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# DO NOT EDIT
# Autogenerated from the notebook robust_models_1.ipynb.
# Edit the notebook and then sync the output with this file.
#
# flake8: noqa
# DO NOT EDIT
#!/usr/bin/env python3
# coding: utf-8
# # M-Estimators for Robust Linear Modeling
from statsmodels.compat import lmap
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
import statsmodels.api as sm
# * An M-estimator minimizes the function
#
# $$Q(e_i, \rho) = \sum_i~\rho \left (\frac{e_i}{s}\right )$$
#
# where $\rho$ is a symmetric function of the residuals
#
# * The effect of $\rho$ is to reduce the influence of outliers
# * $s$ is an estimate of scale.
# * The robust estimates $\hat{\beta}$ are computed by the iteratively re-
# weighted least squares algorithm
# * We have several choices available for the weighting functions to be
# used
norms = sm.robust.norms
def plot_weights(support, weights_func, xlabels, xticks):
fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(111)
ax.plot(support, weights_func(support))
ax.set_xticks(xticks)
ax.set_xticklabels(xlabels, fontsize=16)
ax.set_ylim(-.1, 1.1)
return ax
# ### Andrew's Wave
help(norms.AndrewWave.weights)
a = 1.339
support = np.linspace(-np.pi * a, np.pi * a, 100)
andrew = norms.AndrewWave(a=a)
plot_weights(support, andrew.weights, ['$-\pi*a$', '0', '$\pi*a$'],
[-np.pi * a, 0, np.pi * a])
# ### Hampel's 17A
help(norms.Hampel.weights)
c = 8
support = np.linspace(-3 * c, 3 * c, 1000)
hampel = norms.Hampel(a=2., b=4., c=c)
plot_weights(support, hampel.weights, ['3*c', '0', '3*c'], [-3 * c, 0, 3 * c])
# ### Huber's t
help(norms.HuberT.weights)
t = 1.345
support = np.linspace(-3 * t, 3 * t, 1000)
huber = norms.HuberT(t=t)
plot_weights(support, huber.weights, ['-3*t', '0', '3*t'], [-3 * t, 0, 3 * t])
# ### Least Squares
help(norms.LeastSquares.weights)
support = np.linspace(-3, 3, 1000)
lst_sq = norms.LeastSquares()
plot_weights(support, lst_sq.weights, ['-3', '0', '3'], [-3, 0, 3])
# ### Ramsay's Ea
help(norms.RamsayE.weights)
a = .3
support = np.linspace(-3 * a, 3 * a, 1000)
ramsay = norms.RamsayE(a=a)
plot_weights(support, ramsay.weights, ['-3*a', '0', '3*a'], [-3 * a, 0, 3 * a])
# ### Trimmed Mean
help(norms.TrimmedMean.weights)
c = 2
support = np.linspace(-3 * c, 3 * c, 1000)
trimmed = norms.TrimmedMean(c=c)
plot_weights(support, trimmed.weights, ['-3*c', '0', '3*c'],
[-3 * c, 0, 3 * c])
# ### Tukey's Biweight
help(norms.TukeyBiweight.weights)
c = 4.685
support = np.linspace(-3 * c, 3 * c, 1000)
tukey = norms.TukeyBiweight(c=c)
plot_weights(support, tukey.weights, ['-3*c', '0', '3*c'], [-3 * c, 0, 3 * c])
# ### Scale Estimators
# * Robust estimates of the location
x = np.array([1, 2, 3, 4, 500])
# * The mean is not a robust estimator of location
x.mean()
# * The median, on the other hand, is a robust estimator with a breakdown
# point of 50%
np.median(x)
# * Analogously for the scale
# * The standard deviation is not robust
x.std()
# Median Absolute Deviation
#
# $$ median_i |X_i - median_j(X_j)|) $$
# Standardized Median Absolute Deviation is a consistent estimator for
# $\hat{\sigma}$
#
# $$\hat{\sigma}=K \cdot MAD$$
#
# where $K$ depends on the distribution. For the normal distribution for
# example,
#
# $$K = \Phi^{-1}(.75)$$
stats.norm.ppf(.75)
print(x)
sm.robust.scale.mad(x)
np.array([1, 2, 3, 4, 5.]).std()
# * The default for Robust Linear Models is MAD
# * another popular choice is Huber's proposal 2
np.random.seed(12345)
fat_tails = stats.t(6).rvs(40)
kde = sm.nonparametric.KDEUnivariate(fat_tails)
kde.fit()
fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(111)
ax.plot(kde.support, kde.density)
print(fat_tails.mean(), fat_tails.std())
print(stats.norm.fit(fat_tails))
print(stats.t.fit(fat_tails, f0=6))
huber = sm.robust.scale.Huber()
loc, scale = huber(fat_tails)
print(loc, scale)
sm.robust.mad(fat_tails)
sm.robust.mad(fat_tails, c=stats.t(6).ppf(.75))
sm.robust.scale.mad(fat_tails)
# ### Duncan's Occupational Prestige data - M-estimation for outliers
from statsmodels.graphics.api import abline_plot
from statsmodels.formula.api import ols, rlm
prestige = sm.datasets.get_rdataset("Duncan", "carData", cache=True).data
print(prestige.head(10))
fig = plt.figure(figsize=(12, 12))
ax1 = fig.add_subplot(211, xlabel='Income', ylabel='Prestige')
ax1.scatter(prestige.income, prestige.prestige)
xy_outlier = prestige.loc['minister', ['income', 'prestige']]
ax1.annotate('Minister', xy_outlier, xy_outlier + 1, fontsize=16)
ax2 = fig.add_subplot(212, xlabel='Education', ylabel='Prestige')
ax2.scatter(prestige.education, prestige.prestige)
ols_model = ols('prestige ~ income + education', prestige).fit()
print(ols_model.summary())
infl = ols_model.get_influence()
student = infl.summary_frame()['student_resid']
print(student)
print(student.loc[np.abs(student) > 2])
print(infl.summary_frame().loc['minister'])
sidak = ols_model.outlier_test('sidak')
sidak.sort_values('unadj_p', inplace=True)
print(sidak)
fdr = ols_model.outlier_test('fdr_bh')
fdr.sort_values('unadj_p', inplace=True)
print(fdr)
rlm_model = rlm('prestige ~ income + education', prestige).fit()
print(rlm_model.summary())
print(rlm_model.weights)
# ### Hertzprung Russell data for Star Cluster CYG 0B1 - Leverage Points
# * Data is on the luminosity and temperature of 47 stars in the direction
# of Cygnus.
dta = sm.datasets.get_rdataset("starsCYG", "robustbase", cache=True).data
from matplotlib.patches import Ellipse
fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(
111,
xlabel='log(Temp)',
ylabel='log(Light)',
title='Hertzsprung-Russell Diagram of Star Cluster CYG OB1')
ax.scatter(*dta.values.T)
# highlight outliers
e = Ellipse((3.5, 6), .2, 1, alpha=.25, color='r')
ax.add_patch(e)
ax.annotate(
'Red giants',
xy=(3.6, 6),
xytext=(3.8, 6),
arrowprops=dict(facecolor='black', shrink=0.05, width=2),
horizontalalignment='left',
verticalalignment='bottom',
clip_on=True, # clip to the axes bounding box
fontsize=16,
)
# annotate these with their index
for i, row in dta.loc[dta['log.Te'] < 3.8].iterrows():
ax.annotate(i, row, row + .01, fontsize=14)
xlim, ylim = ax.get_xlim(), ax.get_ylim()
from IPython.display import Image
Image(filename='star_diagram.png')
y = dta['log.light']
X = sm.add_constant(dta['log.Te'], prepend=True)
ols_model = sm.OLS(y, X).fit()
abline_plot(model_results=ols_model, ax=ax)
rlm_mod = sm.RLM(y, X, sm.robust.norms.TrimmedMean(.5)).fit()
abline_plot(model_results=rlm_mod, ax=ax, color='red')
# * Why? Because M-estimators are not robust to leverage points.
infl = ols_model.get_influence()
h_bar = 2 * (ols_model.df_model + 1) / ols_model.nobs
hat_diag = infl.summary_frame()['hat_diag']
hat_diag.loc[hat_diag > h_bar]
sidak2 = ols_model.outlier_test('sidak')
sidak2.sort_values('unadj_p', inplace=True)
print(sidak2)
fdr2 = ols_model.outlier_test('fdr_bh')
fdr2.sort_values('unadj_p', inplace=True)
print(fdr2)
# * Let's delete that line
l = ax.lines[-1]
l.remove()
del l
weights = np.ones(len(X))
weights[X[X['log.Te'] < 3.8].index.values - 1] = 0
wls_model = sm.WLS(y, X, weights=weights).fit()
abline_plot(model_results=wls_model, ax=ax, color='green')
# * MM estimators are good for this type of problem, unfortunately, we do
# not yet have these yet.
# * It's being worked on, but it gives a good excuse to look at the R cell
# magics in the notebook.
yy = y.values[:, None]
xx = X['log.Te'].values[:, None]
# **Note**: The R code and the results in this notebook has been converted
# to markdown so that R is not required to build the documents. The R
# results in the notebook were computed using R 3.5.1 and robustbase 0.93.
# ```ipython
# %load_ext rpy2.ipython
#
# %R library(robustbase)
# %Rpush yy xx
# %R mod <- lmrob(yy ~ xx);
# %R params <- mod$coefficients;
# %Rpull params
# ```
# ```ipython
# %R print(mod)
# ```
# ```
# Call:
# lmrob(formula = yy ~ xx)
# \--> method = "MM"
# Coefficients:
# (Intercept) xx
# -4.969 2.253
# ```
params = [-4.969387980288108, 2.2531613477892365] # Computed using R
print(params[0], params[1])
abline_plot(intercept=params[0], slope=params[1], ax=ax, color='red')
# ### Exercise: Breakdown points of M-estimator
np.random.seed(12345)
nobs = 200
beta_true = np.array([3, 1, 2.5, 3, -4])
X = np.random.uniform(-20, 20, size=(nobs, len(beta_true) - 1))
# stack a constant in front
X = sm.add_constant(X, prepend=True) # np.c_[np.ones(nobs), X]
mc_iter = 500
contaminate = .25 # percentage of response variables to contaminate
all_betas = []
for i in range(mc_iter):
y = np.dot(X, beta_true) + np.random.normal(size=200)
random_idx = np.random.randint(0, nobs, size=int(contaminate * nobs))
y[random_idx] = np.random.uniform(-750, 750)
beta_hat = sm.RLM(y, X).fit().params
all_betas.append(beta_hat)
all_betas = np.asarray(all_betas)
se_loss = lambda x: np.linalg.norm(x, ord=2)**2
se_beta = lmap(se_loss, all_betas - beta_true)
# #### Squared error loss
np.array(se_beta).mean()
all_betas.mean(0)
beta_true
se_loss(all_betas.mean(0) - beta_true)
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