"""James-Stein"""
import numpy as np
import pandas as pd
import scipy
from scipy import optimize
from category_encoders.ordinal import OrdinalEncoder
import category_encoders.utils as util
from sklearn.utils.random import check_random_state
__author__ = 'Jan Motl'
[docs]class JamesSteinEncoder(util.BaseEncoder, util.SupervisedTransformerMixin):
"""James-Stein estimator.
Supported targets: binomial and continuous. For polynomial target support, see PolynomialWrapper.
For feature value `i`, James-Stein estimator returns a weighted average of:
1. The mean target value for the observed feature value `i`.
2. The mean target value (regardless of the feature value).
This can be written as::
JS_i = (1-B)*mean(y_i) + B*mean(y)
The question is, what should be the weight `B`?
If we put too much weight on the conditional mean value, we will overfit.
If we put too much weight on the global mean, we will underfit.
The canonical solution in machine learning is to perform cross-validation.
However, Charles Stein came with a closed-form solution to the problem.
The intuition is: If the estimate of `mean(y_i)` is unreliable (`y_i` has high variance),
we should put more weight on `mean(y)`. Stein put it into an equation as::
B = var(y_i) / (var(y_i)+var(y))
The only remaining issue is that we do not know `var(y)`, let alone `var(y_i)`.
Hence, we have to estimate the variances. But how can we reliably estimate the
variances, when we already struggle with the estimation of the mean values?!
There are multiple solutions:
1. If we have the same count of observations for each feature value `i` and all
`y_i` are close to each other, we can pretend that all `var(y_i)` are identical.
This is called a pooled model.
2. If the observation counts are not equal, it makes sense to replace the variances
with squared standard errors, which penalize small observation counts::
SE^2 = var(y)/count(y)
This is called an independent model.
James-Stein estimator has, however, one practical limitation - it was defined
only for normal distributions. If you want to apply it for binary classification,
which allows only values {0, 1}, it is better to first convert the mean target value
from the bound interval <0,1> into an unbounded interval by replacing mean(y)
with log-odds ratio::
log-odds_ratio_i = log(mean(y_i)/mean(y_not_i))
This is called binary model. The estimation of parameters of this model is, however,
tricky and sometimes it fails fatally. In these situations, it is better to use beta
model, which generally delivers slightly worse accuracy than binary model but does
not suffer from fatal failures.
Parameters
----------
verbose: int
integer indicating verbosity of the output. 0 for none.
cols: list
a list of columns to encode, if None, all string columns will be encoded.
drop_invariant: bool
boolean for whether or not to drop encoded columns with 0 variance.
return_df: bool
boolean for whether to return a pandas DataFrame from transform (otherwise it will be a numpy array).
handle_missing: str
options are 'return_nan', 'error' and 'value', defaults to 'value', which returns the prior probability.
handle_unknown: str
options are 'return_nan', 'error' and 'value', defaults to 'value', which returns the prior probability.
model: str
options are 'pooled', 'beta', 'binary' and 'independent', defaults to 'independent'.
randomized: bool,
adds normal (Gaussian) distribution noise into training data in order to decrease overfitting (testing data are untouched).
sigma: float
standard deviation (spread or "width") of the normal distribution.
Example
-------
>>> from category_encoders import *
>>> import pandas as pd
>>> from sklearn.datasets import fetch_openml
>>> bunch = fetch_openml(name="house_prices", as_frame=True)
>>> display_cols = ["Id", "MSSubClass", "MSZoning", "LotFrontage", "YearBuilt", "Heating", "CentralAir"]
>>> y = bunch.target
>>> X = pd.DataFrame(bunch.data, columns=bunch.feature_names)[display_cols]
>>> enc = JamesSteinEncoder(cols=['CentralAir', 'Heating']).fit(X, y)
>>> numeric_dataset = enc.transform(X)
>>> print(numeric_dataset.info())
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 1460 entries, 0 to 1459
Data columns (total 7 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 Id 1460 non-null float64
1 MSSubClass 1460 non-null float64
2 MSZoning 1460 non-null object
3 LotFrontage 1201 non-null float64
4 YearBuilt 1460 non-null float64
5 Heating 1460 non-null float64
6 CentralAir 1460 non-null float64
dtypes: float64(6), object(1)
memory usage: 80.0+ KB
None
References
----------
.. [1] Parametric empirical Bayes inference: Theory and applications, equations 1.19 & 1.20, from
https://www.jstor.org/stable/2287098
.. [2] Empirical Bayes for multiple sample sizes, from
http://chris-said.io/2017/05/03/empirical-bayes-for-multiple-sample-sizes/
.. [3] Shrinkage Estimation of Log-odds Ratios for Comparing Mobility Tables, from
https://journals.sagepub.com/doi/abs/10.1177/0081175015570097
.. [4] Stein's paradox and group rationality, from
http://www.philos.rug.nl/~romeyn/presentation/2017_romeijn_-_Paris_Stein.pdf
.. [5] Stein's Paradox in Statistics, from
http://statweb.stanford.edu/~ckirby/brad/other/Article1977.pdf
"""
prefit_ordinal = True
encoding_relation = util.EncodingRelation.ONE_TO_ONE
def __init__(self, verbose=0, cols=None, drop_invariant=False, return_df=True, handle_unknown='value',
handle_missing='value', model='independent', random_state=None, randomized=False, sigma=0.05):
super().__init__(verbose=verbose, cols=cols, drop_invariant=drop_invariant, return_df=return_df,
handle_unknown=handle_unknown, handle_missing=handle_missing)
self.ordinal_encoder = None
self.mapping = None
self.random_state = random_state
self.randomized = randomized
self.sigma = sigma
self.model = model
def _fit(self, X, y, **kwargs):
self.ordinal_encoder = OrdinalEncoder(
verbose=self.verbose,
cols=self.cols,
handle_unknown='value',
handle_missing='value'
)
self.ordinal_encoder = self.ordinal_encoder.fit(X)
X_ordinal = self.ordinal_encoder.transform(X)
# Training
if self.model == 'independent':
self.mapping = self._train_independent(X_ordinal, y)
elif self.model == 'pooled':
self.mapping = self._train_pooled(X_ordinal, y)
elif self.model == 'beta':
self.mapping = self._train_beta(X_ordinal, y)
elif self.model == 'binary':
# The label must be binary with values {0,1}
unique = y.unique()
if len(unique) != 2:
raise ValueError("The target column y must be binary. But the target contains " + str(len(unique)) + " unique value(s).")
if y.isna().any():
raise ValueError("The target column y must not contain missing values.")
if np.max(unique) < 1:
raise ValueError("The target column y must be binary with values {0, 1}. Value 1 was not found in the target.")
if np.min(unique) > 0:
raise ValueError("The target column y must be binary with values {0, 1}. Value 0 was not found in the target.")
# Perform the training
self.mapping = self._train_log_odds_ratio(X_ordinal, y)
else:
raise ValueError("model='" + str(self.model) + "' is not a recognized option")
def _transform(self, X, y=None):
X = self.ordinal_encoder.transform(X)
if self.handle_unknown == 'error':
if X[self.cols].isin([-1]).any().any():
raise ValueError('Unexpected categories found in dataframe')
# Loop over columns and replace nominal values with WOE
X = self._score(X, y)
return X
def _more_tags(self):
tags = super()._more_tags()
tags["predict_depends_on_y"] = True
return tags
def _train_pooled(self, X, y):
# Implemented based on reference [1]
# Initialize the output
mapping = {}
# Calculate global statistics
prior = y.mean()
target_var = y.var()
global_count = len(y)
for switch in self.ordinal_encoder.category_mapping:
col = switch.get('col')
values = switch.get('mapping')
# Calculate sum and count of the target for each unique value in the feature col
stats = y.groupby(X[col]).agg(['mean', 'count'])
# See: Computer Age Statistical Inference: Algorithms, Evidence, and Data Science (Bradley Efron & Trevor Hastie, 2016)
# Equations 7.19 and 7.20.
# Note: The equations assume normal distribution of the label. But our label is p(y|x),
# which is definitely not normally distributed as probabilities are bound to lie on interval 0..1.
# We make this approximation because Efron does it as well.
# Equation 7.19
# Explanation of the equation:
# https://stats.stackexchange.com/questions/191444/variance-in-estimating-p-for-a-binomial-distribution
# if stats['count'].var() > 0:
# warnings.warn('The pooled model assumes that each category is observed exactly N times. This was violated in "' + str(col) +'" column. Consider comparing the accuracy of this model to "independent" model.')
# This is a parametric estimate of var(p) in the binomial distribution.
# We do not use it because we also want to support non-binary targets.
# The difference in the estimates is small.
# variance = prior * (1 - prior) / stats['count'].mean()
# This is a squared estimate of standard error of the mean:
# https://en.wikipedia.org/wiki/Standard_error
variance = target_var/(stats['count'].mean())
# Equation 7.20
SSE = ((stats['mean']-prior)**2).sum() # Sum of Squared Errors
if SSE > 0: # We have to avoid division by zero
B = ((len(stats['count'])-3)*variance) / SSE
B = B.clip(0, 1)
estimate = prior + (1 - B) * (stats['mean'] - prior)
else:
estimate = stats['mean']
# Ignore unique values. This helps to prevent overfitting on id-like columns
# This works better than: estimate[stats['count'] == 1] = prior
if len(stats['mean']) == global_count:
estimate[:] = prior
if self.handle_unknown == 'return_nan':
estimate.loc[-1] = np.nan
elif self.handle_unknown == 'value':
estimate.loc[-1] = prior
if self.handle_missing == 'return_nan':
estimate.loc[values.loc[np.nan]] = np.nan
elif self.handle_missing == 'value':
estimate.loc[-2] = prior
# Store the estimate for transform() function
mapping[col] = estimate
return mapping
def _train_independent(self, X, y):
# Implemented based on reference [2]
# Initialize the output
mapping = {}
# Calculate global statistics
prior = y.mean()
global_count = len(y)
global_var = y.var()
for switch in self.ordinal_encoder.category_mapping:
col = switch.get('col')
values = switch.get('mapping')
# Calculate sum and count of the target for each unique value in the feature col
stats = y.groupby(X[col]).agg(['mean', 'var'])
i_var = stats['var'].fillna(0) # When we do not have more than 1 sample, assume 0 variance
unique_cnt = len(X[col].unique())
# See: Parametric Empirical Bayes Inference: Theory and Applications (Morris, 1983)
# Equations 1.19 and 1.20.
# Note: The equations assume normal distribution of the label. But our label is p(y|x),
# which is definitely not normally distributed as probabilities are bound to lie on interval 0..1.
# Nevertheless, it seems to perform surprisingly well. This is in agreement with:
# Data Analysis with Stein's Estimator and Its Generalizations (Efron & Morris, 1975)
# The equations are similar to James-Stein estimator, as listed in:
# Stein's Paradox in Statistics (Efron & Morris, 1977)
# Or:
# Computer Age Statistical Inference: Algorithms, Evidence, and Data Science (Efron & Hastie, 2016)
# Equations 7.19 and 7.20.
# The difference is that they have equal count of observations per estimated variable, while we generally
# do not have that. Nice discussion about that is given at:
# http://chris-said.io/2017/05/03/empirical-bayes-for-multiple-sample-sizes/
smoothing = i_var / (global_var + i_var) * (unique_cnt-3) / (unique_cnt-1)
smoothing = 1 - smoothing
smoothing = smoothing.clip(lower=0, upper=1) # Smoothing should be in the interval <0,1>
estimate = smoothing*(stats['mean']) + (1-smoothing)*prior
# Ignore unique values. This helps to prevent overfitting on id-like columns
if len(stats['mean']) == global_count:
estimate[:] = prior
if self.handle_unknown == 'return_nan':
estimate.loc[-1] = np.nan
elif self.handle_unknown == 'value':
estimate.loc[-1] = prior
if self.handle_missing == 'return_nan':
estimate.loc[values.loc[np.nan]] = np.nan
elif self.handle_missing == 'value':
estimate.loc[-2] = prior
# Store the estimate for transform() function
mapping[col] = estimate
return mapping
def _train_log_odds_ratio(self, X, y):
# Implemented based on reference [3]
# Initialize the output
mapping = {}
# Calculate global statistics
global_sum = y.sum()
global_count = y.count()
# Iterative estimation of mu and sigma as given on page 9.
# This problem is traditionally solved with Newton-Raphson method:
# https://en.wikipedia.org/wiki/Newton%27s_method
# But we just use sklearn minimizer.
def get_best_sigma(sigma, mu_k, sigma_k, K):
global mu # Ugly. But I want to be able to read it once the optimization ends.
w_k = 1. / (sigma ** 2 + sigma_k ** 2) # Weights depends on sigma
mu = sum(w_k * mu_k) / sum(w_k) # Mu transitively depends on sigma
total = sum(w_k * (mu_k - mu) ** 2) # We want this to be close to K-1
loss = abs(total - (K - 1))
return loss
for switch in self.ordinal_encoder.category_mapping:
col = switch.get('col')
values = switch.get('mapping')
# Calculate sum and count of the target for each unique value in the feature col
stats = y.groupby(X[col]).agg(['sum', 'count']) # Count of x_{i,+} and x_i
# Create 2x2 contingency table
crosstable = pd.DataFrame()
crosstable['E-A-'] = global_count - stats['count'] + stats['sum'] - global_sum
crosstable['E-A+'] = stats['count'] - stats['sum']
crosstable['E+A-'] = global_sum - stats['sum']
crosstable['E+A+'] = stats['sum']
index = crosstable.index
crosstable = np.array(crosstable, dtype=np.float32) # The argument unites the types into float
# Count of contingency tables.
K = len(crosstable)
# Ignore id-like columns. This helps to prevent overfitting.
if K == global_count:
estimate = pd.Series(0, index=values)
else:
if K > 1: # We want to avoid division by zero in y_k calculation
# Estimate log-odds ratios with Yates correction as listed on page 5.
mu_k = np.log((crosstable[:, 0] + 0.5) * (crosstable[:, 3] + 0.5) / ((crosstable[:, 1] + 0.5) * (crosstable[:, 2] + 0.5)))
# Standard deviation estimate for 2x2 contingency table as given in equation 2.
# The explanation of the equation is given in:
# https://stats.stackexchange.com/questions/266098/how-do-i-calculate-the-standard-deviation-of-the-log-odds
sigma_k = np.sqrt(np.sum(1. / (crosstable + 0.5), axis=1))
# Estimate the sigma and mu. Sigma is non-negative.
result = scipy.optimize.minimize(get_best_sigma, x0=1e-4, args=(mu_k, sigma_k, K), bounds=[(0, np.inf)], method='TNC', tol=1e-12, options={'gtol': 1e-12, 'ftol': 1e-12, 'eps': 1e-12})
sigma = result.x[0]
# Empirical Bayes follows equation 7.
# However, James-Stein estimator behaves perversely when K < 3. Hence, we clip the B into interval <0,1>.
# Literature reference for the clipping:
# Estimates of Income for Small Places: An Application of James-Stein Procedures to Census Data (Fay & Harriout, 1979),
# page 270.
B = (K - 3) * sigma_k ** 2 / ((K - 1) * (sigma ** 2 + sigma_k ** 2))
B = B.clip(0, 1)
y_k = mu + (1 - B) * (mu_k - mu)
# Convert Numpy vector back into Series
estimate = pd.Series(y_k, index=index)
else:
estimate = pd.Series(0, index=values)
if self.handle_unknown == 'return_nan':
estimate.loc[-1] = np.nan
elif self.handle_unknown == 'value':
estimate.loc[-1] = 0
if self.handle_missing == 'return_nan':
estimate.loc[values.loc[np.nan]] = np.nan
elif self.handle_missing == 'value':
estimate.loc[-2] = 0
# Store the estimate for transform() function
mapping[col] = estimate
return mapping
def _train_beta(self, X, y):
# Implemented based on reference [4]
# Initialize the output
mapping = {}
# Calculate global statistics
prior = y.mean()
global_count = len(y)
for switch in self.ordinal_encoder.category_mapping:
col = switch.get('col')
values = switch.get('mapping')
# Calculate sum and count of the target for each unique value in the feature col
stats = y.groupby(X[col]).agg(['mean', 'count'])
# See: Stein's paradox and group rationality (Romeijn, 2017), page 14
smoothing = stats['count'] / (stats['count'] + global_count)
estimate = smoothing*(stats['mean']) + (1-smoothing)*prior
# Ignore unique values. This helps to prevent overfitting on id-like columns
if len(stats['mean']) == global_count:
estimate[:] = prior
if self.handle_unknown == 'return_nan':
estimate.loc[-1] = np.nan
elif self.handle_unknown == 'value':
estimate.loc[-1] = prior
if self.handle_missing == 'return_nan':
estimate.loc[values.loc[np.nan]] = np.nan
elif self.handle_missing == 'value':
estimate.loc[-2] = prior
# Store the estimate for transform() function
mapping[col] = estimate
return mapping
# todo this score function is copied 4 times
def _score(self, X, y):
for col in self.cols:
# Score the column
X[col] = X[col].map(self.mapping[col])
# Randomization is meaningful only for training data -> we do it only if y is present
if self.randomized and y is not None:
random_state_generator = check_random_state(self.random_state)
X[col] = (X[col] * random_state_generator.normal(1., self.sigma, X[col].shape[0]))
return X