# Source code for pyro.ops.stats

```# Copyright (c) 2017-2019 Uber Technologies, Inc.

import math
import numbers

import torch
from torch.fft import irfft, rfft

from .tensor_utils import next_fast_len

def _compute_chain_variance_stats(input):
# compute within-chain variance and variance estimator
# input has shape N x C x sample_shape
N = input.size(0)
chain_var = input.var(dim=0)
var_within = chain_var.mean(dim=0)
var_estimator = (N - 1) / N * var_within
if input.size(1) > 1:
chain_mean = input.mean(dim=0)
var_between = chain_mean.var(dim=0)
var_estimator = var_estimator + var_between
else:
# to make rho_k is the same as autocorrelation when num_chains == 1
# in computing effective_sample_size
var_within = var_estimator
return var_within, var_estimator

[docs]def gelman_rubin(input, chain_dim=0, sample_dim=1):
"""
Computes R-hat over chains of samples. It is required that
``input.size(sample_dim) >= 2`` and ``input.size(chain_dim) >= 2``.

:param torch.Tensor input: the input tensor.
:param int chain_dim: the chain dimension.
:param int sample_dim: the sample dimension.
:returns torch.Tensor: R-hat of ``input``.
"""
assert input.dim() >= 2
assert input.size(sample_dim) >= 2
assert input.size(chain_dim) >= 2
# change input.shape to 1 x 1 x input.shape
# then transpose sample_dim with 0, chain_dim with 1
sample_dim = input.dim() + sample_dim if sample_dim < 0 else sample_dim
chain_dim = input.dim() + chain_dim if chain_dim < 0 else chain_dim
assert chain_dim != sample_dim
input = input.reshape((1, 1) + input.shape)
input = input.transpose(0, sample_dim + 2).transpose(1, chain_dim + 2)

var_within, var_estimator = _compute_chain_variance_stats(input)
rhat = (var_estimator / var_within).sqrt()
return rhat.squeeze(max(sample_dim, chain_dim)).squeeze(min(sample_dim, chain_dim))

[docs]def split_gelman_rubin(input, chain_dim=0, sample_dim=1):
"""
Computes R-hat over chains of samples. It is required that
``input.size(sample_dim) >= 4``.

:param torch.Tensor input: the input tensor.
:param int chain_dim: the chain dimension.
:param int sample_dim: the sample dimension.
:returns torch.Tensor: split R-hat of ``input``.
"""
assert input.dim() >= 2
assert input.size(sample_dim) >= 4
# change input.shape to 1 x 1 x input.shape
# then transpose chain_dim with 0, sample_dim with 1
sample_dim = input.dim() + sample_dim if sample_dim < 0 else sample_dim
chain_dim = input.dim() + chain_dim if chain_dim < 0 else chain_dim
assert chain_dim != sample_dim
input = input.reshape((1, 1) + input.shape)
input = input.transpose(0, chain_dim + 2).transpose(1, sample_dim + 2)

N_half = input.size(1) // 2
new_input = torch.stack([input[:, :N_half], input[:, -N_half:]], dim=1)
new_input = new_input.reshape((-1, N_half) + input.shape[2:])
split_rhat = gelman_rubin(new_input)
return split_rhat.squeeze(max(sample_dim, chain_dim)).squeeze(
min(sample_dim, chain_dim)
)

[docs]def autocorrelation(input, dim=0):
"""
Computes the autocorrelation of samples at dimension ``dim``.

Reference: https://en.wikipedia.org/wiki/Autocorrelation#Efficient_computation

:param torch.Tensor input: the input tensor.
:param int dim: the dimension to calculate autocorrelation.
:returns torch.Tensor: autocorrelation of ``input``.
"""
# https://github.com/stan-dev/math/blob/develop/stan/math/prim/mat/fun/autocorrelation.hpp
N = input.size(dim)
M = next_fast_len(N)
M2 = 2 * M

# transpose dim with -1 for Fourier transform
input = input.transpose(dim, -1)

centered_signal = input - input.mean(dim=-1, keepdim=True)

# Fourier transform
freqvec = torch.view_as_real(rfft(centered_signal, n=M2))
# take square of magnitude of freqvec (or freqvec x freqvec*)
freqvec_gram = freqvec.pow(2).sum(-1)
# inverse Fourier transform
autocorr = irfft(freqvec_gram, n=M2)

# truncate and normalize the result, setting autocorrelation to 1 for all
# constant channels
autocorr = autocorr[..., :N]
autocorr = autocorr / torch.tensor(
range(N, 0, -1), dtype=input.dtype, device=input.device
)
variance = autocorr[..., :1]
constant = (variance == 0).expand_as(autocorr)
autocorr = autocorr / variance.clamp(min=torch.finfo(variance.dtype).tiny)
autocorr[constant] = 1

# transpose back to original shape
return autocorr.transpose(dim, -1)

[docs]def autocovariance(input, dim=0):
"""
Computes the autocovariance of samples at dimension ``dim``.

:param torch.Tensor input: the input tensor.
:param int dim: the dimension to calculate autocorrelation.
:returns torch.Tensor: autocorrelation of ``input``.
"""
return autocorrelation(input, dim) * input.var(dim, unbiased=False, keepdim=True)

def _cummin(input):
"""
Computes cummulative minimum of input at dimension ``dim=0``.

:param torch.Tensor input: the input tensor.
:returns torch.Tensor: accumulate min of `input` at dimension `dim=0`.
"""
# FIXME: is there a better trick to find accumulate min of a sequence?
N = input.size(0)
input_tril = input.unsqueeze(0).repeat((N,) + (1,) * input.dim())
torch.ones(N, N, dtype=input.dtype, device=input.device)
.triu(diagonal=1)
.reshape((N, N) + (1,) * (input.dim() - 1))
)
return input_tril.min(dim=1)

[docs]def effective_sample_size(input, chain_dim=0, sample_dim=1):
"""
Computes effective sample size of input.

Reference:

 `Introduction to Markov Chain Monte Carlo`,
Charles J. Geyer

 `Stan Reference Manual version 2.18`,
Stan Development Team

:param torch.Tensor input: the input tensor.
:param int chain_dim: the chain dimension.
:param int sample_dim: the sample dimension.
:returns torch.Tensor: effective sample size of ``input``.
"""
assert input.dim() >= 2
assert input.size(sample_dim) >= 2
# change input.shape to 1 x 1 x input.shape
# then transpose sample_dim with 0, chain_dim with 1
sample_dim = input.dim() + sample_dim if sample_dim < 0 else sample_dim
chain_dim = input.dim() + chain_dim if chain_dim < 0 else chain_dim
assert chain_dim != sample_dim
input = input.reshape((1, 1) + input.shape)
input = input.transpose(0, sample_dim + 2).transpose(1, chain_dim + 2)

N, C = input.size(0), input.size(1)
# find autocovariance for each chain at lag k
gamma_k_c = autocovariance(input, dim=0)  # N x C x sample_shape

# find autocorrelation at lag k (from Stan reference)
var_within, var_estimator = _compute_chain_variance_stats(input)
rho_k = (var_estimator - var_within + gamma_k_c.mean(dim=1)) / var_estimator
rho_k = 1  # correlation at lag 0 is always 1

# initial positive sequence (formula 1.18 in ) applied for autocorrelation
Rho_k = rho_k if N % 2 == 0 else rho_k[:-1]
Rho_k = Rho_k.reshape((N // 2, 2) + Rho_k.shape[1:]).sum(dim=1)

# separate the first index
Rho_init = Rho_k

if Rho_k.size(0) > 1:
# Theoretically, Rho_k is positive, but due to noise of correlation computation,
# Rho_k might not be positive at some point. So we need to truncate (ignore first index).
Rho_positive = Rho_k[1:].clamp(min=0)

# Now we make the initial monotone (decreasing) sequence.
Rho_monotone = _cummin(Rho_positive)

# Formula 1.19 in 
tau = -1 + 2 * Rho_init + 2 * Rho_monotone.sum(dim=0)
else:
tau = -1 + 2 * Rho_init

n_eff = C * N / tau
return n_eff.squeeze(max(sample_dim, chain_dim)).squeeze(min(sample_dim, chain_dim))

[docs]def resample(input, num_samples, dim=0, replacement=False):
"""
Draws ``num_samples`` samples from ``input`` at dimension ``dim``.

:param torch.Tensor input: the input tensor.
:param int num_samples: the number of samples to draw from ``input``.
:param int dim: dimension to draw from ``input``.
:returns torch.Tensor: samples drawn randomly from ``input``.
"""
weights = torch.ones(input.size(dim), dtype=input.dtype, device=input.device)
indices = torch.multinomial(weights, num_samples, replacement)
return input.index_select(dim, indices)

[docs]def quantile(input, probs, dim=0):
"""
Computes quantiles of ``input`` at ``probs``. If ``probs`` is a scalar,
the output will be squeezed at ``dim``.

:param torch.Tensor input: the input tensor.
:param list probs: quantile positions.
:param int dim: dimension to take quantiles from ``input``.
:returns torch.Tensor: quantiles of ``input`` at ``probs``.
"""
if isinstance(probs, (numbers.Number, list, tuple)):
probs = torch.tensor(probs, dtype=input.dtype, device=input.device)
sorted_input = input.sort(dim)
max_index = input.size(dim) - 1
indices = probs * max_index
# because indices is float, we interpolate the quantiles linearly from nearby points
indices_below = indices.long()
indices_above = (indices_below + 1).clamp(max=max_index)
quantiles_above = sorted_input.index_select(dim, indices_above)
quantiles_below = sorted_input.index_select(dim, indices_below)
weights_above = indices - indices_below.type_as(indices)
weights_below = 1 - weights_above
quantiles = weights_below * quantiles_below + weights_above * quantiles_above
return quantiles if probs.shape != torch.Size([]) else quantiles.squeeze(dim)

[docs]def pi(input, prob, dim=0):
"""
Computes percentile interval which assigns equal probability mass
to each tail of the interval.

:param torch.Tensor input: the input tensor.
:param float prob: the probability mass of samples within the interval.
:param int dim: dimension to calculate percentile interval from ``input``.
:returns torch.Tensor: quantiles of ``input`` at ``probs``.
"""
return quantile(input, [(1 - prob) / 2, (1 + prob) / 2], dim)

[docs]def hpdi(input, prob, dim=0):
"""
Computes "highest posterior density interval" which is the narrowest
interval with probability mass ``prob``.

:param torch.Tensor input: the input tensor.
:param float prob: the probability mass of samples within the interval.
:param int dim: dimension to calculate percentile interval from ``input``.
:returns torch.Tensor: quantiles of ``input`` at ``probs``.
"""
sorted_input = input.sort(dim)
mass = input.size(dim)
index_length = int(prob * mass)
intervals_left = sorted_input.index_select(
dim,
torch.tensor(range(mass - index_length), dtype=torch.long, device=input.device),
)
intervals_right = sorted_input.index_select(
dim,
torch.tensor(range(index_length, mass), dtype=torch.long, device=input.device),
)
intervals_length = intervals_right - intervals_left
index_start = intervals_length.argmin(dim)
indices = torch.stack([index_start, index_start + index_length], dim)

def _weighted_mean(input, log_weights, dim=0, keepdim=False):
dim = input.dim() + dim if dim < 0 else dim
log_weights = log_weights.reshape([-1] + (input.dim() - dim - 1) * )
max_log_weight = log_weights.max(dim=0)
relative_probs = (log_weights - max_log_weight).exp()
return (input * relative_probs).sum(dim=dim, keepdim=keepdim) / relative_probs.sum()

def _weighted_variance(input, log_weights, dim=0, keepdim=False, unbiased=True):
# Ref: https://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Frequency_weights
deviation_squared = (
input - _weighted_mean(input, log_weights, dim, keepdim=True)
).pow(2)
correction = log_weights.size(0) / (log_weights.size(0) - 1.0) if unbiased else 1.0
return _weighted_mean(deviation_squared, log_weights, dim, keepdim) * correction

[docs]def waic(input, log_weights=None, pointwise=False, dim=0):
"""
Computes "Widely Applicable/Watanabe-Akaike Information Criterion" (WAIC) and
its corresponding effective number of parameters.

Reference:

 `WAIC and cross-validation in Stan`,
Aki Vehtari, Andrew Gelman

:param torch.Tensor input: the input tensor, which is log likelihood of a model.
:param torch.Tensor log_weights: weights of samples along ``dim``.
:param int dim: the sample dimension of ``input``.
:returns tuple: tuple of WAIC and effective number of parameters.
"""
if log_weights is None:
log_weights = torch.zeros(
input.size(dim), dtype=input.dtype, device=input.device
)

# computes log pointwise predictive density: formula (3) of 
dim = input.dim() + dim if dim < 0 else dim
weighted_input = input + log_weights.reshape([-1] + (input.dim() - dim - 1) * )
lpd = torch.logsumexp(weighted_input, dim=dim) - torch.logsumexp(log_weights, dim=0)

# computes the effective number of parameters: formula (6) of 
p_waic = _weighted_variance(input, log_weights, dim)

# computes expected log pointwise predictive density: formula (4) of 
elpd = lpd - p_waic
waic = -2 * elpd
return (waic, p_waic) if pointwise else (waic.sum(), p_waic.sum())

[docs]def fit_generalized_pareto(X):
"""
Given a dataset X assumed to be drawn from the Generalized Pareto
Distribution, estimate the distributional parameters k, sigma using a
variant of the technique described in reference , as described in
reference .

References
 'A new and efficient estimation method for the generalized Pareto distribution.'
Zhang, J. and Stephens, M.A. (2009).
 'Pareto Smoothed Importance Sampling.'
Aki Vehtari, Andrew Gelman, Jonah Gabry

:param torch.Tensor: the input data X
:returns tuple: tuple of floats (k, sigma) corresponding to the fit parameters
"""
if not isinstance(X, torch.Tensor) or X.dim() != 1:
raise ValueError("Input X must be a 1-dimensional torch tensor")

X = X.double()
X = torch.sort(X, descending=False)

N = X.size(0)
M = 30 + int(math.sqrt(N))

# b = k / sigma
bs = 1.0 - math.sqrt(M) / (torch.arange(1, M + 1, dtype=torch.double) - 0.5).sqrt()
bs /= 3.0 * X[int(N / 4 - 0.5)]
bs += 1 / X[-1]

ks = torch.log1p(-bs.unsqueeze(-1) * X).mean(-1)
Ls = N * (torch.log(-bs / ks) - (ks + 1.0))

weights = torch.exp(Ls - Ls.unsqueeze(-1))
weights = 1.0 / weights.sum(-1)

not_small_weights = weights > 1.0e-30
weights = weights[not_small_weights]
bs = bs[not_small_weights]
weights /= weights.sum()

b = (bs * weights).sum().item()
k = torch.log1p(-b * X).mean().item()
sigma = -k / b
k = k * N / (N + 10.0) + 5.0 / (N + 10.0)

return k, sigma

[docs]def crps_empirical(pred, truth):
"""
Computes negative Continuous Ranked Probability Score CRPS*  between a
set of samples ``pred`` and true data ``truth``. This uses an ``n log(n)``
time algorithm to compute a quantity equal that would naively have
complexity quadratic in the number of samples ``n``::

CRPS* = E|pred - truth| - 1/2 E|pred - pred'|
= (pred - truth).abs().mean(0)
- (pred - pred.unsqueeze(1)).abs().mean([0, 1]) / 2

Note that for a single sample this reduces to absolute error.

**References**

 Tilmann Gneiting, Adrian E. Raftery (2007)
`Strictly Proper Scoring Rules, Prediction, and Estimation`
https://www.stat.washington.edu/raftery/Research/PDF/Gneiting2007jasa.pdf

:param torch.Tensor pred: A set of sample predictions batched on rightmost dim.
This should have shape ``(num_samples,) + truth.shape``.
:param torch.Tensor truth: A tensor of true observations.
:return: A tensor of shape ``truth.shape``.
:rtype: torch.Tensor
"""
if pred.shape[1:] != (1,) * (pred.dim() - truth.dim() - 1) + truth.shape:
raise ValueError(
"Expected pred to have one extra sample dim on left. "
"Actual shapes: {} versus {}".format(pred.shape, truth.shape)
)
opts = dict(device=pred.device, dtype=pred.dtype)
num_samples = pred.size(0)
if num_samples == 1:
return (pred - truth).abs()

pred = pred.sort(dim=0).values
diff = pred[1:] - pred[:-1]
weight = torch.arange(1, num_samples, **opts) * torch.arange(
num_samples - 1, 0, -1, **opts
)
weight = weight.reshape(weight.shape + (1,) * (diff.dim() - 1))

return (pred - truth).abs().mean(0) - (diff * weight).sum(0) / num_samples**2
```