Method and system for training a neural network with adaptive weight updating and adaptive pruning in principal component space

0Associated
Cases 
0Associated
Defendants 
0Accused
Products 
49Forward
Citations 
0
Petitions 
4
Assignments
First Claim
1. A neural network having a plurality of weights for receiving a sequence of signal inputs x_{t},x_{t+1}, x_{t+2} . . . , each input x_{t} comprising n signal components x_{1} (t), x_{2} (t1), . . . , x_{n} (t(n1)) and for generating an output signal that anticipates the behavior of said input signal for a number of time samples ahead, said neural network comprising:
 transformation means for transforming a set of n signal inputs into a set of principal components having a saliency assigned to each of said principal component;
pruning means, coupled to said transformation means, for pruning a number of said principal components that correspond to the smallest saliencies, where the number of said principal components is limited by a sum of said saliencies of said pruned principal components to be less than or equal to a predefined threshold level, leaving a remaining set of principal components;
first computing means, coupled to said pruning means, for computing the output signal using said set of remaining principal components; and
wherein said neural network an updating means, coupled to said first computing means, for updating the weights of the neural network adaptively based on an error between a target output and the output signal.
4 Assignments
0 Petitions
Accused Products
Abstract
A signal processing system and method for accomplishing signal processing using a neural network that incorporates adaptive weight updating and adaptive pruning for tracking nonstationary signal is presented. The method updates the structural parameters of the neural network in principal component space (eigenspace) for every new available input sample. The nonstationary signal is recursively transformed into a matrix of eigenvectors with a corresponding matrix of eigenvalues. The method applies principal component pruning consisting of deleting the eigenmodes corresponding to the smallest saliencies, where a sum of the smallest saliencies is less than a predefined threshold level. Removing eigenmodes with low saliencies reduces the effective number of parameters and generally improves generalization. The output is then computed by using the remaining eigenmodes and the weights of the neural network are updated using adaptive filtering techniques.
61 Citations
View as Search Results
SYSTEMS AND METHODS FOR TRAINING NEURAL NETWORKS BASED ON CONCURRENT USE OF CURRENT AND RECORDED DATA  
Patent #
US 20110161267A1
Filed 07/28/2010

Current Assignee
Georgia Tech Research Corporation

Sponsoring Entity
Georgia Tech Research Corporation

Method, system and computer program for developing cortical algorithms  
Patent #
US 7,493,295 B2
Filed 03/31/2005

Current Assignee
Francisco J. Ayala

Sponsoring Entity
Francisco J. Ayala

Cognitive image filtering  
Patent #
US 7,577,631 B2
Filed 03/21/2002

Current Assignee
Michael J. Feldhake

Sponsoring Entity
Michael J. Feldhake

System and method of automatically maintaining and recycling software components  
Patent #
US 7,590,599 B2
Filed 12/22/2006

Current Assignee
International Business Machines Corporation

Sponsoring Entity
International Business Machines Corporation

COGNITIVE OPERATING SYSTEM  
Patent #
US 20090313190A1
Filed 08/18/2009

Current Assignee
Michael J. Feldhake

Sponsoring Entity
Michael J. Feldhake

DATA PROCESSING APPARATUS, DATA PROCESSING METHOD, AND COMPUTER PROGRAM  
Patent #
US 20090299933A1
Filed 03/27/2009

Current Assignee
Sony Corporation

Sponsoring Entity
Sony Corporation

Fluxgate sensor for calibrating azimuth at slope and calibration method thereof  
Patent #
US 7,363,185 B2
Filed 10/28/2004

Current Assignee
Samsung Electronics Co. Ltd.

Sponsoring Entity
Samsung Electronics Co. Ltd.

SYSTEM AND METHOD OF AUTOMATICALLY MAINTAINING AND RECYCLING SOFTWARE COMPONENTS  
Patent #
US 20080154776A1
Filed 12/22/2006

Current Assignee
International Business Machines Corporation

Sponsoring Entity
International Business Machines Corporation

Flexible Modeling Concept for Discretely Simulating and Implementing, in particular, Material Processing and Distribution Systems  
Patent #
US 20080228453A1
Filed 11/11/2005

Current Assignee
Siemens AG

Sponsoring Entity
Siemens AG

Contextual data mapping, searching and retrieval  
Patent #
US 20080228761A1
Filed 03/10/2008

Current Assignee
1759304 ONTARIO INC.

Sponsoring Entity
1759304 ONTARIO INC.

Classification method and apparatus based on boosting and pruning of multiple classifiers  
Patent #
US 6,456,991 B1
Filed 09/01/1999

Current Assignee
HRL Laboratories LLC

Sponsoring Entity
HRL Laboratories LLC

Speech recognition programming information retrieved from a remote source to a speech recognition system for performing a speech recognition method  
Patent #
US 6,999,927 B2
Filed 10/15/2003

Current Assignee
Sensory Incorporated

Sponsoring Entity
Sensory Incorporated

Method of performing speech recognition across a network  
Patent #
US 7,092,887 B2
Filed 10/15/2003

Current Assignee
Sensory Incorporated

Sponsoring Entity
Sensory Incorporated

Adaptive learning enhancement to automated model maintenance  
Patent #
US 7,092,922 B2
Filed 05/21/2004

Current Assignee
Computer Associates Think Inc.

Sponsoring Entity
Computer Associates Think Inc.

Pulse oximetry methods and apparatus for use within an auditory canal  
Patent #
US 7,107,088 B2
Filed 05/17/2004

Current Assignee
Sarnoff Corporation

Sponsoring Entity
Sarnoff Corporation

Artificial neural network guessing method and game  
Patent #
US 20060230008A1
Filed 04/08/2005

Current Assignee
20Q.NET INC.

Sponsoring Entity
20Q.NET INC.

System and method for developing artificial intelligence  
Patent #
US 7,139,740 B2
Filed 01/13/2004

Current Assignee
Francisco J. Ayala

Sponsoring Entity
Francisco J. Ayala

Method of detecting flaws in the structure of a surface  
Patent #
US 7,149,337 B2
Filed 11/21/2001

Current Assignee
INB Vision AG

Sponsoring Entity
INB Vision AG

Adaptive learning enhancement to automated model maintenance  
Patent #
US 20050033709A1
Filed 05/21/2004

Current Assignee
Computer Associates Think Inc.

Sponsoring Entity
Computer Associates Think Inc.

Pulse oximetry methods and apparatus for use within an auditory canal  
Patent #
US 20050049471A1
Filed 05/17/2004

Current Assignee
Sarnoff Corporation

Sponsoring Entity
Sarnoff Corporation

Processing methods and apparatus for monitoring physiological parameters using physiological characteristics present within an auditory canal  
Patent #
US 20050059870A1
Filed 08/25/2004

Current Assignee
Sarnoff Corporation

Sponsoring Entity
Sarnoff Corporation

Fluxgate sensor for calibrating azimuth at slope and calibration method thereof  
Patent #
US 20050114076A1
Filed 10/28/2004

Current Assignee
Samsung Electronics Co. Ltd.

Sponsoring Entity
Samsung Electronics Co. Ltd.

Visualization and selforganization of multidimensional data through equalized orthogonal mapping  
Patent #
US 6,907,412 B2
Filed 03/23/2001

Current Assignee
CA Inc. dba CA Technologies

Sponsoring Entity
Computer Associates Think Inc.

Use of two or more sensors to detect different nuclear quadrupole resonance signals of a target compound  
Patent #
US 20050206382A1
Filed 02/04/2005

Current Assignee
E. I. du Pont de Nemours and Company

Sponsoring Entity
E. I. du Pont de Nemours and Company

Combinatorial approach for supervised neural network learning  
Patent #
US 6,954,744 B2
Filed 08/29/2001

Current Assignee
Honeywell International Inc.

Sponsoring Entity
Honeywell International Inc.

METHOD, SYSTEM AND COMPUTER PROGRAM FOR DEVELOPING CORTICAL ALGORITHMS  
Patent #
US 20050256814A1
Filed 03/31/2005

Current Assignee
Francisco J. Ayala

Sponsoring Entity
Francisco J. Ayala

Method for training a neural network  
Patent #
US 6,968,327 B1
Filed 08/24/2000

Current Assignee
WILEX AG

Sponsoring Entity
WILEX AG

Method of performing speech recognition across a network  
Patent #
US 20040083098A1
Filed 10/15/2003

Current Assignee
Sensory Incorporated

Sponsoring Entity
Sensory Incorporated

Speech recognition method  
Patent #
US 20040083103A1
Filed 10/15/2003

Current Assignee
Sensory Incorporated

Sponsoring Entity
Sensory Incorporated

System and method for developing artificial intelligence  
Patent #
US 20040143559A1
Filed 01/13/2004

Current Assignee
Francisco J. Ayala

Sponsoring Entity
Francisco J. Ayala

Coin validation  
Patent #
US 6,799,670 B1
Filed 03/22/2002

Current Assignee
MICROSYSTEM CONTROLS PTY LTD

Sponsoring Entity
MICROSYSTEM CONTROLS PTY LTD

Cognitive operating system  
Patent #
US 20030049589A1
Filed 03/21/2002

Current Assignee
Michael J. Feldhake

Sponsoring Entity
Michael J. Feldhake

Speech recognition in consumer electronic products  
Patent #
US 6,665,639 B2
Filed 01/16/2002

Current Assignee
Sensory Incorporated

Sponsoring Entity
Sensory Incorporated

Method of detecting flaws in the structure of a surface  
Patent #
US 20020072874A1
Filed 11/21/2001

Current Assignee
INB Vision AG

Sponsoring Entity
INB Vision AG

System for discovering implicit relationships in data and a method of using the same  
Patent #
US 6,466,929 B1
Filed 11/12/1999

Current Assignee
University of Delaware

Sponsoring Entity
University of Delaware

Visualization and selforganization of multidimensional data through equalized orthogonal mapping  
Patent #
US 6,212,509 B1
Filed 05/02/2000

Current Assignee
CA Inc. dba CA Technologies

Sponsoring Entity
Computer Associates Think Inc.

Method and apparatus for machine learning  
Patent #
US 6,249,781 B1
Filed 05/05/1999

Current Assignee
Verizon Laboratories Incorporated

Sponsoring Entity
Verizon Laboratories Incorporated

Method and system for training an artificial neural network  
Patent #
US 6,269,351 B1
Filed 03/31/1999

Current Assignee
Mantra Technologies Incorporated

Sponsoring Entity
DRYKEN TECHNOLOGIES

Method for predicting the therapeutic outcome of a treatment  
Patent #
US 6,317,731 B1
Filed 05/11/2000

Current Assignee
Advanced Biological Laboratories SA

Sponsoring Entity
Luciano Joanne Sylvia

Data processing apparatus, data processing method, and computer program for setting targets in a range and taking action to attain the target  
Patent #
US 8,195,586 B2
Filed 03/27/2009

Current Assignee
Sony Corporation

Sponsoring Entity
Sony Corporation

Contextual data mapping, searching and retrieval  
Patent #
US 8,266,145 B2
Filed 03/10/2008

Current Assignee
1759304 ONTARIO INC.

Sponsoring Entity
1759304 ONTARIO INC.

Systems and methods for training neural networks based on concurrent use of current and recorded data  
Patent #
US 8,489,528 B2
Filed 07/28/2010

Current Assignee
Georgia Tech Research Corporation

Sponsoring Entity
Georgia Tech Research Corporation

Flexible modeling concept for discretely simulating and implementing, in particular, material processing and distribution systems  
Patent #
US 8,775,137 B2
Filed 11/11/2005

Current Assignee
Siemens AG

Sponsoring Entity
Siemens AG

ULTRASONIC DIAGNOSTIC APPARATUS, IMAGE PROCESSING APPARATUS, AND IMAGE PROCESSING METHOD  
Patent #
US 20150366540A1
Filed 06/05/2015

Current Assignee
Canon Medical Systems Corporation

Sponsoring Entity
Canon Medical Systems Corporation

Apparatus, method and article to facilitate automatic detection and removal of fraudulent user information in a network environment  
Patent #
US 10,277,710 B2
Filed 10/12/2017

Current Assignee
Plentyoffish Media ULC

Sponsoring Entity
Plentyoffish Media ULC

Systems and methods for training and employing a machine learning system in providing service level upgrade offers  
Patent #
US 10,387,795 B1
Filed 03/30/2015

Current Assignee
Plentyoffish Media ULC

Sponsoring Entity
Plentyoffish Media Incorporated

Intelligent control with hierarchical stacked neural networks  
Patent #
US 10,417,563 B1
Filed 04/07/2017

Current Assignee
Michael Lamport Commons, Mitzi Sturgeon White

Sponsoring Entity
Michael Lamport Commons, Mitzi Sturgeon White

System and method for hyperparameter analysis for multilayer computational structures  
Patent #
US 10,534,994 B1
Filed 11/11/2015

Current Assignee
Cadence Design Systems Incorporated

Sponsoring Entity
Cadence Design Systems Incorporated

Apparatus, method and article to effect electronic message reply rate matching in a network environment  
Patent #
US 10,540,607 B1
Filed 12/08/2014

Current Assignee
Plentyoffish Media ULC

Sponsoring Entity
Plentyoffish Media Incorporated

Method for operating an optimal weight pruning apparatus for designing artificial neural networks  
Patent #
US 5,636,326 A
Filed 07/07/1995

Current Assignee
Ricoh Corporation

Sponsoring Entity
Ricoh Corporation

Neural network having an optimized transfer function for each neuron  
Patent #
US 5,280,564 A
Filed 02/18/1992

Current Assignee
Honda Giken Kogyo Kabushiki Kaisha

Sponsoring Entity
Honda Giken Kogyo Kabushiki Kaisha

Neural network process measurement and control  
Patent #
US 5,282,261 A
Filed 08/03/1990

Current Assignee
Rockwell Automation Technologies Incorporated

Sponsoring Entity
E. I. du Pont de Nemours and Company

Adaptive neuron modelan architecture for the rapid learning of nonlinear topological transformations  
Patent #
US 5,371,834 A
Filed 08/28/1992

Current Assignee
The United States of America As Represented By The Secretary of Agriculture

Sponsoring Entity
The United States of America As Represented By The Secretary of Agriculture

Neural network for processing both spatial and temporal data with time based backpropagation  
Patent #
US 5,253,329 A
Filed 12/26/1991

Current Assignee
The United States of America As Represented By The Secretary of Agriculture

Sponsoring Entity
The United States of America As Represented By The Secretary of Agriculture

Stable adaptive neural network controller  
Patent #
US 5,268,834 A
Filed 06/24/1991

Current Assignee
Massachusetts Institute of Technology

Sponsoring Entity
Massachusetts Institute of Technology

Recurrent neural network with variable size intermediate layer  
Patent #
US 5,129,039 A
Filed 07/10/1991

Current Assignee
Sony Corporation

Sponsoring Entity
Sony Corporation

Cellular neural network  
Patent #
US 5,140,670 A
Filed 10/05/1989

Current Assignee
Regents of the University of California

Sponsoring Entity
Regents of the University of California

Universal process control using artificial neural networks  
Patent #
US 5,159,660 A
Filed 10/11/1990

Current Assignee
WESTERN THUNDER A CORP. OF CA

Sponsoring Entity
WESTERN THUNDER A CORP. OF CA

Method and procedure for neural control of dynamic processes  
Patent #
US 5,175,678 A
Filed 08/15/1990

Current Assignee
ELSAG INTERNATIONAL B.V.

Sponsoring Entity
ELSAG INTERNATIONAL B.V.

Fire alarm system  
Patent #
US 5,168,262 A
Filed 07/18/1990

Current Assignee
NOHMI BOSAI KABUSHIKI KAISHA

Sponsoring Entity
NOHMI BOSAI KABUSHIKI KAISHA

Neural network with back propagation controlled through an output confidence measure  
Patent #
US 5,052,043 A
Filed 05/07/1990

Current Assignee
Eastman Kodak Company

Sponsoring Entity
Eastman Kodak Company

21 Claims
 1. A neural network having a plurality of weights for receiving a sequence of signal inputs x_{t},x_{t+1}, x_{t+2} . . . , each input x_{t} comprising n signal components x_{1} (t), x_{2} (t1), . . . , x_{n} (t(n1)) and for generating an output signal that anticipates the behavior of said input signal for a number of time samples ahead, said neural network comprising:
transformation means for transforming a set of n signal inputs into a set of principal components having a saliency assigned to each of said principal component; pruning means, coupled to said transformation means, for pruning a number of said principal components that correspond to the smallest saliencies, where the number of said principal components is limited by a sum of said saliencies of said pruned principal components to be less than or equal to a predefined threshold level, leaving a remaining set of principal components; first computing means, coupled to said pruning means, for computing the output signal using said set of remaining principal components; and wherein said neural network an updating means, coupled to said first computing means, for updating the weights of the neural network adaptively based on an error between a target output and the output signal.
 2. A neural network having a plurality of weights for receiving a sequence of signal inputs x_{t},x_{t+1}, x_{t+2}. . . , each input x_{t} comprising n signal components x_{1} (t), x_{2} (t1), . . . , x_{n} (t(n1)) and for generating an output signal that anticipates the behavior of said input signal for a number of time samples ahead, said neural network comprising:
transformation means for transforming a set of n signal inputs into a set of principal components having a saliency assigned to each of said principal component; pruning means coupled to said transformation means, for pruning a number of said principal components that correspond to the smallest saliencies, where the number of said principal components is limited by a sum of said saliencies of said pruned principal components to be less than or equal to a predefined threshold level, leaving a remaining set of principal components; first computing means, coupled to said pruning means, for computing the output signal using said set of remaining principal components; and updating means, coupled to said first computing means, for updating the weights of the neural network adaptively based on an error between a target output and the output signal, wherein said transformation means includes an estimation means for recursively estimating a current set of principal components from a set of principal components of a previously transformed set of n signal inputs.  View Dependent Claims (3, 4, 5, 6, 7, 8, 9)
 4. The neural network of claim 2, wherein said estimation means estimates said current set of principal components by directly calculating a matrix Q_{t} and a matrix Λ
 _{t}, where Q_{t} is a matrix of eigenvectors and Λ
_{t} is a matrix of eigenvalues.
 _{t}, where Q_{t} is a matrix of eigenvectors and Λ
 5. The neural network of claim 2, wherein said saliencies are calculated in accordance to the formula
 space="preserve" listingtype="equation">s.sub.i (t)= v.sub.i x.sub.i !.sup.T v.sub.i x.sub.i !=v.sub.t.sup.T v.sub.i x.sub.i.sub.2
where x_{t} is the KarhunenLoeve expansion of x_{t} and v_{i} is a p×
1 vector of W_{t}^{T} defined as W_{t}^{T} = v_{1},v_{2}, . . . , v_{n} !.
 6. The neural network of claim 2, wherein said saliencies are calculated in accordance to the formula
 space="preserve" listingtype="equation">s.sub.i (t)=v.sub.t.sup.T v.sub.i x.sub.i.sup.2
where x_{t} is the KarhunenLoeve expansion of x_{t}, v_{i} is a p×
1 vector of W_{t}^{T} defined as W_{t}^{T} = v_{1},v_{2}, . . . , v_{n} !, x_{i}^{2} is defined as x_{i}^{2} (t)=μ
x_{i}^{2} (t1)+(1μ
)x_{i}^{2} (t) and μ
is a forgetting factor.
 space="preserve" listingtype="equation">s.sub.i =λ
.sub.i v.sub.i.sup.T v.sub.i,
_{i} is the ith element on the diagonal of Λ
_{t} and v_{i} is a p×
1 vector of W_{t}^{T} defined as W_{t}^{T} = v_{1},v_{2}, . . . , v_{n} !.
second computing means for computing an output in principal component space; identifying means, coupled to said second computing means, for identifying said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level; and third computing means, coupled to said identifying means, for computing a pruning vector from said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level.
identifying means for identifying said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level; and fourth computing means, coupled to said identifying means, for computing a weight matrix in regular space from said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level.
 10. A method of signal processing, utilizing a neural network having a plurality of weights, for receiving a sequence of signal inputs x_{t}, x_{t+1},x_{t+2} . . . , each input x_{t} comprising n signal components x_{1} (t), x_{2} (t1), . . . , x_{n} (t(n1)) and for generating an output signal that anticipates the behavior of said input signal for a number of time samples ahead, said method comprising the steps of:
(a) transforming a set of n signal inputs into a set of principal components having a saliency assigned to each of said principal component; (b) pruning a number of said principal components that correspond to the smallest saliencies, where the number of said pruned principal components is limited by a sum of said saliencies of said pruned principal components to be less than or equal to a predefined threshold level, leaving a remaining set of principal components; (c) computing said output signal using said remaining set of principal components; and (d) updating the weights of the neural network adaptively based on an error between a target output and the output signal.
 11. A method of signal processing, utilizing a neural network having a plurality of weights, for receiving a sequence of signal inputs x_{t},x_{t+1},x_{t+2}. . . each input x_{t} comprising n signal components x_{1} (t),x_{2} (t1), . . . , x_{n} (t(n1)) and for generating an output signal that anticipates the behavior of said input signal for a number of time samples ahead, said method comprising the steps of:
(a) transforming a set of n signal inputs into a set of principal components having a saliency assigned to each of said principal component; (b) pruning a number of said principal components that correspond to the smallest saliencies, where the number of said pruned principal components is limited by a sum of said saliencies of said pruned principal components to be less than or equal to a predefined threshold level, leaving a remaining set of principal components; (c) computing said output signal using said remaining set of principal components; and (d) updating the weights of the network adaptively based on an error between a target output and the output signal wherein said transformation step includes an estimation step for recursively estimating a current set of principal components from a set of principal components of a previously transformed set of n signal inputs.  View Dependent Claims (12, 13, 14, 15, 16, 17, 18, 19)
 14. The method of claim 11 wherein said estimating step estimates said current set of principal components by directly calculating a matrix Q_{t} and a matrix Λ
 _{t}, where Q_{t} is a matrix of eigenvectors and Λ
_{t} is a matrix of eigenvalues.
 _{t}, where Q_{t} is a matrix of eigenvectors and Λ
 15. The method of claim 11, wherein said saliencies are calculated in accordance to the formula
 space="preserve" listingtype="equation">s.sub.i (t)= v.sub.i x.sub.i !.sup.T v.sub.i x.sub.i !=v.sub.t.sup.T v.sub.i x.sub.i.sup.2
where x_{t} is the KarhunenLoeve expansion of x_{t} and v_{i} is a p×
1 vector of W_{t}^{T} defined as W_{t}^{T} = v_{1},v_{2}, . . . , v_{n} !.
 16. The method of claim 11, wherein said saliencies are calculated in accordance to the formula
 space="preserve" listingtype="equation">s.sub.i (t)=v.sub.t.sup.T v.sub.i x.sub.i.sup.2
where xx_{t} is the KarhunenLoeve expansion of x_{t}, v_{i} is a p×
1 vector of W_{t}^{T} defined as W_{t}^{T} = v_{1},v_{2}, . . . , v_{n} !, x_{i} ^{2} is defined as x_{i}^{2} (t)=μ
x_{i}^{2} (t1)+(1μ
)x_{i}^{2} (t) and μ
is a forgetting factor.
 space="preserve" listingtype="equation">s.sub.i =λ
.sub.i v.sub.i.sup.T v.sub.i,
_{i} is the ith element on the diagonal of Λ
_{t} and v_{i} is a p×
1 vector of W_{t}^{T} defined as W_{t}^{T} = v_{1},v_{2}, . . . , v_{n} !.
computing an output in principal component space; identifying said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level; and computing a pruning vector from said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level.
identifying said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level; and computing a weight matrix in regular space from said principal components that correspond to the smallest saliencies, where a sum of said smallest saliencies is less than a predefined threshold level.
 20. A signal processing system having a neural network for receiving a sequence of signal inputs x_{t}, x_{t+1}, x_{t+2} . . . , each input x_{t} comprising n signal components x_{1} (t), x_{2} (t1), . . . , x_{n} (t(n1)) and generating an output signal that anticipates the behavior of said input signal for a number of time samples ahead, said neural network having a plurality of hierarchically connected nodes forming a plurality of layers, each of said layer consisting of at least one node, said nodes being interconnected with a plurality of weights, said signal processing system comprising:
transformation means for transforming a set of n signal inputs into a set of principal components having a saliency assigned to each of said principal component; pruning means, coupled to said transformation means, for pruning a number of said principal components that correspond to the smallest saliencies, where the number of said pruned principal components is limited by a sum of said saliencies of said pruned principal components to be less than or equal to a predefined threshold level, leaving a remaining set of principal components; computing means, coupled to said pruning means, for computing the output signal of a layer of the neural network using said set of remaining principal components; and updating means, coupled to said computing means, for updating the weights of the neural network adaptively based on an error between a target output and the output signal.  View Dependent Claims (21)
1 Specification
This application is a continuation of patent application Ser. No. 08/448,770 entitled "METHOD AND SYSTEM FOR TRAINING A NEURAL NETWORK WITH ADAPTIVE WEIGHT UPDATING AND ADAPTIVE PRUNING IN PRINCIPAL COMPONENT SPACE" filed on May 24, 1995, now abandoned.
The present invention relates generally to the field of neural information processing and, more particularly, to a signal processing system and method for accomplishing signal processing with a neural network that incorporates adaptive weight updating and adaptive pruning for tracking nonstationary signals.
Over the years, neural network modeling has been developed to solve problems ranging from natural language understanding to visual processing. A neural network is a computational model composed of neurons (or simply nodes) and connections between the nodes. The strength of each connection is expressed by a numerical value called a weight, which can be modified. Similarly, the "firing" of each node is associated with a threshold numerical value, which is referred to as the nodes'"'"' activation. The activation of a given node is based on the activations of the nodes that have connections directed at that node and the weights on those connections. In general, a neural network incorporates some special nodes called input nodes with their activation externally set, while other nodes are distinguished as output nodes.
In contrast to conventional computers, which are programmed to perform specific tasks, most neural networks can be taught, or trained. As such, a rule that updates the activations is typically referred to as the update rule. Similarly, learning in a neural network is generally accomplished using a learning rule to adjust the weights.
A typical neural network model has a set of input patterns and a set of output patterns. The role of the neural network is to perform a function that associates each input pattern with an output pattern. A learning process, such as "error backpropagation", uses the statistical properties of a training set of input/output patterns to generalize outputs from new inputs.
Error backpropagation is a supervised learning process through which a neural network learns optimal weights. Error backpropagation compares the responses of the output nodes to a desired response, and adjusts the weights in the network so that if the same input is presented to the network again, the network'"'"'s response will be closer to the desired response.
Referring to FIG. 1, the learning rule of error backpropagation is applied to a multilayer neural network having an input layer 130, an intermediate layer or socalled hidden layer 140 and an output layer 150. The output values of all nodes n_{h} 112 in the input layer 130 are distributed as an input value to each of the node n_{i} 114 in the intermediate layer 140. The output value of each of the nodes in the intermediate layer 140 is distributed as an input value to every node n_{j} 116 in the output layer 150. Each node n_{j} 116 produces a value which is the total sum net of output values O_{i} of a node n_{i} coupled to the output node n_{j} by a coupling weight w_{ji}, transformed by a predetermined function f. This same concept applies to the intermediate node n_{i} 114 with respect to input node n_{h} 112. In other words, when the values within a pattern p are provided as an input value to each node n_{h} 112, an output value O_{pi} and O_{pj} for each node n_{i} 114 and n_{j} 116 respectively, can be expressed by the following formulas: ##EQU1##
Hence, the network acquires output value O_{pj} of the output node n_{j} 116 of the output layer 150 by sequentially computing the output values of the input n_{j} 116, each corresponding to a node from the input layer 130 towards the output layer 150.
The process of learning in accordance with error backpropagation consists of updating coupling weights w_{ji} and w_{ih}, so that the total sum E_{p} of the square errors between the output value O_{pj} of each node of the output layer 150 on applying the input pattern p and the desired output t_{pj}, is minimized. Hence, the total network error E for the input pattern p is defined by: ##EQU2## This algorithm is sequentially applied from the output layer 150 back toward the input layer 130. The network error with respect to any weight, e.g. weight w_{ji}, is given by the gradient ∂E_{p} /∂w_{ji} of the total network error E_{p} with respect to a change in that weight. Hence, the error δ_{j}, for each output node n_{j} 116 can be determined as a function of the corresponding actual value O_{pj} and target value t_{pj} and the difference therebetween for that node, as follows:
δ.sub.j =t.sub.pj (1t.sub.pj)(O.sub.pj t.sub.pj) (3)
and for an intermediate node n_{i} 114, as follows: ##EQU3## After the neural errors are determined, these errors are propagated, via leads 170, back toward the network input nodes.
The coupling weights of both the output layer 150 and the intermediate layer 140 are adjusted according to the following learning rules 180 for n_{j} 116:
Δw.sub.ji (n+1)=ηδ.sub.j t.sub.pj +αΔw.sub.ji (n) (5)
and for each intermediate node n_{i} 114:
Δw.sub.ih (n+1)=ηδ.sub.i t.sub.pi αΔw.sub.ih (n)(6)
In the above formulas, η represents the rate of learning, which is a constant, and it determines how fast the network weights converge during network training. Coefficient a represents a stabilization factor for reducing the error oscillations and accelerating the convergence thereof. Both coefficients η and α can be empirically determined from the number of nodes, layers, input values or output values. This weight adjustment process is repeated until the patterns in the training set are exhausted or when the final error value falls below a predefined upper bound E_{max}. For a detailed explanation of error backpropagation in neural networks, see S. Haykin, Neural Networks, IEEE Press, (1994).
However, error backpropagation is limited in that this technique does not provide any information concerning the optimal number of nodes in the neural network. For example, if the neural network has a predefined number of nodes, the error backpropagation will continue to update the weights for all nodes regardless of whether all the nodes are necessary to achieve the desired response. The effect to the output of having too many nodes will be "overfitting", which leads to poor performance on an outofsample data set. Conversely, if the number of nodes defining the network is too few, the neural network will not be optimal because the network will be missing vital information.
To address this network optimization issue, techniques have been developed to assess the need to add or remove a node from a neural network. However, these techniques are generally not well suited for signals whose statistical properties change over time. Such signals are known as "nonstationary signals". For example, if a node is added to a neural network, it will require several time steps to acquire the necessary information to train the weight for this new node. Since the statistical properties of nonstationary signals may change rapidly, the neural network may no longer be of optimal size by the time the new node is trained.
Real world signals such as financial, physiological and geographical data often are nonstationary. Because the number of parameters in a network is a crucial factor in it'"'"'s ability to generalize, it is the goal of an appropriate model to track the nonstationary signals by adaptively (online) updating its parameters. Ideally, this means updating, in response to changes in the input signal, "structural parameters" such as the effective number of hidden nodes (intermediate layer nodes) within the network.
However, traditional methods generally do not provide this capability. For a nonstationary signal, it is not appropriate to fix the model parameters after training on a representative data set.
In addition, weight updates make use of the gradient (∂E/∂w) of the error E with respect to the weights. Generally, this gradient can be directly computed from the neural network by an error backpropagation process. However, such a gradient cannot be computed with respect to the number of nodes. Since these parameters are in the form of integers, it would not be possible to compute the gradient of the error with respect to parameters, which are required for gradientbased optimization methods.
Therefore, a need exists in the art for a system and method capable of adaptively updating the structural parameter of a neural network for every new available sample of data for tracking nonstationary signals.
The present invention overcomes the disadvantages associated with the prior art by providing a signal processing system and method that updates structural parameters of the neural network system. Specifically, the invention updates the structural parameters of the neural network in principal component space for every new available input sample. The method of the present invention referred generally as adaptive eigenpruning and adaptive weight updating, consists of six steps which are applied to each new available sample of the input signal. Since the present invention can be applied to any layer of a neural network, the term input signal may represent an input signal to an input layer of a neural network or an input signal to a hidden layer of a neural network.
The first step transforms a nonstationary signal (input signal) to principal component space where the nonstationary signal is transformed into a matrix of eigenvectors with a corresponding matrix of eigenvalues. In statistical literature, this first step is known as performing a principal component analysis (PCA) on the signal. This is a transformation that projects the input signal into a different space used to determine the resonance of the input signal.
However, performing a principal component transformation directly on every new signal input is computationally expensive, so that once a transformation is completed, the inventive method employs recursive estimation techniques for estimating eigenvectors and eigenvalues. The adaptive principal component extraction (APEX) or the LEArning machine for adaptive feature extraction via Principal component analysis (LEAP) are just two examples of such extraction techniques.
The second step transforms the nonstationary input signal (in general, a vector signal) to its principal component space (which is hereinafter referred to as "eigenspace"). The goal of this transformation is to make the components of the input signal mutually orthogonal. The advantage of such a representation is that the effects of the orthogonal components (the "eigenmodes") of the input signal on the filter or neural network output signal can be analyzed individually without taking the other eigenmodes into account.
In the third step, the method computes an "unpruned" output signal by multiplying the orthogonalized input signal with the filter or neural network weight matrix.
In the fourth step, the method selects components of the weighted input signal for pruning from the output signal. The selection procedure identifies the eigenmodes that are revealed in eigenspace to be of minimal influence on the output signal of the system. This step is derives an upperbound on the modeling error introduced by deleting the eigenmodes. This error upperbound is defined as the saliency for the ith eigenmode.
In the fifth step, the method completes the "eigenpruning" by subtracting eigenmodes with small saliencies from the output signal. Since eigenpruning and recomputing the error upperbounds are performed for each new input sample, this method is known as adaptive eigenpruning. Removing eigenmodes reduces the effective number of parameters and generally improves generalization, i.e., performance on an outofsample data set.
Finally, the sixth step applies standard filtering techniques such as the Transform Domain Adaptive Filtering (TDAF) to update the weights of the filter or neural network.
Specifically, the present invention applies the six steps mechanism to every input sample, thereby adaptively updating the weights and effective number of nodes in a neural network for every input sample of a nonstationary signal.
The teachings of the present invention can be readily understood by considering the following detailed description in conjunction with the accompanying drawings, in which:
FIG. 1 is a block diagram of a neural network applying a prior art error back propagation process;
FIG. 2 is a block diagram of a signal processing system that contains a neural network that embodies the teachings of the present invention;
FIG. 3 is a flowchart showing the process of adaptive weight updating and adaptive eigenpruning;
FIG. 4 is a block diagram of a neural network applying a spectral decomposition of y_{t} ;
FIG. 5 is a block diagram of FIG. 3 showing the process of adaptive weight updating and adaptive eigenpruning; and
FIG. 6 is a block diagram for the mechanism of recursive estimation of the eigenvalue and eigenvector matrices.
After considering the following description, those skilled in the art will realize that the teachings of this invention can be readily utilized to predict nonstationary signals (including, but not limited to financial signals). In general, this invention discloses a method and system for updating a layer in a neural network by adaptively updating the weights and effective number of nodes of the network. This is accomplished by applying the concept of adaptive eigenpruning to a neural network.
FIG. 2 depicts a signal processing system that utilizes the present inventions. This general signal processing system consists of a signal receiving section 210, a signal processing section 220, a processor 230, a monitor 240 and a keyboard 250.
Signal receiving section 210 serves to receive input data signals, such as financial data. Signal receiving section 210 consists of a data receiving section 211, a data storage section 212, and input/output (I/O) switch 214. Data receiving section 211 may include a number of devices such as a modem and an analogtodigital converter. A modem is a wellknown device that comprises a modulator and a demodulator for sending and receiving binary data over a telephone line, while an analogtodigital converter converts analog signals into a digital form. Hence, input signals are received "online" through signal receiving section 210 and, if necessary, are converted to a digital form.
The data storage section 212 serves to store input signals received by data receiving section 211. Data storage section 212 may incorporate a number of devices such as a disk drive, semiconductor memory or other storage media. These storage devices provide a method for applying a delay to the input signals and store input signals for processing at a later time. Finally, the stored input signals are presented to the signal processing section 220 through input/output switch 214, which channels the input signal from signal receiving section 210 to signal processing section 220. The I/O switch 214 also channels the input signals between data receiving section 211 and data storage section 212.
The signal processing system of the present invention includes a signal processing section 220 for producing an output signal O_{p} from input signal patterns p. The signal processing section 220 consists of a neural network 222 with a learning section 221. The neural network 222 includes at least an input layer 224 and an output layer 228. The neural network may optionally include a number of intermediate layers 226 (also known as hidden layers). Each layer includes at least one node.
A desired (target) output signal t_{p} and the output signal O_{p} are applied to learning section 221. Applying a learning signal as described below, learning section 221 causes neural network 222 to undergo learning by updating the parameters w in such a manner that each w is sequentially and repeatedly updated.
The neural network 222 of signal processing section 220 computes, in accordance with the sequences of steps shown by the flow charts of FIG. 3 and 6, an output signal O_{p} from an input pattern p. Output signal O_{p} is an output value that correlates to a prediction of the value of input pattern p for n samples ahead.
The processor 230 of the signal processing system receives the output signal O_{p} and performs additional data processing such as determining the direction or trend of the input pattern p. Furthermore, the processor 230 is coupled to a monitor 240 and a keyboard 250 for displaying data and receiving inputs respectively.
The method of adaptively updating the dimensions of a neural network for every new available input sample of data for tracking nonstationary signals are described with reference to method 300 of FIG. 3. Method 300 starts by taking an input vector x_{t} = x_{1} (t), x_{2} (t), . . . , x_{n} (t)!^{T} 310 and transforms it into principal component space in step 320. This step multiplies the vector of elements representing the input signal by a matrix so as to transform the coordinates in which the vector is represented to ones that display the maximum variance along their axes, thus removing correlations between pairs of coordinates.
To illustrate, the method assumes a signal plus noise model with the signal generated by a function linear in the weights which can be represented by y_{t} =W_{0} x_{t} +e_{t}. This, in turn, is modeled by:
y.sub.t =Wx.sub.t (7)
Depending on the particular application, x_{f} may represent one of a number of different signals, including an input vector signal, an output signal from a layer in a neural network or a regression vector on y_{t} itself. It should be noted that throughout this specification, vectors are represented with an underscore, while matrices are represented by capitals.
However, the dimension of x_{f} may be time varying, which makes it difficult to estimate the dimension of x_{t}. Hence, choosing a fixed dimensional choice is inappropriate. The covariance matrix of input x_{t} is defined as R_{t} =E x_{t} x_{t}^{T} !, which can be estimated by:
R.sub.t =Σ.sub.n=0.sup.t μ.sup.tn x.sub.n x.sub.n.sup.T(8)
Mu (μ) is defined as a forgetting (decaying) factor which causes exponential decay of the temporal weighing of x_{t} in R_{t}. Where the vector signals are nonstationary, μ is less than 1. Specifically, the forgetting factor μ is commonly chosen in the interval 0.950<μ<0.999.
The eigendecomposition of R_{t}, also called principal component extraction of x_{t}, can be defined by:
R.sub.t Q.sub.t =Q.sub.t Λ.sub.t (9)
where Q_{t} = q_{1},q_{2}, . . . , q_{n} ! is the n×n orthogonal eigenvector matrix (the adjective orthogonal implies that Q_{t} Q_{t}^{T} =1), and Λ_{t} =diag Ex_{t}^{2} (t),Ex_{t}^{2} (t), . . . , Ex_{n}^{2} (t)! is a n×n diagonal matrix of the eigenvalues of R_{t}. Hence, by solving for the unique solution of Q_{t} and Λ_{t}, the inventive method achieves eigendecomposition. For a detailed discussion of this decomposition, see e.g. S. Haykin, Adaptive Filter Theory, PrenticeHall, (1991, chapter 4). It should be noted that step 320 is only performed for t≧n. Namely, Q_{t} and Λ_{t} do not exist when t<n.
However, it is computationally expensive to calculate Q_{t} and Λ_{t} for each new signal input x_{t}. Hence, the method employs recursive estimation of the eigenvalues and eigenvectors as described below with reference to process 600 of FIG. 6.
Continuing with method 300 of FIG. 3, once Q_{t} and Λ_{t} have been eigendecomposed from the vector x_{t}, the next step applies principal component pruning (eigenpruning). Eigenpruning is based upon the concept disclosed by Levin et al. in Fast Pruning Using Principal Components, Advances in Neural Information Processing Systems (NIPS)6 pp. 3542 (1994). This step consists of eliminating eigenmodes that are revealed in eigenspace to have minimal influence on the output of the system. In fact, the advantage of eigenspace decomposition is that it offers the ability to identify the independent contribution of each individual eigenmode on the error of the network. The process of eigenpruning is described below collectively in steps 330370 in FIG. 3.
In step 330, x_{t} is computed by:
x.sub.t =Q.sub.t.sup.T x.sub.t (10)
The term x_{t} is referred to as the KarhunenLoeve expansion of x_{t}. The goal of this transformation is to bring x_{t} to its principal component space. This transformation is effectively a rotation of x_{t}.
In step 340, y_{t}, the "unpruned" output of a layer of the neural network is computed by:
y.sub.t =W.sub.t.sup.T x.sub.t (11)
If y_{t} is a p×1 vector, then W_{t}^{T} is a p×n weight matrix of a layer in the neural network. W_{t}^{T} is defined as W_{t}^{T} = v_{1},v_{2}, . . . , v_{n} !, where each v_{t} is a p×1 vector. Hence, y_{t} can also be expressed as y_{t} =Σ_{i=1}^{n} v_{t} x_{t}. This is known as the spectral decomposition of y_{t} and is illustrated in the neural network of FIG. 4. There are "n" vector terms (eigenmodes) in the spectral sum. Since x_{t} 410 is in principal component space, it can be demonstrated that each eigenmode v_{i} x_{i} 420 contributes independently to the error in the output y_{t} 430 in a layer of the neural network. Using an available desired output signal vector d_{t}, the error is defined as e_{t} =d_{t} y_{t}. If y_{t} is from the output layer of a neural network, a desired output signal d_{t} is usually available. If y_{t} holds activities for an intermediate (or hidden) layer in a neural network, a desired output signal d, can be estimated by the backpropagation process. For a detailed exposition on computing desired signals for the hidden layers of a neural network by means of the backpropagation algorithm, see S. Haykin, Neural Networks, IEEE Press, (1994, chapter 6).
The next step in the eigenpruning process is performed in step 350 of FIG. 3. Eigenpruning is implemented by calculating the "saliency" of each eigenmode. Saliency is a quantity that measures the effect of the i^{th} eigenmode on the output error. Eigenpruning consists of deleting the corresponding eigenmodes with the smallest saliencies such that the sum of the deleted saliencies is less than or equal to a predefined percent of the total sum of the saliencies. Removing eigenmodes with low saliencies reduces the effective number of parameters defining the network and improves generalization. In the embodiment of the present invention, one percent (1%) is chosen. Although this percentage works well with the training data that were used to train the neural network, other heuristics could be used as well.
There are three embodiments of the present invention with regard to the calculation of saliency for each eigenmode. In the first embodiment, since the contribution of v_{i} x_{i} to the error is independent from other eigenmodes, it follows that the maximum squared error introduced by each eigenmode v_{i} x_{i} is:
s.sub.i (t)= v.sub.i x.sub.i !.sup.T v.sub.i x.sub.i !=v.sub.t.sup.T v.sub.i x.sub.i.sup.2 (12)
The saliency s_{i} (t) is an upper bound for the increase of the squared error when v_{i} x_{i} is subtracted from y_{t}.
In the second embodiment, saliency is alternatively expressed as:
s.sub.i (t)=v.sub.t.sup.T v.sub.i x.sub.i.sup.2, 13)
where x_{i}^{2} is a lowpass filtered version of x_{i}^{2}, namely x_{i}^{2} (t)=μx_{i}^{2} (t1)+(1μ)x_{i}^{2} (t), where μ is the forgetting factor discussed above.
In the third embodiment of the present invention, saliency is defined as:
s.sub.i =λ.sub.i v.sub.i.sup.T v.sub.i, (14)
where λ_{i} is the ith element on the diagonal of Λ_{t}.
To illustrate, since Q_{t} is orthonormal, which implies Q_{t}^{T} =Q_{t}^{1}, y_{t} can be expressed as:
y.sub.t =W.sub.t Q.sub.t Q.sub.t.sup.T x.sub.t =W.sub.t x.sub.t =Σ.sub.i x.sub.i v.sub.i (15)
where W_{t} =W_{t} Q_{t} and x_{t} =Q_{t}^{T} x_{t}. Both x_{i} and v_{i} (denote the ith column of W_{t} by v_{i}) lie in the space spanned by the columns of Q_{t}. This third embodiment of saliency represents the modeling error (y_{t} y_{t})^{2} introduced by deleting the contribution of the term x_{i} v_{i}. Lambda (λ_{i}) is the ith eigenvalue in the eigenvalue matrix Λ_{t}.
The third embodiment of the present invention is different from the other two embodiments in that, the third embodiment computes saliency by using the ith eigenvalue λ_{i} from the eigenvalue matrix Λ_{t}, whereas the first two embodiments use the current input signal x_{i}. The difference is that λ_{i} is a smoothed average of x_{i}^{2}.
With s_{i} calculated in step 350 for each eigenmode using one of the three embodiments discussed above, the method then identifies a set of indices (Π) of eigenmodes with "small" saliencies. As discussed above, Π may represent the set of indices for which the cumulative sum of the saliencies is less than or equal to a predefined percent of the total sum (over "n" eigenmodes) of the saliencies. When Π has been identified, a p×1 pruning vector y_{t}^{p} is computed as:
y.sub.t.sup.p =Σ.sub.iεΠ v.sub.i x.sub.i. (16)
The pruning vector y_{t}^{p} represents the set of eigenmodes that have minimal influence on the actual output signal of a layer of the neural network. As such, this pruning vector y_{t}^{p} is used in step 360 to calculate the actual output.
In step 360, the method computes the output y_{t} of a layer as:
y.sub.t =y.sub.t y.sub.t.sup.p. (17)
Since y_{t} represents the "unpruned" output of a layer of the neural network, then it follows that y_{t} =Σ_{i}εΠ v_{i} x_{i}, where y_{t} is a pruned version of y_{t} as computed in step 340. Step 360 effectively completes the eigenpruning process and generates the output signal for a layer of the neural network.
Finally, in step 370 the method updates the weights of the neural network by using standard adaptive filtering techniques such as the Transform Domain Adaptive Filtering (TDAF). If an error vector e_{t} =d_{t} y_{t} is available (or possibly has been estimated by the backpropagation process), then the weights of the corresponding layer of the neural network are updated by the following formula:
W.sub.t+1 =W.sub.t ηΛ.sub.t.sup.1 x.sub.t e.sub.t.sup.T, (TDAF)(18)
where η is a learning rate parameter and x_{t} =Q_{t}^{T} x_{t}.
It should be noted that other standard adaptive filtering techniques exist and that the use of the TDAF algorithm in the present invention is only illustrative. For a detailed discussion of the Transform Domain Adaptive Filtering (TDAF), see Marshall et al., The Use of Orthogonal Transforms for Improving Performance of Adaptive Filters, IEEE Transactions on Circuits and Systems, vol.36, no.4, (1989).
After computing step 370, the method queries at step 380, whether additional input samples exist. If the query is answered affirmatively, the process reverts back to step 320 along path 390 and the process of adaptive eigenpruning and adaptive weight updating is repeated for the next input sample of the nonstationary signal. If the query is answered negatively, the process ends. An alternative perspective of the process 300 of FIG. 3 is provided in FIG. 5 in the form of a block diagram.
In FIG. 3, the weights and the effective number of nodes of the neural network are updated for every available input sample of a nonstationary signal. However, the direct computation of the matrices Q_{t} and Λ_{t} for every input vector x_{t} is inherently expensive, especially if n is a high number. Hence, it is necessary to employ techniques that can be used to recursively estimate the eigenvalues and eigenvectors.
There are two embodiments of the present invention for recursive estimation of the eigenvector matrix Q_{t} and eigenvalue matrix Λ_{t}. These embodiments incorporate different extraction techniques and they are described with reference to FIG. 6. As discussed, the eigendecomposition of step 320 of FIG. 3 becomes computationally expensive when eigendecomposition is computed directly for each new input sample. Hence, step 320 includes steps to recursively estimate the eigenvector matrix Q_{t} and eigenvalue matrix Λ_{t} as new input samples are received.
Referring to FIG. 6, in step 610, a decision is made as to whether direct eigendecomposition should be performed. This decision is based on whether eigendecomposition has been performed for the previous input vector x_{t}.
In step 615, if eigendecomposition has not been performed, a decision is made as to whether t≧n. As noted above, when t<n, Q_{t} and Λ_{t} do not exist and eigendecomposition is not available at this point. Since eigendecomposition is performed on a n×n matrix of x_{t}, this causes the process to end at this point. If t≧n, then the process proceeds to step 620, which is the identical step as described in step 320 for the direct eigendecomposition of x_{t}.
In the first embodiment of the present invention, if Q_{t} and Λ_{t} are available from the eigendecomposition of the previous signal input x_{t1}, then recursive estimation of Q_{t} and Λ_{t} are computed with respect to the new input sample x_{t} by updating the covariance matrix R_{t} in step 630. By using the covariance matrix R_{t1}, the covariance matrix R_{t} can be computed recursively as:
R.sub.t =μR.sub.t1 +(1μ)x.sub.t x.sub.t.sup.T. (19)
In step 640, an eigenspace decomposition on R_{t} is then subsequently performed to obtain Q_{t} and Λ_{t}.
The advantage of accurately tracking Q_{t} and Λ_{t} for the purpose of estimating the eigenvalues and eigenvectors recursively is significant. This step dramatically reduces the computational overhead of eigendecomposing the new vector x_{t} directly as disclosed above in step 320. The saving is realized from having to only compute x_{t} x_{t}^{T} which is only the product of a n×1 vector with a 1×n transpose vector. Using the covariance matrix of R_{t1} obtained from the previous eigendecomposition of x_{t1}, R_{t} can be computed inexpensively.
To illustrate, the recursive eigenpruning process requires the computation of the eigenvector matrix Q_{t} and eigenvalue matrix Λ_{t} for each new input sample. The straightforward method is to perform a standard eigendecomposition of R_{t}. The number of elementary arithmetic manipulations (such as multiplications and additions) required for the eigendecomposition of R_{t} is on the order of n^{3}. This means that if the dimension of R_{t} is n×n (say n=100), it will require n^{3} (1,000,000 for n=100) arithmetic manipulations to compute the eigenvector matrix Q_{t} and eigenvalue matrix Λ_{t}. Clearly, if these computations are performed for every input sample, a large computational resource will be required. However, with the use of extraction techniques, the saving in computational steps equates approximately to reducing n^{3} computational steps to that of n^{2} computational steps. As n increases, this saving in computational steps becomes extremely significant.
In a second embodiment of the present invention, algorithms such as APEX or LEAP are used to directly compute Q_{t} and Λ_{t} without updating the covariance matrix R_{t}. These algorithms are described in detailed in the articles by Kung S. Y., and Diamantaras., A Neural Network Learning Algorithm for Adaptive Principal Component Extraction, Proc. of IEEE Conf. on Acoustic, Speech and signal Processing (ICASSP), pp. 861864, (1990) and Chen H. and Liu R., An Online Unsupervised Learning Machine for Adaptive Feature Extraction, IEEE trans. on Circuits and Systems II, vol. 41, no.2, pp. 8798, (1994). Namely, given the first m1 principal components, the algorithm can produce the mth component iteratively. The use of these extraction techniques will further reduce the total number of computational steps by removing step 630, where the covariance matrix R_{t} is recursively updated.
These algorithms have been developed to reduce the computational load for the updating of Q_{t} and Λ_{t}. The key to these "fast" eigendecomposition algorithms is that Q_{t} and Λ_{t} are updated for each input sample directly from the data stream of x_{t} without computing R_{t}. Both APEX and LEAP are implemented in neural network hardware and/or software and their performance is comparable. Both methods consist of two mechanisms. The first mechanism computes the correlations between the various components of input signal x_{t}. In APEX and LEAP, this mechanism is implemented by the "Hebbian" ("correlation") updating rule. The Hebbian algorithm finds the dominant correlations.
The second mechanism constrains the first mechanism such that the various correlations found by the Hebbian rule are orthogonal to each other. APEX uses an orthogonalizing learning rule that is referred to as antiHebbian learning, whereas LEAP implements the orthogonalizing process by the GramSchmidt rule. The mechanisms for correlation learning (Hebbian rule) and orthogonalization (antiHebbian and GramSchmidt rule) are standard and widely applied.
It should be noted that some of these algorithms were premised on the fact that the input signal is stationary. However, the advantage of applying these techniques to nonstationary signals coupled with the concept of adaptive eigenpruning is significant. As discussed above, the concept of eigenpruning provides an accurate method to update the weights and the effective number of nodes for a neural network, but it is computationally expensive when applied to a nonstationary signal for each input sample. Hence, the use of extraction techniques provides a powerful and efficient solution to reduce the number of computational steps necessary to implement the adaptive eigenpruning process for a nonstationary signal.
Thus, a novel neural network for processing nonstationary signals has been disclosed. However, many modifications and variations of the present invention will become apparent to those skilled in the art.
Specifically, the method of the present invention is implemented completely in eigenspace. However, it is possible to selectively modify various steps of process 300 in different space. To illustrate, eigenpruning can be accomplished by setting the values in a corresponding column in the eigenvector matrix Q_{t} to zeros, which correlates to eigenmodes with small saliencies. After the eigenvector matrix Q(t) is eigenpruned, the weights of the network are updated. The eigenpruned weight matrix is computed as:
W.sub.ep (t)=W(t)Q.sub.ep (t)Q.sub.ep.sup.T (t), (20)
where ep represents eigenpruned. Using the updated weight matrix W_{ep} (t), a model output can be computed.
The model output is computed as:
y.sub.ep (t)=W.sub.ep (t)x(t). (21)
Finally, the weights of the neural network are updated using standard adaptive filtering techniques such as the Least Mean Squares (LMS). This filtering technique is used in place of the TDAF and the weights of the corresponding layer of the neural network are updated by the following formula:
W.sub.t+1 =W.sub.t +2ηε.sub.k x.sub.t,(LMS) (22)
where η is a learning rate parameter and ε_{k} is the error signal. Hence, many modifications of the present invention are possible.
The neural network according to the present invention was tested to perform a number of simulations. In one simulation, the tickbytick values from May 1985 of the U.S. dollar versus Swiss Franc exchange rate were used as the nonstationary input signal. This time series contains successive tick values that were separated by a few minutes, but the sampling period was not a constant. The time series was processed to the extent that the maximal absolute value was rescaled to one and the mean value was rescaled to zero.
The goal is to predict the value of the exchange rate five ticks ahead, while making use of past values of the exchange rate. The neural network was trained using a training set of 1,000 samples and the next set of 1,000 samples was used as testing data. A two layer feedforward network with 40 `tanh` hidden nodes and one (1) linear output node was employed. In conjunction, a 10dimensional vector x(t) = y(t5), y(t6), . . . , y(t14)!^{T} was used as an input and y(t) as a target output. The goal of the network was to predict the exchange rate five (5) ticks ahead, while making use of the exchange rate of the last ten (10) ticks. The network was loaded with weights w_{o}, the weights from the trained unpruned network, and the weights were adapted online as the network moved over the test data, while the network employed adaptive weight updating and adaptive eigenpruning. A learning rate η=0.05 and forgetting factor λ=0.98 were chosen.
The results are very impressive. Instead of focusing on the predicted values of the exchange rate, the fraction of correctly estimated directions of the exchange rate was computed. In other words, the direction of the exchange rate was predicted. The direction of the exchange rate is computed as: ##EQU4## The area θ, θ! is referred to as the deadzone. For θ=0, i.e. no deadzone, the adaptive network correctly estimated 72% of the exchange rate directional movements. In fact, when a dead zone of θ=0.017 is used, conditional on that a change took place (dir(t)≢0), the adaptive network predicted 80% of the exchange rate directional movements.
There has thus been shown and described a novel neural network for processing nonstationary signals such as financial data. Many changes, modifications, variations and other uses and applications of the subject invention will, however, become apparent to those skilled in the art after considering this specification and the accompanying drawings which disclose the embodiments thereof. All such changes, modifications, variations and other uses and applications which do not depart from the spirit and scope of the invention are deemed to be covered by the invention, which is to be limited only by the claims which follow.