class LinearMixedModel(object):
r"""Class representing a linear mixed model.
.. include:: ../_templates/experimental.rst
:class:`LinearMixedModel` represents a linear model of the form
.. math::
y \sim \mathrm{N}(X \beta, \, \sigma^2 K + \tau^2 I)
where
- :math:`\mathrm{N}` is a :math:`n`-dimensional normal distribution.
- :math:`y` is a known vector of :math:`n` observations.
- :math:`X` is a known :math:`n \times p` design matrix for :math:`p` fixed effects.
- :math:`K` is a known :math:`n \times n` positive semi-definite kernel.
- :math:`I` is the :math:`n \times n` identity matrix.
- :math:`\beta` is a :math:`p`-parameter vector of fixed effects.
- :math:`\sigma^2` is the variance parameter on :math:`K`.
- :math:`\tau^2` is the variance parameter on :math:`I`.
In particular, the residuals for the :math:`i^\mathit{th}` and :math:`j^\mathit{th}`
observations have covariance :math:`\sigma^2 K_{ij}` for :math:`i \neq j`.
This model is equivalent to a
`mixed model <https://en.wikipedia.org/wiki/Mixed_model>`__
of the form
.. math::
y = X \beta + Z u + \epsilon
by setting :math:`K = ZZ^T` where
- :math:`Z` is a known :math:`n \times r` design matrix for :math:`r` random effects.
- :math:`u` is a :math:`r`-vector of random effects drawn from :math:`\mathrm{N}(0, \sigma^2 I)`.
- :math:`\epsilon` is a :math:`n`-vector of random errors drawn from :math:`\mathrm{N}(0, \tau^2 I)`.
However, :class:`LinearMixedModel` does not itself realize :math:`K` as a linear kernel
with respect to random effects, nor does it take :math:`K` explicitly as input. Rather,
via the eigendecomposion :math:`K = U S U^T`, the the class leverages a third, decorrelated
form of the model
.. math::
Py \sim \mathrm{N}(PX \beta, \, \sigma^2 (\gamma S + I))
where
- :math:`P = U^T: \mathbb{R}^n \rightarrow \mathbb{R}^n` is an orthonormal transformation
that decorrelates the observations. The rows of :math:`P` are an eigenbasis for :math:`K`.
- :math:`S` is the :math:`n \times n` diagonal matrix of corresponding eigenvalues.
- :math:`\gamma = \frac{\sigma^2}{\tau^2}` is the ratio of variance parameters.
Hence, the triple :math:`(Py, PX, S)` determines the probability
of the observations for any choice of model parameters, and is
therefore sufficient for inference.
This triple, with S encoded as a vector, is the default
("full-rank") initialization of the class.
:class:`LinearMixedModel` also provides an efficient strategy to fit the
model above with :math:`K` replaced by its rank-:math:`r` approximation
:math:`K_r = P_r^T S_r P_r` where
- :math:`P_r: \mathbb{R}^n \rightarrow \mathbb{R}^r` has orthonormal rows
consisting of the top :math:`r` eigenvectors of :math:`K`.
- :math:`S_r` is the :math:`r \times r` diagonal matrix of corresponding
non-zero eigenvalues.
For this low-rank model, the quintuple :math:`(P_r y, P_r X, S_r, y, X)`
is similarly sufficient for inference and corresponds to the "low-rank"
initialization of the class. Morally, :math:`y` and :math:`X` are
required for low-rank inference because the diagonal :math:`\gamma S + I`
is always full-rank.
If :math:`K` actually has rank :math:`r`, then :math:`K = K_r`
and the low-rank and full-rank models are equivalent.
Hence low-rank inference provides a more efficient, equally-exact
algorithm for fitting the full-rank model.
This situation arises, for example, when :math:`K` is the linear kernel
of a mixed model with fewer random effects than observations.
Even when :math:`K` has full rank, using a lower-rank approximation may
be an effective from of regularization, in addition to boosting
computational efficiency.
**Initialization**
The class may be initialized directly or with one of two methods:
- :meth:`from_kinship` takes :math:`y`, :math:`X`, and :math:`K` as ndarrays.
The model is always full-rank.
- :meth:`from_random_effects` takes :math:`y` and :math:`X` as ndarrays and
:math:`Z` as an ndarray or block matrix. The model is full-rank if and
only if :math:`n \leq m`.
Direct full-rank initialization takes :math:`Py`, :math:`PX`, and :math:`S`
as ndarrays. The following class attributes are set:
.. list-table::
:header-rows: 1
* - Attribute
- Type
- Value
* - `low_rank`
- bool
- ``False``
* - `n`
- int
- Number of observations :math:`n`
* - `f`
- int
- Number of fixed effects :math:`p`
* - `r`
- int
- Effective number of random effects, must equal :math:`n`
* - `py`
- ndarray
- Rotated response vector :math:`P y` with shape :math:`(n)`
* - `px`
- ndarray
- Rotated design matrix :math:`P X` with shape :math:`(n, p)`
* - `s`
- ndarray
- Eigenvalues vector :math:`S` of :math:`K` with shape :math:`(n)`
* - `p_path`
- str
- Path at which :math:`P` is stored as a block matrix
Direct low-rank initialization takes :math:`P_r y`, :math:`P_r X`, :math:`S_r`,
:math:`y`, and :math:`X` as ndarrays. The following class attributes are set:
.. list-table::
:header-rows: 1
* - Attribute
- Type
- Value
* - `low_rank`
- bool
- ``True``
* - `n`
- int
- Number of observations :math:`n`
* - `f`
- int
- Number of fixed effects :math:`p`
* - `r`
- int
- Effective number of random effects, must be less than :math:`n`
* - `py`
- ndarray
- Projected response vector :math:`P_r y` with shape :math:`(r)`
* - `px`
- ndarray
- Projected design matrix :math:`P_r X` with shape :math:`(r, p)`
* - `s`
- ndarray
- Eigenvalues vector :math:`S_r` of :math:`K_r` with shape :math:`(r)`
* - `y`
- ndarray
- Response vector with shape :math:`(n)`
* - `x`
- ndarray
- Design matrix with shape :math:`(n, p)`
* - `p_path`
- str
- Path at which :math:`P` is stored as a block matrix
**Fitting the model**
:meth:`fit` uses `restricted maximum likelihood
<https://en.wikipedia.org/wiki/Restricted_maximum_likelihood>`__ (REML)
to estimate :math:`(\beta, \sigma^2, \tau^2)`.
This is done by numerical optimization of the univariate function
:meth:`compute_neg_log_reml`, which itself optimizes REML constrained to a
fixed ratio of variance parameters. Each evaluation of
:meth:`compute_neg_log_reml` has computational complexity
.. math::
\mathit{O}(rp^2 + p^3).
:meth:`fit` adds the following attributes at this estimate.
.. list-table::
:header-rows: 1
* - Attribute
- Type
- Value
* - `beta`
- ndarray
- :math:`\beta`
* - `sigma_sq`
- float
- :math:`\sigma^2`
* - `tau_sq`
- float
- :math:`\tau^2`
* - `gamma`
- float
- :math:`\gamma = \frac{\sigma^2}{\tau^2}`
* - `log_gamma`
- float
- :math:`\log{\gamma}`
* - `h_sq`
- float
- :math:`\mathit{h}^2 = \frac{\sigma^2}{\sigma^2 + \tau^2}`
* - `h_sq_standard_error`
- float
- asymptotic estimate of :math:`\mathit{h}^2` standard error
**Testing alternative models**
The model is also equivalent to its augmentation
.. math::
y \sim \mathrm{N}\left(x_\star\beta_\star + X \beta, \, \sigma^2 K + \tau^2 I\right)
by an additional covariate of interest :math:`x_\star` under the
null hypothesis that the corresponding fixed effect parameter
:math:`\beta_\star` is zero. Similarly to initialization, full-rank testing
of the alternative hypothesis :math:`\beta_\star \neq 0` requires
:math:`P x_\star`, whereas the low-rank testing requires :math:`P_r x_\star`
and :math:`x_\star`.
After running :meth:`fit` to fit the null model, one can test each of a
collection of alternatives using either of two implementations of the
likelihood ratio test:
- :meth:`fit_alternatives_numpy` takes one or two ndarrays. It is a pure Python
method that evaluates alternatives serially on master.
- :meth:`fit_alternatives` takes one or two paths to block matrices. It
evaluates alternatives in parallel on the workers.
Per alternative, both have computational complexity
.. math::
\mathit{O}(rp + p^3).
Parameters
----------
py: :class:`ndarray`
Projected response vector :math:`P_r y` with shape :math:`(r)`.
px: :class:`ndarray`
Projected design matrix :math:`P_r X` with shape :math:`(r, p)`.
s: :class:`ndarray`
Eigenvalues vector :math:`S` with shape :math:`(r)`.
y: :class:`ndarray`, optional
Response vector with shape :math:`(n)`.
Include for low-rank inference.
x: :class:`ndarray`, optional
Design matrix with shape :math:`(n, p)`.
Include for low-rank inference.
p_path: :obj:`str`, optional
Path at which :math:`P` has been stored as a block matrix.
"""
@typecheck_method(py=np.ndarray,
px=np.ndarray,
s=np.ndarray,
y=nullable(np.ndarray),
x=nullable(np.ndarray),
p_path=nullable(str))
def __init__(self, py, px, s, y=None, x=None, p_path=None):
if y is None and x is None:
low_rank = False
elif y is not None and x is not None:
low_rank = True
else:
raise ValueError(
'for low-rank, set both y and x; for full-rank, do not set y or x.'
)
_check_dims(py, 'py', 1)
_check_dims(px, 'px', 2)
_check_dims(s, 's', 1)
r = s.size
f = px.shape[1]
if py.size != r:
raise ValueError("py and s must have the same size")
if px.shape[0] != r:
raise ValueError(
"px must have the same number of rows as the size of s")
if low_rank:
_check_dims(y, 'y', 1)
_check_dims(x, 'x', 2)
n = y.size
if n <= r:
raise ValueError("size of y must be larger than the size of s")
if x.shape[0] != n:
raise ValueError(
"x must have the same number of rows as the size of y")
if x.shape[1] != f:
raise ValueError("px and x must have the same number columns")
else:
n = r
if p_path is not None:
n_rows, n_cols = BlockMatrix.read(p_path).shape
if n_cols != n:
raise ValueError(
"LinearMixedModel: Number of columns in the block "
f"matrix at 'p_path' ({n_cols}) must equal "
f"the size of 'y' ({n})")
if n_rows != r:
raise ValueError(
"LinearMixedModel: Number of rows in the block "
f"matrix at 'p_path' ({n_rows}) must equal "
f"the size of 'py' ({r})")
self.low_rank = low_rank
self.n = n
self.f = f
self.r = r
self.py = py
self.px = px
self.s = s
self.y = y
self.x = x
self.p_path = p_path
self._check_dof()
self.beta = None
self.sigma_sq = None
self.tau_sq = None
self.gamma = None
self.log_gamma = None
self.h_sq = None
self.h_sq_standard_error = None
self.optimize_result = None
self._fitted = False
if low_rank:
self._yty = y @ y
self._xty = x.T @ y
self._xtx = x.T @ x
self._dof = n - f
self._d = None
self._ydy = None
self._xdy = None
self._xdx = None
self._dof_alt = n - (f + 1)
self._d_alt = None
self._ydy_alt = None
self._xdy_alt = np.zeros(f + 1)
self._xdx_alt = np.zeros((f + 1, f + 1))
self._residual_sq = None
self._scala_model = None
def _reset(self):
self._fitted = False
self.beta = None
self.sigma_sq = None
self.tau_sq = None
self.gamma = None
self.log_gamma = None
self.h_sq = None
self.h_sq_standard_error = None
self.optimize_result = None
def compute_neg_log_reml(self, log_gamma, return_parameters=False):
r"""Compute negative log REML constrained to a fixed value
of :math:`\log{\gamma}`.
This function computes the triple :math:`(\beta, \sigma^2, \tau^2)` with
:math:`\gamma = \frac{\sigma^2}{\tau^2}` at which the restricted
likelihood is maximized and returns the negative of the restricted log
likelihood at these parameters (shifted by the constant defined below).
The implementation has complexity :math:`\mathit{O}(rp^2 + p^3)` and is
inspired by `FaST linear mixed models for genome-wide association studies (2011)
<https://www.nature.com/articles/nmeth.1681>`__.
The formulae follow from `Bayesian Inference for Variance Components Using Only Error Contrasts (1974)
<http://faculty.dbmi.pitt.edu/day/Bioinf2132-advanced-Bayes-and-R/previousDocuments/Bioinf2132-documents-2016/2016-11-22/Harville-1974.pdf>`__.
Harville derives that for fixed covariance :math:`V`, the restricted
likelihood of the variance parameter :math:`V` in the model
.. math::
y \sim \mathrm{N}(X \beta, \, V)
is given by
.. math::
(2\pi)^{-\frac{1}{2}(n - p)}
\det(X^T X)^\frac{1}{2}
\det(V)^{-\frac{1}{2}}
\det(X^T V^{-1} X)^{-\frac{1}{2}}
e^{-\frac{1}{2}(y - X\hat\beta)^T V^{-1}(y - X\hat\beta)}.
with
.. math::
\hat\beta = (X^T V^{-1} X)^{-1} X^T V^{-1} y.
In our case, the variance is
.. math::
V = \sigma^2 K + \tau^2 I = \sigma^2 (K + \gamma^{-1} I)
which is determined up to scale by any fixed value of the ratio
:math:`\gamma`. So for input :math:`\log \gamma`, the
negative restricted log likelihood is minimized at
:math:`(\hat\beta, \hat\sigma^2)` with :math:`\hat\beta` as above and
.. math::
\hat\sigma^2 = \frac{1}{n - p}(y - X\hat\beta)^T (K + \gamma^{-1} I)^{-1}(y - X\hat\beta).
For :math:`\hat V` at this :math:`(\hat\beta, \hat\sigma^2, \gamma)`,
the exponent in the likelihood reduces to :math:`-\frac{1}{2}(n-p)`, so
the negative restricted log likelihood may be expressed as
.. math::
\frac{1}{2}\left(\log \det(\hat V) + \log\det(X^T \hat V^{-1} X)\right) + C
where
.. math::
C = \frac{1}{2}\left(n - p + (n - p)\log(2\pi) - \log\det(X^T X)\right)
only depends on :math:`X`. :meth:`compute_neg_log_reml` returns the value of
the first term, omitting the constant term.
Parameters
----------
log_gamma: :obj:`float`
Value of :math:`\log{\gamma}`.
return_parameters:
If ``True``, also return :math:`\beta`, :math:`\sigma^2`,
and :math:`\tau^2`.
Returns
-------
:obj:`float` or (:obj:`float`, :class:`ndarray`, :obj:`float`, :obj:`float`)
If `return_parameters` is ``False``, returns (shifted) negative log REML.
Otherwise, returns (shifted) negative log REML, :math:`\beta`, :math:`\sigma^2`,
and :math:`\tau^2`.
"""
from scipy.linalg import solve, LinAlgError
gamma = np.exp(log_gamma)
d = 1 / (self.s + 1 / gamma)
logdet_d = np.sum(np.log(d)) + (self.n - self.r) * log_gamma
if self.low_rank:
d -= gamma
dpy = d * self.py
ydy = self.py @ dpy + gamma * self._yty
xdy = self.px.T @ dpy + gamma * self._xty
xdx = (self.px.T * d) @ self.px + gamma * self._xtx
else:
dpy = d * self.py
ydy = self.py @ dpy
xdy = self.px.T @ dpy
xdx = (self.px.T * d) @ self.px
try:
beta = solve(xdx, xdy, assume_a='pos')
residual_sq = ydy - xdy.T @ beta
sigma_sq = residual_sq / self._dof
tau_sq = sigma_sq / gamma
neg_log_reml = (np.linalg.slogdet(xdx)[1] - logdet_d +
self._dof * np.log(sigma_sq)) / 2
self._d, self._ydy, self._xdy, self._xdx = d, ydy, xdy, xdx # used in fit
if return_parameters:
return neg_log_reml, beta, sigma_sq, tau_sq
else:
return neg_log_reml
except LinAlgError as e:
raise Exception(
'linear algebra error while solving for REML estimate') from e
@typecheck_method(log_gamma=nullable(numeric),
bounds=tupleof(numeric),
tol=float,
maxiter=int)
def fit(self, log_gamma=None, bounds=(-8.0, 8.0), tol=1e-8, maxiter=500):
r"""Find the triple :math:`(\beta, \sigma^2, \tau^2)` maximizing REML.
This method sets the attributes `beta`, `sigma_sq`, `tau_sq`, `gamma`,
`log_gamma`, `h_sq`, and `h_sq_standard_error` as described in the
top-level class documentation.
If `log_gamma` is provided, :meth:`fit` finds the REML solution
with :math:`\log{\gamma}` constrained to this value. In this case,
`h_sq_standard_error` is ``None`` since `h_sq` is not estimated.
Otherwise, :meth:`fit` searches for the value of :math:`\log{\gamma}`
that minimizes :meth:`compute_neg_log_reml`, and also sets the attribute
`optimize_result` of type `scipy.optimize.OptimizeResult
<https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html>`__.
Parameters
----------
log_gamma: :obj:`float`, optional
If provided, the solution is constrained to have this value of
:math:`\log{\gamma}`.
bounds: :obj:`float`, :obj:`float`
Lower and upper bounds for :math:`\log{\gamma}`.
tol: :obj:`float`
Absolute tolerance for optimizing :math:`\log{\gamma}`.
maxiter: :obj:`float`
Maximum number of iterations for optimizing :math:`\log{\gamma}`.
"""
if self._fitted:
self._reset()
fit_log_gamma = True if log_gamma is None else False
if fit_log_gamma:
from scipy.optimize import minimize_scalar
self.optimize_result = minimize_scalar(self.compute_neg_log_reml,
method='bounded',
bounds=bounds,
options={
'xatol': tol,
'maxiter': maxiter
})
if self.optimize_result.success:
if self.optimize_result.x - bounds[0] < 0.001:
raise Exception(
"failed to fit log_gamma: optimum within 0.001 of lower bound."
)
elif bounds[1] - self.optimize_result.x < 0.001:
raise Exception(
"failed to fit log_gamma: optimum within 0.001 of upper bound."
)
else:
self.log_gamma = self.optimize_result.x
else:
raise Exception(
f'failed to fit log_gamma:\n {self.optimize_result}')
else:
self.log_gamma = log_gamma
_, self.beta, self.sigma_sq, self.tau_sq = self.compute_neg_log_reml(
self.log_gamma, return_parameters=True)
self.gamma = np.exp(self.log_gamma)
self.h_sq = self.sigma_sq / (self.sigma_sq + self.tau_sq)
self._residual_sq = self.sigma_sq * self._dof
self._d_alt = self._d
self._ydy_alt = self._ydy
self._xdy_alt[1:] = self._xdy
self._xdx_alt[1:, 1:] = self._xdx
if fit_log_gamma:
self.h_sq_standard_error = self._estimate_h_sq_standard_error()
self._fitted = True
def _estimate_h_sq_standard_error(self):
epsilon = 1e-4 # parabolic interpolation radius in log_gamma space
lg = self.log_gamma + np.array([-epsilon, 0.0, epsilon])
h2 = 1 / (1 + np.exp(-lg))
nll = [self.compute_neg_log_reml(lgi) for lgi in lg]
if nll[1] > nll[0] or nll[1] > nll[2]:
i = 0 if nll[1] > nll[0] else 2
raise Exception(
f'Minimum of negative log likelihood fit as {nll[1]} at log_gamma={lg[1]},'
f'\n but found smaller value of {nll[i]} at log_gamma={lg[i]}.'
f'\n Investigate by plotting the negative log likelihood function.'
)
# Asymptotically near MLE, nLL = a * h2^2 + b * h2 + c with a = 1 / (2 * se^2)
# By Lagrange interpolation:
a = ((h2[2] * (nll[1] - nll[0]) + h2[1] * (nll[0] - nll[2]) + h2[0] *
(nll[2] - nll[1])) / ((h2[1] - h2[0]) * (h2[0] - h2[2]) *
(h2[2] - h2[1])))
return 1 / np.sqrt(2 * a)
def h_sq_normalized_lkhd(self):
r"""Estimate the normalized likelihood of :math:`\mathit{h}^2` over the
discrete grid of percentiles.
Examples
--------
Plot the estimated normalized likelihood function:
>>> import matplotlib.pyplot as plt # doctest: +SKIP
>>> plt.plot(range(101), model.h_sq_normalized_lkhd()) # doctest: +SKIP
Notes
-----
This method may be used to visualize the approximate posterior on
:math:`\mathit{h}^2` under a flat prior.
The resulting ndarray ``a`` has length 101 with ``a[i]`` equal to the
maximum likelihood over all :math:`\beta` and :math:`\sigma^2` with
:math:`\mathit{h}^2` constrained to ``i / 100``. The values for
``1 <= i <= 99`` are normalized to sum to 1, and ``a[0]`` and ``a[100]``
are set to ``nan``.
Returns
-------
:class:`ndarray` of :obj:`float64`
Normalized likelihood values for :math:`\mathit{h}^2`.
"""
log_lkhd = np.zeros(101, dtype=np.float64)
log_lkhd[0], log_lkhd[100] = np.nan, np.nan
for h2 in range(1, 100):
gamma = h2 / (100.0 - h2)
log_lkhd[h2] = -self.compute_neg_log_reml(np.log(gamma))
log_lkhd -= np.max(log_lkhd[1:-1])
lkhd = np.exp(log_lkhd)
lkhd /= np.sum(lkhd[1:-1])
return lkhd
@typecheck_method(pa_t_path=str,
a_t_path=nullable(str),
partition_size=nullable(int))
def fit_alternatives(self, pa_t_path, a_t_path=None, partition_size=None):
r"""Fit and test alternative model for each augmented design matrix in parallel.
Notes
-----
The alternative model is fit using REML constrained to the value of
:math:`\gamma` set by :meth:`fit`.
The likelihood ratio test of fixed effect parameter :math:`\beta_\star`
uses (non-restricted) maximum likelihood:
.. math::
\chi^2 = 2 \log\left(\frac{
\max_{\beta_\star, \beta, \sigma^2}\mathrm{N}
(y \, | \, x_\star \beta_\star + X \beta; \sigma^2(K + \gamma^{-1}I)}
{\max_{\beta, \sigma^2} \mathrm{N}
(y \, | \, x_\star \cdot 0 + X \beta; \sigma^2(K + \gamma^{-1}I)}
\right)
The p-value is given by the tail probability under a chi-squared
distribution with one degree of freedom.
The resulting table has the following fields:
.. list-table::
:header-rows: 1
* - Field
- Type
- Value
* - `idx`
- int64
- Index of augmented design matrix.
* - `beta`
- float64
- :math:`\beta_\star`
* - `sigma_sq`
- float64
- :math:`\sigma^2`
* - `chi_sq`
- float64
- :math:`\chi^2`
* - `p_value`
- float64
- p-value
:math:`(P_r A)^T` and :math:`A^T` (if given) must have the same number
of rows (augmentations). These rows are grouped into partitions for
parallel processing. The number of partitions equals the ceiling of
``n_rows / partition_size``, and should be at least the number or cores
to make use of all cores. By default, there is one partition per row of
blocks in :math:`(P_r A)^T`. Setting the partition size to an exact
(rather than approximate) divisor or multiple of the block size reduces
superfluous shuffling of data.
The number of columns in each block matrix must be less than :math:`2^{31}`.
Warning
-------
The block matrices must be stored in row-major format, as results
from :meth:`.BlockMatrix.write` with ``force_row_major=True`` and from
:meth:`.BlockMatrix.write_from_entry_expr`. Otherwise, this method
will produce an error message.
Parameters
----------
pa_t_path: :obj:`str`
Path to block matrix :math:`(P_r A)^T` with shape :math:`(m, r)`.
Each row is a projected augmentation :math:`P_r x_\star` of :math:`P_r X`.
a_t_path: :obj:`str`, optional
Path to block matrix :math:`A^T` with shape :math:`(m, n)`.
Each row is an augmentation :math:`x_\star` of :math:`X`.
Include for low-rank inference.
partition_size: :obj:`int`, optional
Number of rows to process per partition.
Default given by block size of :math:`(P_r A)^T`.
Returns
-------
:class:`.Table`
Table of results for each augmented design matrix.
"""
from hail.table import Table
self._check_dof(self.f + 1)
if self.low_rank and a_t_path is None:
raise ValueError('model is low-rank so a_t is required.')
elif not (self.low_rank or a_t_path is None):
raise ValueError('model is full-rank so a_t must not be set.')
if self._scala_model is None:
self._set_scala_model()
backend = Env.spark_backend('LinearMixedModel.fit_alternatives')
jfs = backend.fs._jfs
if partition_size is None:
block_size = Env.hail().linalg.BlockMatrix.readMetadata(
jfs, pa_t_path).blockSize()
partition_size = block_size
elif partition_size <= 0:
raise ValueError(
f'partition_size must be positive, found {partition_size}')
jpa_t = Env.hail().linalg.RowMatrix.readBlockMatrix(
jfs, pa_t_path, partition_size)
if a_t_path is None:
maybe_ja_t = None
else:
maybe_ja_t = Env.hail().linalg.RowMatrix.readBlockMatrix(
jfs, a_t_path, partition_size)
return Table._from_java(
backend._jbackend.pyFitLinearMixedModel(self._scala_model, jpa_t,
maybe_ja_t))
@typecheck_method(pa=np.ndarray,
a=nullable(np.ndarray),
return_pandas=bool)
def fit_alternatives_numpy(self, pa, a=None, return_pandas=False):
r"""Fit and test alternative model for each augmented design matrix.
Notes
-----
This Python-only implementation runs serially on master. See
the scalable implementation :meth:`fit_alternatives` for documentation
of the returned table.
Parameters
----------
pa: :class:`ndarray`
Projected matrix :math:`P_r A` of alternatives with shape :math:`(r, m)`.
Each column is a projected augmentation :math:`P_r x_\star` of :math:`P_r X`.
a: :class:`ndarray`, optional
Matrix :math:`A` of alternatives with shape :math:`(n, m)`.
Each column is an augmentation :math:`x_\star` of :math:`X`.
Required for low-rank inference.
return_pandas: :obj:`bool`
If true, return pandas dataframe. If false, return Hail table.
Returns
-------
:class:`.Table` or :class:`.pandas.DataFrame`
Table of results for each augmented design matrix.
"""
self._check_dof(self.f + 1)
if not self._fitted:
raise Exception("null model is not fit. Run 'fit' first.")
n_cols = pa.shape[1]
assert pa.shape[0] == self.r
if self.low_rank:
assert a.shape[0] == self.n and a.shape[1] == n_cols
data = [(i, ) + self._fit_alternative_numpy(pa[:, i], a[:, i])
for i in range(n_cols)]
else:
data = [(i, ) + self._fit_alternative_numpy(pa[:, i], None)
for i in range(n_cols)]
df = pd.DataFrame.from_records(
data, columns=['idx', 'beta', 'sigma_sq', 'chi_sq', 'p_value'])
if return_pandas:
return df
else:
return Table.from_pandas(df, key='idx')
def _fit_alternative_numpy(self, pa, a):
from scipy.linalg import solve, LinAlgError
from scipy.stats.distributions import chi2
gamma = self.gamma
dpa = self._d_alt * pa
# single thread => no need to copy
ydy = self._ydy_alt
xdy = self._xdy_alt
xdx = self._xdx_alt
if self.low_rank:
xdy[0] = self.py @ dpa + gamma * (self.y @ a)
xdx[0, 0] = pa @ dpa + gamma * (a @ a)
xdx[0, 1:] = self.px.T @ dpa + gamma * (self.x.T @ a)
else:
xdy[0] = self.py @ dpa
xdx[0, 0] = pa @ dpa
xdx[0, 1:] = self.px.T @ dpa
try:
beta = solve(xdx, xdy, assume_a='pos') # only uses upper triangle
residual_sq = ydy - xdy.T @ beta
sigma_sq = residual_sq / self._dof_alt
chi_sq = self.n * np.log(
self._residual_sq / residual_sq) # division => precision
p_value = chi2.sf(chi_sq, 1)
return beta[0], sigma_sq, chi_sq, p_value
except LinAlgError:
return tuple(4 * [float('nan')])
def _set_scala_model(self):
from hail.utils.java import Env
from hail.linalg import _jarray_from_ndarray, _breeze_from_ndarray
if not self._fitted:
raise Exception("null model is not fit. Run 'fit' first.")
self._scala_model = Env.hail().stats.LinearMixedModel.pyApply(
self.gamma, self._residual_sq, _jarray_from_ndarray(self.py),
_breeze_from_ndarray(self.px), _jarray_from_ndarray(self._d_alt),
self._ydy_alt, _jarray_from_ndarray(self._xdy_alt),
_breeze_from_ndarray(self._xdx_alt),
_jarray_from_ndarray(self.y) if self.low_rank else None,
_breeze_from_ndarray(self.x) if self.low_rank else None)
def _check_dof(self, f=None):
if f is None:
f = self.f
dof = self.n - f
if dof <= 0:
raise ValueError(
f"{self.n} {plural('observation', self.n)} with {f} fixed {plural('effect', f)} "
f"implies {dof} {plural('degree', dof)} of freedom. Must be positive."
)
@classmethod
@typecheck_method(y=np.ndarray,
x=np.ndarray,
k=np.ndarray,
p_path=nullable(str),
overwrite=bool)
def from_kinship(cls, y, x, k, p_path=None, overwrite=False):
r"""Initializes a model from :math:`y`, :math:`X`, and :math:`K`.
Examples
--------
>>> from hail.stats import LinearMixedModel
>>> y = np.array([0.0, 1.0, 8.0, 9.0])
>>> x = np.array([[1.0, 0.0],
... [1.0, 2.0],
... [1.0, 1.0],
... [1.0, 4.0]])
>>> k = np.array([[ 1. , -0.8727875 , 0.96397335, 0.94512946],
... [-0.8727875 , 1. , -0.93036112, -0.97320323],
... [ 0.96397335, -0.93036112, 1. , 0.98294169],
... [ 0.94512946, -0.97320323, 0.98294169, 1. ]])
>>> model, p = LinearMixedModel.from_kinship(y, x, k)
>>> model.fit()
>>> model.h_sq # doctest: +SKIP_OUTPUT_CHECK
0.2525148830695317
>>> model.s # doctest: +SKIP_OUTPUT_CHECK
array([3.83501295, 0.13540343, 0.02454114, 0.00504248])
Truncate to a rank :math:`r=2` model:
>>> r = 2
>>> s_r = model.s[:r]
>>> p_r = p[:r, :]
>>> model_r = LinearMixedModel(p_r @ y, p_r @ x, s_r, y, x)
>>> model.fit()
>>> model.h_sq # doctest: +SKIP_OUTPUT_CHECK
0.25193197591429695
Notes
-----
This method eigendecomposes :math:`K = P^T S P` on the master and
returns ``LinearMixedModel(p @ y, p @ x, s)`` and ``p``.
The performance of eigendecomposition depends critically on the
number of master cores and the NumPy / SciPy configuration, viewable
with ``np.show_config()``. For Intel machines, we recommend installing
the `MKL <https://anaconda.org/anaconda/mkl>`__ package for Anaconda.
`k` must be positive semi-definite; symmetry is not checked as only the
lower triangle is used.
Parameters
----------
y: :class:`ndarray`
:math:`n` vector of observations.
x: :class:`ndarray`
:math:`n \times p` matrix of fixed effects.
k: :class:`ndarray`
:math:`n \times n` positive semi-definite kernel :math:`K`.
p_path: :obj:`str`, optional
Path at which to write :math:`P` as a block matrix.
overwrite: :obj:`bool`
If ``True``, overwrite an existing file at `p_path`.
Returns
-------
model: :class:`LinearMixedModel`
Model constructed from :math:`y`, :math:`X`, and :math:`K`.
p: :class:`ndarray`
Matrix :math:`P` whose rows are the eigenvectors of :math:`K`.
"""
_check_dims(y, "y", 1)
_check_dims(x, "x", 2)
_check_dims(k, "k", 2)
n = k.shape[0]
if k.shape[1] != n:
raise ValueError("from_kinship: 'k' must be a square matrix")
if y.shape[0] != n:
raise ValueError("from_kinship: 'y' and 'k' must have the same "
"number of rows")
if x.shape[0] != n:
raise ValueError("from_kinship: 'x' and 'k' must have the same "
"number of rows")
s, u = hl.linalg._eigh(k)
if s[0] < -1e12 * s[-1]:
raise Exception("from_kinship: smallest eigenvalue of 'k' is"
f"negative: {s[0]}")
# flip singular values to descending order
s = np.flip(s, axis=0)
u = np.fliplr(u)
p = u.T
if p_path:
BlockMatrix.from_numpy(p).write(p_path, overwrite=overwrite)
model = LinearMixedModel(p @ y, p @ x, s, p_path=p_path)
return model, p
@classmethod
@typecheck_method(y=np.ndarray,
x=np.ndarray,
z=oneof(np.ndarray, hl.linalg.BlockMatrix),
p_path=nullable(str),
overwrite=bool,
max_condition_number=float,
complexity_bound=int)
def from_random_effects(cls,
y,
x,
z,
p_path=None,
overwrite=False,
max_condition_number=1e-10,
complexity_bound=8192):
r"""Initializes a model from :math:`y`, :math:`X`, and :math:`Z`.
Examples
--------
>>> from hail.stats import LinearMixedModel
>>> y = np.array([0.0, 1.0, 8.0, 9.0])
>>> x = np.array([[1.0, 0.0],
... [1.0, 2.0],
... [1.0, 1.0],
... [1.0, 4.0]])
>>> z = np.array([[0.0, 0.0, 1.0],
... [0.0, 1.0, 2.0],
... [1.0, 2.0, 4.0],
... [2.0, 4.0, 8.0]])
>>> model, p = LinearMixedModel.from_random_effects(y, x, z)
>>> model.fit()
>>> model.h_sq # doctest: +SKIP_OUTPUT_CHECK
0.38205307244271675
Notes
-----
If :math:`n \leq m`, the returned model is full rank.
If :math:`n > m`, the returned model is low rank. In this case only,
eigenvalues less than or equal to `max_condition_number` times the top
eigenvalue are dropped from :math:`S`, with the corresponding
eigenvectors dropped from :math:`P`. This guards against precision
loss on left eigenvectors computed via the right gramian :math:`Z^T Z`
in :meth:`BlockMatrix.svd`.
In either case, one can truncate to a rank :math:`r` model as follows.
If `p` is an ndarray:
>>> p_r = p[:r, :] # doctest: +SKIP
>>> s_r = model.s[:r] # doctest: +SKIP
>>> model_r = LinearMixedModel(p_r @ y, p_r @ x, s_r, y, x) # doctest: +SKIP
If `p` is a block matrix:
>>> p[:r, :].write(p_r_path) # doctest: +SKIP
>>> p_r = BlockMatrix.read(p_r_path) # doctest: +SKIP
>>> s_r = model.s[:r] # doctest: +SKIP
>>> model_r = LinearMixedModel(p_r @ y, p_r @ x, s_r, y, x, p_r_path) # doctest: +SKIP
This method applies no standardization to `z`.
Warning
-------
If `z` is a block matrix, then ideally `z` should be the result of
directly reading from disk (and possibly a transpose). This is most
critical if :math:`n > m`, because in this case multiplication by `z`
will result in all preceding transformations being repeated
``n / block_size`` times, as explained in :class:`.BlockMatrix`.
At least one dimension must be less than or equal to 46300.
See the warning in :meth:`.BlockMatrix.svd` for performance
considerations.
Parameters
----------
y: :class:`ndarray`
:math:`n` vector of observations :math:`y`.
x: :class:`ndarray`
:math:`n \times p` matrix of fixed effects :math:`X`.
z: :class:`ndarray` or :class:`BlockMatrix`
:math:`n \times m` matrix of random effects :math:`Z`.
p_path: :obj:`str`, optional
Path at which to write :math:`P` as a block matrix.
Required if `z` is a block matrix.
overwrite: :obj:`bool`
If ``True``, overwrite an existing file at `p_path`.
max_condition_number: :obj:`float`
Maximum condition number. Must be greater than 1e-16.
complexity_bound: :obj:`int`
Complexity bound for :meth:`.BlockMatrix.svd` when `z` is a block
matrix.
Returns
-------
model: :class:`LinearMixedModel`
Model constructed from :math:`y`, :math:`X`, and :math:`Z`.
p: :class:`ndarray` or :class:`.BlockMatrix`
Matrix :math:`P` whose rows are the eigenvectors of :math:`K`.
The type is block matrix if `z` is a block matrix and
:meth:`.BlockMatrix.svd` of `z` returns :math:`U` as a block matrix.
"""
z_is_bm = isinstance(z, BlockMatrix)
if z_is_bm and p_path is None:
raise ValueError("from_random_effects: 'p_path' required when 'z'"
"is a block matrix.")
if max_condition_number < 1e-16:
raise ValueError(
"from_random_effects: 'max_condition_number' must "
f"be at least 1e-16, found {max_condition_number}")
_check_dims(y, "y", 1)
_check_dims(x, "x", 2)
_check_dims(z, "z", 2)
n, m = z.shape
if y.shape[0] != n:
raise ValueError("from_random_effects: 'y' and 'z' must have the "
"same number of rows")
if x.shape[0] != n:
raise ValueError("from_random_effects: 'x' and 'z' must have the "
"same number of rows")
if z_is_bm:
u, s0, _ = z.svd(complexity_bound=complexity_bound)
p = u.T
p_is_bm = isinstance(p, BlockMatrix)
else:
u, s0, _ = hl.linalg._svd(z, full_matrices=False)
p = u.T
p_is_bm = False
s = s0**2
low_rank = n > m
if low_rank:
assert np.all(np.isfinite(s))
r = int(np.searchsorted(-s, -max_condition_number * s[0]))
if r < m:
info(
f'from_random_effects: model rank reduced from {m} to {r} '
f'due to ill-condition.'
f'\n Largest dropped eigenvalue was {s[r]}.')
s = s[:r]
p = p[:r, :]
if p_path is not None:
if p_is_bm:
p.write(p_path, overwrite=overwrite)
p = BlockMatrix.read(p_path)
else:
BlockMatrix.from_numpy(p).write(p_path, overwrite=overwrite)
if p_is_bm:
py, px = (p @ y.reshape(n, 1)).to_numpy().flatten(), (
p @ x).to_numpy()
else:
py, px = p @ y, p @ x
if low_rank:
model = LinearMixedModel(py, px, s, y, x, p_path)
else:
model = LinearMixedModel(py, px, s, p_path=p_path)
return model, p
# checks agreement of model initialization
def _same(self, other, tol=1e-6, up_to_sign=True):
def same_rows_up_to_sign(a, b, atol):
assert a.shape[0] == b.shape[0]
return all(
np.allclose(a[i], b[i], atol=atol)
or np.allclose(-a[i], b[i], atol=atol)
for i in range(a.shape[0]))
close = same_rows_up_to_sign if up_to_sign else np.allclose
if self.low_rank != other.low_rank:
print(f'different low_rank: {self.low_rank}, {other.low_rank}')
return False
same = True
if not close(self.py, other.py, atol=tol):
print(f'different py:\n{self.py}\n{other.py}')
same = False
if not close(self.px, other.px, atol=tol):
print(f'different px:\n{self.px}\n{other.px}')
same = False
if not np.allclose(self.s, other.s, atol=tol):
print(f'different s:\n{self.s}\n{other.s}')
same = False
if self.low_rank and not close(self.y, other.y, atol=tol):
print(f'different y:\n{self.y}\n{other.y}')
same = False
if self.low_rank and not close(self.x, other.x, atol=tol):
print(f'different x\n{self.x}\n{other.x}')
same = False
if self.p_path != other.p_path:
print(f'different p_path:\n{self.p_path}\n{other.p_path}')
same = False
return same
from functools import reduce
import hail as hl
from hail.expr.functions import _ndarray
from hail.expr.functions import array as aarray
from hail.expr.types import HailType, tfloat64, ttuple, tndarray
from hail.typecheck import typecheck, nullable, oneof, tupleof, sequenceof
from hail.expr.expressions import (expr_int32, expr_int64, expr_tuple,
expr_any, expr_array, expr_ndarray,
expr_numeric, Int64Expression, cast_expr,
construct_expr)
from hail.expr.expressions.typed_expressions import NDArrayNumericExpression
from hail.ir import NDArrayQR, NDArrayInv, NDArrayConcat, NDArraySVD, Apply
tsequenceof_nd = oneof(sequenceof(expr_ndarray()), expr_array(expr_ndarray()))
shape_type = oneof(expr_int64, tupleof(expr_int64), expr_tuple())
def array(input_array, dtype=None):
"""Construct an :class:`.NDArrayExpression`
Examples
--------
>>> hl.eval(hl.nd.array([1, 2, 3, 4]))
array([1, 2, 3, 4], dtype=int32)
>>> hl.eval(hl.nd.array([[1, 2, 3], [4, 5, 6]]))
array([[1, 2, 3],
[4, 5, 6]], dtype=int32)
from functools import reduce
import hail as hl
from hail.expr.functions import _ndarray
from hail.expr.functions import array as aarray
from hail.expr.types import HailType, tfloat64, ttuple, tndarray
from hail.typecheck import typecheck, nullable, oneof, tupleof, sequenceof
from hail.expr.expressions import (expr_int32, expr_int64, expr_tuple,
expr_any, expr_array, expr_ndarray,
expr_numeric, Int64Expression, cast_expr,
construct_expr)
from hail.expr.expressions.typed_expressions import NDArrayNumericExpression
from hail.ir import NDArrayQR, NDArrayInv, NDArrayConcat
tsequenceof_nd = oneof(sequenceof(expr_ndarray()), tupleof(expr_ndarray()),
expr_array(expr_ndarray()))
shape_type = oneof(expr_int64, tupleof(expr_int64), expr_tuple())
def array(input_array, dtype=None):
"""Construct an :class:`.NDArrayExpression`
Examples
--------
>>> hl.eval(hl.nd.array([1, 2, 3, 4]))
array([1, 2, 3, 4], dtype=int32)
>>> hl.eval(hl.nd.array([[1, 2, 3], [4, 5, 6]]))
array([[1, 2, 3],
[4, 5, 6]], dtype=int32)
class Function(object):
def __init__(self, f, param_types, ret_type, name, type_args=()):
self._f = f
self._name = name
self._type_args = type_args
self._param_types = param_types
self._ret_type = ret_type
def __call__(self, *args):
return self._f(*args)
@typecheck(f=anytype,
param_types=hail_type,
_name=nullable(str),
type_args=tupleof(hail_type))
def define_function(f, *param_types, _name=None, type_args=()):
mname = _name if _name is not None else Env.get_uid()
param_names = [Env.get_uid() for _ in param_types]
body = f(*(construct_expr(Ref(pn), pt)
for pn, pt in zip(param_names, param_types)))
ret_type = body.dtype
r = CSERenderer(stop_at_jir=True)
code = r(body._ir)
jbody = body._ir.parse(code,
ref_map=dict(zip(param_names, param_types)),
ir_map=r.jirs)
Env.hail().expr.ir.functions.IRFunctionRegistry.pyRegisterIR(
mname, [ta._parsable_string() for ta in type_args], param_names,
start : int or :class:`.Expression` of type :py:data:`.tint32`
Start of range.
stop : int or :class:`.Expression` of type :py:data:`.tint32`
End of range.
step : int or :class:`.Expression` of type :py:data:`.tint32`
Step of range.
Returns
-------
:class:`.NDArrayNumericExpression`
A 1-dimensional ndarray from `start` to `stop` by `step`.
"""
return array(hl.range(start, stop, step))
@typecheck(shape=oneof(expr_int64, tupleof(expr_int64), expr_tuple()),
value=expr_any,
dtype=nullable(HailType))
def full(shape, value, dtype=None):
"""Creates a hail :class:`.NDArrayNumericExpression` full of the specified value.
Examples
--------
Create a 5 by 7 NDArray of type :py:data:`.tfloat64` 9s.
>>> hl.nd.full((5, 7), 9)
It is possible to specify a type other than :py:data:`.tfloat64` with the `dtype` argument.
>>> hl.nd.full((5, 7), 9, dtype=hl.tint32)
