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, 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) # 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