = arima(ts, order=(1, 1, 1))
print(result.coefficients, result.aic)
best = auto_arima(ts)
print(best.order, best.aic)

# --- GAM ---
from pystatistics.gam import gam, s

x = np.linspace(0, 2 * np.pi, 200)
y = np.sin(x) + np.random.randn(200) * 0.3
result = gam(y, smooths=[s('x1')], smooth_data={'x1': x})
print(result.edf, result.gcv)
print(result.summary())

Modules

Module	Status	Description
regression/ LM	Complete	Linear models (OLS) with CPU QR and GPU Cholesky
regression/ GLM	Complete	Generalized linear models (Gaussian, Binomial, Poisson, Gamma, Negative Binomial) via IRLS
mvnmle/	Complete	Multivariate normal MLE with missing data (Direct + EM)
descriptive/	Complete	Descriptive statistics, correlation, quantiles, skewness, kurtosis
hypothesis/	Complete	t-test, chi-squared, Fisher exact, Wilcoxon, KS, proportions, F-test, p.adjust
montecarlo/	Complete	Bootstrap (ordinary, balanced, parametric), permutation tests, 5 CI methods, batched GPU solver
survival/	Complete	Survival analysis: Kaplan-Meier, log-rank test, Cox PH (CPU), discrete-time (GPU)
anova/	Complete	ANOVA: one-way, factorial, ANCOVA, repeated measures, Type I/II/III SS, Tukey/Bonferroni/Dunnett, Levene's test
mixed/ LMM/GLMM	Complete	Linear and generalized linear mixed models (random intercepts/slopes, nested/crossed, REML/ML, Satterthwaite df, GLMM Laplace)
ordinal/	Complete	Proportional odds (cumulative link) models matching R MASS::polr
multinomial/	Complete	Multinomial logit (softmax) regression matching R nnet::multinom
multivariate/	Complete	PCA and maximum likelihood factor analysis with varimax/promax rotation
timeseries/	Complete	ACF, PACF, ADF, KPSS, ETS, ARIMA, SARIMA, auto_arima, decompose, STL
gam/	Complete	Generalized additive models with penalized regression splines matching R mgcv::gam

See docs/ROADMAP.md for detailed scope, GPU applicability, and implementation priority for each module.

Architecture

Every module follows the same pattern:

DataSource -> Design -> fit() -> Backend.solve() -> Result[Params] -> Solution

• CPU backends are the gold standard, validated against R to rtol = 1e-10.
• GPU backends are validated against CPU backends per the tolerances above.
• Two-tier validation ensures correctness at any scale: Python-CPU vs R, then Python-GPU vs Python-CPU.

Installation

pip install pystatistics

# With GPU support (requires PyTorch)
pip install pystatistics[gpu]

# Development
pip install pystatistics[dev]

License

MIT

Author

Hai-Shuo ([e-mail])