# Source code for statsmodels.nonparametric.smoothers_lowess

```# -*- 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, 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
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
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.

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)``.

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)

# 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 missing in ['drop', 'raise']:
# Cut out missing values
if all_valid:
y = endog
x = exog
else:
if missing == 'drop':
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])

res = _lowess(y, x, frac=frac, it=it, delta=delta)
_, 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(y)
yfitted_.fill(np.nan)
yfitted_[sort_index] = yfitted
yfitted = yfitted_
else:
yfitted = yfitted

if not all_valid:
yfitted_ = np.empty_like(endog)
yfitted_.fill(np.nan)