In [1]:

import numpy as np
from ipywidgets import interact
from bokeh.plotting import figure, show, output_notebook
from bokeh.layouts import gridplot
from bokeh.io import push_notebook

output_notebook()

BokehJS 0.12.5 successfully loaded.

algorithm¶

In [2]:

def local_regression(x0, X, Y, tau):
    # add bias term
    x0 = np.r_[1, x0]
    X = np.c_[np.ones(len(X)), X]
    
    # fit model: normal equations with kernel
    xw = X.T * radial_kernel(x0, X, tau)
    beta = np.linalg.pinv(xw @ X) @ xw @ Y
    
    # predict value
    return x0 @ beta

In [3]:

def radial_kernel(x0, X, tau):
    return np.exp(np.sum((X - x0) ** 2, axis=1) / (-2 * tau * tau))

data¶

In [4]:

n = 1000

# generate dataset
X = np.linspace(-3, 3, num=n)
Y = np.log(np.abs(X ** 2 - 1) + .5)

# jitter X
X += np.random.normal(scale=.1, size=n)

fit & plot models¶

In [5]:

def plot_lwr(tau):
    # prediction
    domain = np.linspace(-3, 3, num=300)
    prediction = [local_regression(x0, X, Y, tau) for x0 in domain]

    plot = figure(plot_width=400, plot_height=400)
    plot.title.text = 'tau=%g' % tau
    plot.scatter(X, Y, alpha=.3)
    plot.line(domain, prediction, line_width=2, color='red')
    
    return plot

In [6]:

show(gridplot([
    [plot_lwr(10.), plot_lwr(1.)],
    [plot_lwr(0.1), plot_lwr(0.01)]
]))

interactive model¶

In [7]:

def interactive_update(tau):
    model.data_source.data['y'] = [local_regression(x0, X, Y, tau) for x0 in domain]
    push_notebook()

domain = np.linspace(-3, 3, num=100)
prediction = [local_regression(x0, X, Y, 1.) for x0 in domain]

plot = figure()
plot.scatter(X, Y, alpha=.3)
model = plot.line(domain, prediction, line_width=2, color='red')
show(plot, notebook_handle=True)

Out[7]:

<Bokeh Notebook handle for In[7]>

In [8]:

interact(interactive_update, tau=(0.01, 3., 0.01))

Out[8]:

<function __main__.interactive_update>

In [ ]: