M-Estimators for Robust Linear Modeling

In [1]:
%matplotlib inline

from __future__ import print_function
from statsmodels.compat import lmap
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

import statsmodels.api as sm
/build/statsmodels-hMUIZz/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.
  from pandas.core import datetools
  • 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
In [2]:
norms = sm.robust.norms
In [3]:
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

In [4]:
help(norms.AndrewWave.weights)
Help on function weights in module statsmodels.robust.norms:

weights(self, z)
    Andrew's wave weighting function for the IRLS algorithm
    
    The psi function scaled by z
    
    Parameters
    ----------
    z : array-like
        1d array
    
    Returns
    -------
    weights : array
        weights(z) = sin(z/a)/(z/a)     for \|z\| <= a*pi
    
        weights(z) = 0                  for \|z\| > a*pi

In [5]:
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

In [6]:
help(norms.Hampel.weights)
Help on function weights in module statsmodels.robust.norms:

weights(self, z)
    Hampel weighting function for the IRLS algorithm
    
    The psi function scaled by z
    
    Parameters
    ----------
    z : array-like
        1d array
    
    Returns
    -------
    weights : array
        weights(z) = 1                            for \|z\| <= a
    
        weights(z) = a/\|z\|                        for a < \|z\| <= b
    
        weights(z) = a*(c - \|z\|)/(\|z\|*(c-b))      for b < \|z\| <= c
    
        weights(z) = 0                            for \|z\| > c

In [7]:
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

In [8]:
help(norms.HuberT.weights)
Help on function weights in module statsmodels.robust.norms:

weights(self, z)
    Huber's t weighting function for the IRLS algorithm
    
    The psi function scaled by z
    
    Parameters
    ----------
    z : array-like
        1d array
    
    Returns
    -------
    weights : array
        weights(z) = 1          for \|z\| <= t
    
        weights(z) = t/\|z\|      for \|z\| > t

In [9]:
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

In [10]:
help(norms.LeastSquares.weights)
Help on function weights in module statsmodels.robust.norms:

weights(self, z)
    The least squares estimator weighting function for the IRLS algorithm.
    
    The psi function scaled by the input z
    
    Parameters
    ----------
    z : array-like
        1d array
    
    Returns
    -------
    weights : array
        weights(z) = np.ones(z.shape)

In [11]:
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

In [12]:
help(norms.RamsayE.weights)
Help on function weights in module statsmodels.robust.norms:

weights(self, z)
    Ramsay's Ea weighting function for the IRLS algorithm
    
    The psi function scaled by z
    
    Parameters
    ----------
    z : array-like
        1d array
    
    Returns
    -------
    weights : array
        weights(z) = exp(-a*\|z\|)

In [13]:
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

In [14]:
help(norms.TrimmedMean.weights)
Help on function weights in module statsmodels.robust.norms:

weights(self, z)
    Least trimmed mean weighting function for the IRLS algorithm
    
    The psi function scaled by z
    
    Parameters
    ----------
    z : array-like
        1d array
    
    Returns
    -------
    weights : array
        weights(z) = 1             for \|z\| <= c
    
        weights(z) = 0             for \|z\| > c

In [15]:
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

In [16]:
help(norms.TukeyBiweight.weights)
Help on function weights in module statsmodels.robust.norms:

weights(self, z)
    Tukey's biweight weighting function for the IRLS algorithm
    
    The psi function scaled by z
    
    Parameters
    ----------
    z : array-like
        1d array
    
    Returns
    -------
    weights : array
        psi(z) = (1 - (z/c)**2)**2          for \|z\| <= R
    
        psi(z) = 0                          for \|z\| > R

In [17]:
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
In [18]:
x = np.array([1, 2, 3, 4, 500])
  • The mean is not a robust estimator of location
In [19]:
x.mean()
Out[19]:
102.0
  • The median, on the other hand, is a robust estimator with a breakdown point of 50%
In [20]:
np.median(x)
Out[20]:
3.0
  • Analagously for the scale
  • The standard deviation is not robust
In [21]:
x.std()
Out[21]:
199.00251254695254

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,

$$K = \Phi^{-1}(.75)$$
In [22]:
stats.norm.ppf(.75)
Out[22]:
0.67448975019608171
In [23]:
print(x)
[  1   2   3   4 500]
In [24]:
sm.robust.scale.stand_mad(x)
/build/statsmodels-hMUIZz/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/robust/scale.py:49: FutureWarning: stand_mad is deprecated and will be removed in 0.7.0. Use mad instead.
  "instead.", FutureWarning)
Out[24]:
1.482602218505602
In [25]:
np.array([1,2,3,4,5.]).std()
Out[25]:
1.4142135623730951
  • The default for Robust Linear Models is MAD
  • another popular choice is Huber's proposal 2
In [26]:
np.random.seed(12345)
fat_tails = stats.t(6).rvs(40)
In [27]:
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);
In [28]:
print(fat_tails.mean(), fat_tails.std())
0.0688231044811 1.34716332297
In [29]:
print(stats.norm.fit(fat_tails))
(0.068823104481087499, 1.3471633229698652)
In [30]:
print(stats.t.fit(fat_tails, f0=6))
(6, 0.039009187170278181, 1.0564230978488927)
In [31]:
huber = sm.robust.scale.Huber()
loc, scale = huber(fat_tails)
print(loc, scale)
0.04048984333271795 1.1557140047569665
In [32]:
sm.robust.stand_mad(fat_tails)
/build/statsmodels-hMUIZz/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/robust/scale.py:49: FutureWarning: stand_mad is deprecated and will be removed in 0.7.0. Use mad instead.
  "instead.", FutureWarning)
Out[32]:
1.1153350011654151
In [33]:
sm.robust.stand_mad(fat_tails, c=stats.t(6).ppf(.75))
/build/statsmodels-hMUIZz/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/robust/scale.py:49: FutureWarning: stand_mad is deprecated and will be removed in 0.7.0. Use mad instead.
  "instead.", FutureWarning)
Out[33]:
1.0483916565928972
In [34]:
sm.robust.scale.mad(fat_tails)
Out[34]:
1.1153350011654151

Duncan's Occupational Prestige data - M-estimation for outliers

In [35]:
from statsmodels.graphics.api import abline_plot
from statsmodels.formula.api import ols, rlm
In [36]:
prestige = sm.datasets.get_rdataset("Duncan", "car", cache=True).data
In [37]:
print(prestige.head(10))
            type  income  education  prestige
accountant  prof      62         86        82
pilot       prof      72         76        83
architect   prof      75         92        90
author      prof      55         90        76
chemist     prof      64         86        90
minister    prof      21         84        87
professor   prof      64         93        93
dentist     prof      80        100        90
reporter      wc      67         87        52
engineer    prof      72         86        88
In [38]:
fig = plt.figure(figsize=(12,12))
ax1 = fig.add_subplot(211, xlabel='Income', ylabel='Prestige')
ax1.scatter(prestige.income, prestige.prestige)
xy_outlier = prestige.ix['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);
/usr/lib/python3/dist-packages/ipykernel_launcher.py:4: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated
  after removing the cwd from sys.path.
In [39]:
ols_model = ols('prestige ~ income + education', prestige).fit()
print(ols_model.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:               prestige   R-squared:                       0.828
Model:                            OLS   Adj. R-squared:                  0.820
Method:                 Least Squares   F-statistic:                     101.2
Date:                Sat, 30 Sep 2017   Prob (F-statistic):           8.65e-17
Time:                        07:23:18   Log-Likelihood:                -178.98
No. Observations:                  45   AIC:                             364.0
Df Residuals:                      42   BIC:                             369.4
Df Model:                           2                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     -6.0647      4.272     -1.420      0.163     -14.686       2.556
income         0.5987      0.120      5.003      0.000       0.357       0.840
education      0.5458      0.098      5.555      0.000       0.348       0.744
==============================================================================
Omnibus:                        1.279   Durbin-Watson:                   1.458
Prob(Omnibus):                  0.528   Jarque-Bera (JB):                0.520
Skew:                           0.155   Prob(JB):                        0.771
Kurtosis:                       3.426   Cond. No.                         163.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [40]:
infl = ols_model.get_influence()
student = infl.summary_frame()['student_resid']
print(student)
accountant            0.303900
pilot                 0.340920
architect             0.072256
author                0.000711
chemist               0.826578
minister              3.134519
professor             0.768277
dentist              -0.498082
reporter             -2.397022
engineer              0.306225
undertaker           -0.187339
lawyer               -0.303082
physician             0.355687
welfare.worker       -0.411406
teacher               0.050510
conductor            -1.704032
contractor            2.043805
factory.owner         1.602429
store.manager         0.142425
banker                0.508388
bookkeeper           -0.902388
mail.carrier         -1.433249
insurance.agent      -1.930919
store.clerk          -1.760491
carpenter             1.068858
electrician           0.731949
RR.engineer           0.808922
machinist             1.887047
auto.repairman        0.522735
plumber              -0.377954
gas.stn.attendant    -0.666596
coal.miner            1.018527
streetcar.motorman   -1.104485
taxi.driver           0.023322
truck.driver         -0.129227
machine.operator      0.499922
barber                0.173805
bartender            -0.902422
shoe.shiner          -0.429357
cook                  0.127207
soda.clerk           -0.883095
watchman             -0.513502
janitor              -0.079890
policeman             0.078847
waiter               -0.475972
Name: student_resid, dtype: float64
In [41]:
print(student.ix[np.abs(student) > 2])
minister      3.134519
reporter     -2.397022
contractor    2.043805
Name: student_resid, dtype: float64
/usr/lib/python3/dist-packages/ipykernel_launcher.py:1: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated
  """Entry point for launching an IPython kernel.
In [42]:
print(infl.summary_frame().ix['minister'])
dfb_Intercept      0.144937
dfb_income        -1.220939
dfb_education      1.263019
cooks_d            0.566380
dffits             1.433935
dffits_internal    1.303510
hat_diag           0.173058
standard_resid     2.849416
student_resid      3.134519
Name: minister, dtype: float64
/usr/lib/python3/dist-packages/ipykernel_launcher.py:1: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated
  """Entry point for launching an IPython kernel.
In [43]:
sidak = ols_model.outlier_test('sidak')
sidak.sort('unadj_p', inplace=True)
print(sidak)
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-43-88381d4a1ea2> in <module>()
      1 sidak = ols_model.outlier_test('sidak')
----> 2 sidak.sort('unadj_p', inplace=True)
      3 print(sidak)

/usr/lib/python3/dist-packages/pandas/core/generic.py in __getattr__(self, name)
   3079             if name in self._info_axis:
   3080                 return self[name]
-> 3081             return object.__getattribute__(self, name)
   3082 
   3083     def __setattr__(self, name, value):

AttributeError: 'DataFrame' object has no attribute 'sort'
In [44]:
fdr = ols_model.outlier_test('fdr_bh')
fdr.sort('unadj_p', inplace=True)
print(fdr)
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-44-69e9cb113222> in <module>()
      1 fdr = ols_model.outlier_test('fdr_bh')
----> 2 fdr.sort('unadj_p', inplace=True)
      3 print(fdr)

/usr/lib/python3/dist-packages/pandas/core/generic.py in __getattr__(self, name)
   3079             if name in self._info_axis:
   3080                 return self[name]
-> 3081             return object.__getattribute__(self, name)
   3082 
   3083     def __setattr__(self, name, value):

AttributeError: 'DataFrame' object has no attribute 'sort'
In [45]:
rlm_model = rlm('prestige ~ income + education', prestige).fit()
print(rlm_model.summary())
                    Robust linear Model Regression Results                    
==============================================================================
Dep. Variable:               prestige   No. Observations:                   45
Model:                            RLM   Df Residuals:                       42
Method:                          IRLS   Df Model:                            2
Norm:                          HuberT                                         
Scale Est.:                       mad                                         
Cov Type:                          H1                                         
Date:                Sat, 30 Sep 2017                                         
Time:                        07:23:18                                         
No. Iterations:                    18                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     -7.1107      3.879     -1.833      0.067     -14.713       0.492
income         0.7015      0.109      6.456      0.000       0.489       0.914
education      0.4854      0.089      5.441      0.000       0.311       0.660
==============================================================================

If the model instance has been used for another fit with different fit
parameters, then the fit options might not be the correct ones anymore .
In [46]:
print(rlm_model.weights)
accountant            1.000000
pilot                 1.000000
architect             1.000000
author                1.000000
chemist               1.000000
minister              0.344596
professor             1.000000
dentist               1.000000
reporter              0.441669
engineer              1.000000
undertaker            1.000000
lawyer                1.000000
physician             1.000000
welfare.worker        1.000000
teacher               1.000000
conductor             0.538445
contractor            0.552262
factory.owner         0.706169
store.manager         1.000000
banker                1.000000
bookkeeper            1.000000
mail.carrier          0.690764
insurance.agent       0.533499
store.clerk           0.618656
carpenter             0.935848
electrician           1.000000
RR.engineer           1.000000
machinist             0.570360
auto.repairman        1.000000
plumber               1.000000
gas.stn.attendant     1.000000
coal.miner            0.963821
streetcar.motorman    0.832870
taxi.driver           1.000000
truck.driver          1.000000
machine.operator      1.000000
barber                1.000000
bartender             1.000000
shoe.shiner           1.000000
cook                  1.000000
soda.clerk            1.000000
watchman              1.000000
janitor               1.000000
policeman             1.000000
waiter                1.000000
dtype: float64

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.
In [47]:
dta = sm.datasets.get_rdataset("starsCYG", "robustbase", cache=True).data
In [48]:
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.ix[dta['log.Te'] < 3.8].iterrows():
    ax.annotate(i, row, row + .01, fontsize=14)
xlim, ylim = ax.get_xlim(), ax.get_ylim()
/usr/lib/python3/dist-packages/ipykernel_launcher.py:15: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated
  from ipykernel import kernelapp as app
In [49]:
from IPython.display import Image
Image(filename='star_diagram.png')
---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
<ipython-input-49-6b65229361a7> in <module>()
      1 from IPython.display import Image
----> 2 Image(filename='star_diagram.png')

/usr/lib/python3/dist-packages/IPython/core/display.py in __init__(self, data, url, filename, format, embed, width, height, retina, unconfined, metadata)
    755         self.unconfined = unconfined
    756         self.metadata = metadata
--> 757         super(Image, self).__init__(data=data, url=url, filename=filename)
    758 
    759         if retina:

/usr/lib/python3/dist-packages/IPython/core/display.py in __init__(self, data, url, filename)
    392         self.filename = None if filename is None else unicode_type(filename)
    393 
--> 394         self.reload()
    395         self._check_data()
    396 

/usr/lib/python3/dist-packages/IPython/core/display.py in reload(self)
    777         """Reload the raw data from file or URL."""
    778         if self.embed:
--> 779             super(Image,self).reload()
    780             if self.retina:
    781                 self._retina_shape()

/usr/lib/python3/dist-packages/IPython/core/display.py in reload(self)
    410         """Reload the raw data from file or URL."""
    411         if self.filename is not None:
--> 412             with open(self.filename, self._read_flags) as f:
    413                 self.data = f.read()
    414         elif self.url is not None:

FileNotFoundError: [Errno 2] No such file or directory: 'star_diagram.png'
In [50]:
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)
Out[50]:
In [51]:
rlm_mod = sm.RLM(y, X, sm.robust.norms.TrimmedMean(.5)).fit()
abline_plot(model_results=rlm_mod, ax=ax, color='red')
Out[51]:
  • Why? Because M-estimators are not robust to leverage points.
In [52]:
infl = ols_model.get_influence()
In [53]:
h_bar = 2*(ols_model.df_model + 1 )/ols_model.nobs
hat_diag = infl.summary_frame()['hat_diag']
hat_diag.ix[hat_diag > h_bar]
/usr/lib/python3/dist-packages/ipykernel_launcher.py:3: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated
  This is separate from the ipykernel package so we can avoid doing imports until
Out[53]:
10    0.194103
19    0.194103
29    0.198344
33    0.194103
Name: hat_diag, dtype: float64
In [54]:
sidak2 = ols_model.outlier_test('sidak')
sidak2.sort('unadj_p', inplace=True)
print(sidak2)
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-54-a6c575f5e012> in <module>()
      1 sidak2 = ols_model.outlier_test('sidak')
----> 2 sidak2.sort('unadj_p', inplace=True)
      3 print(sidak2)

/usr/lib/python3/dist-packages/pandas/core/generic.py in __getattr__(self, name)
   3079             if name in self._info_axis:
   3080                 return self[name]
-> 3081             return object.__getattribute__(self, name)
   3082 
   3083     def __setattr__(self, name, value):

AttributeError: 'DataFrame' object has no attribute 'sort'
In [55]:
fdr2 = ols_model.outlier_test('fdr_bh')
fdr2.sort('unadj_p', inplace=True)
print(fdr2)
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-55-cf26bbbd14f6> in <module>()
      1 fdr2 = ols_model.outlier_test('fdr_bh')
----> 2 fdr2.sort('unadj_p', inplace=True)
      3 print(fdr2)

/usr/lib/python3/dist-packages/pandas/core/generic.py in __getattr__(self, name)
   3079             if name in self._info_axis:
   3080                 return self[name]
-> 3081             return object.__getattribute__(self, name)
   3082 
   3083     def __setattr__(self, name, value):

AttributeError: 'DataFrame' object has no attribute 'sort'
  • Let's delete that line
In [56]:
del ax.lines[-1]
In [57]:
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')
---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
<ipython-input-57-e3ec53a40864> in <module>()
      2 weights[X[X['log.Te'] < 3.8].index.values - 1] = 0
      3 wls_model = sm.WLS(y, X, weights=weights).fit()
----> 4 abline_plot(model_results=wls_model, ax=ax, color='green')

/build/statsmodels-hMUIZz/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/graphics/regressionplots.py in abline_plot(intercept, slope, horiz, vert, model_results, ax, **kwargs)
    689 
    690     data_y = [x[0]*slope+intercept, x[1]*slope+intercept]
--> 691     ax.set_xlim(x)
    692     #ax.set_ylim(y)
    693 

/usr/lib/python3/dist-packages/matplotlib/axes/_base.py in set_xlim(self, left, right, emit, auto, **kw)
   2910 
   2911         if emit:
-> 2912             self.callbacks.process('xlim_changed', self)
   2913             # Call all of the other x-axes that are shared with this one
   2914             for other in self._shared_x_axes.get_siblings(self):

/usr/lib/python3/dist-packages/matplotlib/cbook.py in process(self, s, *args, **kwargs)
    547             for cid, proxy in list(six.iteritems(self.callbacks[s])):
    548                 try:
--> 549                     proxy(*args, **kwargs)
    550                 except ReferenceError:
    551                     self._remove_proxy(proxy)

/usr/lib/python3/dist-packages/matplotlib/cbook.py in __call__(self, *args, **kwargs)
    414             mtd = self.func
    415         # invoke the callable and return the result
--> 416         return mtd(*args, **kwargs)
    417 
    418     def __eq__(self, other):

/build/statsmodels-hMUIZz/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/graphics/regressionplots.py in update_datalim(self, ax)
    701             children = ax.get_children()
    702             abline = [children[i] for i in range(len(children))
--> 703                       if isinstance(children[i], ABLine2D)][0]
    704             x = ax.get_xlim()
    705             y = [x[0]*slope+intercept, x[1]*slope+intercept]

IndexError: list index out of range
  • MM estimators are good for this type of problem, unfortunately, we don't 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.
In [58]:
yy = y.values[:,None]
xx = X['log.Te'].values[:,None]
In [59]:
%load_ext rpy2.ipython

%R library(robustbase)
%Rpush yy xx
%R mod <- lmrob(yy ~ xx);
%R params <- mod$coefficients;
%Rpull params
---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
<ipython-input-59-a5161b228cba> in <module>()
----> 1 get_ipython().magic('load_ext rpy2.ipython')
      2 
      3 get_ipython().magic('R library(robustbase)')
      4 get_ipython().magic('Rpush yy xx')
      5 get_ipython().magic('R mod <- lmrob(yy ~ xx);')

/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py in magic(self, arg_s)
   2156         magic_name, _, magic_arg_s = arg_s.partition(' ')
   2157         magic_name = magic_name.lstrip(prefilter.ESC_MAGIC)
-> 2158         return self.run_line_magic(magic_name, magic_arg_s)
   2159 
   2160     #-------------------------------------------------------------------------

/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py in run_line_magic(self, magic_name, line)
   2077                 kwargs['local_ns'] = sys._getframe(stack_depth).f_locals
   2078             with self.builtin_trap:
-> 2079                 result = fn(*args,**kwargs)
   2080             return result
   2081 

<decorator-gen-62> in load_ext(self, module_str)

/usr/lib/python3/dist-packages/IPython/core/magic.py in <lambda>(f, *a, **k)
    186     # but it's overkill for just that one bit of state.
    187     def magic_deco(arg):
--> 188         call = lambda f, *a, **k: f(*a, **k)
    189 
    190         if callable(arg):

/usr/lib/python3/dist-packages/IPython/core/magics/extension.py in load_ext(self, module_str)
     35         if not module_str:
     36             raise UsageError('Missing module name.')
---> 37         res = self.shell.extension_manager.load_extension(module_str)
     38 
     39         if res == 'already loaded':

/usr/lib/python3/dist-packages/IPython/core/extensions.py in load_extension(self, module_str)
     81             if module_str not in sys.modules:
     82                 with prepended_to_syspath(self.ipython_extension_dir):
---> 83                     __import__(module_str)
     84             mod = sys.modules[module_str]
     85             if self._call_load_ipython_extension(mod):

ImportError: No module named 'rpy2'
In [60]:
%R print(mod)
ERROR:root:Line magic function `%R` not found.
In [61]:
print(params)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-61-73ac4b936803> in <module>()
----> 1 print(params)

NameError: name 'params' is not defined
In [62]:
abline_plot(intercept=params[0], slope=params[1], ax=ax, color='green')
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-62-e1a9f35b3320> in <module>()
----> 1 abline_plot(intercept=params[0], slope=params[1], ax=ax, color='green')

NameError: name 'params' is not defined

Exercise: Breakdown points of M-estimator

In [63]:
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
In [64]:
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)
In [65]:
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

In [66]:
np.array(se_beta).mean()
Out[66]:
0.44502948730686182
In [67]:
all_betas.mean(0)
Out[67]:
array([ 2.99711706,  0.99898147,  2.49909344,  2.99712918, -3.99626521])
In [68]:
beta_true
Out[68]:
array([ 3. ,  1. ,  2.5,  3. , -4. ])
In [69]:
se_loss(all_betas.mean(0) - beta_true)
Out[69]:
3.2360913286754188e-05