2009:11:10 Magdi A. Mohamed 1/18 Q-Metrics in Theory and Practice PRESENTATION TO UNIVERSITY OF FLORIDA – LOUISVILLE, FL 2009:11:10 d =-1 = 1 d =0 = 1 d -1,0) = 1 d e = d p=2 = 1 Dimension1 Dimension2 d t = d p=1 = 1 d p=infinity = 1 x=(x 1 , x 2 ) y=(y 1 , y 2 ) Q-Metrics for Different Lambda Values Graph of d(x,y)=1 in 2-Dimensional Space
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter Lambda, which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
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2009:11:10
Magdi A. Mohamed 1/18
Q-Metricsin Theory and Practice
PRESENTATION TOUNIVERSITY OF FLORIDA – LOUISVILLE, FL
2009:11:10
d=-1 = 1
d=0 = 1
d-1,0) = 1
de = dp=2 = 1Dimension1
Dimension2
dt = dp=1 = 1
dp=infinity = 1
x=(x1,x2)
y=(y1,y2)
Q-Metrics for Different Lambda Values
Graph of d(x,y)=1 in 2-Dimensional Space
2009:11:10
Magdi A. Mohamed 2/18
Q-Measure ConceptFuzzy Measure AxiomsLet be non-intersecting sets
• Boundary conditions:
• Monotonicity:
• Continuity:
guaranteed for discrete spaces
Probability Measure (1933)replaces monotonicity by additivity:
Sugeno -Measure (1975)adds one more axiom:
for a unique that satisfies g(X)=1
Q-Measure Extensions (2003)for any choice of >-1, !=0, define:
where fi [0,1] are density generators
Convergence Behavior of Q-Measures
0.875
0.9
0.925
0.95
0.975
1
1.025
1.05
0 2 4 6 8 10 12
Iteration, n
Scali
ng
Facto
r, f
n
Case 1
Case 2
Case 3
0 -1, ,1)1(
1)1(
)(
Xx
i
Ax
i
i
i
f
f
Aq
XBA ,
)()( 2121 AmAmAA
1)(,0)( Xmm
)()()( BpApBAp
)()()()()( BgAgBgAgBAg
2009:11:10
Magdi A. Mohamed 3/18
Q-Measuresin a nutshell
X
x2 x3
x4
A
x1
x6 x5
x7 x8 x9
B=Ac
q-measures providemore expressive and
computationally attractivenonlinear models
foruncertainty
management
q(A)
q(Ac)
=0probability
>0plausibility
<0belief
0)(
0)(
0)(
)(
)(
0)(
1
0)(
0)(
Aq
Aq
Aq
Bq
Aq
BAf
Bf
Af
BA
XBA
when modeling a complex system,
it’s an oversimplificationto assume that the
interdependency among information sources is
linear
2009:11:10
Magdi A. Mohamed 4/18
Q-filter ComputationsN=5 Tap Case - Nonlinearity, Adaptivity, and Model Capacity
Case StudiesCDMA Data Filtering for Cognitive Radio
Linear Filter Equalization
)(tI f
)(tQ f
)(te
)(tI
)(tQ
)(tS)(tSuplink
Training[065,504 samples]
Testing[200,000 samples]
Real
RMS = 31.31
Correlation = 99.14%
RMS = 31.25
Correlation = 99.11%
ImaginaryRMS=20.49
Correlation = 99.55%
RMS = 20.54
Correlation = 99.52%
Existing Linear Filter (Target)- 63 coefficients
Q-Filter Solution- 7 coefficients
Solution Comparison Performance Comparison
Q-Filter Performance (Real)
-600
-400
-200
0
200
400
600
800
1
14
27
40
53
66
79
92
10
5
11
8
13
1
14
4
15
7
17
0
18
3
19
6
20
9
22
2
23
5
24
8
Time
Sig
na
l
Q-Filter
Target
2009:11:10
Magdi A. Mohamed 6/18
Q-Metric ConceptMetric AxiomsA function d(x,y) defined for x and y in a set X is a metric provided that: • d(x,y) > 0, and d(x,y) = 0 iff x=y• d(x,y) = d(y,x)• d(x,y) + d(y,z) > d(x,z)The pair (X,d) is called a Metric Space
P-Metrics, dp (x,y) Defined, for 1 < p < infinity, by:
dp(x,y) = [ sum { |xi-yi|p } ](1/p)
Manhattan (Taxi-Cab) Distance, dt (x,y) Same as p-metric with p=1
Euclidean Distance, de (x,y) Same as p-metric with p=2
Mahalanobis Distance, dm (x,y) Defined using covariance matrix A, by:
dm(x,y) = (x-y)’ A-1 (x-y)
Q-Metrics Definition, d (x,y)
For xx,y X=[0,1]n and [-1,0) define:
We call the pair (X, d) a Q-Metric Space
Graph of d(x,y)=1 in 2-D Space
/ 1 1 ),( 1
n
iii yxyxd
d=-1 = 1
d=0 = 1
d-1,0) = 1
de = dp=2 = 1Dimension1
Dimension2
dt = dp=1 = 1
dp=infinity = 1
x=(x1,x2)
y=(y1,y2)
2009:11:10
Magdi A. Mohamed 7/18
Q-Metric Based SVMNonlinear Classification and Regression Cases
NovelQMB-SVC
NovelQMB-SVR
ConventionalRBF-SVC
ConventionalRBF-SVR
2009:11:10
Magdi A. Mohamed 8/18
Q-Aggregates Conceptthe math behind the effect
Aggregation Operator AxiomsA function
h: [0,1]n -> [0,1], n > 2, is an aggregation operator provided that:
• h(0, 0, …, 0) = 0• h(1, 1, …, 1) = 1• h is monotonic non-decreasing in all its
arguments• h is continuous• h is symmetric in all its arguments
Generalized MeansDefined, for -infinity < < infinity, by:
h(a1, …, an) = [ (a1 + … + an
) / n ](1/)
Q-Aggregate Definition For ai [0,1], n > 2, & define:
11
1 1 ..., ,
1
11
n
i
n
ii
n
aaah
EXISTING NOVEL
2009:11:10
Magdi A. Mohamed 9/18
Aggregation Operationsprior art and q-aggregate coverage
• Veracity operator can behave as a transform from x to e(-x).
• Computationally efficient since it only requires accumulations, multiplications and a division. No exponential function calculation.
xnxxv 1,, 1,0 0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
X
notX
Evidence
Sugeno Negation Operator:
otherwize ,0
1 ,1
1xif
x
x
xn ),1[
]1,0[
x
Veracity Operator:
2009:11:10
Magdi A. Mohamed 15/18
The New Q-RBF Neural Networks
Notes:• Use more powerful metrics different from a fixed Lp or other
classical type of metrics.
• Use better aggregation operation in classification problem than simple linear weighted averaging.
• Negation and veracity functions are more computationally attractive, with low-cost than e(-x), suitable for hardware implementations, particularly in embedded platforms.