# Source code for statsmodels.nonparametric.kde

"""
Univariate Kernel Density Estimators

References
----------
Racine, Jeff. (2008) "Nonparametric Econometrics: A Primer," Foundation and
Trends in Econometrics: Vol 3: No 1, pp1-88.
http://dx.doi.org/10.1561/0800000009

https://en.wikipedia.org/wiki/Kernel_%28statistics%29

Silverman, B.W.  Density Estimation for Statistics and Data Analysis.
"""
import numpy as np
from scipy import integrate, stats

from statsmodels.sandbox.nonparametric import kernels
from statsmodels.tools.validation import array_like, float_like

from . import bandwidths
from .kdetools import forrt, revrt, silverman_transform
from .linbin import fast_linbin

# Kernels Switch for estimators

kernel_switch = dict(
gau=kernels.Gaussian,
epa=kernels.Epanechnikov,
uni=kernels.Uniform,
tri=kernels.Triangular,
biw=kernels.Biweight,
triw=kernels.Triweight,
cos=kernels.Cosine,
cos2=kernels.Cosine2,
)

def _checkisfit(self):
try:
self.density
except Exception:
raise ValueError("Call fit to fit the density first")

# Kernel Density Estimator Class
[docs]class KDEUnivariate(object): """ Univariate Kernel Density Estimator. Parameters ---------- endog : array_like The variable for which the density estimate is desired. Notes ----- If cdf, sf, cumhazard, or entropy are computed, they are computed based on the definition of the kernel rather than the FFT approximation, even if the density is fit with FFT = True. KDEUnivariate is much faster than KDEMultivariate, due to its FFT-based implementation. It should be preferred for univariate, continuous data. KDEMultivariate also supports mixed data. See Also -------- KDEMultivariate kdensity, kdensityfft Examples -------- >>> import statsmodels.api as sm >>> import matplotlib.pyplot as plt >>> nobs = 300 >>> np.random.seed(1234) # Seed random generator >>> dens = sm.nonparametric.KDEUnivariate(np.random.normal(size=nobs)) >>> dens.fit() >>> plt.plot(dens.cdf) >>> plt.show() """ def __init__(self, endog): self.endog = array_like(endog, "endog", ndim=1, contiguous=True)
[docs] def fit( self, kernel="gau", bw="normal_reference", fft=True, weights=None, gridsize=None, adjust=1, cut=3, clip=(-np.inf, np.inf), ): """ Attach the density estimate to the KDEUnivariate class. Parameters ---------- kernel : str The Kernel to be used. Choices are: - "biw" for biweight - "cos" for cosine - "epa" for Epanechnikov - "gau" for Gaussian. - "tri" for triangular - "triw" for triweight - "uni" for uniform bw : str, float, callable The bandwidth to use. Choices are: - "scott" - 1.059 * A * nobs ** (-1/5.), where A is min(std(x),IQR/1.34) - "silverman" - .9 * A * nobs ** (-1/5.), where A is min(std(x),IQR/1.34) - "normal_reference" - C * A * nobs ** (-1/5.), where C is calculated from the kernel. Equivalent (up to 2 dp) to the "scott" bandwidth for gaussian kernels. See bandwidths.py - If a float is given, its value is used as the bandwidth. - If a callable is given, it's return value is used. The callable should take exactly two parameters, i.e., fn(x, kern), and return a float, where: * x - the clipped input data * kern - the kernel instance used fft : bool Whether or not to use FFT. FFT implementation is more computationally efficient. However, only the Gaussian kernel is implemented. If FFT is False, then a 'nobs' x 'gridsize' intermediate array is created. gridsize : int If gridsize is None, max(len(x), 50) is used. cut : float Defines the length of the grid past the lowest and highest values of x so that the kernel goes to zero. The end points are -/+ cut*bw*{min(x) or max(x)} adjust : float An adjustment factor for the bw. Bandwidth becomes bw * adjust. Returns ------- KDEUnivariate The instance fit, """ if isinstance(bw, str): self.bw_method = bw else: self.bw_method = "user-given" if not callable(bw): bw = float_like(bw, "bw") endog = self.endog if fft: if kernel != "gau": msg = "Only gaussian kernel is available for fft" raise NotImplementedError(msg) if weights is not None: msg = "Weights are not implemented for fft" raise NotImplementedError(msg) density, grid, bw = kdensityfft( endog, kernel=kernel, bw=bw, adjust=adjust, weights=weights, gridsize=gridsize, clip=clip, cut=cut, ) else: density, grid, bw = kdensity( endog, kernel=kernel, bw=bw, adjust=adjust, weights=weights, gridsize=gridsize, clip=clip, cut=cut, ) self.density = density self.support = grid self.bw = bw self.kernel = kernel_switch[kernel](h=bw) # we instantiate twice, # should this passed to funcs? # put here to ensure empty cache after re-fit with new options self.kernel.weights = weights if weights is not None: self.kernel.weights /= weights.sum() self._cache = {} return self
@cache_readonly def cdf(self): """ Returns the cumulative distribution function evaluated at the support. Notes ----- Will not work if fit has not been called. """ _checkisfit(self) kern = self.kernel if kern.domain is None: # TODO: test for grid point at domain bound a, b = -np.inf, np.inf else: a, b = kern.domain def func(x, s): return kern.density(s, x) support = self.support support = np.r_[a, support] gridsize = len(support) endog = self.endog probs = [ integrate.quad(func, support[i - 1], support[i], args=endog)[0] for i in range(1, gridsize) ] return np.cumsum(probs) @cache_readonly def cumhazard(self): """ Returns the hazard function evaluated at the support. Notes ----- Will not work if fit has not been called. """ _checkisfit(self) return -np.log(self.sf) @cache_readonly def sf(self): """ Returns the survival function evaluated at the support. Notes ----- Will not work if fit has not been called. """ _checkisfit(self) return 1 - self.cdf @cache_readonly def entropy(self): """ Returns the differential entropy evaluated at the support Notes ----- Will not work if fit has not been called. 1e-12 is added to each probability to ensure that log(0) is not called. """ _checkisfit(self) def entr(x, s): pdf = kern.density(s, x) return pdf * np.log(pdf + 1e-12) kern = self.kernel if kern.domain is not None: a, b = self.domain else: a, b = -np.inf, np.inf endog = self.endog # TODO: below could run into integr problems, cf. stats.dist._entropy return -integrate.quad(entr, a, b, args=(endog,))[0] @cache_readonly def icdf(self): """ Inverse Cumulative Distribution (Quantile) Function Notes ----- Will not work if fit has not been called. Uses scipy.stats.mstats.mquantiles. """ _checkisfit(self) gridsize = len(self.density) return stats.mstats.mquantiles(self.endog, np.linspace(0, 1, gridsize))
[docs] def evaluate(self, point): """ Evaluate density at a point or points. Parameters ---------- point : {float, ndarray} Point(s) at which to evaluate the density. """ _checkisfit(self) return self.kernel.density(self.endog, point)
# Kernel Density Estimator Functions def kdensity( x, kernel="gau", bw="normal_reference", weights=None, gridsize=None, adjust=1, clip=(-np.inf, np.inf), cut=3, retgrid=True, ): """ Rosenblatt-Parzen univariate kernel density estimator. Parameters ---------- x : array_like The variable for which the density estimate is desired. kernel : str The Kernel to be used. Choices are - "biw" for biweight - "cos" for cosine - "epa" for Epanechnikov - "gau" for Gaussian. - "tri" for triangular - "triw" for triweight - "uni" for uniform bw : str, float, callable The bandwidth to use. Choices are: - "scott" - 1.059 * A * nobs ** (-1/5.), where A is min(std(x),IQR/1.34) - "silverman" - .9 * A * nobs ** (-1/5.), where A is min(std(x),IQR/1.34) - "normal_reference" - C * A * nobs ** (-1/5.), where C is calculated from the kernel. Equivalent (up to 2 dp) to the "scott" bandwidth for gaussian kernels. See bandwidths.py - If a float is given, its value is used as the bandwidth. - If a callable is given, it's return value is used. The callable should take exactly two parameters, i.e., fn(x, kern), and return a float, where: * x - the clipped input data * kern - the kernel instance used weights : array or None Optional weights. If the x value is clipped, then this weight is also dropped. gridsize : int If gridsize is None, max(len(x), 50) is used. adjust : float An adjustment factor for the bw. Bandwidth becomes bw * adjust. clip : tuple Observations in x that are outside of the range given by clip are dropped. The number of observations in x is then shortened. cut : float Defines the length of the grid past the lowest and highest values of x so that the kernel goes to zero. The end points are -/+ cut*bw*{min(x) or max(x)} retgrid : bool Whether or not to return the grid over which the density is estimated. Returns ------- density : ndarray The densities estimated at the grid points. grid : ndarray, optional The grid points at which the density is estimated. Notes ----- Creates an intermediate (gridsize x nobs) array. Use FFT for a more computationally efficient version. """ x = np.asarray(x) if x.ndim == 1: x = x[:, None] clip_x = np.logical_and(x > clip[0], x < clip[1]) x = x[clip_x] nobs = len(x) # after trim if gridsize is None: gridsize = max(nobs, 50) # do not need to resize if no FFT # handle weights if weights is None: weights = np.ones(nobs) q = nobs else: # ensure weights is a numpy array weights = np.asarray(weights) if len(weights) != len(clip_x): msg = "The length of the weights must be the same as the given x." raise ValueError(msg) weights = weights[clip_x.squeeze()] q = weights.sum() # Get kernel object corresponding to selection kern = kernel_switch[kernel]() if callable(bw): bw = float(bw(x, kern)) # user passed a callable custom bandwidth function elif isinstance(bw, str): bw = bandwidths.select_bandwidth(x, bw, kern) # will cross-val fit this pattern? else: bw = float_like(bw, "bw") bw *= adjust a = np.min(x, axis=0) - cut * bw b = np.max(x, axis=0) + cut * bw grid = np.linspace(a, b, gridsize) k = ( x.T - grid[:, None] ) / bw # uses broadcasting to make a gridsize x nobs # set kernel bandwidth kern.seth(bw) # truncate to domain if ( kern.domain is not None ): # will not work for piecewise kernels like parzen z_lo, z_high = kern.domain domain_mask = (k < z_lo) | (k > z_high) k = kern(k) # estimate density k[domain_mask] = 0 else: k = kern(k) # estimate density k[k < 0] = 0 # get rid of any negative values, do we need this? dens = np.dot(k, weights) / (q * bw) if retgrid: return dens, grid, bw else: return dens, bw def kdensityfft( x, kernel="gau", bw="normal_reference", weights=None, gridsize=None, adjust=1, clip=(-np.inf, np.inf), cut=3, retgrid=True, ): """ Rosenblatt-Parzen univariate kernel density estimator Parameters ---------- x : array_like The variable for which the density estimate is desired. kernel : str ONLY GAUSSIAN IS CURRENTLY IMPLEMENTED. "bi" for biweight "cos" for cosine "epa" for Epanechnikov, default "epa2" for alternative Epanechnikov "gau" for Gaussian. "par" for Parzen "rect" for rectangular "tri" for triangular bw : str, float, callable The bandwidth to use. Choices are: - "scott" - 1.059 * A * nobs ** (-1/5.), where A is min(std(x),IQR/1.34) - "silverman" - .9 * A * nobs ** (-1/5.), where A is min(std(x),IQR/1.34) - "normal_reference" - C * A * nobs ** (-1/5.), where C is calculated from the kernel. Equivalent (up to 2 dp) to the "scott" bandwidth for gaussian kernels. See bandwidths.py - If a float is given, its value is used as the bandwidth. - If a callable is given, it's return value is used. The callable should take exactly two parameters, i.e., fn(x, kern), and return a float, where: * x - the clipped input data * kern - the kernel instance used weights : array or None WEIGHTS ARE NOT CURRENTLY IMPLEMENTED. Optional weights. If the x value is clipped, then this weight is also dropped. gridsize : int If gridsize is None, min(len(x), 512) is used. Note that the provided number is rounded up to the next highest power of 2. adjust : float An adjustment factor for the bw. Bandwidth becomes bw * adjust. clip : tuple Observations in x that are outside of the range given by clip are dropped. The number of observations in x is then shortened. cut : float Defines the length of the grid past the lowest and highest values of x so that the kernel goes to zero. The end points are -/+ cut*bw*{x.min() or x.max()} retgrid : bool Whether or not to return the grid over which the density is estimated. Returns ------- density : ndarray The densities estimated at the grid points. grid : ndarray, optional The grid points at which the density is estimated. Notes ----- Only the default kernel is implemented. Weights are not implemented yet. This follows Silverman (1982) with changes suggested by Jones and Lotwick (1984). However, the discretization step is replaced by linear binning of Fan and Marron (1994). This should be extended to accept the parts that are dependent only on the data to speed things up for cross-validation. References ---------- Fan, J. and J.S. Marron. (1994) Fast implementations of nonparametric curve estimators. Journal of Computational and Graphical Statistics. 3.1, 35-56. Jones, M.C. and H.W. Lotwick. (1984) Remark AS R50: A Remark on Algorithm AS 176. Kernal Density Estimation Using the Fast Fourier Transform. Journal of the Royal Statistical Society. Series C. 33.1, 120-2. Silverman, B.W. (1982) `Algorithm AS 176. Kernel density estimation using the Fast Fourier Transform. Journal of the Royal Statistical Society. Series C. 31.2, 93-9. """ x = np.asarray(x) # will not work for two columns. x = x[np.logical_and(x > clip[0], x < clip[1])] # Get kernel object corresponding to selection kern = kernel_switch[kernel]() if callable(bw): bw = float(bw(x, kern)) # user passed a callable custom bandwidth function elif isinstance(bw, str): # if bw is None, select optimal bandwidth for kernel bw = bandwidths.select_bandwidth(x, bw, kern) # will cross-val fit this pattern? else: bw = float_like(bw, "bw") bw *= adjust nobs = len(x) # after trim # 1 Make grid and discretize the data if gridsize is None: gridsize = np.max((nobs, 512.0)) gridsize = 2 ** np.ceil(np.log2(gridsize)) # round to next power of 2 a = np.min(x) - cut * bw b = np.max(x) + cut * bw grid, delta = np.linspace(a, b, int(gridsize), retstep=True) RANGE = b - a # TODO: Fix this? # This is the Silverman binning function, but I believe it's buggy (SS) # weighting according to Silverman # count = counts(x,grid) # binned = np.zeros_like(grid) #xi_{k} in Silverman # j = 0 # for k in range(int(gridsize-1)): # if count[k]>0: # there are points of x in the grid here # Xingrid = x[j:j+count[k]] # get all these points # # get weights at grid[k],grid[k+1] # binned[k] += np.sum(grid[k+1]-Xingrid) # binned[k+1] += np.sum(Xingrid-grid[k]) # j += count[k] # binned /= (nobs)*delta**2 # normalize binned to sum to 1/delta # NOTE: THE ABOVE IS WRONG, JUST TRY WITH LINEAR BINNING binned = fast_linbin(x, a, b, gridsize) / (delta * nobs) # step 2 compute FFT of the weights, using Munro (1976) FFT convention y = forrt(binned) # step 3 and 4 for optimal bw compute zstar and the density estimate f # do not have to redo the above if just changing bw, ie., for cross val # NOTE: silverman_transform is the closed form solution of the FFT of the # gaussian kernel. Not yet sure how to generalize it. zstar = silverman_transform(bw, gridsize, RANGE) * y # 3.49 in Silverman # 3.50 w Gaussian kernel f = revrt(zstar) if retgrid: return f, grid, bw else: return f, bw