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 :
out: numpy array :
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 true 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 _lowess_tricube function applied to |x_i-x_j|.
If iter>0, 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.
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 >>> from sm.nonparametric import 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 >>> from sm.nonparametric import 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)