Transcript Metody Inteligencji Obliczeniowej

Computational intelligence
for data understanding
Włodzisław Duch
Department of Informatics,
Nicolaus Copernicus University, Toruń, Poland
Google: W. Duch
Best Summer Course’08
Plan
What is this tutorial about ?
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How to discover knowledge in data;
how to create comprehensible models of data;
how to evaluate new data;
how to understand what CI methods do.
AI, CI & Data Mining.
Forms of useful knowledge.
Integration of different methods in GhostMiner.
Exploration & Visualization.
Rule-based data analysis .
Neurofuzzy models.
Neural models, understanding what they do.
Similarity-based models, prototype rules.
Case studies.
DM future: k-separability and meta-learning.
From data to expert systems.
AI, CI & DM
Artificial Intelligence: symbolic models of knowledge.
• Higher-level cognition: reasoning, problem solving,
planning, heuristic search for solutions.
• Machine learning, inductive, rule-based methods.
• Technology: expert systems.
Computational Intelligence, Soft Computing:
methods inspired by many sources:
• biology – evolutionary, immune, neural computing
• statistics, patter recognition
• probability – Bayesian networks
• logic – fuzzy, rough …
Perception, object recognition.
Data Mining, Knowledge Discovery in Databases.
• discovery of interesting patterns, rules, knowledge.
• building predictive data models.
CI definition
Computational Intelligence. An International Journal (1984)
+ 10 other journals with “Computational Intelligence”,
D. Poole, A. Mackworth & R. Goebel,
Computational Intelligence - A Logical Approach.
(OUP 1998), GOFAI book, logic and reasoning.
CI should:
• be problem-oriented, not method oriented;
• cover all that CI community is doing now, and is likely to do in future;
• include AI – they also think they are CI ...
CI: science of solving (effectively) non-algorithmizable
problems.
Problem-oriented definition, firmly anchored in computer sci/engineering.
AI: focused problems requiring higher-level cognition, the rest of CI is
more focused on problems related to perception/action/control.
What can we learn?
Good part of CI is about learning.
What can we learn?
Neural networks are universal approximators and evolutionary
algorithms solve global optimization problems – so everything
can be learned? Not quite ...
Duda, Hart & Stork, Ch. 9, No Free Lunch + Ugly Duckling Theorems:
• Uniformly averaged over all target functions the expected error for all
learning algorithms [predictions by economists] is the same.
• Averaged over all target functions no learning algorithm yields
generalization error that is superior to any other.
• There is no problem-independent or “best” set of features.
“Experience with a broad range of techniques is the best insurance for
solving arbitrary new classification problems.”
What is there to learn?
Brains ... what is in EEG? What happens in the brain?
Industry: what happens?
Genetics, proteins ...
Forms of useful knowledge
AI/Machine Learning camp:
Neural nets are black boxes.
Unacceptable! Symbolic rules forever.
But ... knowledge accessible to humans is in:
• symbols,
• similarity to prototypes (intuition),
• images, visual representations.
What type of explanation is satisfactory?
Interesting question for cognitive scientists but ...
in different fields answers are different!
Forms of knowledge
• Humans remember examples of each category and
refer to such examples – as similarity-based or
nearest-neighbors methods do.
• Humans create prototypes out of many examples
– same as Gaussian classifiers, RBF networks,
neurofuzzy systems.
• Logical rules are the highest form of summarization of
knowledge, require good linguistic variables.
3 types of explanation presented here:
• exemplar-based: prototypes and similarity;
• logic-based: symbols and rules;
• visualization-based: maps, diagrams, relations ...
GhostMiner Philosophy
GhostMiner, data mining tools from our lab + Fujitsu:
http://www.fqs.pl/ghostminer/
• Separate the process of model building (hackers) and knowledge
discovery, from model use (lamers) =>
GhostMiner Developer & GhostMiner Analyzer
• There is no free lunch – provide different type of tools for
knowledge discovery.
Decision tree, neural, neurofuzzy, similarity-based, SVM,
committees.
• Provide tools for visualization of data.
• Support the process of knowledge discovery/model building
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and evaluating, organizing it into projects.
Many other interesting DM packages of this sort exists:
Weka, Yale, Orange, Knime ...
168 packages on the-data-mine.com list!
Wine data example
Chemical analysis of wine from grapes grown in the same
region in Italy, but derived from three different cultivars.
Task: recognize the source of wine sample.
13 quantities measured, all features are continuous:
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alcohol content
ash content
magnesium content
flavanoids content
proanthocyanins phenols content
OD280/D315 of diluted wines
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malic acid content
alkalinity of ash
total phenols content
nonanthocyanins phenols
content
color intensity
hue
proline.
Exploration and visualization
General info about the data
Exploration: data
Inspect the data
Exploration: data statistics
Distribution of feature values
Proline has very large values, most methods will benefit from data
standardization before further processing.
Exploration: data standardized
Standardized data: unit standard deviation, about 2/3 of all data should
fall within [mean-std,mean+std]
Other options: normalize to [-1,+1],
or normalize rejecting p% of extreme values.
Exploration: 1D histograms
Distribution of feature values in classes
Some features are more useful than the others.
Exploration: 1D/3D histograms
Distribution of feature values in classes, 3D
Exploration: 2D projections
Projections on selected 2D
Projections on selected 2D
Visualize data
Hard to imagine relations in more than 3D.
Use parallel coordinates and other methods.
Linear methods: PCA, FDA, PP ... use input combinations.
SOM mappings: popular for visualization, but rather inaccurate, there
is no measure of distortions.
Measure of topographical distortions:
map all Xi points from Rn to xi points in Rm, m < n, and ask:
how well are Rij = D(Xi, Xj) distances reproduced
by distances rij = d(xi,xj) ?
Use m = 2 for visualization,
use higher m for dimensionality reduction.
Sequences of the Globin family
226 protein sequences of the Globin family; similarity matrix
S(proteini,proteinj) shows high similarity values (dark spots) within
subgroups, MDS shows cluster structure of the data (from Klock &
Buhmann 1997); vector rep. of proteins is not easy.
Visualize data: MDS
Multidimensional scaling: invented in psychometry by Torgerson
(1952), re-invented by Sammon (1969) and myself (1994) …
Minimize measure of topographical distortions moving the x
coordinates.
2
1
S1  x  
R  rij  x  
2   ij
 Rij i j
MDS
Large distances
Sammon
intermediate
i j
1
S2  x  
 Rij

i j
i j
S3  x  
1
 Rij
i j
1  r  x  
ij
Rij
 1  r  x 
i j
2
ij
Rij 
2
MDS local
local structure
as important as
large scale
Visualize data: Wine
3 clusters are clearly distinguished, 2D is fine.
The green outlier can be identified easily.
Decision trees
Simplest things should be done first:
use decision tree to find logical rules.
Test single attribute, find good point to split the data, separating
vectors from different classes.
DT advantages: fast, simple, easy to understand, easy to program,
many good algorithms.
Tree for 3 kinds of iris flowers,
petal and sepal leafs
measured in cm.
Decision borders
Univariate trees:
test the value of a single attribute x < a.
or for nomial features select a subset of values.
Multivariate trees:
test on combinations of attributes W.X < a.
Result:
feature space is divided into
large hyperrectangular areas
with decision borders
perpendicular to axes.
Splitting criteria
Most popular: information gain, used in C4.5 and other trees.
Which attribute is better?
Which should be at the top
of the tree?
Look at entropy reduction,
or information gain index.
E(S )  Pt lg2 Pt  Pf lg2 Pf
S
S
G( S , A)  E( S ) 
E ( S ) 
E ( S )
S
S
CART trees use Gini index
of node purity (Renyi quadratic
entropy):
C
Gini  node   1   Pi 2
i 1
Non-Bayesian selection
Bayesian MAP selection: choose max a posteriori P(C|X)
P(C,A1)
P(C,A2)
A=0
0.0100
0.0900
0.0300
0.1300
A=1
0.4900
0.4100
0.4700
0.3700
P(C0)=0.5
P(C1)=0.5
P(C|X)=P(C,X)/P(X)
MAP is here equivalent to a majority classifier (MC):
given A=x, choose maxC P(C,A=x)
MC(A1)=0.58, S+=0.98, S-=0.18, AUC=0.58, MI= 0.058
MC(A2)=0.60, S+=0.94, S-=0.26, AUC=0.60, MI= 0.057
MC(A1)<MC(A2), AUC(A1)<AUC(A2), but MI(A1)>MI(A2) !
Problem: for binary features non-optimal decisions are taken!
SSV decision tree
Separability Split Value tree:
based on the separability criterion.
Define subsets of data D using a binary test f(X,s) to split the data
into left and right subset D = LS  RS.
LS  s, f , D   X  D : f ( X, s )  T
RS  s, f , D   D  LS  s, f , D 
SSV criterion: separate as many pairs of vectors from different classes
as possible; minimize the number of separated from the same class.
SSV ( s)  2 LS  s, f , D  Dc  RS  s, f , D 
cC

 D  Dc 
 min LS  s, f , D  Dc , RS  s, f , D  Dc
cC

SSV – complex tree
Trees may always learn to achieve 100% accuracy.
Very few vectors are left in the leaves – splits are not reliable and will
overfit the data!
SSV – simplest tree
Pruning finds the nodes that should be removed to increase
generalization – accuracy on unseen data.
Trees with 7 nodes left: 15 errors/178 vectors.
SSV – logical rules
Trees may be converted to logical rules.
Simplest tree leads to 4 logical rules:
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2.
3.
4.
if proline > 719 and flavanoids > 2.3 then class 1
if proline < 719 and OD280 > 2.115 then class 2
if proline > 719 and flavanoids < 2.3 then class 3
if proline < 719 and OD280 < 2.115 then class 3
How accurate are such rules?
Not 15/178 errors, or 91.5% accuracy!
Run 10-fold CV and average the results.
85±10%?
Run 10X and average
85±10%±2%? Run again ...
SSV – optimal trees/rules
Optimal: estimate how well rules will generalize.
Use stratified crossvalidation for training;
use beam search for better results.
1. if OD280/D315 > 2.505 and proline > 726.5 then class 1
2. if OD280/D315 < 2.505 and hue > 0.875 and malic-acid <
2.82 then class 2
3. if OD280/D315 > 2.505 and proline < 726.5 then class 2
4. if OD280/D315 < 2.505 and hue > 0.875 and malic-acid >
2.82 then class 3
5. if OD280/D315 < 2.505 and hue < 0.875 then class 3
Note 6/178 errors, or 91.5% accuracy!
Run 10-fold CV: results are 85±10%? Run 10X!
Logical rules
Crisp logic rules: for continuous x use linguistic variables (predicate
functions).
sk(x)  True [Xk x  X'k], for example:
small(x) = True{x|x < 1}
medium(x) = True{x|x  [1,2]}
large(x)
= True{x|x > 2}
Linguistic variables are used in crisp (prepositional, Boolean) logic
rules:
IF small-height(X) AND has-hat(X) AND has-beard(X)
THEN (X is a Brownie)
ELSE IF ... ELSE ...
Crisp logic decisions
Crisp logic is based on rectangular membership
functions:
True/False values jump from 0 to 1.
Step functions are used for partitioning of the
feature space.
Very simple hyper-rectangular decision borders.
Expressive power of crisp logical rules is very
limited!
Similarity cannot be captured by rules.
Logical rules - advantages
Logical rules, if simple enough, are preferable.
• Rules may expose limitations of black box solutions.
• Only relevant features are used in rules.
• Rules may sometimes be more accurate than NN and other CI
methods.
• Overfitting is easy to control, rules usually have small number
of parameters.
• Rules forever !?
A logical rule about logical rules is:
IF the number of rules is relatively small
AND the accuracy is sufficiently high.
THEN rules may be an optimal choice.
Logical rules - limitations
Logical rules are preferred but ...
• Only one class is predicted p(Ci|X,M) = 0 or 1; such
black-and-white picture may be inappropriate in many applications.
• Discontinuous cost function allow only non-gradient optimization
methods, more expensive.
• Sets of rules are unstable: small change in the dataset leads to a
large change in structure of sets of rules.
• Reliable crisp rules may reject some cases as unclassified.
• Interpretation of crisp rules may be misleading.
• Fuzzy rules remove some limitations, but are not so
comprehensible.
Fuzzy inputs vs. fuzzy rules
Crisp rule Ra(x) = Q(xa) applied to uncertain input with uniform input
uncertainty U(x;Dx)=1 in [xDx, xDx] and zero outside is true to the
degree given by a semi-linear function S(x;Dx):
Input uncertainty and the probability
that Ra(x) rule is true.
For other input uncertainties similar
relations hold!
For example, triangular U(x):
leads to sigmoidal S(x) function.
For more input conditions rules are
true to the degree described by soft
trapezoidal functions, difference of
two sigmoidal functions.
Crisp rules + input uncertainty  fuzzy rules for crisp inputs = MLP !
From rules to probabilities
Data has been measured with unknown error.
Assume Gaussian distribution:
x  Gx  G( y; x, sx )
x – fuzzy number with Gaussian membership function.
A set of logical rules R is used for fuzzy input vectors:
Monte Carlo simulations for arbitrary system => p(Ci|X)
Analytical evaluation p(C|X) is based on cumulant function:
 a  x 
1
  a  x    G  y; x, sx  dy  1  erf 
      (a  x) 
2 

 sx 2  
a
  2.4 / 2sx
Error function is identical to
logistic f. < 0.02
Rules - choices
Simplicity vs. accuracy (but not too accurate!).
Confidence vs. rejection rate (but not too much!).
 p
p  true | predicted   
 p
p
p
p r  p

p r  p
p is a hit; p false alarm; p is a miss.
Accuracy (overall)
Error rate
Rejection rate
Sensitivity
Specificity
A(M) = p+ p
L(M) = p+ p
R(M)=p+r+pr= 1L(M)A(M)
S+(M)= p+|+ = p++ /p+
S(M)= p| = p /p
Rules – error functions
The overall accuracy is equal to a combination of sensitivity and
selectivity weighted by the a priori probabilities:
A(M) = pS(M)+pS(M)
Optimization of rules for the C+ class; accuracy-rejection tradeoff:
large g means no errors but high rejection rate.
E(M;g)= gL(M)A(M)= g (p+p)  (p+p)
minM E(M;g)  minM {(1+g)L(M)+R(M)}
Optimization with different costs of errors
minM E(M;a) = minM {p+ a p} =
minM {p1S(M))  pr(M) + a [p1S(M))  pr(M)]}
ROC curves
ROC curves display S+ vs. (1S) for different models (classifiers)
or different confidence thresholds:
Ideal classifier: below some
threshold S+ = 1 (all positive
cases recognized) for 1-S= 0
(no false alarms) .
Useless classifier (blue): same
number of true positives as
false alarms for any threshold.
Reasonable classifier (red):
no errors until some threshold that allows for recognition of 0.5 positive
cases, no errors if 1-S > 0.6; slowly rising errors in between.
Good measure of quality: high AUC, Area Under ROC Curve.
AUC = 0.5 is random guessing, AUC = 1 is perfect prediction.
Gaussian fuzzification of crisp rules
Very important case: Gaussian input uncertainty.
Rule Ra(x) = {xa} is fulfilled by Gx with probability:

p  Ra (Gx )  T    G  y; x, sx  dy     ( x  a) 
a
Error function is approximated by logistic function;
assuming error distribution (x)1 x)),
for s2=1.7 approximates Gauss < 3.5%
Rule Rab(x) = {b> x a} is fulfilled by Gx with
probability:
b
p  Rab (Gx )  T    G  y; x, sx  dy     ( x  a)      ( x  b) 
a
Soft trapezoids and NN
The difference between two sigmoids makes a soft trapezoidal
membership functions.
Conclusion:
fuzzy logic with soft trapezoidal membership functions
(x)  (x-b)  to a crisp logic + Gaussian uncertainty of inputs.
Optimization of rules
Fuzzy: large receptive fields, rough estimations.
Gx – uncertainty of inputs, small receptive fields.
Minimization of the number of errors – difficult, non-gradient, but
now Monte Carlo or analytical p(C|X;M).
2
1
E { X }; R, sx     p  Ci | X ; M     C ( X ), Ci  
2 X i
•
Gradient optimization works for large number of parameters.
•
Parameters sx are known for some features, use them as
optimization parameters for others!
Probabilities instead of 0/1 rule outcomes.
Vectors that were not classified by crisp rules have now non-zero
probabilities.
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Mushrooms
The Mushroom Guide: no simple rule for mushrooms; no rule like:
‘leaflets three, let it be’ for Poisonous Oak and Ivy.
8124 cases, 51.8% are edible, the rest non-edible.
22 symbolic attributes, up to 12 values each, equivalent to 118 logical
features, or 2118=3.1035 possible input vectors.
Odor: almond, anise, creosote, fishy, foul, musty, none, pungent,
spicy
Spore print color: black, brown, buff, chocolate, green, orange,
purple, white, yellow.
Safe rule for edible mushrooms:
odor=(almond.or.anise.or.none)  spore-print-color =  green
48 errors, 99.41% correct
This is why animals have such a good sense of smell!
What does it tell us about odor receptors?
Mushrooms rules
To eat or not to eat, this is the question! Not any more ...
A mushroom is poisonous if:
R1) odor =  (almond  anise  none);
120 errors, 98.52%
R2) spore-print-color = green
48 errors, 99.41%
R3) odor = none  stalk-surface-below-ring = scaly
 stalk-color-above-ring =  brown
8 errors, 99.90%
R4) habitat = leaves  cap-color = white
no errors!
R1 + R2 are quite stable, found even with 10% of data;
R3 and R4 may be replaced by other rules, ex:
R'3): gill-size=narrow  stalk-surface-above-ring=(silky  scaly)
R'4): gill-size=narrow  population=clustered
Only 5 of 22 attributes used! Simplest possible rules?
100% in CV tests - structure of this data is completely clear.
Recurrence of breast cancer
Institute of Oncology, University Medical Center, Ljubljana.
286 cases, 201 no (70.3%), 85 recurrence cases (29.7%)
9 symbolic features: age (9 bins), tumor-size (12 bins), nodes involved
(13 bins), degree-malignant (1,2,3), area, radiation, menopause, nodecaps.
no-recurrence,40-49,premeno,25-29,0-2,?,2, left, right_low, yes
Many systems tried, 65-78% accuracy reported. Single rule:
IF (nodes-involved  [0,2]  degree-malignant = 3 THEN recurrence
ELSE no-recurrence
77% accuracy, only trivial knowledge in the data: highly malignant
cancer involving many nodes is likely to strike back.
Neurofuzzy system
Fuzzy: mx0,1 (no/yes) replaced by a degree mx[0,1].
Triangular, trapezoidal, Gaussian or other membership f.
M.f-s in many
dimensions:
Feature Space Mapping (FSM) neurofuzzy system.
Neural adaptation, estimation of probability density distribution (PDF)
using single hidden layer network
(RBF-like), with nodes realizing separable functions:
G  X ; P    Gi  X i ; Pi 
i 1
FSM
Initialize using clusterization or decision trees.
Triangular & Gaussian f. for fuzzy rules.
Rectangular functions for crisp rules.
Between 9-14 rules with triangular membership functions are
created; accuracy in 10xCV tests about 96±4.5%
Similar results obtained with Gaussian functions.
Rectangular functions: simple rules are created, many nearly
equivalent descriptions of this data exist.
If proline > 929.5 then class 1 (48 cases, 45 correct
+ 2 recovered by other rules).
If color < 3.79285 then class 2 (63 cases, 60 correct)
Interesting rules, but overall accuracy is only 88±9%
Prototype-based rules
C-rules (Crisp), are a special case of F-rules (fuzzy rules).
F-rules (fuzzy rules) are a special case of P-rules (Prototype).
P-rules have the form:
IF P = arg minR D(X,R) THAN Class(X)=Class(P)
D(X,R) is a dissimilarity (distance) function, determining decision
borders around prototype P.
P-rules are easy to interpret!
IF
X=You are most similar to the P=Superman
THAN You are in the Super-league.
IF
X=You are most similar to the P=Weakling
THAN You are in the Failed-league.
“Similar” may involve different features or D(X,P).
P-rules
Euclidean distance leads to a Gaussian fuzzy membership functions +
product as T-norm.
D  X, P    d  X i , Pi   Wi  X i  Pi 
i
mP  X   e
 D  X ,P 
2
i
e

 d  X i ,Pi 
i
 e
Wi  X i  Pi 
2
i
  mi  X i , Pi 
i
Manhattan function => m(X;P)=exp{|X-P|}
Various distance functions lead to different MF.
Ex. data-dependent distance functions, for symbolic data:


DVDM  X, Y      p  C j | X i   p  C j | Yi  
i  j



DPDF  X, Y      p  X i | C j   p  C j | Yi  
i  j

Promoters
DNA strings, 57 aminoacids, 53 + and 53 - samples
tactagcaatacgcttgcgttcggtggttaagtatgtataatgcgcgggcttgtcgt
Euclidean distance, symbolic
s =a, c, t, g replaced by x=1, 2, 3, 4
PDF distance, symbolic
s=a, c, t, g replaced by p(s|+)
P-rules
New distance functions from info theory => interesting MF.
MF => new distance function, with local D(X,R) for each cluster.
Crisp logic rules: use Chebyshev distance (L norm):
DCh(X,P) = ||XP|| = maxi Wi |XiPi|
DCh(X,P) = const => rectangular contours.
Chebyshev distance with thresholds P
IF DCh(X,P)  P THEN C(X)=C(P)
is equivalent to a conjunctive crisp rule
IF X1[P1P/W1,P1P/W1] …XN [PN P/WN,PNP/WN]
THEN C(X)=C(P)
Decision borders
D(P,X)=const and decision borders D(P,X)=D(Q,X).
Euclidean distance from 3
prototypes, one per class.
Minkovski a=20 distance from
3 prototypes.
P-rules for Wine
Manhattan distance:
6 prototypes kept,
4 errors, f2 removed
Chebyshev distance:
15 prototypes kept, 5 errors, f2,
f8, f10 removed
Euclidean distance:
11 prototypes kept,
7 errors
Many other solutions.
Scatterograms for hypothyroid
Shows images of training vectors mapped by
neural network; for more than 2 classes
either linear projections, or several 2D
scatterograms, or parallel coordinates.
Good for:
•
analysis of the learning process;
•
comparison of network solutions;
•
stability of the network;
•
analysis of the effects of regularization;
•
evaluation of confidence by perturbation of
the query vector.
...
Details: W. Duch, IJCNN 2003
Neural networks
• MLP – Multilayer Perceptrons, most popular NN models.
Use soft hyperplanes for discrimination.
Results are difficult to interpret, complex decision borders.
Prediction, approximation: infinite number of classes.
• RBF – Radial Basis Functions.
RBF with Gaussian functions are equivalent to fuzzy systems with
Gaussian membership functions, but …
No feature selection => complex rules.
Other radial functions => not separable!
Use separable functions, not radial => FSM.
• Many methods to convert MLP NN to logical rules.
What NN really do?
• Common opinion: NN are black boxes.
NN provide complex mappings that may involve various kinks
and discontinuities, but NN have the power!
•
Solution 1 (common): extract rules approximating NN mapings.
•
Solution 2 (new): visualize neural mapping.
RBF network for fuzzy XOR, using 4
Gaussian nodes:
rows for =1/7,1 and 7
left column: scatterogram of the hidden
node activity in 4D.
middle columns: parallel coordinate view
right column: output view (2D)
Wine example
• MLP with 2 hidden nodes, SCG training, regularization a=0.5
•
After 3 iterations: output, parallel, hidden.
After convergence + with noise var=0.05 added
Rules from MLPs
Why is it difficult?
Multi-layer perceptron (MLP) networks: stack many perceptron units,
performing threshold logic:
M-of-N rule: IF (M conditions of N are true) THEN ...
Problem: for N inputs number of subsets is 2N.
Exponentially growing number of possible conjunctive rules.
MLP2LN
Converts MLP neural networks into a network performing logical
operations (LN).
Input
layer
Output:
one node
per class.
Aggregation:
better features
Linguistic units:
windows, filters
Rule units:
threshold logic
MLP2LN training
Constructive algorithm: add as many nodes as needed.
Optimize cost function:
1
p
p 2
E ( W )   ( F ( Xi ; W)  ti )  minimize errors +
2 p i
1
2
2
W
 ij 
enforce zero connections +
i j
2
W

2
i j
2
ij
(Wij  1) (Wij  1)
2
2
leave only +1 and -1 weights
makes interpretation easy.
L-units
Create linguistic variables.
Numerical representation for R-nodes
Vsk=(1,1,1,...) for sk=low
Vsk=(1,1,1,...) for sk=normal
L-units: 2 thresholds as adaptive
parameters;
logistic (x), or tanh(x)[1, 1]
L( X )  S1 W1x  b  S2 W2 x  b '
Product of bi-central functions is
logical rule, used by IncNet NN.
Soft trapezoidal functions change
into rectangular filters (Parzen
windows).
4 types, depending on signs Si.
Iris example
Network after training:
iris setosa: q=1
(0,0,0;0,0,0;+1,0,0;+1,0,0)
iris versicolor: q=2
(0,0,0;0,0,0;0,+1,0;0,+1,0)
iris virginica: q=1
(0,0,0;0,0,0;0,0,+1;0,0,+1)
Rules:
If (x3=s  x4=s) setosa
If (x3=mx4=m) versicolor
If (x3=l  x4=l) virginica
3 errors only (98%).
Learning dynamics
Decision regions shown every 200 training epochs in x3, x4 coordinates;
borders are optimally placed with wide margins.
Thyroid screening
Clinical
findings
Garavan Institute, Sydney,
Australia
15 binary, 6 continuous
Training: 93+191+3488
Validate: 73+177+3178

Determine important
clinical factors

Calculate prob. of
each diagnosis.
Age
sex
…
…
Hidden
units
Final
diagnoses
Normal
Hypothyroid
TSH
T4U
T3
TT4
TBG
Hyperthyroid
Thyroid – some results.
Accuracy of diagnoses obtained with several systems – rules are
accurate.
Method
Rules/Features Training % Test %
MLP2LN optimized
4/6
99.9
99.36
CART/SSV Decision Trees
3/5
99.8
99.33
Best Backprop MLP
-/21
100
98.5
Naïve Bayes
-/-
97.0
96.1
k-nearest neighbors
-/-
-
93.8
Thyroid – output visualization.
2D – plot scatterogram of the vectors transformed by the network.
3D – display it inside the cube, use perspective.
ND – make linear projection of network outputs on the polygon vertices
Data mining packages
GhostMiner, data mining tools from our lab + Fujitsu:
http://www.fqspl.com.pl/ghostminer/
• Separate the process of model building (hackers) and knowledge
discovery, from model use (lamers) => GM Developer & Analyzer
• No free lunch => provide different type of tools for knowledge
•
discovery: decision tree, neural, neurofuzzy, similarity-based, SVM,
committees, tools for visualization of data.
Support the process of knowledge discovery/model building and
evaluating, organizing it into projects.
• Many other interesting DM packages of this sort exists:
Weka, Yale, Orange, Knime ...
168 packages on the-data-mine.com list!
• We are building Intemi, completely new tools.
Are we really
so good?
Surprise!
Almost nothing
can be learned
using such tools!
SVM for parity
Parity with growing number of dimensions: 5x10CV results, Gaussian
kernel SVM and MLP with optimized parameters (C, , # neurons<20).
Q1: How to characterize complexity of Boolean functions?
Non-separability is not sufficient.
Q2: What is the simplest model for a given type of data?
Q3: How to learn such model in an automatic way?
Can Data Mining packages help?
Hundreds of components ... transforming, visualizing ...
Visual “knowledge flow” to
link components, or script
languages (XML) to define
complex experiments.
RM/Yale 3.3: type # components
Data preprocessing
74
Experiment operations
35
Learning methods
114
Metaoptimization schemes
17
Postprocessing
5
Performance validation
14
Visualization, presentation, plugin extensions ...
Why solid foundations are needed
Hundreds of components ... already 351 359 400 combinations!
Our treasure box is full! We can publish forever!
Is this really what we need?
No, we would like to …
press the button and wait for the truth!
Computer power is with us, meta-learning should
find all interesting data models
= sequences of transformations/procedures.
Many considerations: optimal cost solutions, various costs of using
feature subsets; models that are simple & easy to understand; various
representation of knowledge: crisp, fuzzy or prototype rules,
visualization, confidence in predictions ...
Similarity-based framework
(Dis)similarity:
• more general than feature-based description,
• no need for vector spaces (structured objects),
• more general than fuzzy approach (F-rules are reduced to P-rules),
• includes nearest neighbor algorithms, MLPs, RBFs, separable
function networks, SVMs, kernel methods and many others!
Similarity-Based Methods (SBMs) are organized in a framework:
p(Ci|X;M) posterior classification probability or y(X;M) approximators,
models M are parameterized in increasingly sophisticated way.
A systematic search (greedy, beam, evolutionary) in the space of all
SBM models is used to select optimal combination of parameters and
procedures, opening different types of optimization channels, trying to
discover appropriate bias for a given problem.
Results: several candidate models are created, even very limited version
gives best results in 7 out of 12 Stalog problems.
SBM framework components
•
•
•
•
•
•
•
•
•
Pre-processing: objects O => features X, or (diss)similarities D(O,O’).
Calculation of similarity between features d(xi,yi) and objects D(X,Y).
Reference (or prototype) vector R selection/creation/optimization.
Weighted influence of reference vectors G(D(Ri,X)), i=1..k.
Functions/procedures to estimate p(C|X;M) or y(X;M).
Cost functions E[DT;M] and model selection/validation procedures.
Optimization procedures for the whole model Ma.
Search control procedures to create more complex models Ma+1.
Creation of ensembles of (local, competent) models.
• M={X(O), d(.,.), D(.,.), k, G(D), {R}, {pi(R)}, E[.], K(.), S(.,.)}, where:
• S(Ci,Cj) is a matrix evaluating similarity of the classes;
a vector of observed probabilities pi(X) instead of hard labels.
The kNN model p(Ci|X;kNN) = p(Ci|X;k,D(.),{DT});
the RBF model: p(Ci|X;RBF) = p(Ci|X;D(.),G(D),{R}),
MLP, SVM and many other models may all be “re-discovered”.
Meta-learning in SBM scheme
k-NN
k-NN67.5/76.6%
67.5/76.6%
+selection,
+selectio
67.5/76.6 %
n,
+d(x,y);
+k
67.5/76.6%
+d(x,y);
+kopt;
opt;
Canberra 89.9/90.7%
67.5/76.6
Canberra
67.5/76.6
%
+
s
=(0,0,1,0,1,1);
+i
%
89.9/90.7 %
71.6/64.4 %
si=(0,0,1,0,1,
1);
+d(x,y)
+ s+
+d(x,y)
+ selection;
71.6/64.4
%
i=(1,0,1,0.6,0.9,1);
+d(x,y)
si
+d(x,y)
+ sel. or
Canberra 74.6/72.9 %
Canberra 89.9/90.7 %
opt k;
Start from kNN, k=1, all data & features, Euclidean distance, end with a
model that is a novel combination of procedures and parameterizations.
Heterogeneous systems
Problems requiring different scales (multiresolution).
2-class problems, two situations:
C1 inside the sphere, C2 outside.
MLP: at least N+1 hyperplanes, O(N2) parameters.
RBF: 1 Gaussian, O(N) parameters.
C1 in the corner defined by (1,1 ... 1) hyperplane, C2 outside.
MLP: 1 hyperplane, O(N) parameters.
RBF: many Gaussians, O(N2) parameters, poor approx.
Combination: needs both hyperplane and hypersphere!
Logical rule: IF x1>0 & x2>0 THEN C1 Else C2
is not represented properly neither by MLP nor RBF!
Different types of functions in one model, first step beyond inspirations
from single neurons => heterogeneous models.
Heterogeneous everything
Homogenous systems: one type of “building blocks”, same type of
decision borders, ex: neural networks, SVMs, decision trees, kNNs
Committees combine many models together, but lead to complex
models that are difficult to understand.
Ockham razor: simpler systems are better.
Discovering simplest class structures, inductive bias of the data,
requires Heterogeneous Adaptive Systems (HAS).
HAS examples:
NN with different types of neuron transfer functions.
k-NN with different distance functions for each prototype.
Decision Trees with different types of test criteria.
1. Start from large networks, use regularization to prune.
2. Construct network adding nodes selected from a candidate pool.
3. Use very flexible functions, force them to specialize.
Taxonomy of NN activation functions
Taxonomy of NN output functions
Perceptron: implements logical rule x> for x with Gaussian uncertainty.
Taxonomy - TF
HAS decision trees
Decision trees select the best feature/threshold value for univariate
and multivariate trees:
X i  k or T  X; W,k   Wi X i  k
i
Decision borders: hyperplanes.
Introducing tests based on La Minkovsky metric.
T  X; R,R   X  R a   X i  Ri
1/ a
 R
i
For L2 spherical decision border are produced.
For L∞ rectangular border are produced.
Many choices, for example Fisher Linear Discrimination decision trees.
For large databases first clusterize data to get candidate references R.
SSV HAS DT example
SSV HAS tree in GhostMiner 3.0, Wisconsin breast cancer (UCI)
699 cases, 9 features (cell parameters, 1..10)
Classes: benign 458 (65.5%) & malignant 241 (34.5%).
Single rule gives simplest known description of this data:
IF ||X-R303|| < 20.27 then malignant
else benign
coming most often in 10xCV
97.4% accuracy; good prototype for malignant case!
Gives simple thresholds, that’s what MDs like the most!
Best 10CV around
97.5±1.8% (Naïve Bayes + kernel, or SVM)
SSV without distances: 96.4±2.1%
C 4.5 gives
94.7±2.0%
Several simple rules of similar accuracy but different specificity or
sensitivity may be created using HAS DT.
Need to select or weight features and select good prototypes.
More meta-learning
Meta-learning: learning how to learn, replace experts who search for
best models making a lot of experiments.
Search space of models is too large to explore it exhaustively, design
system architecture to support knowledge-based search.
•
•
•
•
Abstract view, uniform I/O, uniform results management.
Directed acyclic graphs (DAG) of boxes representing scheme
placeholders and particular models, interconnected through I/O.
Configuration level for meta-schemes, expanded at runtime level.
An exercise in software engineering for data mining!
Intemi, Intelligent Miner
Meta-schemes: templates with placeholders
•
•
•
•
•
May be nested; the role decided by the input/output types.
Machine learning generators based on meta-schemes.
Granulation level allows to create novel methods.
Complexity control: Length + log(time)
A unified meta-parameters description, defining the range of
sensible values and the type of the parameter changes.
How much can we learn?
Linearly separable or almost separable problems are relatively
simple – deform planes or add dimensions to make data separable.
How to define “slightly non-separable”?
There is only separable and the vast realm of the rest.
Neurons learning complex logic
Boole’an functions are difficult to learn, n bits but 2n nodes =>
combinatorial complexity; similarity is not useful, for parity all
neighbors are from the wrong class. MLP networks have difficulty to
learn functions that are highly non-separable.
Ex. of 2-4D
parity
problems.
Neural logic
can solve it
without
counting; find
a good point
of view.
Projection on W=(111 ... 111) gives clusters with 0, 1, 2 ... n bits;
solution requires abstract imagination + easy categorization.
Easy and difficult problems
Linear separation: good goal if simple topological
deformation of decision borders is sufficient.
Linear separation of such data is possible in higher dimensional
spaces; this is frequently the case in pattern recognition problems.
RBF/MLP networks with one hidden layer solve such problems.
Difficult problems: disjoint clusters, complex logic.
Continuous deformation is not sufficient; networks with localized
functions need exponentially large number of nodes.
Boolean functions: for n bits there are K=2n binary vectors that can be
represented as vertices of n-dimensional hypercube.
Each Boolean function is identified by K bits.
BoolF(Bi) = 0 or 1 for i=1..K, leads to the 2K Boolean functions.
Ex: n=2 functions, vectors {00,01,10,11},
Boolean functions {0000, 0001 ... 1111}, ex. 0001 = AND, 0110 = OR,
each function is identified by number from 0 to 15 = 2K-1.
Boolean functions
n=2, 16 functions, 12 separable, 4 not separable.
n=3, 256 f, 104 separable (41%), 152 not separable.
n=4, 64K=65536, only 1880 separable (3%)
n=5, 4G, but << 1% separable ... bad news!
Existing methods may learn some non-separable functions,
but most functions cannot be learned !
Example: n-bit parity problem; many papers in top journals.
No off-the-shelf systems are able to solve such problems.
For all parity problems SVM is below base rate!
Such problems are solved only by special neural architectures or
special classifiers – if the type of function is known.
 n 
But parity is still trivial ... solved by y  cos    bi 
 i 1 
Abstract imagination
Transformation of data to a space where clustering is easy.
Intuitive answers, as propositional rules may be difficult to formulate.
Here fuzzy XOR (stimuli color/size/shape are not identical) has
been transformed by two groups of neurons that react to similarity.
Network-based intuition: they know the answer, but cannot say it ...
If image of the data forms non-separable clusters in the inner
(hidden) space network outputs will be often wrong.
3-bit parity in 2D and 3D
Output is mixed, errors are at base level (50%), but in the
hidden space ...
Conclusion: separability in the hidden space is perhaps too much to
desire ... inspection of clusters is sufficient for perfect classification;
add second Gaussian layer to capture this activity;
train second RBF on the data (stacking), reducing number of clusters.
Goal of learning
If simple topological deformation of decision borders is sufficient
linear separation is possible in higher dimensional spaces,
“flattening” non-linear decision borders; this is frequently the case
in pattern recognition problems.
RBF/MLP networks with one hidden layer solve the problem.
For complex logic this is not sufficient; networks with localized
functions need exponentially large number of nodes.
Such situations arise in AI reasoning problems, real perception, object
recognition, text analysis, bioinformatics ...
Linear separation is too difficult, set an easier goal.
Linear separation: projection on 2 half-lines in the kernel space:
line y=WX, with y<0 for class – and y>0 for class +.
Simplest extension: separation into k-intervals.
For parity: find direction W with minimum # of intervals, y=W.X
3D case
3-bit functions: X=[b1b2b3], from [0,0,0] to [1,1,1]
f(b1,b2,b3) and f(b1,b2,b3) are symmetric (color change)
8 cube vertices, 28=256 Boolean functions.
0 to 8 red vertices: 1, 8, 28, 56, 70, 56, 28, 8, 1 functions.
For arbitrary direction W index projection W.X gives:
k=1 in 2 cases, all 8 vectors in 1 cluster (all black or all white)
k=2 in 14 cases, 8 vectors in 2 clusters (linearly separable)
k=3 in 42 cases, clusters B R B or W R W
k=4 in 70 cases, clusters R W R W or W R W R
Symmetrically, k=5-8 for 70, 42, 14, 2.
Most logical functions have 4 or 5-separable projections.
Learning = find best projection for each function.
Number of k=1 to 4-separable functions is: 2, 102, 126 and 26
126 of all functions may be learned using 3-separability.
4D case
4-bit functions: X=[b1b2b3b4], from [0,0,0,0] to [1,1,1,1]
16 cube vertices, 216=65636=64K functions.
Random initialization of a single perceptron has 39.2% chance of
creating 8 or 9 clusters for the 4-bit data.
Learning optimal directions W finds:
k=1 in 2 cases, all 16 vectors in 1 cluster (all black or all white)
k=2 in 2.9% cases (or 1880), 16 vectors in 2 clusters (linearly sep)
k=3 in 22% of all cases, clusters B R B or W R W
k=4 in 45% of all cases, clusters R W R W or W R W R
k=5 in 29% of all cases.
Hypothesis: for n-bits highest k=n+1 ?
For 5-bits there are 32 vertices and already 232=4G=4.3.109 functions.
Most are 5-separable, less than 1% is linearly separable!
Biological justification
•
•
•
•
•
•
•
Cortical columns may learn to respond to stimuli with complex logic
resonating in different way.
The second column will learn without problems that such different
reactions have the same meaning: inputs xi and training targets yj.
are same => Hebbian learning DWij ~ xi yj => identical weights.
Effect: same line y=W.X projection, but inhibition turns off one
perceptron when the other is active.
Simplest solution: oscillators based on combination of two neurons
(W.X-b) – (W.X-b’) give localized projections!
We have used them in MLP2LN architecture for extraction of logical
rules from data.
Note: k-sep. learning is not a multistep output neuron, targets are
not known, same class vectors may appear in different intervals!
We need to learn how to find intervals and how to assign them to
classes; new algorithms are needed to learn it!
Network solution
Can one learn a simplest model for arbitrary Boolean function?
2-separable (linearly separable) problems are easy;
non separable problems may be broken into k-separable, k>2.
(y+1)
X1
X2
y=W.
X
X3
X4
Blue: sigmoidal
neurons with threshold,
brown – linear neurons.
+ (y+ )
2
1

1+
1

1
(y+4)
+
1
+
1
+
+
1
1
Neural architecture for
k=4 intervals, or
4-separable problems.
QPC Projection Pursuit
What is needed to learn data with complex logic?
• cluster non-local areas in the X space, use W.X
• capture local clusters after transformation, use G(W.X)
What will solve it? Projected clusters!
1. A class of constructive neural network solution with G(W.X)
functions with special training algorithms.
2. Maximize the leave-one-out error after projection: take localized
function G, count in a soft way cases from the same class as X.
Q  W  G  W   X  X '   H CX , CX' 
X

X'





H CX , CX'  2 CX , CX'  1 1, 1
Classification:
Regression:
~ |YXYX’|
Projection may be done directly to 1 or 2D for visualization,
or higher D for dimensionality reduction, if W has D columns, for .
Parity n=9
Simple gradient learning; QPC index shown below.
Learning hard functions
Training almost perfect for parity, with linear growth in the number of
vectors for k-sep. solution created by the constructive neural algorithm.
Real data
Simple data – similar results, but much simpler models.
Transformation-based framework
Find simplest model that is suitable for a given data, creating non-sep.
that is easy to handle: simpler models generalize better, interpretation.
Compose transformations (neural layers), for example:
•
•
•
•
•
Matching pursuit network for signal decomposition, QPC index.
PCA network, with each node computing principal component.
LDA nets, each node computes LDA direction (including FDA).
ICA network, nodes computing independent components.
KL, or Kullback-Leibler network with orthogonal or non-orthogonal
components; max. of mutual information is a special case
• c2 and other statistical tests for dependency to aggregate features.
• Factor analysis network, computing common and unique factors.
Evolving Transformation Systems (Goldfarb 1990-2006), giving
unified paradigm for inductive learning and structural representations.
Linear separability
SVM visualization of Leukemia microarray data.
Approximate separability
SVM visualization of Heart dataset, overlapping clusters, information in
the data is insufficient for perfect classification.
Interval transformation
Parity data: k-separability is much easier to achieve than full
linear separability.
Rules
QPC visualization of Monks dataset, two logical rules are needed.
Complex distribution
QPC visualization of concentric rings in 2D with strong noise in another
2D; nearest neighbor or combinations of ellipsoidal densities.
Summary
• Challenging data cannot be handled with existing DM tools.
• Similarity-based framework enables meta-learning as search in the
•
•
•
•
•
•
model space, heterogeneous systems add fine granularity.
No off-shelf classifiers are able to learn difficult Boolean functions.
Visualization of hidden neuron’s shows that frequently perfect but
non-separable solutions are found despite base-rate outputs.
Linear separability is not the best goal of learning, other targets that
allow for easy handling of final non-linearities should be defined.
k-separability defines complexity classes for non-separable data.
Transformation-based learning shows the need for componentbased approach to DM, discovery of simplest models.
Most known and new learning methods result from such framework,
neurocognitive inspirations extend it even further.
Many interesting new ideas arise from this line of thinking.
Psychometry
Use CI to find knowledge, create Expert System.
MMPI (Minnesota Multiphasic Personality Inventory) psychometric
test.
Printed forms are scanned or computerized version of the test is
used.
•
Raw data: 550 questions, ex:
I am getting tired quickly: Yes - Don’t know - No
•
Results are combined into 10 clinical scales and 4 validity scales
using fixed coefficients.
•
Each scale measures tendencies towards hypochondria,
schizophrenia, psychopathic deviations, depression, hysteria,
paranoia etc.
Psychometry: goal
• There is no simple correlation between single values and final
diagnosis.
• Results are displayed in form of a histogram, called ‘a
psychogram’. Interpretation depends on the experience and skill
of an expert, takes into account correlations between peaks.
Goal: an expert system providing evaluation and interpretation of MMPI
tests at an expert level.
Problem: experts agree only about 70% of the time;
alternative diagnosis and personality changes over time are
important.
Psychometric data
1600 cases for woman, same number for men.
27 classes:
norm, psychopathic, schizophrenia, paranoia, neurosis, mania,
simulation, alcoholism, drug addiction, criminal tendencies,
abnormal behavior due to ...
Extraction of logical rules: 14 scales = features.
Define linguistic variables and use FSM, MLP2LN, SSV - giving
about 2-3 rules/class.
Psychometric results
Method
Data
N. rules
Accuracy
+Gx%
C 4.5
♀
55
93.0
93.7
♂
61
92.5
93.1
♀
69
95.4
97.6
♂
98
95.9
96.9
FSM
10-CV for FSM is 82-85%, for C4.5 is 79-84%.
Input uncertainty +Gx around 1.5% (best ROC) improves FSM results
to 90-92%.
Psychometric Expert
Probabilities for different classes.
For greater uncertainties more classes are
predicted.
Fitting the rules to the conditions:
typically 3-5 conditions per rule, Gaussian
distributions around measured values that fall
into the rule interval are shown in green.
Verbal interpretation of each case, rule and
scale dependent.
Visualization
Probability of classes versus input
uncertainty.
Detailed input probabilities around the
measured values vs. change in the single
scale; changes over time define ‘patients
trajectory’.
Interactive multidimensional scaling:
zooming on the new case to inspect its
similarity to other cases.
Summary
Computational intelligence methods: neural, decision trees, similaritybased & other, help to understand the data.
Understanding data: achieved by rules, prototypes, visualization.
Small is beautiful => simple is the best!
Simplest possible, but not simpler - regularization of models;
accurate but not too accurate - handling of uncertainty;
high confidence, but not paranoid - rejecting some cases.
• Challenges:
hierarchical systems – higher-order rules, missing information;
discovery of theories rather than data models;
reasoning in complex domains/objects;
integration with image/signal analysis;
applications in data mining, bioinformatics, signal processing, vision ...
The End
We are slowly addressing the challenges.
The methods used here (+ many more) are included in the
Ghostminer, data mining software developed by my group,
in collaboration with FQS, Fujitsu Kyushu Systems
http://www.fqs.pl/ghostminer/
Papers describing in details some of the ideas presented here
may be accessed through my home page:
Google: W Duch
or
http://www.is.umk.pl/~duch