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 distibutions: 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 distibutions 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.

%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

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',
                            httpsource={'chunkbytes': 1024*1024, 'limitbytes': 33554432, 'parallel': 64}
                            ) 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',
                          httpsource={'chunkbytes': 1024*1024, 'limitbytes': 33554432, 'parallel': 64}
                          ) 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 distibutions using ML (see below).

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))
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)
        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.

# pay attention, actually we have very few data
len(original), len(target)
(1000000, 21441)
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

draw_distributions(original_train, target_train, original_weights_train)
KS over  hSPD  =  0.5221121691542183
KS over  pt_b  =  0.21843595024713525
KS over  pt_phi  =  0.40291782089568656
KS over  vchi2_b  =  0.40338992039175237
KS over  mu_pt_sum  =  0.21843595024713525

test part for target distribution

draw_distributions(original_test, target_test, original_weights_test)
KS over  hSPD  =  0.5158094963626292
KS over  pt_b  =  0.21419534750949443
KS over  pt_phi  =  0.40059494459977607
KS over  vchi2_b  =  0.41175584070219473
KS over  mu_pt_sum  =  0.21419534750949443

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.

  1. Simple and fast!

  2. Very few (typically, one or two) variables

  3. Reweighting one variable may bring disagreement in others

  4. Which variable to use in reweighting?

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  =  0.4040534148895752
KS over  pt_b  =  0.12002145775868311
KS over  pt_phi  =  0.27647759413374207
KS over  vchi2_b  =  0.35389961651977664
KS over  mu_pt_sum  =  0.12002145775868311

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 functiion (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).

reweighter = reweight.GBReweighter(n_estimators=250, learning_rate=0.1, max_depth=3, min_samples_leaf=1000,
                                   gb_args={'subsample': 0.4})
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  =  0.0476266801308195
KS over  pt_b  =  0.04107472628768527
KS over  pt_phi  =  0.02732110401850485
KS over  vchi2_b  =  0.02464858170446163
KS over  mu_pt_sum  =  0.04107472628768527

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.)

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))
No reweight   KS: 0.5158094963626292
Bins reweight KS: 0.4040534148895752
GB Reweight   KS: 0.0476266801308195
check_ks_of_expression('hSPD * pt_phi')
No reweight   KS: 0.08869513187855049
Bins reweight KS: 0.11471684755288236
GB Reweight   KS: 0.031108499692095226
check_ks_of_expression('hSPD * pt_phi * vchi2_b')
No reweight   KS: 0.3753859947778093
Bins reweight KS: 0.3432208109797275
GB Reweight   KS: 0.021069159559333306
check_ks_of_expression('pt_b * pt_phi / hSPD ')
No reweight   KS: 0.4722377299010102
Bins reweight KS: 0.3779251510100421
GB Reweight   KS: 0.04147639424867178
check_ks_of_expression('hSPD * pt_b * vchi2_b / pt_phi')
No reweight   KS: 0.49342849356537355
Bins reweight KS: 0.4033963425945809
GB Reweight   KS: 0.04488629200530708


Let’s check how well a classifier is able to distinguish these distributions.

For this puprose 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 distibutions 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.

from sklearn.ensemble import GradientBoostingClassifier
from sklearn.metrics import roc_auc_score
from sklearn.model_selection import train_test_split

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 = GradientBoostingClassifier(subsample=0.3, n_estimators=50).fit(Xtr, Ytr, sample_weight=Wtr)

    print(name, roc_auc_score(Yts, clf.predict_proba(Xts)[:, 1], sample_weight=Wts))
original 0.9311815343367776
bins 0.9041915299217058
gb_weights 0.46997399476563595


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.

plt.hist(weights['gb_weights'], bins=50)
plt.title('predicted weights')
Text(0.5, 1.0, 'predicted weights')
np.max(weights['gb_weights']), np.sum(weights['gb_weights'])
(787.303476124238, 71148.85787815567)

With such a high weight for a single event, this does not look desireable. 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.

# define base reweighter
reweighter_base = reweight.GBReweighter(n_estimators=80,
                                        learning_rate=0.01, 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  =  0.3091205360840006
KS over  pt_b  =  0.18091429410583681
KS over  pt_phi  =  0.30773999103507643
KS over  vchi2_b  =  0.2983588490443341
KS over  mu_pt_sum  =  0.18091429410583681

GB discrimination for reweighting rule

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 = GradientBoostingClassifier(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 0.9369179965204988
2-folding 0.8271570683493737
plt.hist(weights['2-folding'], bins=50)
plt.title('predicted weights')
Text(0.5, 1.0, 'predicted weights')