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Advanced Analysis Techniques in HEP

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A reasonable man adapts himself to the world.

An unreasonable man persists to adapts the world to himself.

So, all progress depends on the unreasonable one.

- Bernard Shaw

Pushpa Bhat

Fermilab

ACAT2000

Fermilab, IL

October 2000

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Introduction
- Intelligent Detectors
- Moving intelligence closer to action

- Optimal Analysis Methods
- The Neural Network Revolution
- New Searches & Precision Measurements
- Discovery reach for the Higgs Boson
- Measuring Top quark mass, Higgs mass

- Sophisticated Approaches
- Probabilistic Approach to Data Analysis
- Summary

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

World before

Experiment

World After

Experiment

Express Analysis

Data Collection

Data Transformation

Data Interpretation

Feature Extraction

Global Decision

Data Collection

Data

Organization

Reduction

Analysis

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Data analysis starts when a high energy event occurs
- Transform electronic data into useful “physics” information in real-time
- Move intelligence closer to action!

- Algorithm-specific hardware
- Neural Networks in Silicon

- Configurable hardware
- FPGAs, DSPs – Implement “smart” algorithms in hardware

- Innovative data management on-line + “smart” algorithms in hardware
- Data in RAM disk & AI algorithms in FPGAs

- Expert Systems for Control & Monitoring

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Particle Identification
- e-ID, t-ID, b-ID, e/g, q/g

- Signal/Background Event Classification
- Signals of new physics are rare and small
(Finding a “jewel” in a hay-stack)

- Signals of new physics are rare and small
- Parameter Estimation
- t mass, H mass, track parameters, for example

- Function Approximation
- Correction functions, tag rates, fake rates

- Data Exploration
- Knowledge Discovery via data-mining
- Data-driven extraction of information, latent structure analysis

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- The measurements being multivariate, the optimal methods of analyses are necessarily multivariate
- Discriminant Analysis: Partition multidimensional variable space, identify boundaries
- Cluster Analysis: Assign objects to groups based on similarity
- Examples
- Fisher linear discriminant, Gaussian classifier
- Kernel-based methods, K-nearest neighbor (clustering) methods
- Adaptive/AI methods

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

x1

D(x1,x2)=2.014x1 + 1.592x2

x2

x2

x1

- Because they are optimal!

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Mth Order Polynomial Fit

M=1

M=3

M=10

x2

x2

x2

x1

x1

x1

- Also, they need to have optimal flexibility/complexity

Flexible

Highly flexible

Simple

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Keep it simple

As simple as possible

Not any simpler

- Einstein

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

defines decision boundaries

that minimize the probability

of misclassification

Posterior probability

So, the problem mathematically reduces to that of calculating

r(x), the Bayes Discriminant Function or probability densities

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Histogramming:
The basic problem of non-parametric density estimation is very simple!

- Histogram data in M bins in each of the d feature variables
- Md bins Curse Of Dimensionality
- In high dimensions, we would either require a huge number of data points or most of the bins would be empty leading to an estimated density of zero.
- But, the variables are generally correlated and hence tend to be restricted to a sub-space Intrinsic Dimensionality

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Akin to Histogramming but adopts importance sampling
- Place in d-dimensional space a hypercube of side h centered on each data point x,
The estimate will have discontinuities

Can be smoothed out using different forms for kernel functions H(u). A common choice is a multivariate kernel

N = Number of data points

H(u) = 1 if xn in the hypercube

= 0 otherwise

h=smoothing

parameter

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Place a hyper-sphere centered at each data point x and allow the radius to grow to a volume V until it contains K data points. Then, density at x
- If our data set contains Nk points in class Ck and N points in total, then

N = Number of data points

Kk = # of points in volume

V for class Ck

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Output of a feed forward neural network can approximate the Bayesian posterior probability p(s|x,y)

Directly without estimating class-conditional probabilities

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Consider the sum

Where

di = 1 for signal

= 0 for background

= vector of parameters

Then

in the limit of large data samples and provided that the

function n(x,y,) is flexible enough.

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

x1

x2

DNN

x3

x4

- NN estimates a mapping function without requiring a mathematical description of how the output formally depends on the input.
- The “hidden” transformation functions, g, adapt themselves to the data as part of the training process. The number of such functions need to grow only as the complexity of the problem grows.

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Discriminant variables

shaded = top

The Discriminants

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Measuring the Top Quark Mass

Background-rich

DØ Lepton+jets

Signal-rich

mt = 173.3 ± 5.6(stat.) ± 6.2 (syst.) GeV/c2

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Strategy for Discovering the Higgs Boson at the Tevatron

P.C. Bhat, R. Gilmartin, H. Prosper, PRD 62 (2000)

hep-ph/0001152

- MH = GeV/c2MH < 225 GeV/c2 at 95% C.L.

Hints from the Analysis of Precision Data

LEP Electroweak Group, http://www.cern.ch/LEPEWWG/plots/summer99

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Signal Processes
- Backgrounds
- Event generation
- WH, ZH, ZZ and Top with PYTHIA
- Wbb, Zbb with CompHEP, fragmentation with PYTHIA

- Detector modeling
- SHW (http://www.physics.rutgers.edu/~jconway/soft/shw/shw.html)
- Trigger, Tracking, Jet-finding
- b-tagging (double b-tag efficiency ~ 45%)
- Di-jet mass resolution ~ 14%

- SHW (http://www.physics.rutgers.edu/~jconway/soft/shw/shw.html)

- (Scaled down to 10% for RunII Higgs Studies)

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

WH Results from NN Analysis

MH = 100 GeV/c2

WH vs Wbb

WH

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

A good chance of discovery up to MH= 130 GeV/c2 with 20-30fb-1

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Use mjj and HT (= Etjets ) to train NNs to predict the Higgs boson mass

13.8%

12.2%

13.1%

11..3%

13%

11%

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Committees of Networks
- Performance can be better than the best single network

- Stacks of Networks
- Control both bias and variance

- Mixture of Experts
- Decompose complex problems

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- Provides probabilistic information on each parameter of a model (SUSY, for example) via marginalization over other parameters
- Bayesian method enables straight-forward and meaningful model comparisons. It also allows treatment of all uncertainties in a consistent manner.
- Mathematically linked to adaptive algorithms such as Neural Networks (NN)
- Hybrid methods involving NN for probability density estimation and Bayesian treatement can be very powerful

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

- We are building very sophisticated equipment and will record unprecedented amounts of data in the coming decade
- Use of advanced “optimal” analysis techniques will be crucial to achieve the physics goals
- Multivariate methods, particularly Neural Network techniques, have already made impact on discoveries and precision measurements and will be the methods of choice in future analyses
- Hybrid methods combining “intelligent” algorithms and probabilistic approach will be the wave of the future

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

r(x,y) = constant

defines an optimal

decision boundary

Feature space

S =

B =

Conventional cuts

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

Probabilistic Approach to Data Analysis

(The Wave of the future)

Bayesian Methods

Bayesian Analysis

Posterior

Likelihood

Prior

M model

A uninteresting parameters

p interesting parameters

d data

Bayesian Analysis of Multi-source Data

P.C. Bhat, H. Prosper, S. Snyder, Phys. Lett. B 407(1997) 73

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat

S=80 WH events, assume background distribution described by Wbb.

Results

S/B = 1/10 Mfit= 114 +/- 11GeV/c2

S/B = 1/5 Mfit= 114 +/- 7GeV/c2

ACAT2000 Oct. 16-20, 2000 Pushpa Bhat