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.

1. 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

3. Q-Measures in a nutshell q-measures provide more expressive and computationally attractive nonlinear models for uncertainty management when modeling a complex system, it’s an oversimplification to assume that the interdependency among information sources is linear X x 2 x 3 x 4 A x 1 x 6 x 5 x 7 x 8 x 9 B=A c q(A) q(A c )  =0 probability  >0 plausibility  <0 belief       

4. Q-filter Computations N=5 Tap Case - Nonlinearity, Adaptivity, and Model Capacity h 5 <h 2 < h 1 <h 3 < h 4 h 4 x 1 x 2 x 3 x 4 x 5 f 1 f 2 f 3 f 4 f 5 Window Slots Signal Value h 1 h 2 h 3 h 5 Density Generators i  h 5 h 2 h 1 h 3 h 4 Threshold Nonlinearity Controller  h(x i ) q(A  ) q({x 4 }) q({x 4 , x 3 }) q({x 4 , x 3 , x 1 }) q({x 4 , x 3 , x 1 , x 2 }) q({x 4 , x 3 , x 1 , x 2 , x 5 })=1.0 Case Adaptive Weight  A  q(  )=0.0 Total area is the Q-filter output value

7. Q-Metric Based SVM Nonlinear Classification and Regression Cases Novel QMB-SVC Novel QMB-SVR Conventional RBF-SVC Conventional RBF-SVR

9. Aggregation Operations prior art and q-aggregate coverage Intersection / Conjunction Operations Averaging / Compensative Operations Union / Disjunction Operations Generalized Means Scheize/Sklar Scheize/Sklar Hamacher Hamacher Frank Frank Yager Yager Dubois/Prade Dubois/Prade Dombi Dombi  p p   s s w w     Q-Aggregates  - 1 + inf 2003 1982 1980 1980 1979 1978 1961 - inf + inf - inf + inf + inf - inf + inf + inf + inf + inf 0 0 0 0 min max

10. QFS Supervised Learning for EKG case study S - Q Q RMS=0.128 S Q Q Q Q - RMS=0.032 A A S Q Q Q RMS=0.044 -

12. Weighted Q-Metrics Recursive Weighted Q-Metrics Calculation Algorithm:

13. Weighted Q-Aggregate Recursive Weighted Q-Aggregate Calculation Algorithm:

16. Case Studies Regression : RF Positioning Classification : Driver Maneuver 2x4x1 neural network Q-RBF RMS = 0.077 BP RMS = 0.110 Q-RBF confusion matrix BP confusion matrix 5x3x2 neural network Q-RBF clearly has better classification results than BP. 2628 223 2 482 2613 1 2 1 2720 131 2 1076 2019 1 2 1

17. Q-Aggregates: Union / Average DOMAINS

18. Q-Aggregates: Intersection / Average DOMAINS