7: Demonstration of distribution reweighting
Reweighting in HEP is used to minimize difference between real data and Monte-Carlo simulation.
Known process is used, for which real data can be obtained.
Target of reweighting: assign weights to MC such that MC and real data distributions coincide.
hep_ml.reweight contains methods to reweight distributions. Typically we use reweighting of monte-carlo to fight drawbacks of simulation, though there are many applications.
In this example we reweight multidimensional distributions: original and target, the aim is to find new weights for original distribution, such that these multidimensional distributions will coincide.
Here we have a toy example without a real physics meaning.
Pay attention: equality of distributions for each feature \(\neq\) equality of multivariate distributions.
All samples are divided into training and validation part. Training part is used to fit reweighting rule and test part is used to estimate reweighting quality.
[1]:
from xgboost import XGBClassifier
%matplotlib inline
import numpy as np
import pandas as pd
import uproot
from hep_ml import reweight
from matplotlib import pyplot as plt
Downloading data
[2]:
columns = ['hSPD', 'pt_b', 'pt_phi', 'vchi2_b', 'mu_pt_sum']
with uproot.open('https://starterkit.web.cern.ch/starterkit/data/advanced-python-2019/MC_distribution.root',
) as original_file:
original_tree = original_file['tree']
original = original_tree.arrays(library='pd')
with uproot.open('https://starterkit.web.cern.ch/starterkit/data/advanced-python-2019/RD_distribution.root',
) as target_file:
target_tree = target_file['tree']
target = target_tree.arrays(library='pd')
original_weights = np.ones(len(original))
prepare train and test samples
train part is used to train reweighting rule
test part is used to evaluate reweighting rule comparing the following things:
Kolmogorov-Smirnov distances for 1d projections
n-dim distributions using ML (see below).
[3]:
from sklearn.model_selection import train_test_split
# divide original samples into training ant test parts
original_train, original_test = train_test_split(original)
# divide target samples into training ant test parts
target_train, target_test = train_test_split(target)
original_weights_train = np.ones(len(original_train))
original_weights_test = np.ones(len(original_test))
[4]:
from hep_ml.metrics_utils import ks_2samp_weighted
hist_settings = {'bins': 100, 'density': True, 'alpha': 0.7}
def draw_distributions(original, target, new_original_weights):
plt.figure(figsize=[15, 7])
for id, column in enumerate(columns, 1):
xlim = np.percentile(np.hstack([target[column]]), [0.01, 99.99])
plt.subplot(2, 3, id)
plt.hist(original[column], weights=new_original_weights, range=xlim, **hist_settings)
plt.hist(target[column], range=xlim, **hist_settings)
plt.title(column)
print('KS over ', column, ' = ', ks_2samp_weighted(original[column], target[column],
weights1=new_original_weights, weights2=np.ones(len(target), dtype=float)))
Original distributions
KS = Kolmogorov-Smirnov distance: a measure of how well two distributions agree, the lower the distance, the better the agreement. In this case we want a low KS value.
[5]:
# pay attention, actually we have very few data
len(original), len(target)
[5]:
(1000000, 21441)
[6]:
draw_distributions(original, target, original_weights)
KS over hSPD = 0.5203540728277889
KS over pt_b = 0.21639364439970188
KS over pt_phi = 0.4020113592414034
KS over vchi2_b = 0.40466385087324064
KS over mu_pt_sum = 0.21639364439970188
train part of original distribution
[7]:
draw_distributions(original_train, target_train, original_weights_train)
KS over hSPD = 0.5202707363183977
KS over pt_b = 0.21493039800782998
KS over pt_phi = 0.40249413930364264
KS over vchi2_b = 0.4045062387996485
KS over mu_pt_sum = 0.21493039800782998
test part for target distribution
[8]:
draw_distributions(original_test, target_test, original_weights_test)
KS over hSPD = 0.5217258854691396
KS over pt_b = 0.22360259839551955
KS over pt_phi = 0.4022615060621957
KS over vchi2_b = 0.40645434247421147
KS over mu_pt_sum = 0.22360259839551955
Bins-based reweighting in n dimensions
Typical way to reweight distributions is based on bins.
Usually histogram reweighting is used, in each bin the weight of original distribution is multiplied by:
\(m_{bin} = \frac{w_{target}}{w_{original}}\)
where \(w_{target}\) and \(w_{original}\) are the total weight of events in each bin for target and original distributions.
Simple and fast!
Very few (typically, one or two) variables
Reweighting one variable may bring disagreement in others
Which variable to use in reweighting?
[9]:
bins_reweighter = reweight.BinsReweighter(n_bins=20, n_neighs=1.)
bins_reweighter.fit(original_train, target_train)
bins_weights_test = bins_reweighter.predict_weights(original_test)
# validate reweighting rule on the test part comparing 1d projections
draw_distributions(original_test, target_test, bins_weights_test)
KS over hSPD = 0.4139893516055126
KS over pt_b = 0.12833678561871548
KS over pt_phi = 0.28206013153448334
KS over vchi2_b = 0.3449341728390707
KS over mu_pt_sum = 0.12833678561871548
Gradient Boosted Reweighter
This algorithm is inspired by gradient boosting and is able to fight curse of dimensionality. It uses decision trees and special loss function (ReweightLossFunction).
A classifier is trained to discriminate between real data and MC. This means we are able to reweight in several variables rather than just one. GBReweighter from hep_ml is able to handle many variables and requires less data (for the same performance).
GBReweighter supports negative weights (to reweight MC to splotted real data).
[10]:
reweighter = reweight.GBReweighter(n_estimators=50, learning_rate=0.1, max_depth=3, min_samples_leaf=1000,
gb_args={'subsample': 0.7})
reweighter.fit(original_train, target_train)
gb_weights_test = reweighter.predict_weights(original_test)
# validate reweighting rule on the test part comparing 1d projections
draw_distributions(original_test, target_test, gb_weights_test)
KS over hSPD = 0.04057034169281315
KS over pt_b = 0.043103939398626334
KS over pt_phi = 0.03509226207921043
KS over vchi2_b = 0.024237946124960996
KS over mu_pt_sum = 0.043103939398626334
Comparing some simple expressions:
The most interesting is checking some other variables in multidimensional distributions (those are expressed via original variables). Here we can check the KS distance in multidimensional distributions. (The lower the value, the better agreement of distributions.)
[11]:
def check_ks_of_expression(expression):
col_original = original_test.eval(expression, engine='python')
col_target = target_test.eval(expression, engine='python')
w_target = np.ones(len(col_target), dtype='float')
print('No reweight KS:', ks_2samp_weighted(col_original, col_target,
weights1=original_weights_test, weights2=w_target))
print('Bins reweight KS:', ks_2samp_weighted(col_original, col_target,
weights1=bins_weights_test, weights2=w_target))
print('GB Reweight KS:', ks_2samp_weighted(col_original, col_target,
weights1=gb_weights_test, weights2=w_target))
[12]:
check_ks_of_expression('hSPD')
No reweight KS: 0.5217258854691396
Bins reweight KS: 0.4139893516055126
GB Reweight KS: 0.04057034169281315
[13]:
check_ks_of_expression('hSPD * pt_phi')
No reweight KS: 0.09428859951519122
Bins reweight KS: 0.11901534428342514
GB Reweight KS: 0.024008743177935665
[14]:
check_ks_of_expression('hSPD * pt_phi * vchi2_b')
No reweight KS: 0.3741215597842718
Bins reweight KS: 0.3411622138334073
GB Reweight KS: 0.029107023946130517
[15]:
check_ks_of_expression('pt_b * pt_phi / hSPD ')
No reweight KS: 0.483328838276331
Bins reweight KS: 0.38844984008999384
GB Reweight KS: 0.058923845427925586
[16]:
check_ks_of_expression('hSPD * pt_b * vchi2_b / pt_phi')
No reweight KS: 0.49551554374257784
Bins reweight KS: 0.39765854325367306
GB Reweight KS: 0.02673277862696727
GB-discrimination
Let’s check how well a classifier is able to distinguish these distributions.
For this purpose we split the data into train and test, then we train a classifier to distinguish between the real data and MC distributions.
We can use the ROC Area Under Curve as a measure of performance. If ROC AUC = 0.5 on the test sample, the distributions are identical, if ROC AUC = 1.0, they are ideally separable. So we want a ROC AUC as close to 0.5 as possible to know that we cannot separate the reweighted distributions.
[17]:
from sklearn.metrics import roc_auc_score
from sklearn.model_selection import train_test_split
from xgboost import XGBClassifier
data = np.concatenate([original_test, target_test])
labels = np.array([0] * len(original_test) + [1] * len(target_test))
weights = {}
weights['original'] = original_weights_test
weights['bins'] = bins_weights_test
weights['gb_weights'] = gb_weights_test
for name, new_weights in weights.items():
W = np.concatenate([new_weights / new_weights.sum() * len(target_test), [1] * len(target_test)])
Xtr, Xts, Ytr, Yts, Wtr, Wts = train_test_split(data, labels, W, random_state=42, train_size=0.51)
clf = XGBClassifier(subsample=0.8, n_estimators=50).fit(Xtr, Ytr, sample_weight=Wtr)
print(name, roc_auc_score(Yts, clf.predict_proba(Xts)[:, 1], sample_weight=Wts))
original 0.9281898250334095
bins 0.9026183756235051
gb_weights 0.5621824720048235
Great!
To the classifier, the two datasets seem now nearly undistingishable. Can we improve that? The GBReweighter is quite sensible to its hyperparameters, especially to n_estimators. Feel free to go back and increase it (to e.g. 250). What happens?
Answer: we get an even lower score! 0.49! Yeey! Or wait…
What did just happen?
Our algorithm completely overfitted the distribution. The problem is that while for the classification we have a true label for each event, we do not have a true weight for each event, we just have a true distribution so a “true local density” of the events. Or something like that.
As powerful as the GBReweighter is, as much does it need some care to be taken when choosing the hyperparameters as it can easily overfit. And, worse, the overfitting cannot be spotted so easily.
While this is a topic on its own, whatever you do with the GBReweighter, be sure to really validate your result.
A hint of what may goes wrong is given when plotting the weights.
[18]:
plt.hist(weights['gb_weights'], bins=50)
plt.yscale('log')
plt.title('predicted weights')
[18]:
Text(0.5, 1.0, 'predicted weights')
[19]:
np.max(weights['gb_weights']), np.sum(weights['gb_weights'])
[19]:
(np.float64(446.6012189014473), np.float64(69109.23050514855))
With such a high weight for a single event, this does not look desirable. And be aware of ad-hoc solutions: just clipping or removing weights is completely wrong as this would disturb the distribution completely.
A good way to proceed is to play around with the hyperparameters in order to avoid overfitting until the weights distribution looks “reasonable”. Especially we don’t want to have high weights in there if avoidable.
How to tune
First find an appropriate number of estimators.
max_depth basically determines the order of correlations to be taken into account.
n_estimators has a tradeoff vs learning_rate. Increasing the former by a factor of n and reducing the latter by a factor of 1 / n keeps the reweighter with the same capability (e.g. overfitting) but tends to smooth out things. So use this at the end as more estimators take more time.
Folding reweighter
FoldingReweighter uses k folding in order to obtain unbiased weights for the whole distribution.
The hyperparameters have been adjusted here. Be aware that n_estimators=80 with learning_rate=0.01 reads as n_estimators=8 and learning_rate=0.1 (in the above). So we greatly reduced the number of estimators.
[20]:
# define base reweighter
reweighter_base = reweight.GBReweighter(n_estimators=40,
learning_rate=0.02, max_depth=4, min_samples_leaf=100,
gb_args={'subsample': 0.8})
reweighter = reweight.FoldingReweighter(reweighter_base, n_folds=2)
# it is not needed divide data into train/test parts; reweighter can be train on the whole samples
reweighter.fit(original, target)
# predict method provides unbiased weights prediction for the whole sample
# folding reweighter contains two reweighters, each is trained on one half of samples
# during predictions each reweighter predicts another half of samples not used in training
folding_weights = reweighter.predict_weights(original)
draw_distributions(original, target, folding_weights)
KFold prediction using folds column
KS over hSPD = 0.3041597647416483
KS over pt_b = 0.180480045266332
KS over pt_phi = 0.3074360022568821
KS over vchi2_b = 0.29790572565369344
KS over mu_pt_sum = 0.180480045266332
GB discrimination for reweighting rule
[21]:
data = np.concatenate([original, target])
labels = np.array([0] * len(original) + [1] * len(target))
weights = {}
weights['original'] = original_weights
weights['2-folding'] = folding_weights
for name, new_weights in weights.items():
W = np.concatenate([new_weights / new_weights.sum() * len(target), [1] * len(target)])
Xtr, Xts, Ytr, Yts, Wtr, Wts = train_test_split(data, labels, W, random_state=42, train_size=0.51)
clf = XGBClassifier(subsample=0.6, n_estimators=30).fit(Xtr, Ytr, sample_weight=Wtr)
print(name, roc_auc_score(Yts, clf.predict_proba(Xts)[:, 1], sample_weight=Wts))
original 0.9358950292314191
2-folding 0.8270886510400624
[22]:
plt.hist(weights['2-folding'], bins=50)
plt.yscale('log')
plt.title('predicted weights')
[22]:
Text(0.5, 1.0, 'predicted weights')
