Transcript Pattern Analysis - Rensselaer Polytechnic Institute

Pattern Analysis
Prof. Bennett
Math Model of Learning and
Discovery 2/14/05
Based on Chapter 1 of
Shawe-Taylor and Cristianini
Outline
What is pattern analysis?
Illustrate issues via example
Pattern definitions
Examples of practical tasks
Pattern algorithms
Summary
Pattern Analysis
The automatic detection of patterns in
data from the same source.
Make predictions of new data coming
from the same source.
Data may take many forms:
images, text, records of commercial
transactions, genome sequences, family
tree
Data Driven Analysis
D
Mercury 0.24
Venus 0.62
Earth
1.00
Mars
1.88
Jupiter 11.90
Saturn 29.30
P
0.39
0.72
1.00
1.53
5.31
9.55
P2
0.058
0.38
1.00
3.53
142.0
870.0
Kepler Analyzed Brahe’s Planetary Motion Data
P = Period D = Average Distance from Sun
P3
0.059
0.39
1.00
3.58
141.00
871.00
Found “Regularities”
Observed P3= D2
Developed three laws of planetary motion.
Compressible:
Data can be represented by one column
Predictable:
Discovering hidden relations allow us to
predict other columns.
Third Law is exact.
Data Representation I
Nonlinear Model of D and P
D  P 0
2
3
2
3
ˆ
ˆ
Linear Model of D  D and P  P
2
3
ˆ
ˆ
D  P  D  P 0
Data Representation II
Assume we know plane of orbit, so we
can represent positions as (x,y) pairs
Also know orbit is ellipse
c1 x  c2 y  c3 xy  c4 x  c5 y  c6  0
2
2
Data Representation
c1 x  c2 y  c3 xy  c4 x  c5 y  c6  0
2
2
Pattern is nonlinear function of x,y
2
2
Pattern is linear function of x , y , xy, x, y
Linear relationships are easier to find.
Set of Hypotheses
c1 x  c2 y  c3 xy  c4 x  c5 y  c6  0
2
2
Hypothesis Ellipse compute
c1 , c2 , c3 , c4 , c5 , c6
Hypothesis Circle compute
c1 , c2 , c6
UNDERFITS
Set of Hypotheses
Hypothesis any continuous function
OVERFITS!!!
Depends on size of hypothesis class
Use domain knowledge to limit hypotheses
Approximate Pattern
Noisy Data
Typical Pattern Analysis
Approximate not exact.
Data has errors and omissions.
Cannot predict graduate school
performance from GRE’s and grades
alone.
Best Representation/Model unknown.
Make approximate predictions – need to
address how accurate estimates are.
Definition: Exact Pattern
A general exact pattern, f, for data
source S satisfies
f ( x)  0
for all data x from source S
Approximate Pattern
A general approximate pattern, f, for
data source S satisfies
f ( x)  0
for all data x from source S
Statistical Pattern
A general statistical pattern, f, for data
source S generated iid according to
distribution D satisfies
ED f ( x)  Ex f ( x)  0
for all data x from source S
Two and Multiclass
Classification
Example – Character Recognition
two class - is it an A or not?
multiclass – what letter is it ?
f (z )  L( y, g (x))  0
g is prediction function
L is loss function
y  {1,1} or y  {1, 2,3, 4,..., N }
Regression
Example –Determine drug bioavailability
through the intestine. Estimate
apparent permeability as assayed via
intestinal cell line.
f (z )  L( y, g (x))  0
g is prediction function
L is loss function
yR

Density Estimation
Estimate the probability that a particular
event occurs, p(x). Use it to detect
improbably events like fraud.
f ( x)  0
 f(x)dx  1
x
E ( ln( f ( x))  E (  ln( p( x))  0
Kullback-Lieber divergence
Principal Component Analysis
Find a projection of the data that
captures the major variance in the data.
Eigenfaces - capture essential qualities
of faces to help ID and reduce storage
needs.
Projection
PV : X  V
Residual
f ( x)  PV ( x)  x
2
Minimize expected value of residual
Pattern Analysis Algorithm
A Pattern Analysis Algorithm
input = finite set of data from source S
a.k.a. the training set
output = detector function f
or no patterns detected
Pattern Algorithm Issues
Efficiency and Scalability – memory and
CPU requirements, large data sets
Robustness – find approximate patterns
on noisy data
Stability - discover genuine patterns,
find same problems on different views
of the dataset
Stability
Generalization –
Find pattern on future data
Pattern may exist by chance for finite
sample
Provide statistical guarantee that
pattern truly exist with caveat that with
small probability that algorithm may
have been mislead.
Example
Observe that for state agency that all 20
babies adopted in last 10 years from country
x are girls.
Pattern, only girls are available for adoption
from that country.
With probability p=(0.5)220 could observe data
even if chance of girls and boys equally likely.
So with chance p, we were mislead.
Statistical Learning Theory
Produce a pattern based on a finite
sample. Provide bounds on the
probability that pattern approximately
represents a true pattern with some
probability.
Probably Approximately Correct
Recoding Strategy
With proper representation, the problem
can become easier (linear model
works).
Develop general purpose linear learning
methods.
Change recoding using “kernel
functions”
Key Ideas
Patterns are regularities in data from a
specified source
Algorithm takes finite sample and computes
pattern

Efficiency, robustness, and stability
Representation -- Kernels
Strategy = Generic Algorithms + Recoding
Many Learning Tasks in this framework