# -*- coding: utf-8 -*-
"""Lowess - wrapper for cythonized extension
Author : Chris Jordan-Squire
Author : Carl Vogel
Author : Josef Perktold
"""
import numpy as np
from ._smoothers_lowess import lowess as _lowess
[docs]def lowess(endog, exog, frac=2.0/3.0, it=3, delta=0.0, xvals=None, is_sorted=False,
missing='drop', return_sorted=True):
'''LOWESS (Locally Weighted Scatterplot Smoothing)
A lowess function that outs smoothed estimates of endog
at the given exog values from points (exog, endog)
Parameters
----------
endog : 1-D numpy array
The y-values of the observed points
exog : 1-D numpy array
The x-values of the observed points
frac : float
Between 0 and 1. The fraction of the data used
when estimating each y-value.
it : int
The number of residual-based reweightings
to perform.
delta : float
Distance within which to use linear-interpolation
instead of weighted regression.
xvals: 1-D numpy array
Values of the exogenous variable at which to evaluate the regression.
If supplied, cannot use delta.
is_sorted : bool
If False (default), then the data will be sorted by exog before
calculating lowess. If True, then it is assumed that the data is
already sorted by exog. If xvals is specified, then it too must be
sorted if is_sorted is True.
missing : str
Available options are 'none', 'drop', and 'raise'. If 'none', no nan
checking is done. If 'drop', any observations with nans are dropped.
If 'raise', an error is raised. Default is 'drop'.
return_sorted : bool
If True (default), then the returned array is sorted by exog and has
missing (nan or infinite) observations removed.
If False, then the returned array is in the same length and the same
sequence of observations as the input array.
Returns
-------
out : {ndarray, float}
The returned array is two-dimensional if return_sorted is True, and
one dimensional if return_sorted is False.
If return_sorted is True, then a numpy array with two columns. The
first column contains the sorted x (exog) values and the second column
the associated estimated y (endog) values.
If return_sorted is False, then only the fitted values are returned,
and the observations will be in the same order as the input arrays.
If xvals is provided, then return_sorted is ignored and the returned
array is always one dimensional, containing the y values fitted at
the x values provided by xvals.
Notes
-----
This lowess function implements the algorithm given in the
reference below using local linear estimates.
Suppose the input data has N points. The algorithm works by
estimating the `smooth` y_i by taking the frac*N closest points
to (x_i,y_i) based on their x values and estimating y_i
using a weighted linear regression. The weight for (x_j,y_j)
is tricube function applied to abs(x_i-x_j).
If it > 1, then further weighted local linear regressions
are performed, where the weights are the same as above
times the _lowess_bisquare function of the residuals. Each iteration
takes approximately the same amount of time as the original fit,
so these iterations are expensive. They are most useful when
the noise has extremely heavy tails, such as Cauchy noise.
Noise with less heavy-tails, such as t-distributions with df>2,
are less problematic. The weights downgrade the influence of
points with large residuals. In the extreme case, points whose
residuals are larger than 6 times the median absolute residual
are given weight 0.
`delta` can be used to save computations. For each `x_i`, regressions
are skipped for points closer than `delta`. The next regression is
fit for the farthest point within delta of `x_i` and all points in
between are estimated by linearly interpolating between the two
regression fits.
Judicious choice of delta can cut computation time considerably
for large data (N > 5000). A good choice is ``delta = 0.01 * range(exog)``.
If `xvals` is provided, the regression is then computed at those points
and the fit values are returned. Otherwise, the regression is run
at points of `exog`.
Some experimentation is likely required to find a good
choice of `frac` and `iter` for a particular dataset.
References
----------
Cleveland, W.S. (1979) "Robust Locally Weighted Regression
and Smoothing Scatterplots". Journal of the American Statistical
Association 74 (368): 829-836.
Examples
--------
The below allows a comparison between how different the fits from
lowess for different values of frac can be.
>>> import numpy as np
>>> import statsmodels.api as sm
>>> lowess = sm.nonparametric.lowess
>>> x = np.random.uniform(low = -2*np.pi, high = 2*np.pi, size=500)
>>> y = np.sin(x) + np.random.normal(size=len(x))
>>> z = lowess(y, x)
>>> w = lowess(y, x, frac=1./3)
This gives a similar comparison for when it is 0 vs not.
>>> import numpy as np
>>> import scipy.stats as stats
>>> import statsmodels.api as sm
>>> lowess = sm.nonparametric.lowess
>>> x = np.random.uniform(low = -2*np.pi, high = 2*np.pi, size=500)
>>> y = np.sin(x) + stats.cauchy.rvs(size=len(x))
>>> z = lowess(y, x, frac= 1./3, it=0)
>>> w = lowess(y, x, frac=1./3)
'''
endog = np.asarray(endog, float)
exog = np.asarray(exog, float)
# Whether xvals argument was provided
given_xvals = (xvals is not None)
# Inputs should be vectors (1-D arrays) of the
# same length.
if exog.ndim != 1:
raise ValueError('exog must be a vector')
if endog.ndim != 1:
raise ValueError('endog must be a vector')
if endog.shape[0] != exog.shape[0] :
raise ValueError('exog and endog must have same length')
if xvals is not None:
xvals = np.ascontiguousarray(xvals)
if xvals.ndim != 1:
raise ValueError('exog_predict must be a vector')
if missing in ['drop', 'raise']:
mask_valid = (np.isfinite(exog) & np.isfinite(endog))
all_valid = np.all(mask_valid)
if all_valid:
y = endog
x = exog
else:
if missing == 'drop':
x = exog[mask_valid]
y = endog[mask_valid]
else:
raise ValueError('nan or inf found in data')
elif missing == 'none':
y = endog
x = exog
all_valid = True # we assume it's true if missing='none'
else:
raise ValueError("missing can only be 'none', 'drop' or 'raise'")
if not is_sorted:
# Sort both inputs according to the ascending order of x values
sort_index = np.argsort(x)
x = np.array(x[sort_index])
y = np.array(y[sort_index])
if not given_xvals:
# If given no explicit x values, we use the x-values in the exog array
xvals = exog
xvalues = x
xvals_all_valid = all_valid
if missing == 'drop':
xvals_mask_valid = mask_valid
else:
if delta != 0.0:
raise ValueError("Cannot have non-zero 'delta' and 'xvals' values")
# TODO: allow this again
mask_valid = np.isfinite(xvals)
if missing == "raise":
raise ValueError("NaN values in xvals with missing='raise'")
elif missing == 'drop':
xvals_mask_valid = mask_valid
xvalues = xvals
xvals_all_valid = True if missing == "none" else np.all(mask_valid)
# With explicit xvals, we ignore 'return_sorted' and always
# use the order provided
return_sorted = False
if missing in ['drop', 'raise']:
xvals_mask_valid = np.isfinite(xvals)
xvals_all_valid = np.all(xvals_mask_valid)
if xvals_all_valid:
xvalues = xvals
else:
if missing == 'drop':
xvalues = xvals[xvals_mask_valid]
else:
raise ValueError("nan or inf found in xvals")
if not is_sorted:
sort_index = np.argsort(xvalues)
xvalues = np.array(xvalues[sort_index])
else:
xvals_all_valid = True
y = np.ascontiguousarray(y)
x = np.ascontiguousarray(x)
if not given_xvals:
# Run LOWESS on the data points
res, _ = _lowess(y, x, x, np.ones_like(x),
frac=frac, it=it, delta=delta, given_xvals=False)
else:
# First run LOWESS on the data points to get the weights of the data points
# using it-1 iterations, last iter done next
if it > 0:
_, weights = _lowess(y, x, x, np.ones_like(x),
frac=frac, it=it-1, delta=delta, given_xvals=False)
else:
weights = np.ones_like(x)
xvalues = np.ascontiguousarray(xvalues, dtype=float)
# Then run once more using those supplied weights at the points provided by xvals
# No extra iterations are performed here since weights are fixed
res, _ = _lowess(y, x, xvalues, weights,
frac=frac, it=0, delta=delta, given_xvals=True)
_, yfitted = res.T
if return_sorted:
return res
else:
# rebuild yfitted with original indices
# a bit messy: y might have been selected twice
if not is_sorted:
yfitted_ = np.empty_like(xvalues)
yfitted_.fill(np.nan)
yfitted_[sort_index] = yfitted
yfitted = yfitted_
else:
yfitted = yfitted
if not xvals_all_valid:
yfitted_ = np.empty_like(xvals)
yfitted_.fill(np.nan)
yfitted_[xvals_mask_valid] = yfitted
yfitted = yfitted_
# we do not need to return exog anymore
return yfitted