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Ph.D. Final Exam Neural Network Ensonification Emulation: Training And Application

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### Ph.D. Final ExamNeural Network Ensonification Emulation: Training And Application

JAE-BYUNG JUNG

Department of Electrical Engineering

University of Washington

August 8, 2001

Overview

- Review of adaptive sonar
- Neural network training for varying output nodes
- On-line training
- Batch-mode training
- Neural network inversion
- Sensitivity analysis
- Maximal area coverage problem
- Conclusions and ideas for future works

INTRODUCTION- Sonar Surveillance

- Software model emulating acoustic propagation
- Computationally intensive
- Not suitable for real time control

Control

Sonar

Surveillance

System

Sonar

Performance

Map

Environment

Sonar Data 1

- The physical range-depth output surveillance area is sampled by 30 ranges from 0 to 6 km at steps of 0.2 km and 13 depths from 0 to 200m at steps of 15m
- Data Size : 2,500 pattern vectors. (2,000 patterns are used for training the neural network, and 500 patterns are not used for training and reserved for testing.

Sonar Data 2

- The wider surveillance area is considered including 75 sampled ranges from 0 to 15 km at steps of 0.2 km and 20 sampled depths from 0 to 400m at steps of 20m
- The shape of SE map varies depending on different bathometry
- Data Size : 8,000 pattern vectors. (5,000 patterns are used for training the neural network, and 3,000 patterns are not used for training and reserved for testing)

Neural Network Replacement

- Fast Reproduction of SE Map
- Inversion (Derivative Existence)
- Real-time Control (Optimization)

Training NN

- High dimensionality of output space
- Multi-Layered Perceptrons
- Neural Smithing (e.g. input data jittering, pattern clipping, and weight decay …)
- Widely varying bathymetry
- Adaptive training strategy for flexible output dimensionality

MLP Training

Multilayer perceptrons (MLP’s) typically use a fixed network topology for all training patterns in a data set.

MLP Training with varying output

We consider the case where the dimension of the output nodes can vary from training pattern to training pattern

Flexible Dimensionality

- Generally, MLP must have a fixed network topology

A single neural network can not handle flexible network topology

- A modular neural network structure that has local experts for different dimension specific training patterns

It becomes increasingly difficult, however, to implement a large number of neural networks as the number of local experts increases

Flexible Dimensionality

- Let’s define a new output vector, O, for transformation to fix output dimensionality as

where,O(n) is the nth actual output training pattern vector,OA(n) is an arbitrary output vector for O(n) and filled with arbitrary “don’t care” constant numbers

- The dimensionality becomes enlarged to be spanned and is fixed as

where N is the number of training pattern vectors, Span(·) represents a dimensional span from each output vector to the maximally expandable dimensions over N different pattern vectors, and D(·) represents a dimensionality of the output vector

Flexible Dimensionality

- Train a single neural network using a fixed-dimensional output vector O(n) by

1) Filling arbitrary constant value into OA(n)

High spatial freq. components are washed out

2) Smearing neighborhood pixels to OA(n)

Still need to train unnecessary part

(longer training time)

- OA(n) can be ignored when O(n) is projected onto O(n) in the testing phase.

Inputs : I(n)={IC(n), IP(n)}

IP(n) : profile inputs

describe output profile

assign each output neuron to either O(n) or OA(n)

IC(n) : characteristic inputs

contain the other input parameters

Outputs : O’(n) ={O(n), OA(n)}

OA(n) : “Don’t care” category

The weights associated with these neurons are not updated for the nth pattern vector

O(n) : Normal weight correction

The other weights associated with O(n) are updated with step size modification

Don’t Care TrainingAdvantages

Significantly reduced training time by not correcting weights in don’t care category

Boundary problem is solved

Less training vectors required

Focus on active nodes only

Drawbacks

Rough weight space due to irregular weight correction

Possibly leading to local minima

Don’t Care TrainingStep Size Modification

- Even amount of opportunity for weight correction to every output neuron
- Statistical information : From the training data set, frequency of weight correction associated with each output neuron are updated.

Step Size Modification

- On-line training
- Batch-mode training

MSE (Mean Squared Error)

MSE can not represent the training performance well due to the different vector size(dimensionality)

Average MSE : pixel wise representation of MSE

Performance ComparisonPerformance Comparison

: Training of neural networks

Performance Comparison

: Generalization performance from testing error

Contributions:Training

- A novel neural network learning algorithm for data sets with varying output node dimension is proposed.
- Selective weight update
- Fast convergence
- Improved accuracy
- Step size modification
- Good generalization
- Improved accuracy

NN Training

: Finding Wfrom given input-output relationship

NN Inversion

: Finding I from given target output T

I

I

W

W

O

O

T

T

Inversion of neural network- W : NN Weight
- I : Input
- O : Output
- T : Target

When we want to find a subset of input vector, i, so that minimize the objective function, E(i), which can be denoted as

E(i) = 0.5(ti – oi)2

where oi is the neural network output for input I, and ti is the desired output for input i.

If is the kthcomponent of the vector it, then gradient descent suggests the recursion

where is the step size and t is the iteration index

Iteration for inversion in the equation can be solved as follows

Inversion of neural networkwhere, for any neuron,

Single element inversion

- The subset of outputs to be inverted is confined in one output pixel at a time during an inversion process while other outputs are floated.
- A single input control parameter is achieved during the iterative inversion process while 4 environmental parameters are clamped (fixed) to specific values [wind speed = 7m/s, sound speed at surface = 1500m/s, sound speed at bottom = 1500m/s, and bottom type = 9(soft mud)].

Multiple parameter inversion and maximizing the target area

- Multiple output SE values can be inverted at a time
- The output target area is tiled with 2x2 pixel regions. Thus, 2x2 output pixel groups are inverted one at a time to find out the best combination of these 5 input parameters to satisfy the corresponding SE values

Pre-Clustering for NN Inversion

Pre-clustering of data set from the output space

Separate training of partitioned data sets

(Local Experts)

Training Improvement

Inversion Improvement

UnsupervisedClustering

Partitioning a collection of data points into a number of subgroups

- When we know the number of prototypes
- K-NN, Fuzzy C-Means, …
- When no a priori information is available.
- ART, Kohonen SOFM, …

Adaptive Resonance Theory

- Unsupervised learning network developed by Carpenter and Grossberg in 1987
- ART1 is designed for clustering binary vectors
- ART2 accepts continuous-valued vectors

- F1 layer is an input processing field comprising the input portion and the interface portion.
- F2 layer is a cluster unit that is a competitive layer in that the units compete in a winner-take-all mode for the right to learn each input pattern.
- The third layer is a reset mechanism that controls the degree of similarity of patterns placed on the same cluster

Training Phase

Entire Data Set

: Unsupervised Learning

ART2

sub-data

sub-data

sub-data

(cluster K)

(cluster 1)

(cluster 2)

Training

Training

Training

: Supervised Learning

ANN K

ANN 1

ANN 2

Inversion Phase

ART2 Cluster 1

(N-Dimension)

ART2 Cluster 3

(N-Dimension)

ART2 Cluster K

(N-Dimension)

Desired Output

(M-Dimension)

Cluster Selection (Projection)

ANN 1

Inversion

ANN 2

Inversion

ANN K

Inversion

Optimal Input Parameters

Inversion from ART2 Modular Local Experts

- Multiple parameter inversion and maximizing the target area.
- The output target area is tiled with 2x2 pixel regions. Thus, 2x2 output pixel groups are inverted one at a time to find out the best combination of these 5 input parameters to satisfy the corresponding SE values.

Contributions:Inversion

- A new neural network inversion algorithm was proposed whereby several neural networks are inverted in parallel.
- Advantages include the ability to segment the problem into multiple sub-problems which each can be independently modified as changes to the system occur over time.
- The concept is similar to the mixture of experts problem applied to neural network inversion.

Sensitivity Analysis

- Feature selection as neural network is being trained or after the training. Useful to eliminate superfluous input parameters [Rambhia].
- reducing the dimension of the decision space and
- increasing speed and accuracy of the system.
- When implemented in hardware, the non-linearity occuring in the operation of various network component may practically make a network impossible to train significantly [Jiao].
- very important in the investigation of non-ideal effects. (important issue from the view point of engineering).
- Once neural network is trained, it is very important to determine which of the control parameters are critical to the decision making at a certain operating point.

The neural network sensitivity shows how sensitively the output OT responds with respect to the change of the input parameter k.NN Sensitivity

ODC

(Don’t Care Area)

ik

OT

(Target Surveillance Area)

NN Sensitivity

- Chain rule is used to derive

where h represents hidden neurons.

Generally, for n hidden layers

where hn represents neurons in nth hidden layer.

NN Sensitivity – output neuron

1. Output Layers

: find local gradient of oi at output neuron i

neti

oi

hj

wij

i

NN Sensitivity – hidden neuron

2. Hidden Layers

: find gradient at hidden neuron j with respect to OT

netj

hj

wji

i

j(netj)

j

Contributions: Sensitivity Analysis

- Once neural network is trained, especially, it is very important to determine which of the control parameters are critical to the decision making at a certain operating point such that environmental situation or/and control criteria is given.

Multiple Objects Optimization

- Optimization of multiple objects in order to satisfy system’s maximum cooperating performance.
- The composite effort of the system team is significantly more important than a single system’s individual performance

Target Covering Problem

- Multiple rectangular boxes move to cover the circular target area.
- Each box is situated by 4 parameters including 2 position variables, (x0, y0), an orientation, , and an aspect ratio, r.
- The area of each box is fixed.

Target Parameters

where (c1, c2) is center of gravity and R is radius of circular target.

Box 1

Box N

. . .

q

q

x

y

r

x

y

r

1

1

1

1

N

N

N

N

Genetic Algorithm(Optimization)- Optimization deals with problems of function maximization or minimization with several variables usually subject to certain constraints in general.
- While traditional search algorithms commonly impose severe constraints to the functions to be minimized (or maximized) such as continuity, derivative existence or unimodality, genetic algorithms work in a different ways, acting as a global probabilistic search method.
- Chromosomal Representation.

Box and Target Parameters

A = 2000 chromosomes

R = 64

(C1,C2) = (128,128)

GA Parameters

Population Size = 20 chromosomes

Bit Resolution = 8 bits

Probability of Crossover = 0.7

Probability of Mutation = 0.03

Experiment 1 : 2-Box ProblemExperiment 2 : 4-Box Problem

- Box and Target Parameters
- A = 1000
- R = 64
- (C1,C2) = (128,128)
- GA Parameters
- Population Size = 30 chromosomes
- Bit Resolution = 8 bits
- Probability of Crossover = 0.8
- Probability of Mutation = 0.02

MULTIPLE SONAR PING OPTIMIZATION

- Sonar Coverage : When the output pixels above a certain threshold value are only meaningful, those pixels in the output surveillance area are considered as “covered”.
- Maximization of the aggregated sonar coverage from the given number of pings allows minimization of the counter detection by the object for which we are looking.
- Genetic algorithm based approach to find the best combination of control parameters when the environment is given.

Genetic Algorithm : Population

- The sonar control parameter is in the range of [10m 130m] and the required precision is 3 places after the decimal point. So, the required number of bits for a depth is 17. Thus the population size has 40 chromosomes and each has 68(=4x17) bit strings

Contributions:Maximal Area Coverage

- The systems need not be the replications of each other but can, for example, specialize in different aspects of appeasing the fitness function.
- The search can be constrained, for example, a constraint imposing module can be inserted.

Conclusions

- A novel neural network learning algorithm (Don’t care training with step size modification) for data sets with varying output dimension is proposed.
- A new neural network inversion algorithm was proposed whereby several neural networks are inverted in parallel. (the ability to segment the problem into multiple sub-problems)
- The sensitivity of neural network is investigated. Once neural network is trained, especially, it is critical to the decision making at a certain operating point. This can be done through input parameter sensitivity analysis.
- There exist numerous generalizations of the fundamental architecture of maximal area coverage problem that allow application to a larger scope of problems.

Ideas for Future Works

- More work could be done for more accurate training of sonar data. Especially, multi resolution neural networks could help extract discrete detection maps.
- Data pruning using nearest neighbor analysis before training or query-based learning using sensitivity analysis during training could improve the training time or/and accuracy.
- Extensive research for the use of evolutionary algorithms to improve the inversion speed and precision. Particle swarm optimization or genetic algorithms could be considered for more flexibility on imposing feasibility constraints.

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