def log_pdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
norm = np.log(np.sqrt(2)) - np.log(self.scale * np.sqrt(np.pi))
p = norm - (X**2 / (2 * self.variance))
return np.where(X >= 0, p, 1)
def pdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
norm = np.sqrt(2) / (self.scale * np.sqrt(np.pi))
p = norm * np.exp(-X**2 / (2 * self.scale**2))
return np.where(X > 0, p, 0)
def partial_fit(self, X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# first fit
if not hasattr(self, '_n_samples'):
self._n_samples = 0
# Update center and variance
if self._empirical_variance is None:
self._n_samples += X.shape[0] - np.isnan(X).sum()
self._empirical_variance = np.nanvar(X)
else:
# previous values
prev_size = self._n_samples
prev_variance = self._empirical_variance
# new values
curr_size = X.shape[0] - np.isnan(X).sum()
curr_variance = np.nanvar(X)
# update size
self._n_samples = prev_size + curr_size
# update variance
self._empirical_variance = (
(prev_variance * prev_size) +
(curr_variance * curr_size)) / self._n_samples
norm = (1 - (2 / np.pi))
self.scale = _handle_zeros_in_scale(
np.sqrt(self._empirical_variance / norm))
return self
def quantile(self, *q):
"""Quantile Function
Also known as the inverse cumulative Distribution function, this function
takes known quantiles and returns the associated `X` value from the
support domain.
.. math::
\begin{cases}
0 &\text{if } 0 \leq q \lt p
1 &\text{if } p \leq q \lt 1
\end{cases}
Parameters
----------
q : numpy.ndarray, float
The probabilities within domain [0, 1]
Returns
-------
numpy.ndarray
The `X` values from the support domain associated with the input
quantiles.
"""
# check array for numpy structure
q = check_array(q, reduce_args=True, ensure_1d=True)
out = np.ceil(sc.bdtrik(q, 1, self.bias))
return np.where(self.bias >= q, 0, out)
def log_pdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
norm = 2 * self.variance
log_scale = np.log(self.scale) + np.log(np.sqrt(2 * np.pi))
return -((X - self.center)**2) / norm - log_scale
def partial_fit(self, X):
# check_array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True).astype(float)
# identify values outside of support
X[self.support.not_contains(X)] = np.nan
# first fit
if not hasattr(self, "_n_samples"):
self._n_samples = 0
if self._mean is None:
self._n_samples += X.shape[0] - np.isnan(X).sum()
self._mean = np.nanmean(X)
else:
# previous values
prev_size = self._n_samples
prev_mean = self._mean
# new values
curr_size = X.shape[0] - np.isnan(X).sum()
curr_mean = np.nanmean(X)
# update size
self._n_samples = prev_size + curr_size
# update mean
self._mean = ((prev_mean * prev_size) +
(curr_mean * curr_size)) / self._n_samples
self.bias = 1 / self._mean
return self
def log_pmf(self, *X):
"""Log Probability Mass Function
The probability mass function for the Bernoulli distribution is given
by two cases.
.. math::
\begin{cases}
1-p &\text{if } X = 0\\
p &\text{if } X = 1
\end{cases}
where `p` is the :code:`bias` in favor of a positive event
Parameters
----------
X : numpy.ndarray, int
1D dataset which falls within the domain of the given distribution
support. The Bernoulli distribution expects series of 0 or 1 only.
This value is often denoted `k` in the literature.
Returns
-------
numpy.ndarray
The output log transformed probability mass reported elementwise
with respect to the input data.
"""
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
return np.log(self.pmf(X))
def log_pdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
norm = sc.betaln(self.alpha, self.beta)
p = (self.alpha - 1) * np.log(X) + (self.beta - 1) * np.log(1 - X)
return p - norm
def partial_fit(self, X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# first fit
if not hasattr(self, '_n_samples'):
self._n_samples = 0
# Update rate
if self.rate is None:
self._n_samples += X.shape[0] - np.isnan(X).sum()
self.rate = np.nanmean(X)
else:
# previous values
prev_size = self._n_samples
prev_rate = self.rate
# new values
curr_size = X.shape[0] - np.isnan(X).sum()
curr_rate = np.nanmean(X)
# update size
self._n_samples = prev_size + curr_size
# update rate
self.rate = ((prev_rate * prev_size) +
(curr_rate * curr_size)) / self._n_samples
return self
def pdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
norm = sc.beta(self.alpha, self.beta)
p = np.power(X, self.alpha - 1) * np.power(1 - X, self.beta - 1)
return p / norm
def quantile(self, *q):
# check array for numpy structure
q = check_array(q, reduce_args=True, ensure_1d=True)
if self.high_inclusive:
return self.low + q * (self.high - self.low)
return self.low + q * ((self.high - 1) - self.low)
def log_cdf(self, X):
"""Log Cumulative Distribution Function
The cumulative distribution function for the Bernoulli distribution is
given by three cases.
.. math::
\begin{cases}
0 &\text{if } X \leq 0
1 - p &\text{if } 0 \leq X \lt 1\\
1 &\text{if } X \geq 1
\end{cases}
where `p` is the :code:`bias` in favor of a positive event
Parameters
----------
X : numpy.ndarray, int
1D dataset which falls within the domain of the given distribution
support. The Bernoulli distribution expects series of 0 or 1 only.
This value is often denoted `k` in the literature.
Returns
-------
numpy.ndarray
The output log transformed cumulative distribution reported elementwise
with respect to the input data.
"""
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
return np.log(self.cdf(X))
def quantile(self, *q):
# check array for numpy structure
q = check_array(q, reduce_args=True, ensure_1d=True)
vals = np.ceil(sc.pdtrik(q, self.rate))
vals1 = np.maximum(vals - 1, 0)
temp = sc.pdtr(vals1, self.rate)
return np.where(temp >= q, vals1, vals)
def log_pdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# alias parameters
a, b = self.shape, self.rate
return a * np.log(b) + (a - 1) * np.log(X) - b * X - sc.gammaln(a)
def cdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# floor X values
X = np.floor(X)
return sc.bdtr(X, self.n_trials, self.bias)
def cdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# alias parameters
a, b = self.shape, self.rate
return sc.gammainc(a, b * X)
def partial_fit(self, X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
super(ChiSquared, self).partial_fit(X)
self.dof = np.round(2 * self.shape).astype(int)
return self
def cdf(self, *X):
# check array for numpy structure
X = np.floor(
check_array(X, reduce_args=True, ensure_1d=True, dtype=int))
if self.high_inclusive:
return np.clip((X - self.low) / (self.high - self.low), 0, 1)
return np.clip((X - self.low) / ((self.high - 1) - self.low), 0, 1)
def cdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# floor X values
X = np.floor(X)
return sc.betainc(self.n_success, X + 1, self.bias)
def quantile(self, *q):
# check array for numpy structure
q = check_array(q, reduce_args=True, ensure_1d=True)
# alias parameters
a, b = self.shape, self.rate
return sc.gammaincinv(a, q) / b
def quantile(self, *q):
# check array for numpy structure
q = check_array(q, reduce_args=True, ensure_1d=True)
# compute quantile
out = np.ceil(np.log(1 - q) / np.log(1 - self.bias)) - 1
# return safely with bounds check
return np.where(self.support.contains(out), out, np.nan)
def pmf(self, *X):
# check array for numpy structure
# NOTE: feature_axis set to rows to ensure that *args that represent a
# single observtion will be the correct shape. Otherwise, users will
# *have* to pass correct shape for multiple observations (which does not
# effect the final shape)
X = check_array(X, reduce_args=True, atleast_2d=True, feature_axis=0)
return np.exp(self.log_pmf(X))
def log_pdf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
lb = self.low <= X
if self.high_inclusive:
ub = self.high >= X
else:
ub = self.high > X
return np.log(lb * ub) - np.log(self.high - self.low)
def partial_fit(self, X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# first fit
if not hasattr(self, "_n_samples"):
self._n_samples = 0
# update mean
if self.shape is None and self.rate is None:
self._n_samples += X.shape[0] - np.isnan(X).sum()
self._mean = np.nanmean(X)
self._log_mean = np.nanmean(np.log(X))
else:
# previous values
prev_size = self._n_samples
prev_mean = self._mean
prev_log_mean = self._log_mean
# current values
curr_size = X.shape[0] - np.isnan(X).sum()
curr_mean = np.nanmean(X)
curr_log_mean = np.nanmean(np.log(X))
# update size
self._n_samples = prev_size + curr_size
# update mean
self._mean = ((prev_mean * prev_size) +
(curr_mean * curr_size)) / self._n_samples
# update log-mean
self._log_mean = ((prev_log_mean * prev_size) +
(curr_log_mean * curr_size)) / self._n_samples
# solving for shape parameter has no analytical closed form solution
# however shape is numerically well behaved and can be computed with
# some level of numerical stability. Below we estimate parameter `s`
# which aids in the estimation of shape parameter `k`.
s = np.log(self._mean) - self._log_mean
k = (3 - s + np.sqrt((s - 3) * (s - 3) + 24 * s)) / (12 * s)
# this estimation of k is within 1.5% of correct value updated with
# explicit form of Newton-Raphson
k -= (np.log(k) - sc.psi(k) - s) / ((1 / k) - sc.psi(k))
# solve for theta (theta = 1 / self.rate)
theta = self._mean / k
# update parameters
self.shape = k
self.rate = 1 / theta
return self
def log_pmf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True, dtype=int)
lb = self.low <= X
if self.high_inclusive:
ub = self.high >= X
nrange = self.high - self.low
else:
ub = self.high > X
nrange = (self.high - 1) - self.low
return np.log(lb * ub) - np.log(nrange)
def log_pmf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# floor values of X
X = np.floor(X)
# alias n_success
k = self.n_success
# expand all components of log-pmf
(k + X - 1, k)
out = (sc.gammaln(k + X) - (sc.gammaln(k + 1) + sc.gammaln(X)) +
X * nanlog(1 - self.bias) + k * nanlog(self.bias))
return out
def log_pmf(self, *X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# Floor values of X
X = np.floor(X)
# Expand all components of log-pmf
out = (
sc.gammaln(self.n_trials + 1)
- (sc.gammaln(X + 1) + sc.gammaln(self.n_trials - X + 1))
+ sc.xlogy(X, self.bias)
+ sc.xlog1py(self.n_trials - X, -self.bias)
)
return out
def partial_fit(self, X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True).astype(float)
# identify values outside of support
# NOTE: we don't know the "true" upper bounds so we only
# check that values are positive
invalid = X < 0
X[invalid] = np.nan
# first fit
if not hasattr(self, "_n_samples"):
self._n_samples = 0
if self._mean is None and self._variance is None:
self._n_samples += X.shape[0] - np.isnan(X).sum()
self._mean = np.nanmean(X)
self._variance = np.nanvar(X)
else:
# previous values
prev_size = self._n_samples
prev_mean = self._mean
prev_variance = self._variance
# new values
curr_size = X.shape[0] - np.isnan(X).sum()
curr_mean = np.nanmean(X)
curr_variance = np.nanvar(X)
# update size
self._n_samples = prev_size + curr_size
# update mean
self._mean = (
(prev_mean * prev_size) + (curr_mean * curr_size)
) / self._n_samples
# update variance
self._variance = (
(prev_variance * prev_size) + (curr_variance * curr_size)
) / self._n_samples
self.bias = 1 - (self._variance / self._mean)
self.n_trials = np.round(self._mean / self.bias).astype(int)
return self
def quantile(self, *q):
# check array for numpy structure
q = check_array(q, reduce_args=True, ensure_1d=True)
# get the upper value of X (ceiling)
X_up = np.ceil(sc.nbdtrik(q, self.n_success, self.bias))
# get the lower value of X (floor)
X_down = np.maximum(X_up - 1, 0)
# recompute quantiles to validate transformation
q_test = sc.nbdtr(X_down, self.n_success, self.bias)
# when q_test is greater than true, shift output down
out = np.where(q_test >= q, X_down, X_up).astype(int)
# return only in-bound values
return np.where(self.support.contains(out), out, np.nan)
def partial_fit(self, X):
# check array for numpy structure
X = check_array(X, reduce_args=True, ensure_1d=True)
# First fit
if self.low is None and self.high is None:
self.low = np.nanmin(X)
self.high = np.nanmax(X)
else:
# Update distribution support
curr_low, curr_high = np.nanmin(X), np.nanmax(X)
if curr_low < self.low:
self.low = curr_low
if curr_high > self.high:
self.high = curr_high
return self
