LOWESS (Locally Weighted Scatterplot Smoothing)
A lowess function that outs smoothed estimates of endog at the given exog values from points (exog, endog)
endog: 1-D numpy array :
exog: 1-D numpy array :
frac: float :
it: int :
delta: float :
is_sorted : bool
missing : str
return_sorted : bool
out: ndarray, float :
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.
Cleveland, W.S. (1979) “Robust Locally Weighted Regression and Smoothing Scatterplots”. Journal of the American Statistical Association 74 (368): 829-836.
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)