Peter Raeth, Ph.D.

Transcription

1 Peter Raeth, Ph.D. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States. For details see

2 Terms and symbols First-Grade arithmetic in various forms Arithmetic as equations Equations as functions Using functions as basis functions Gaussian radial basis functions Using Gaussians to detect data stream anomalies Two examples Summary 2

3 Sample data value arriving at a given moment in time usually at some fixed interval but not necessarily Next-Sample Predictor given the present data value (Dt), make a reasonable guess at n the next data value (Dt+1) before that value arrives = FiniteSummation i= 0 3

4 A+B=C An elementary equation Adds two values to produce new result A and B must be in the same units feet, pounds, grains, apples, oranges, dollars, etc Very generic Can apply to anything Key to the development of new concepts and capabilities is to expand specifics into generalities and then to compress generalities into specifics. 4

5 A + B + D + + zeta = C Expands on the earlier equation Combines more than two values Still produces new result via addition 5

6 n i= 1 Extends our thinking to Vi adding together (summing) some number of values Says nothing about what values represent Shorthand for previous slides equations i = specific value n = number of values V = set of values sigma = summation symbol 6

7 We will employ an equation as a basis function - simple building-block function hi e ( ci st )( ci st ) wi Our Gaussian is said to be radial since wi is the same on both sides of the center. h = maximum height c = Gaussian center, where maximum height is reached st = sample at a given time t w = Gaussian width (variance), sqrt(w) = standard deviation i refers to a specific Gaussian There can be more than one Gaussian basis function Each can have a different center, width, height Each basis function evaluated at current sample s value, s = st 7

8 c=0 h=5 8

9 c=0 h=5 9

10 1st Gaussian c = -5 h = -1 w=3 2nd Gaussian c=5 h=1 w=3 10

11 First 100 of 8000 samples Range: Not random, but almost - there is some structure, an underlying pattern 11

12 Mean range-relative prediction error Rate of decline directly related to bit-size - recall truncation error mentioned earlier 12

13 Detecting changes in data stream control law Model Initialization Transition Zones (Detections) 13

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15 Image is two-dimensional Algorithm is for one-dimensional data streams Each pixel in image can be taken as an independent one dimensional data stream Image model composed of a number of sub-models equal to the number of pixels Yields pixel-level detections Object-level detections depend on pixel-level detections Methods in machine vision accomplish that 15

16 Model plane moving through indoor scene Highly active and undersampled background Spinning fan appears to keep switching direction 16

17 Natural outdoor scene Person walks across camera s field of view 17

18 Sample data arriving over time, usually at regular intervals Prediction observe temporal relationship of sample data values make reasonable guess at next value before it arrives Detection if prediction error suddenly gets worse, an anomaly exists Decision have to decide what to do when an anomaly is found Action need to carry out the decision Common arithmetic enables the required analysis fully automatic with no human intervention 18

19 Amdahl, G.M. (1967). Validity of the Single-Processor Approach to Achieving Large Scale Computing Capabilities. Proc AFIPS, v 30, Amdahl, G.M. (1988). Limits of Expectation. Journal of Supercomputer Applications, 2(1), Bailey, D.H. (1991, Jun 11). Twelve Ways to Fool the Masses When Giving Performance Results on Parallel Computers. Technical Report # RNR , NASA Ames Research Center, Moffett Field, CA. Raeth, P.G. (2003). Finding Unexpected Events in Staring Continuous-Dwell Sensor Data Streams via Adaptive Prediction. Dissertation presented to the faculty of Nova Southeastern University. Zomaya, A.Y.H. (1996). Parallel and Distributed Computing Handbook. New York, NY: McGraw-Hill. 19