# -*- coding: utf-8 -*-
from __future__ import division
from statsmodels.base.model import Model
from .factor_rotation import rotate_factors, promax
import numpy as np
from numpy.linalg import eigh, inv, norm, matrix_rank
import pandas as pd
from statsmodels.tools.decorators import cache_readonly
from scipy.optimize import minimize
from statsmodels.iolib import summary2
import warnings
try:
import matplotlib.pyplot
missing_matplotlib = False
except ImportError:
missing_matplotlib = True
if not missing_matplotlib:
from .plots import plot_scree, plot_loadings
_opt_defaults = {'gtol': 1e-7}
def _check_args_1(endog, n_factor, corr, nobs):
msg = "Either endog or corr must be provided."
if endog is not None and corr is not None:
raise ValueError(msg)
if endog is None and corr is None:
warnings.warn('Both endog and corr are provided, ' +
'corr will be used for factor analysis.')
if n_factor <= 0:
raise ValueError('n_factor must be larger than 0! %d < 0' %
(n_factor))
if nobs is not None and endog is not None:
warnings.warn("nobs is ignored when endog is provided")
def _check_args_2(endog, n_factor, corr, nobs, k_endog):
if n_factor > k_endog:
raise ValueError('n_factor cannot be greater than the number'
' of variables! %d > %d' %
(n_factor, k_endog))
if np.max(np.abs(np.diag(corr) - 1)) > 1e-10:
raise ValueError("corr must be a correlation matrix")
if corr.shape[0] != corr.shape[1]:
raise ValueError('Correlation matrix corr must be a square '
'(rows %d != cols %d)' % corr.shape)
[docs]class Factor(Model):
"""
Factor analysis
Parameters
----------
endog : array-like
Variables in columns, observations in rows. May be `None` if
`corr` is not `None`.
n_factor : int
The number of factors to extract
corr : array-like
Directly specify the correlation matrix instead of estimating
it from `endog`. If provided, `endog` is not used for the
factor analysis, it may be used in post-estimation.
method : str
The method to extract factors, currently must be either 'pa'
for principal axis factor analysis or 'ml' for maximum
likelihood estimation.
smc : True or False
Whether or not to apply squared multiple correlations (method='pa')
endog_names: str
Names of endogeous variables. If specified, it will be used
instead of the column names in endog
nobs : int
The number of observations, not used if endog is present. Needs to
be provided for inference if endog is None.
missing : 'none', 'drop', or 'raise'
Missing value handling for endog, default is row-wise deletion 'drop'
If 'none', no nan checking is done. If 'drop', any observations with
nans are dropped. If 'raise', an error is raised.
Notes
-----
**Experimental**
Supported rotations: 'varimax', 'quartimax', 'biquartimax',
'equamax', 'oblimin', 'parsimax', 'parsimony', 'biquartimin',
'promax'
If method='ml', the factors are rotated to satisfy condition IC3
of Bai and Li (2012). This means that the scores have covariance
I, so the model for the covariance matrix is L * L' + diag(U),
where L are the loadings and U are the uniquenesses. In addition,
L' * diag(U)^{-1} L must be diagonal.
References
----------
.. [*] Hofacker, C. (2004). Exploratory Factor Analysis, Mathematical
Marketing. http://www.openaccesstexts.org/pdf/Quant_Chapter_11_efa.pdf
.. [*] J Bai, K Li (2012). Statistical analysis of factor models of high
dimension. Annals of Statistics. https://arxiv.org/pdf/1205.6617.pdf
"""
def __init__(self, endog=None, n_factor=1, corr=None, method='pa',
smc=True, endog_names=None, nobs=None, missing='drop'):
_check_args_1(endog, n_factor, corr, nobs)
if endog is not None:
super(Factor, self).__init__(endog, exog=None, missing=missing)
endog = self.endog # after preprocessing like missing, asarray
k_endog = endog.shape[1]
nobs = endog.shape[0]
corr = self.corr = np.corrcoef(endog, rowvar=0)
elif corr is not None:
corr = self.corr = np.asarray(corr)
k_endog = self.corr.shape[0]
self.endog = None
else:
msg = "Either endog or corr must be provided."
raise ValueError(msg)
_check_args_2(endog, n_factor, corr, nobs, k_endog)
self.n_factor = n_factor
self.loadings = None
self.communality = None
self.method = method
self.smc = smc
self.nobs = nobs
self.method = method
self.corr = corr
self.k_endog = k_endog
if endog_names is None:
if hasattr(corr, 'index'):
endog_names = corr.index
if hasattr(corr, 'columns'):
endog_names = corr.columns
self.endog_names = endog_names
@property
def endog_names(self):
"""Names of endogenous variables"""
if self._endog_names is not None:
return self._endog_names
else:
if self.endog is not None:
return self.data.ynames
else:
d = 0
n = self.corr.shape[0] - 1
while n > 0:
d += 1
n //= 10
return [('var%0' + str(d) + 'd') % i
for i in range(self.corr.shape[0])]
@endog_names.setter
def endog_names(self, value):
# Check validity of endog_names:
if value is not None:
if len(value) != self.corr.shape[0]:
raise ValueError('The length of `endog_names` must '
'equal the number of variables.')
self._endog_names = np.asarray(value)
else:
self._endog_names = None
[docs] def fit(self, maxiter=50, tol=1e-8, start=None, opt_method='BFGS',
opt=None, em_iter=3):
"""
Estimate factor model parameters.
Parameters
----------
maxiter : int
Maximum number of iterations for iterative estimation algorithms
tol : float
Stopping critera (error tolerance) for iterative estimation
algorithms
start : array-like
Starting values, currently only used for ML estimation
opt_method : string
Optimization method for ML estimation
opt : dict-like
Keyword arguments passed to optimizer, only used for ML estimation
em_iter : int
The number of EM iterations before starting gradient optimization,
only used for ML estimation.
Returns
-------
results: FactorResults
"""
method = self.method.lower()
if method == 'pa':
return self._fit_pa(maxiter=maxiter, tol=tol)
elif method == 'ml':
return self._fit_ml(start, em_iter, opt_method, opt)
else:
msg = "Unknown factor extraction approach '%s'" % self.method
raise ValueError(msg)
def _fit_pa(self, maxiter=50, tol=1e-8):
"""
Extract factors using the iterative principal axis method
Parameters
----------
maxiter : int
Maximum number of iterations for communality estimation
tol : float
If `norm(communality - last_communality) < tolerance`,
estimation stops
Returns
-------
results : FactorResults instance
"""
R = self.corr.copy() # inplace modification below
# Parameter validation
self.n_comp = matrix_rank(R)
if self.n_factor > self.n_comp:
raise ValueError('n_factor must be smaller or equal to the rank'
' of endog! %d > %d' %
(self.n_factor, self.n_comp))
if maxiter <= 0:
raise ValueError('n_max_iter must be larger than 0! %d < 0' %
(maxiter))
if tol <= 0 or tol > 0.01:
raise ValueError('tolerance must be larger than 0 and smaller than'
' 0.01! Got %f instead' % (tol))
# Initial communality estimation
if self.smc:
c = 1 - 1 / np.diag(inv(R))
else:
c = np.ones(len(R))
# Iterative communality estimation
eigenvals = None
for i in range(maxiter):
# Get eigenvalues/eigenvectors of R with diag replaced by
# communality
for j in range(len(R)):
R[j, j] = c[j]
L, V = eigh(R, UPLO='U')
c_last = np.array(c)
ind = np.argsort(L)
ind = ind[::-1]
L = L[ind]
n_pos = (L > 0).sum()
V = V[:, ind]
eigenvals = np.array(L)
# Select eigenvectors with positive eigenvalues
n = np.min([n_pos, self.n_factor])
sL = np.diag(np.sqrt(L[:n]))
V = V[:, :n]
# Calculate new loadings and communality
A = V.dot(sL)
c = np.power(A, 2).sum(axis=1)
if norm(c_last - c) < tol:
break
self.eigenvals = eigenvals
self.communality = c
self.uniqueness = 1 - c
self.loadings = A
return FactorResults(self)
# Unpacks the model parameters from a flat vector, used for ML
# estimation. The first k_endog elements of par are the square
# roots of the uniquenesses. The remaining elements are the
# factor loadings, packed one factor at a time.
def _unpack(self, par):
return (par[0:self.k_endog]**2,
np.reshape(par[self.k_endog:], (-1, self.k_endog)).T)
# Packs the model parameters into a flat parameter, used for ML
# estimation.
def _pack(self, load, uniq):
return np.concatenate((np.sqrt(uniq), load.T.flat))
[docs] def loglike(self, par):
"""
Evaluate the log-likelihood function.
Parameters
----------
par : ndarray or tuple of 2 ndarray's
The model parameters, either a packed representation of
the model parameters or a 2-tuple containing a `k_endog x
n_factor` matrix of factor loadings and a `k_endog` vector
of uniquenesses.
Returns
-------
loglike : float
"""
if type(par) is np.ndarray:
uniq, load = self._unpack(par)
else:
load, uniq = par[0], par[1]
loadu = load / uniq[:, None]
lul = np.dot(load.T, loadu)
# log|GG' + S|
# Using matrix determinant lemma:
# |GG' + S| = |I + G'S^{-1}G|*|S|
lul.flat[::lul.shape[0]+1] += 1
_, ld = np.linalg.slogdet(lul)
v = np.sum(np.log(uniq)) + ld
# tr((GG' + S)^{-1}C)
# Using Sherman-Morrison-Woodbury
w = np.sum(1 / uniq)
b = np.dot(load.T, self.corr / uniq[:, None])
b = np.linalg.solve(lul, b)
b = np.dot(loadu, b)
w -= np.trace(b)
# Scaled log-likelihood
return -(v + w) / (2*self.k_endog)
[docs] def score(self, par):
"""
Evaluate the score function (first derivative of loglike).
Parameters
----------
par : ndarray or tuple of 2 ndarray's
The model parameters, either a packed representation of
the model parameters or a 2-tuple containing a `k_endog x
n_factor` matrix of factor loadings and a `k_endog` vector
of uniquenesses.
Returns
-------
score : ndarray
"""
if type(par) is np.ndarray:
uniq, load = self._unpack(par)
else:
load, uniq = par[0], par[1]
# Center term of SMW
loadu = load / uniq[:, None]
c = np.dot(load.T, loadu)
c.flat[::c.shape[0]+1] += 1
d = np.linalg.solve(c, load.T)
# Precompute these terms
lud = np.dot(loadu, d)
cu = (self.corr / uniq) / uniq[:, None]
r = np.dot(cu, load)
lul = np.dot(lud.T, load)
luz = np.dot(cu, lul)
# First term
du = 2*np.sqrt(uniq) * (1/uniq - (d * load.T).sum(0) / uniq**2)
dl = 2*(loadu - np.dot(lud, loadu))
# Second term
h = np.dot(lud, cu)
f = np.dot(h, lud.T)
du -= 2*np.sqrt(uniq) * (np.diag(cu) - 2*np.diag(h) + np.diag(f))
dl -= 2*r
dl += 2*np.dot(lud, r)
dl += 2*luz
dl -= 2*np.dot(lud, luz)
# Can't use _pack because we are working with the square root
# uniquenesses directly.
return -np.concatenate((du, dl.T.flat)) / (2*self.k_endog)
# Maximum likelihood factor analysis.
def _fit_ml(self, start, em_iter, opt_method, opt):
"""estimate Factor model using Maximum Likelihood
"""
# Starting values
if start is None:
load, uniq = self._fit_ml_em(em_iter)
start = self._pack(load, uniq)
elif len(start) == 2:
if len(start[1]) != start[0].shape[0]:
msg = "Starting values have incompatible dimensions"
raise ValueError(msg)
start = self._pack(start[0], start[1])
else:
raise ValueError("Invalid starting values")
def nloglike(par):
return -self.loglike(par)
def nscore(par):
return -self.score(par)
# Do the optimization
if opt is None:
opt = _opt_defaults
r = minimize(nloglike, start, jac=nscore, method=opt_method,
options=opt)
if not r.success:
warnings.warn("Fitting did not converge")
par = r.x
uniq, load = self._unpack(par)
if uniq.min() < 1e-10:
warnings.warn("Some uniquenesses are nearly zero")
# Rotate solution to satisfy IC3 of Bai and Li
load = self._rotate(load, uniq)
self.uniqueness = uniq
self.communality = 1 - uniq
self.loadings = load
self.mle_retvals = r
return FactorResults(self)
def _fit_ml_em(self, iter):
"""estimate Factor model using EM algorithm
"""
# Starting values
np.random.seed(3427)
load = 0.1*np.random.normal(size=(self.k_endog, self.n_factor))
uniq = 0.5 * np.ones(self.k_endog)
for k in range(iter):
loadu = load / uniq[:, None]
f = np.dot(load.T, loadu)
f.flat[::f.shape[0]+1] += 1
r = np.linalg.solve(f, loadu.T)
q = np.dot(loadu.T, load)
h = np.dot(r, load)
c = load - np.dot(load, h)
c /= uniq[:, None]
g = np.dot(q, r)
e = np.dot(g, self.corr)
d = np.dot(loadu.T, self.corr) - e
a = np.dot(d, c)
a -= np.dot(load.T, c)
a.flat[::a.shape[0]+1] += 1
b = np.dot(self.corr, c)
load = np.linalg.solve(a, b.T).T
uniq = np.diag(self.corr) - (load * d.T).sum(1)
return load, uniq
def _rotate(self, load, uniq):
"""rotate loadings for MLE
"""
# Rotations used in ML estimation.
load, s, _ = np.linalg.svd(load, 0)
load *= s
if self.nobs is None:
nobs = 1
else:
nobs = self.nobs
cm = np.dot(load.T, load / uniq[:, None]) / nobs
_, f = np.linalg.eig(cm)
load = np.dot(load, f)
return load
[docs]class FactorResults(object):
"""
Factor results class
For result summary, scree/loading plots and factor rotations
Parameters
----------
factor : Factor
Fitted Factor class
Attributes
----------
uniqueness: ndarray
The uniqueness (variance of uncorrelated errors unique to
each variable)
communality: ndarray
1 - uniqueness
loadings : ndarray
Each column is the loading vector for one factor
loadings_no_rot : ndarray
Unrotated loadings, not available under maximum likelihood
analyis.
eigenvalues : ndarray
The eigenvalues for a factor analysis obtained using
principal components; not available under ML estimation.
n_comp : int
Number of components (factors)
nbs : int
Number of observations
fa_method : string
The method used to obtain the decomposition, either 'pa' for
'principal axes' or 'ml' for maximum likelihood.
df : int
Degrees of freedom of the factor model.
Notes
-----
Under ML estimation, the default rotation (used for `loadings`) is
condition IC3 of Bai and Li (2012). Under this rotation, the
factor scores are iid and standardized. If `G` is the canonical
loadings and `U` is the vector of uniquenesses, then the
covariance matrix implied by the factor analysis is `GG' +
diag(U)`.
Status: experimental, Some refactoring will be necessary when new
features are added.
"""
def __init__(self, factor):
self.model = factor
self.endog_names = factor.endog_names
self.loadings_no_rot = factor.loadings
if hasattr(factor, "eigenvals"):
self.eigenvals = factor.eigenvals
self.communality = factor.communality
self.uniqueness = factor.uniqueness
self.rotation_method = None
self.fa_method = factor.method
self.n_comp = factor.loadings.shape[1]
self.nobs = factor.nobs
self._factor = factor
if hasattr(factor, "mle_retvals"):
self.mle_retvals = factor.mle_retvals
p, k = self.loadings_no_rot.shape
self.df = ((p - k)**2 - (p + k)) // 2
# no rotation, overwritten in `rotate`
self.loadings = factor.loadings
self.rotation_matrix = np.eye(self.n_comp)
def __str__(self):
return self.summary().__str__()
[docs] def rotate(self, method):
"""
Apply rotation, inplace modification of this Results instance
Parameters
----------
method : string
Rotation to be applied. Allowed methods are varimax,
quartimax, biquartimax, equamax, oblimin, parsimax,
parsimony, biquartimin, promax.
Returns
-------
None : nothing returned, modifications are inplace
Notes
-----
Warning: 'varimax', 'quartimax' and 'oblimin' are verified against R or
Stata. Some rotation methods such as promax do not produce the same
results as the R or Stata default functions.
See Also
--------
factor_rotation : subpackage that implements rotation methods
"""
self.rotation_method = method
if method not in ['varimax', 'quartimax', 'biquartimax',
'equamax', 'oblimin', 'parsimax', 'parsimony',
'biquartimin', 'promax']:
raise ValueError('Unknown rotation method %s' % (method))
if method in ['varimax', 'quartimax', 'biquartimax', 'equamax',
'parsimax', 'parsimony', 'biquartimin']:
self.loadings, T = rotate_factors(self.loadings_no_rot, method)
elif method == 'oblimin':
self.loadings, T = rotate_factors(self.loadings_no_rot,
'quartimin')
elif method == 'promax':
self.loadings, T = promax(self.loadings_no_rot)
else:
raise ValueError('rotation method not recognized')
self.rotation_matrix = T
def _corr_factors(self):
"""correlation of factors implied by rotation
If the rotation is oblique, then the factors are correlated.
currently not cached
Returns
-------
corr_f : ndarray
correlation matrix of rotated factors, assuming initial factors are
orthogonal
"""
T = self.rotation_matrix
corr_f = T.T.dot(T)
return corr_f
[docs] def factor_score_params(self, method='bartlett'):
"""compute factor scoring coefficient matrix
The coefficient matrix is not cached.
Parameters
----------
method : 'bartlett' or 'regression'
Method to use for factor scoring.
'regression' can be abbreviated to `reg`
Returns
-------
coeff_matrix : ndarray
matrix s to compute factors f from a standardized endog ys.
``f = ys dot s``
Notes
-----
The `regression` method follows the Stata definition.
Method bartlett and regression are verified against Stats.
Two inofficial methods, 'ols' and 'gls', produce similar factor scores
but are not verified.
See Also
--------
`factor_scoring` : compute factor scores using scoring matrix
"""
L = self.loadings
T = self.rotation_matrix.T
#TODO: check row versus column convention for T
uni = 1 - self.communality #self.uniqueness
if method == 'bartlett':
s_mat = np.linalg.inv(L.T.dot(L/(uni[:,None]))).dot((L.T / uni)).T
elif method.startswith('reg'):
corr = self.model.corr
corr_f = self._corr_factors()
# if orthogonal then corr_f is just eye
s_mat = corr_f.dot(L.T.dot(np.linalg.inv(corr))).T
elif method == 'ols':
# not verified
corr = self.model.corr
corr_f = self._corr_factors()
s_mat = corr_f.dot(np.linalg.pinv(L)).T
elif method == 'gls':
# not verified
#s_mat = np.linalg.inv(1*np.eye(L.shape[1]) + L.T.dot(L/(uni[:,None])))
corr = self.model.corr
corr_f = self._corr_factors()
s_mat = np.linalg.inv(np.linalg.inv(corr_f) + L.T.dot(L/(uni[:,None])))
s_mat = s_mat.dot(L.T / uni).T
else:
raise ValueError('method not available, use "bartlett ' +
'or "regression"')
return s_mat
[docs] def factor_scoring(self, endog=None, method='bartlett', transform=True):
"""factor scoring: compute factors for endog
If endog was not provided when creating the factor class, then
a standarized endog needs to be provided here.
Parameters
----------
method : 'bartlett' or 'regression'
Method to use for factor scoring.
'regression' can be abbreviated to `reg`
transform : bool
If transform is true and endog is provided, then it will be
standardized using mean and scale of original data, which has to
be available in this case.
If transform is False, then a provided endog will be used unchanged.
The original endog in the Factor class will
always be standardized if endog is None, independently of `transform`.
Returns
-------
factor_score : ndarray
estimated factors using scoring matrix s and standarized endog ys
``f = ys dot s``
Notes
-----
Status: transform option is experimental and might change.
See Also
--------
`factor_score_params` : scoring matrix
"""
if transform is False and endog is not None:
# no transformation in this case
endog = np.asarray(endog)
else:
# we need to standardize with the original mean and scale
if self.model.endog is not None:
m = self.model.endog.mean(0)
s = self.model.endog.std(ddof=1, axis=0)
if endog is None:
endog = self.model.endog
else:
endog = np.asarray(endog)
else:
raise ValueError('If transform is True, then `endog` needs ' +
'to be available in the Factor instance.')
endog = (endog - m) / s
s_mat = self.factor_score_params(method=method)
factors = endog.dot(s_mat)
return factors
[docs] def summary(self):
summ = summary2.Summary()
summ.add_title('Factor analysis results')
loadings_no_rot = pd.DataFrame(
self.loadings_no_rot,
columns=["factor %d" % (i)
for i in range(self.loadings_no_rot.shape[1])],
index=self.endog_names
)
if hasattr(self, "eigenvals"):
# eigenvals not available for ML method
eigenvals = pd.DataFrame(
[self.eigenvals], columns=self.endog_names, index=[''])
summ.add_dict({'': 'Eigenvalues'})
summ.add_df(eigenvals)
communality = pd.DataFrame([self.communality],
columns=self.endog_names, index=[''])
summ.add_dict({'': ''})
summ.add_dict({'': 'Communality'})
summ.add_df(communality)
summ.add_dict({'': ''})
summ.add_dict({'': 'Pre-rotated loadings'})
summ.add_df(loadings_no_rot)
summ.add_dict({'': ''})
if self.rotation_method is not None:
loadings = pd.DataFrame(
self.loadings,
columns=["factor %d" % (i)
for i in range(self.loadings.shape[1])],
index=self.endog_names
)
summ.add_dict({'': '%s rotated loadings' % (self.rotation_method)})
summ.add_df(loadings)
return summ
[docs] def get_loadings_frame(self, style='display', sort_=True, threshold=0.3,
highlight_max=True, color_max='yellow',
decimals=None):
"""get loadings matrix as DataFrame or pandas Styler
Parameters
----------
style : 'display' (default), 'raw' or 'strings'
Style to use for display
* 'raw' returns just a DataFrame of the loadings matrix, no options are
applied
* 'display' add sorting and styling as defined by other keywords
* 'strings' returns a DataFrame with string elements with optional sorting
and surpressing small loading coefficients.
sort_ : boolean
If True, then the rows of the DataFrame is sorted by contribution of each
factor. applies if style is either 'display' or 'strings'
threshold : float
If the threshold is larger than zero, then loading coefficients are
either colored white (if style is 'display') or replace by empty
string (if style is 'strings').
highlight_max : boolean
This add a background color to the largest coefficient in each row.
color_max : html color
default is 'yellow'. color for background of row maximum
decimals : None or int
If None, then pandas default precision applies. Otherwise values are
rounded to the specified decimals. If style is 'display', then the
underlying dataframe is not changed. If style is 'strings', then
values are rounded before conversion to strings.
Returns
-------
loadings : DataFrame or pandas Styler instance
The return is a pandas Styler instance, if style is 'display' and
at least one of highlight_max, threshold or decimals is applied.
Otherwise, the returned loadings is a DataFrame.
Examples
--------
>>> mod = Factor(df, 3, smc=True)
>>> res = mod.fit()
>>> res.get_loadings_frame(style='display', decimals=3, threshold=0.2)
To get a sorted DataFrame, all styling options need to be turned off:
>>> df_sorted = res.get_loadings_frame(style='display',
... highlight_max=False, decimals=None, threshold=0)
Options except for highlighting are available for plain test or Latex
usage:
>>> lds = res_u.get_loadings_frame(style='strings', decimals=3,
... threshold=0.3)
>>> print(lds.to_latex())
"""
loadings_df = pd.DataFrame(
self.loadings,
columns=["factor %d" % (i)
for i in range(self.loadings.shape[1])],
index=self.endog_names
)
if style not in ['raw', 'display', 'strings']:
msg = "style has to be one of 'raw', 'display', 'strings'"
raise ValueError(msg)
if style == 'raw':
return loadings_df
# add sorting and some formatting
if sort_ is True:
loadings_df2 = loadings_df.copy()
n_f = len(loadings_df2)
high = np.abs(loadings_df2.values).argmax(1)
loadings_df2['high'] = high
loadings_df2['largest'] = np.abs(loadings_df.values[np.arange(n_f), high])
loadings_df2.sort_values(by=['high', 'largest'], ascending=[True, False], inplace=True)
loadings_df = loadings_df2.drop(['high', 'largest'], axis=1)
if style == 'display':
sty = None
if threshold > 0:
def color_white_small(val):
"""
Takes a scalar and returns a string with
the css property `'color: white'` for small values, black otherwise.
takes threshold from outer scope
"""
color = 'white' if np.abs(val) < threshold else 'black'
return 'color: %s' % color
sty = loadings_df.style.applymap(color_white_small)
if highlight_max is True:
def highlight_max(s):
'''
highlight the maximum in a Series yellow.
'''
s = np.abs(s)
is_max = s == s.max()
return ['background-color: '+ color_max if v else '' for v in is_max]
if sty is None:
sty = loadings_df.style
sty = sty.apply(highlight_max, axis=1)
if decimals is not None:
if sty is None:
sty = loadings_df.style
sty.format("{:.%sf}" % decimals)
if sty is None:
return loadings_df
else:
return sty
if style == 'strings':
ld = loadings_df
if decimals is not None:
ld = ld.round(decimals)
ld = ld.astype(str)
if threshold > 0:
ld[loadings_df.abs() < threshold] = ''
return ld
[docs] def plot_scree(self, ncomp=None):
"""
Plot of the ordered eigenvalues and variance explained for the loadings
Parameters
----------
ncomp : int, optional
Number of loadings to include in the plot. If None, will
included the same as the number of maximum possible loadings
Returns
-------
fig : figure
Handle to the figure
"""
if missing_matplotlib:
raise ImportError("Matplotlib missing")
return plot_scree(self.eigenvals, self.n_comp, ncomp)
[docs] def plot_loadings(self, loading_pairs=None, plot_prerotated=False):
"""
Plot factor loadings in 2-d plots
Parameters
----------
loading_pairs : None or a list of tuples
Specify plots. Each tuple (i, j) represent one figure, i and j is
the loading number for x-axis and y-axis, respectively. If `None`,
all combinations of the loadings will be plotted.
plot_prerotated : True or False
If True, the loadings before rotation applied will be plotted. If
False, rotated loadings will be plotted.
Returns
-------
figs : a list of figure handles
"""
if missing_matplotlib:
raise ImportError("Matplotlib missing")
if self.rotation_method is None:
plot_prerotated = True
loadings = self.loadings_no_rot if plot_prerotated else self.loadings
if plot_prerotated:
title = 'Prerotated Factor Pattern'
else:
title = '%s Rotated Factor Pattern' % (self.rotation_method)
var_explained = self.eigenvals / self.n_comp * 100
return plot_loadings(loadings, loading_pairs=loading_pairs,
title=title, row_names=self.endog_names,
percent_variance=var_explained)
[docs] @cache_readonly
def fitted_cov(self):
"""
Returns the fitted covariance matrix.
"""
c = np.dot(self.loadings, self.loadings.T)
c.flat[::c.shape[0]+1] += self.uniqueness
return c
[docs] @cache_readonly
def uniq_stderr(self, kurt=0):
"""
The standard errors of the uniquenesses.
Parameters
----------
kurt: float
Excess kurtosis
Notes
-----
If excess kurtosis is known, provide as `kurt`. Standard
errors are only available if the model was fit using maximum
likelihood. If `endog` is not provided, `nobs` must be
provided to obtain standard errors.
These are asymptotic standard errors. See Bai and Li (2012)
for conditions under which the standard errors are valid.
The standard errors are only applicable to the original,
unrotated maximum likelihood solution.
"""
if self.fa_method.lower() != "ml":
msg = "Standard errors only available under ML estimation"
raise ValueError(msg)
if self.nobs is None:
msg = "nobs is required to obtain standard errors."
raise ValueError(msg)
v = self.uniqueness**2 * (2 + kurt)
return np.sqrt(v / self.nobs)
[docs] @cache_readonly
def load_stderr(self):
"""
The standard errors of the loadings.
Standard errors are only available if the model was fit using
maximum likelihood. If `endog` is not provided, `nobs` must be
provided to obtain standard errors.
These are asymptotic standard errors. See Bai and Li (2012)
for conditions under which the standard errors are valid.
The standard errors are only applicable to the original,
unrotated maximum likelihood solution.
"""
if self.fa_method.lower() != "ml":
msg = "Standard errors only available under ML estimation"
raise ValueError(msg)
if self.nobs is None:
msg = "nobs is required to obtain standard errors."
raise ValueError(msg)
v = np.outer(self.uniqueness, np.ones(self.loadings.shape[1]))
return np.sqrt(v / self.nobs)
