DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS
[0035] While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that there is no intent to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.
[0036] Referring now to FIG. 1, an exemplary analysis system 10 is shown which incorporates various features of the present invention therein. In general, the exemplary analysis system 10 is operable to (i) monitor movement of a patient, and (ii) classify the monitored movement as being characteristic of a certain neurological disorder. More specifically, the exemplary analysis system 10 is operable to (i) collect movement data that is representative of movement of a patient over a collection period, (ii) process the collected movement data, and (iii) generate output that indicates whether the movement of the patient during the collection period is indicative of a person who has Parkinson's disease (PD) or essential tremor (ET). To this end, the exemplary analysis system 10 includes a movement monitoring device 20, a preprocessor 30, a multiplexor 40, and a computational intelligence system 50.
[0037] The movement monitoring device 20 is generally operable to monitor movement of a patient, and produce movement data that generally represents the movement of the patient. More specifically, the movement monitoring device 20 in an exemplary embodiment is operable to produce an analog movement signal having an amplitude that varies with respect to time based upon movement of the patient being monitored. In particular, in an exemplary embodiment, the movement monitoring device 20 is operable to vary the amplitude of the analog movement signal such that the analog movement signal has a resolution of about 10 milliG's (i.e. ten one thousandths of Earth's gravitational force) and a temporal resolution of at least 25 samples per second.
[0038] The preprocessor 30 is operable to receive movement data from the movement monitoring device 20, and process the movement data to obtain preprocessed movement data having a form suitable for processing by the computational intelligence system 50. More specifically, the preprocessor 30 in an exemplary embodiment is operable to receive the movement data from the movement monitoring device 20, extract characteristics of the movement from the movement data, generate data patterns from the extracted characteristics of the movement data, and provide the computational intelligence system 50 with the generated data patterns via the multiplexor 40.
[0039] The multiplexor 40 is operable to receive data patterns from the preprocessor 30 and a training mechanism 60, and provide the computational intelligence system 50 with the data patterns originating from either the preprocessor 30 or the training mechanism 60 based upon the state of an analysis control signal.
[0040] In general, the computational intelligence system 50 is operable to receive preprocessed movement data from the preprocessor 30, and analyze the preprocessed movement data. More specifically, the computational intelligence system 50 in an exemplary embodiment is trained to classify movement based upon a data pattern received from the preprocessor 30 and a predetermined group of neurological disorder classification. In particular, the computational intelligence system 50 of an exemplary embodiment is operable to generate an output that is indicative of an appropriate neurological disorder classification based upon the received data pattern and a predetermined group of neurological disorder classifications that include a normal classification, a Parkinson's disease classification, and an essential tremor classification. In yet another exemplary embodiment, the computational intelligence system 50 is operable to generate an output that is indicative of an appropriate neurological disorder classification based upon the received data pattern and a predetermined group of neurological disorder classifications that include a normal tremor classification, and a nonnormal tremor classification (e.g. a classification encompassing both PD and ET).
[0041] Implementation of the Movement Monitoring Device
[0042] As stated above, the movement monitoring device 20 essentially monitors the movement of a patient and generates movement data that is representative of the monitored movement. To this end, the movement monitoring device 20 in an exemplary embodiment is implemented with a TeleActigraph (TAG) produced by Precision Control Design, Inc. (PCD) of Fort Walton Beach, Fla. In particular, the PCD TAG includes a wristmounted actigraph 80 that samples an analog movement signal at a rate of about 27 Hz with about a 12bit resolution, and transmits the resulting 12bit digital samples via a 300 MHz wireless link to a beltworn unit that is operable to store up to about 5 Megabytes of data or roughly 5 hours and 20 minutes of data. Alternatively, a wristmounted actigraph similar to Precision Control Design's model number BMA32 but lacking a beltworn unit is described in detail in U.S. Pat. No. 5,197,489, the disclosure of which is hereby incorporated by reference.
[0043] As depicted in FIG. 2 and FIG. 3, the exemplary PCD TAG system includes a wristmounted actigraph 80 and a beltworn unit 100. The wristmounted actigraph 80 is constructed to have a shape and size similar to a wristwatch, and the beltworn unit 100 is constructed to have a shape and size similar to a belt with a large belt buckle. More specifically, the beltworn unit 100 includes a flexible band 101 which secures a generally rectangular housing 102 to the waist of a person. Similarly, the actigraph 80 includes a flexible band 81 which secures a generally rectangular housing 82 against the skin surface of a patient being monitored. Typically, the actigraph 80 is mounted on the nondominant wrist of the patient which has been found to correlate well with body activity.
[0044] In use, the actigraph 80 and the beltworn unit 100 are worn by the patient for a predetermined collection period such as a few minutes or several hours. During the collection period, a sensor circuit of the actigraph 80 generates an analog movement signal having a voltage amplitude that is modulated based upon activity of a person wearing the wristmounted actigraph 80. Moreover, an A/D converter of the actigraph 80 samples the analog movement signal at a sampling rate such as 27 Hz in order to obtain 12bit digital samples of the analog movement signal generated by the sensor circuit. Furthermore, the actigraph 80 includes a transmitter which is operable to transfer the generate 12bit digital samples to the beltworn unit 100.
[0045] Data collected by the beltworn unit 100 during the collection period is later downloaded to a computer 83 for further processing and analysis. More specifically, communication between the beltworn unit 100 and the computer 83 is facilitated by an interface unit 84, which is connected to a data port of the computer 83 via a conventional RS232 cable 85 or the like. The interface unit 84 preferably includes a receptacle 86 on its top surface dimensioned to receive the rectangular housing 102 of the beltworn unit 100. An electrical connector located along one side of receptacle 86 engages a connector on housing 102 when the beltworn unit 100 is seated within the receptacle 86. With this arrangement the beltworn unit 100 can be quickly and conveniently installed and removed from interface unit 84.
[0046] The interface unit 84 enables data to be exchanged between the personal computer 83 and the PCD TAG system in both directions. The bidirectional nature of the data exchange enables (i) collected data to be downloaded from the PCD TAG system to the computer 83, and (ii) operating instructions to be uploaded from the computer 83 to the PCD TAG system which control the operation of the actigraph 80 in subsequent data collection assignments. A plurality of controls 90 on the top surface of interface unit 84 assist the operator in accomplishing the downloading and uploading functions.
[0047] General Operation of an Exemplary Analysis System
[0048] Shown in FIG. 4, there is illustrated a flowchart that depicts the general operation of the exemplary analysis system 10. More specifically, the general operation of the analysis system 10 will be described in a manner which assumes the computational intelligence system 50 of the analysis system 10 has been properly trained to classify movement data based upon a group of predetermined neurological classifications that includes a normal classification, a PD classification, and an ET classification. In general, the normal classification corresponds to persons who exhibit movement indicative of normal physiologic tremor, the PD classification corresponds to persons who exhibit movement indicative of Parkinson's disease, and the ET classification corresponds to persons who exhibit movement indicative of essential tremor. Training of an exemplary embodiment of the computational intelligence system 50 to classify movement data based upon a group of predetermined neurological classifications is described in detail with reference to FIGS. 6, 7, 8A, 8B, 9, and 10.
[0049] As depicted in FIG. 4, the analysis system 10 begins in step 402 with the movement monitoring device 20 monitoring movement of a patient suspected of having a neurological disorder, and collecting movement data from the patient that is representative of the movement of the patient over a collection period. In an exemplary embodiment, collection of the movement data is accomplished by providing the PCD TAG system with operating instructions via the interface unit 84 and the computer 83, and then mounting the actigraph 80 of the PCD TAG system to the nondominant wrist of the patient. More specifically, in an exemplary embodiment using the PCD TAG system, the system is set to mode 9 which results in configuring the actigraph 80 for low gain and a bandwidth of about 0.1 to about 14 Hz. In mode 9, the actigraph 80 has been found to generate digital samples between the ranges of about 1230 (−400) to about 2230 (+600) with a zero level of about 1630.
[0050] After mounting the actigraph 80 to the nondominant wrist of the patient, the patient in an exemplary embodiment is asked to perform a series of postural movements with the nondominant wrist so that the PCD TAG system may obtain movement data for different postural tremor types (e.g. postural tremor, rest tremor, and kinetic tremor). In particular, the patient in an exemplary embodiment of the analysis system 10 is requested to sit in a chair in front of a table and hold their nondominant arm in a stationary horizontal position for a period of 1 minute in order to obtain postural tremor data from the patient. Afterwards, the patient is requested to rest their nondominant hand their thigh for a period of 1 minute in order to obtain rest tremor data. After obtaining the rest tremor data, the patient is requested to alternately touch his or her nose and the finger of a person sitting directly in front of them for a period of 1 minute in order to obtain kinetic tremor data.
[0051] During the above intervals of time, the actigraph 80 generates an analog movement signal that is representative of the movement of the patient over the collection period, digitizes the analog movement signal by sampling the analog movement signal at a predetermined sampling rate (e.g. 27 Hz), and stores the digital samples in a memory of the beltworn unit 100. Accordingly, after the collection period is completed, the beltworn unit 100 may be placed in the interface unit 84 so that the computer 83 may download the 12bit digital samples that are representative of the movement of the patient during the collection period.
[0052] In step 404, the preprocessor 30 processes the movement data collected by the movement monitoring device 20 in order to extract characteristics of the movement which the computational intelligence system 50 is trained to analyze. To this end, the preprocessor 30 essentially discards obtained movement data that is likely not representative of movement of the patient, and extracts characteristics of the movement data from the retained movement data. More specifically, the actigraph 80 used to implement the exemplary movement monitoring device 20 requires some time to settle (warm up) once data collection is started. As a result, the first few digital samples produced by the actigraph 80 do not accurately represent activity of the patient. Moreover, the last few digital samples produced by the actigraph 80 also may not accurately represent activity of the patient due to the action required to stop the actigraph 80 from collecting movement data. Furthermore, samples associated with a transition between collecting different postural tremor data are also discarded to better ensure the associated samples are reflective of the desired postural tremor type.
[0053] For example, the preprocessor 30 in an exemplary embodiment discards the first 5 seconds and last 5 seconds worth of samples for each postural tremor type. In other words, assuming that 1 minutes worth of movement data is collected for each postural tremor type, the preprocessor after discarding the first 5 seconds and last 5 seconds of digital sample obtains 50 seconds worth of digital samples for each postural tremor type. In an exemplary embodiment, the actigraph 80 generates digital samples at a rate of approximately 27 Hz. Accordingly, the preprocessor 30 in the exemplary embodiment essentially retains about 1350 digital samples for each postural tremor type.
[0054] The exemplary preprocessor 30 then extracts characteristics of the movement of the patient from the movement data. For example, depending upon the type of information the computational intelligence system 50 was trained to analyze, the preprocessor 30 may extract frequency characteristics of the movement from the movement data by obtaining the power spectral density, the fast Fourier transform, or other frequency transform of the movement data. Moreover, the preprocessor 30 may extract statistical data such as the number of zero crossings during a time window or epoch, the maximum during an epoch, the minimum during an epoch, the average during an epoch, and the average power during an epoch.
[0055] In particular, the preprocessor 30 in an exemplary embodiment calculates power spectral density data from the remaining digital samples for each postural tremor type. More specifically, the exemplary preprocessor 30 for each tremor type performs a one hundred twentyeight (128) point frequency transformation upon a sliding window of greater than two hundred and fiftysix (256) digital samples thus resulting in sixtyfour (64) power spectral density data points for each window or epoch. For example, in an exemplary embodiment, the preprocessor 30 calculates sixtyfour (64) power spectral density data points for each fifty seconds of digital samples.
[0056] Of the sixtyfour (64) power spectral density data points, the exemplary preprocessor 30 discards the first two (2) data points and the last two (2) data points thus resulting in sixty (60) data points for each epoch which span a frequency range between roughly 0.5 Hz and 13 Hz. Assuming three (3) postural tremor types, and sixty (60) power spectral density points per postural tremor type, the exemplary preprocessor 30 obtains a total of 180 power spectral density points per patient. After obtaining power spectral density points that are representative of the movement of a single patient during the collection period, the preprocessor 30 takes the square root of each of the power spectral density points in order to obtain amplitude data points that are proportional to the amplitude of the movement. Furthermore, the preprocessor 30 further divides each of the amplitude data points by the largest amplitude data point in order to normalize the amplitude data points to a range between 0 and 1.
[0057] After extracting characteristics of the movement of the patient from the movement data, the preprocessor 30 in step 406 generates an input pattern A_{k }for the computational intelligence system 50 from the extracted characteristics of the movement. In particular, the preprocessor 30 in an exemplary embodiment generates an input pattern A_{k }having 180 input signal a_{k1}, a_{k2}, . . . a_{k180 }where each input signal a_{kx }is representative of one of the 180 extracted amplitude data points. It should be appreciated by those skilled in the art, that the preprocessor 30 may further include other extracted characteristics of the movement in the input pattern A_{k }such as the number of zero crossings per epoch, and the average power of the movement per epoch if the computational intelligence system 50 is trained to process the additional characteristics.
[0058] In step 408, the computational intelligence system 50 analyzes the preprocessed movement data collected from the patient. To this end, the multiplexor 40 receives the input pattern A_{k }from the preprocessor 30 and applies the input pattern A_{k }to inputs of the computational intelligence system 50. More specifically, in an exemplary embodiment utilizing a trained neural network 100 for the computation intelligence 60, the inputs signal a_{1k}, a_{2k}, . . . a_{180k }of the input pattern A_{k }are applied to the input layer F_{x }of the trained neural network 100. As a result of the input pattern A_{k }being applied to the input layer F_{x}, the trained neural network 100 generates an output pattern Z_{k }that is indicative of an appropriate classification for the movement.
[0059] For example, in an exemplary embodiment, the trained neural network 100 of the computational intelligence system 50 is configured to generate an output pattern Z_{k }having three output signals z_{1k}, z_{2k}, and z_{3k }each having a value between 0 and 1. The closer the first output signal z_{1k }is to 1 the more likely the person has Parkinson's disease and the closer the output signal z_{1k }is to 0 the more likely the person does not have Parkinson's disease. The closer the second output signal z_{2k }is to 1 the more likely the person has essential tremor and the closer the second output signal z_{2k }is to 0 the more likely the person does not have essential tremor. Furthermore, the closer the third output signal z_{3k }is to 1 the more likely the person exhibits normal physiologic tremor characteristics and the closer the third output signal z_{3k }is to 0 the more likely the person does exhibit normal tremor characteristics.
[0060] Finally in step 410, the analysis system 10 provides analysis output results such as an appropriate neurological disorder classification for the movement. More specifically, the analysis system 10 in an exemplary embodiment displays upon a monitor of the computer 83 a graphical plot of the 60 amplitude points for each postural tremor type (180 total), and the output pattern Z_{k }generated by the computational intelligence system 50. Moreover, the analysis system 10 in an exemplary embodiment includes fuzzy logic that receives the output pattern Z_{k }and generates an indication of whether the movement is indicative of (i) people who have Parkinson's disease, (ii) people who have essential tremor, (iii) people who exhibit normal tremor characteristics, or (iv) people with whom the analysis system 10 is unable to reach a conclusive result.
[0061] Alternatively, the neurological disorder classification information determined in step 410 may include one or more levels of severity for one or more previously identified neurological disorders. In such a system, neurological disorder classification information relating to disorder severity can be used in the quantitative evaluation of the efficacy of a treatment.
[0062] Implementation of Computational Intelligence System
[0063] While computational intelligence system 50 will be described in detail with reference to a neural network implementation that is trained and simplified with particle swarm optimization, it should be appreciated by those skilled in the art that the computational intelligence system 50 may be implemented with other computational intelligence paradigms. For example, the computational intelligence system 50 may be implemented with or include other neural network paradigms such as error backpropagation, and learning vector quantization (LVQ). Moreover, the computational intelligence system 50 may be implemented with or include other computational intelligence paradigms such as genetic algorithms, evolutionary algorithms, and fuzzy logic. The computational intelligence system 50 may also be implemented with or include hybrids of the basic computational intelligence paradigms. For example, the computational intelligence system 50 may include a fuzzy expert system whose rules and membership functions are automatically evolved from training data patterns via an LVQ neural network, genetic algorithms, and particle swarm optimization. A method of evolving a fuzzy expert system from training patterns sets is described in detail in Yuhui Shi, et al., Implementation of Evolutionary Fuzzy Systems, IEEE Transactions on Fuzzy Systems, Vol. 7, No. 2, April 1999 at 109, the disclosure of which is hereby incorporated by reference.
[0064] Referring now to FIG. 5, there is illustrated an exemplary neural network 100 for implementing the computational intelligence system 50 of the exemplary analysis system 10. As depicted, the neural network 100 includes an input layer F_{x }of processing elements PEx_{0}, PEx_{1}, . . . PEx_{n}, a hidden layer F_{y }of processing elements PEy_{0}, PEy_{1}, . . . PEy_{q}, and an output layer F_{z }of processing elements PEz_{1}, PEz_{2}, . . . PEz_{p}. In general, each processing element PEx_{h }of the input layer F_{x }is coupled to each processing element PEy_{i }of the hidden layer F_{y }via a matrix of weighted connections W. Moreover, each processing element PEy_{i }of the hidden layer F_{y }is coupled to each processing element PEz_{j }of the output layer F_{z }via a matrix of weighted connections U.
[0065] The exemplary neural network 100 of FIG. 5 is commonly referred to as a fully connected feedforward neural network since a weighted connection exists between each processing element of adjacent layers and no signal path of the neural network 100 passes through a processing element more than once. While the exemplary neural network 100 is a fully connected feedforward neural network, it should be appreciated by those skilled in the art that the neural network 100 could alternatively be implemented with sparsely connected neural network topologies, randomly connected neural network topologies, and/or feedback neural network topologies.
[0066] Referring to FIG. 5 in more detail, the input layer F_{x }of the neural network 100 includes a biasing processing element PEx_{0 }and input processing elements PEx_{1}, PEx_{2}, . . . PEx_{n}. The biasing processing element PEx_{0 }essentially provides an internal bias for each of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q}. To this end, the biasing processing element PEx_{0 }of the input layer F_{x }is operable to generate a constant output signal x_{0 }(e.g. a signal having a value of 1) which is propagated to the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }via weighted connections w_{10}, w_{20}, . . . w_{q0 }of the weighted connections matrix W.
[0067] The input processing elements PEx_{1}, PEx_{2}, . . . PEx_{n }essentially distribute input signals a_{1k}, a_{2k}, . . . a_{nk }of an input pattern A_{k }to the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q}. To this end, each input processing element PEx_{h }is operable to (i) receive a single input signal a_{hk }of the input pattern A_{k}, and (ii) generate a corresponding output signal x_{h }which is propagated to the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }via the weighted connections matrix W. More specifically, each input processing element PEx_{h }of the exemplary neural network 100 is operable to generate an output signal x_{h }that is equal to its respective input signal a_{hk}. Moreover, the weighted connections w_{1h}, w_{2h}, . . . w_{qh }associated with each input processing element PEx_{h }are operable to propagate the generated output signal x_{h }to each of the hidden layer processing elements PEy_{1}, PEy_{2}, . . . PEy_{q}.
[0068] The hidden layer F_{y }of the neural network 100 includes a biasing processing element PEy_{0 }and the hidden processing elements PEy_{1}, PEy_{2}, PEy_{q}. The biasing processing element PEy_{0 }essentially provides an internal bias for each of the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p}. To this end, the biasing processing element PEy_{0 }of the hidden layer F_{y }is operable to generate a constant output signal y_{0 }(e.g. a signal having a value of 1) which is propagated to the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p }via weighted connections u_{01}, u_{02}, . . . u_{0p }of the weighted connections matrix U.
[0069] The hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }essentially generate output signals y_{1}, y_{2}, . . . y_{q }that are a function of the received output signals x_{0}, x_{1}, . . . x_{n }and weighted connections W. More specifically, each hidden processing element PEy_{I }generates an output signal y_{i }that is a function of the received output vector X (i.e. output signals x_{0}, x_{1}, . . . x_{n}) and associated weighted connections vector W_{i }(i.e weighted connections w_{i0}, w_{i1}, . . . W_{in}. Accordingly, each output signal y_{i }of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }may be represented mathematically as follows:
y_{i}=F(X,W_{i}) (1)
[0070] where F represents the processing element function which is also commonly referred to as the activation function of the processing element.
[0071] In implementing the activation function F, each hidden processing element PEy_{i }of the exemplary neural network 100 performs a combinatory function c( ) of its corresponding inputs X and W_{i }and passes the resulting combinatory value c through a threshold function f( ). More specifically, each hidden processing element PEy_{i }of the exemplary neural network 100 performs a linear combination (i.e. dot product) of its corresponding inputs X and W_{i }and passes the resulting dot product value c through a sigmoid threshold function. The following hidden layer signal equation (2) represents the output signal y_{i }as a threshold function f( ) of the combinatory function c( ) where the combinatory function c( ) is implemented as the linear combination of inputs X and W_{i}:
1$\begin{array}{cc}{y}_{i}=f\ue8a0\left(c\ue8a0\left({X}_{i}\ue89e{W}_{i}\right)\right)=f\ue8a0\left(X\xb7{W}_{i}\right)=f\ue8a0\left(\sum _{m=0}^{n}\ue89e\text{}\ue89e{X}_{m}\ue89e{W}_{\mathrm{im}}\right)& \left(2\right)\end{array}$
[0072] The following equation (3) represents the sigmoid threshold function used by each hidden processing element PEy_{i}.
2$\begin{array}{cc}f\ue8a0\left(x\right)=\frac{1}{1+{\uf74d}^{\alpha \ue89e\text{}\ue89ex}}& \left(3\right)\end{array}$
[0073] where α represents a slope factor of the sigmoid function that in essence scales the inputs X and W_{i }of the hidden processing element PEy_{i}.
[0074] The output layer F_{z }of the neural network 100 includes the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p}. The output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p }essentially generate output signals z_{1}, z_{2}, . . . z_{p }that are a function of the received hidden layer output signals y_{0}, y_{1}, . . . y_{q }and the weighted connections matrix U. More specifically, each output processing element PEz_{j }generates an output signal z_{j }that is a function of a received output vector Y (i.e. output signals y_{0}, y_{1}, . . . y_{q}) and associated weighted connection vector U_{j }(i.e. weighted connections u_{j0}, u_{j1}, . . . u_{jp}). Accordingly, each output signal z_{j }of the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p }may be represented mathematically as follows:
z_{j}=F(Y,U_{j}) (4)
[0075] where F represents the activation function of the processing element function.
[0076] In implementing the activation function F, each output processing element PEz_{j }of the exemplary neural network 100 performs a combinatory function c( ) of its corresponding inputs Y and U_{j }and passes the resulting combinatory value c through a threshold function f( ). More specifically, each output processing element PEz_{j }of the exemplary neural network 100 performs a linear combination (i.e. dot product) of its corresponding inputs Y and U_{j }and passes the resulting dot product value c through a sigmoid threshold function. The following output layer signal equation (5) represents the output signal z_{j }as a threshold function f( ) of the combinatory function c( ) where the combinatory function c( ) is implemented as the linear combination of inputs Y and U_{j}:
3$\begin{array}{cc}{z}_{j}=f\ue8a0\left(c\ue8a0\left(Y,{U}_{j}\right)\right)=f\ue8a0\left(Y\xb7{U}_{j}\right)=f\ue8a0\left(\sum _{m=0}^{q}\ue89e\text{}\ue89e{y}_{m}\ue89e{u}_{\mathrm{jm}}\right)& \left(5\right)\end{array}$
[0077] where f( ) represents the above sigmoid threshold function which is presented again with a slope factor of β instead of α so that the slope factors β_{1}, β_{2}, . . . β_{p }of the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p }are easily distinguishable from the slope factors α_{1}, α_{2}: . . . α_{q }of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q}:
4$\begin{array}{cc}f\ue8a0\left(x\right)=\frac{1}{1+{\uf74d}^{\beta \ue89e\text{}\ue89ex}}& \left(6\right)\end{array}$
[0078] During operation, the neural network 100 essentially receives an input pattern A_{k }and generates a respective output pattern Z_{k }based upon the processing element activation functions F and weighted connections matrices W and U. More specifically, the input layer F_{x }receives an input pattern A_{k }of input signals a_{1k}, a_{2k}, . . . a_{nk}, and generates a corresponding input layer signals x_{1}, X_{2}, . . . X_{n }that are propagated to the hidden layer F_{y }via the weighted connections matrix W. Moreover, the input layer F_{x }generates a biasing signal x_{0 }that is also propagated to the hidden layer F_{y }via the weighted connections matrix W. The hidden layer F_{y }then generates hidden layer signals y_{1}, y_{2}, . . . y_{q }that are based upon the received biasing signal x_{0}, the input layer signals x_{1}, x_{2}, . . . x_{n}, the weighted connections matrix W, and the activation functions F of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q}. Moreover, hidden layer F_{y }generates a biasing signal y_{0 }which is propagated along with the hidden layer signals y_{1}, y_{2}, . . . y_{q }to the output layer F_{z }via the weighted connections matrix U. The output layer F_{z }then generates output signals z_{1}, Z_{2}, . . . z_{p }based upon the received biasing signal y_{0}, the hidden layer signals y_{1}, y_{2}, . . . y_{n}, weighted connections matrix U, and the activation functions F of the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p}.
[0079] While the input processing elements PEx_{1}, PEx_{2}, . . . PEx_{n }of the exemplary neural network 100 essentially implement an identity activation function F( ) that generates an output signal that is equal to a received input signal, the input processing elements PEx_{1}, PEx_{2}, . . . PEx_{n }may also be implemented in a manner similar to the processing elements of the hidden layer F_{y }and the output layer F_{z}. Moreover, while the processing elements of the hidden layer F_{y }and the output layer F_{z }utilize a linear combination function and a sigmoid threshold function, it should be appreciated by those skilled in the art that the activation functions F( ) of the processing elements may be implemented in several other known manners. More specifically, the activation function F( ) may use a different combinatory function c( ) and pass the combinatory result c through a different threshold function f( ). For example, TABLE 1 discussed below lists several alternatives for the threshold function f( ). Moreover, it should be appreciated that while the exemplary processing elements combine the received signals and pass the result through a threshold function, the activation function F( ) may be implemented in such a manner as to generate an output signal directly from the received signals without first performing a combinatory function of the received inputs.
[0080] General Operation for Obtaining Training and Testing Data Sets
[0081] Shown in FIG. 6, there is illustrated a flowchart that depicts the general operation of obtaining a training pattern set TRAIN_{SET }and a test pattern set TEST_{SET }for training and testing the computational intelligence system 50 of the analysis system 10. More specifically, the general operation of obtaining a training pattern set TRAIN_{SET }and a test pattern set TEST_{SET }will be described in a manner which assumes the computational intelligence system 50 of the analysis system 10 will be trained to distinguish movement associated with a normal classification, a PD classification, and a ET classification. Training a neural network 100 of the computational intelligence system 50 with the training pattern set TRAIN_{SET }to distinguish between movement associated with a normal classification, a PD classification, and an ET classification is described in detail below with reference to FIGS. 7, 8A, 8B, 9, and 10.
[0082] As depicted in FIG. 6, the analysis system 10 begins in step 602 with the movement monitoring device 20 collecting movement data from a group of people who are known to have Parkinson's disease, a group of people who are known to have essential tremor, and a group of people who are known to exhibit normal physiologic tremor characteristics. In an exemplary embodiment, collection of the movement data is accomplished by providing the PCD TAG system with operating instructions via the interface unit 84 and the computer 83, and then mounting the actigraph 80 to the nondominant wrist of each person for a separate collection period. More specifically, in an exemplary embodiment, the PCD TAG system is set to mode 9 which results in configuring the actigraph 80 for low gain and a bandwidth of about 0.1 to about 14 Hz.
[0083] After mounting the actigraph 80 to the nondominant wrist of a person from one of the target groups, the person is requested to perform a series of postural movements with the nondominant wrist so that the actigraph 80 may obtain movement data representative of different postural tremor types. In particular, the person in an exemplary embodiment of the analysis system 10 is requested to sit in a chair in front of a table and hold their nondominant hand elevated in a stationary horizontal position for a period of 1 minute in order to obtain postural tremor data from the person. Afterwards, the person is requested to rest their nondominant hand on their thigh for a period of 1 minute in order to obtain rest tremor data. After obtaining the rest tremor data, the person is requested to alternately touch his or her nose and the fingertip of a person located directly in front of them for a period of 1 minute in order to obtain kinetic tremor data. During the above time intervals, the actigraph 80 generates an analog movement signal that is representative of the movement of the patient over the collection period, digitizes the analog movement signal by sampling the analog signal at a predetermined sampling rate (e.g. 27 Hz), and stores the digital samples in a memory of the beltworn unit 100. Accordingly, after the collection period is completed, the beltworn unit 100 may be placed in the interface unit 84 so that the computer 83 may download the 12bit digital samples that are representative of movement of the patient.
[0084] The above data collection process is repeated for each person of each group in order to obtain tremor data for each person of each group. In an exemplary embodiment, the computer 83 stores the downloaded digital samples for each person in a separate file. Accordingly, as a result of repeating the above data collection process for each person, the computer 83 obtains a first set of data files corresponding to people known to have Parkinson's disease, a second set of data files corresponding to people known to have essential tremor, and a third set of data files corresponding to people known to exhibit normal tremor characteristics.
[0085] In step 604, the computer 83 generates a training pattern set TRAIN_{SET }and a test pattern set TEST_{SET }from the obtained movement data obtained from the groups of people having known characteristics. In particular, the computer 83 in an exemplary embodiment utilizes a subset of the data files in order to generate the training pattern set TRAIN_{SET }and the remaining data files in order to generate the test pattern set TEST_{SET}. Typically, fewer files are delegated to the creation of the test pattern set than the training pattern set; however, there is no requirement that this be the case. For example, the computer 83 fro each group of data files may delegate 7 data files out of every 10 data files to the creation of the training pattern set TRAIN_{SET }and use the remaining data files for the creation of the test pattern set TEST_{SET}.
[0086] The computer 83 then creates the training pattern set TRAIN_{SET }by generating a separate input pattern A_{k }and expected output pattern B_{K }for each of the data files delegated to the training pattern set TRAIN_{SET}. Similarly, the computer 83 creates the test pattern set TEST_{SET }by generating a separate input pattern A_{k }and expected output pattern B_{K }for each of the data files delegated to the test pattern set TEST_{SET}. More specifically, the computer 83 extracts characteristics from each data file in the same manner as the preprocessor 30 extracted characteristics from the collected movement data in step 404 of FIG. 4, and creates an input pattern A_{k }from the extracted characteristics in the same manner as the preprocessor 30 created an input pattern in step 406 of FIG. 4.
[0087] The computer 83 then finishes the creation of the training pattern set TRAIN_{SET }and the test pattern set TEST_{SET }by associating an expected output pattern B_{k }to each generated input pattern A_{k}. More specifically, if the input pattern A_{k }was generated from a data file corresponding to a person known to have Parkinson's disease, then the computer 83 associates an expected output pattern B_{k }to the input pattern A_{k }which indicates that the person has Parkinson's disease. Similarly, if the input pattern A_{k }was generated from a data file corresponding to a person known to have essential tremor, then the computer 83 associates an expected output pattern B_{k }to the input pattern A_{k }which indicates that the person has essential tremor. Furthermore, if the input pattern A_{k }was generated from a data file corresponding to a person known to exhibit normal tremor characteristics, then the computer 83 associates an expected output pattern B_{k }to the input pattern A_{k }which indicates that the person exhibits normal tremor characteristics. In an exemplary embodiment, the computer 83 associates a first output pattern B_{k }of 100 to Parkinson's disease, a second output B_{K }of 010 to essential tremor, and a third output B_{K }of 001 to normal tremor characteristics.
[0088] The computer 83 in step 606 then stores the input patterns A_{k }and associated expected output pattern B_{k }for each data file in either a training pattern file or a test pattern file. Accordingly, the computer 83 obtains a training pattern file that includes the training pattern set TRAIN_{SET }and a test pattern file that includes the test pattern set TEST_{SET }which may be used to train and test the computational intelligence system 50 of the analysis system 10.
[0089] Exemplary Training Mechanism
[0090] Referring back to FIG. 1, there is depicted a training mechanism 60, a training pattern set 70, and a test pattern set 75. In general, the training mechanism 60 is operable to train the computational intelligence system 50 to classify movement based upon a predetermined group of neurological disorder classifications and test that the trained computational intelligence system 50 satisfies a threshold level of accuracy. To this end, the training mechanism 60 during a training phase essentially evolves parameters of the computational intelligence system 50 based upon the input patterns A_{k }and expected output patterns B_{k }of the training pattern set 70. Moreover, the training mechanism 60 during a testing phase essentially computes a fitness value for the trained computational intelligence system 50 based upon (i) output patterns Z_{k }generated in response to the trained computational intelligence system 50 processing input patterns A_{k }of the test pattern set 75, and (ii) expected output patterns B_{k }of the test pattern set 75.
[0091] An exemplary training mechanism 60 for training a neural network implementation of the computational intelligence system 50 with particle swarm optimization is described in detail below. However, it should be appreciated that other training mechanisms such as error backpropagation, genetic algorithms, and evolutionary algorithms may be used to train the computational intelligence system 50 based upon the training patterns set 70.
[0092] Shown in greater detail in FIG. 7 is an exemplary training mechanism 60 of the analysis system 10 which defines and/or evolves weighted connection matrices W and U, activation functions F, and processing element layers F_{x}, F_{y}, and F_{z }of the exemplary neural network 100 in order to obtain a computational intelligence system 50 trained to classify movement based upon input patterns A_{k }and a group of predetermined neurological classifications. As depicted, the exemplary training mechanism 60 includes a network evolver 702, a network simplifier 704, and a network verifier 706. In general, the network evolver 702 is operable to (i) define an initial neural network architecture for the neural network 100, and (ii) continually adjust the architecture of the neural network 100 until the network evolver 702 obtains a definition for the neural network 100 that meets predefined criteria. More specifically, the network evolver 702 in an exemplary embodiment is operable to (i) apply input patterns A_{k }of a training pattern set TRAIN_{SET }to the input layer F_{x }of the neural network 100, and (ii) adjust parameters of the neural network 100 based upon output patterns Z_{k }generated by the neural network 100 in response to the input patterns A_{k}. As will be explained in more detail in reference to FIGS. 8A8B, the network evolver 702 in an exemplary embodiment includes a particle swarm optimizer which is operable to adjust parameters of the neural network 100 in such a manner so as to achieve a trained neural network 100 that generates appropriate output patterns Z_{k }in response to processing input patterns A_{k}.
[0093] The network simplifier 704 of the exemplary training mechanism 60 is generally operable to simplify the configuration of the neural network 100. As will be explained in more detail with reference to the network simplification method 900 of FIG. 9, the network simplifier 704 in an exemplary embodiment is operable to (i) receive a definition for the neural network 100 from the network evolver 702, (ii) redefine certain processing elements of the neural network 100 such that the processing element implements a simpler activation function, and (iii) remove unnecessary processing elements from the definition of the neural network 100.
[0094] Finally, the network verifier 706 of the exemplary training mechanism 60 is generally operable to verify the accuracy of the obtained simplified network definition for the neural network 100. More specifically, the network verifier 706 in an exemplary embodiment is operable to (i) apply input patterns A_{k }of a test pattern set TEST_{SET }to the input layer F_{x }of the simplified definition of the neural network 100, and (ii) generate a fitness value that is indicative of how well the trained neural network 100 as defined by obtained simplified definition is able to produce appropriate output patterns Z_{k }in response to processing input patterns A_{k}. In an exemplary embodiment, the network verifier 706 generates the fitness value for the trained neural network by calculating an average sumsquared error between the generated output patterns Z_{k }and expected patterns B_{k }of the test pattern set TEST_{SET }(See, below equation (8) for details on performing an average sumsquared error calculation.)
[0095] From the produced fitness value for the simplified network definition, the exemplary training mechanism 60 is operable to determine whether the neural network 100 has been successfully trained. In particular, the exemplary training mechanism 60 in an exemplary embodiment determines that the neural network 100 has been successfully trained if the fitness value (e.g. average sumsquared error) for the simplified definition has a predetermined relationship to (e.g. less than) a fitness threshold value FIT_{THR }(e.g. 0.01).
[0096] FIGS. 8A8B show a flowchart of a network evolution method 800 that illustrates in detail the operation of the network evolver 702 of the exemplary training mechanism 60. As illustrated in FIGS. 8A8B, the network evolver 702 begins in step 802 by initializing an iteration counter ITER and determining various parameters to be used in evolving the neural network 100. More specifically, the network evolver 702 in an exemplary embodiment initializes the iteration counter ITER by setting the iteration counter to a value of 1.
[0097] Moreover, the network evolver 702 in an exemplary embodiment obtains various parameter values from a configuration file that define (i) a number of particles P# (e.g. 20) for a particle swarm S, (ii) a maximum particle velocity value V_{MAX }(e.g. 10.0), (iii) a maximum position value POS_{MAX }(e.g. 10.0), (iv) a dynamic initial lower limit value LOWER_{0 }(e.g. −0.5), (v) a dynamic initial upper limit value UPPER_{0 }(e.g. 0.5), (vi) a starting inertia weight wi_{0 }(e.g. 0.9), (vii) a slope upper limit SLOPE_{ULIM }(e.g. 90), an error cutoff E_{CUT }(e.g. 0.025), (vii) a maximum number of iterations ITER_{MAX }(e.g. 1000), a training pattern set TRAIN_{SET }(e.g. a file name of a file which includes training pairs of input patterns A_{k }and corresponding expected output patterns B_{k}), a number of pattern pairs PAT# in the training pattern set TRAIN_{SET }(e.g. 800), a test pattern set TEST_{SET }(e.g. a file name of a file which includes testing pairs of input patterns A_{k }and corresponding expected output patterns B_{k}), and a number of pattern pairs TPAT# in the test pattern set TEST_{SET }(e.g. 500).
[0098] After obtaining the above parameters, the network evolver 702 in step 804 generates an initial topology for the neural network 100. More specifically, the network evolver 702 in an exemplary embodiment defines an appropriate three layer, fully connected, feedforward network topology for the neural network 100. (See FIG. 5.) To this end, the network evolver 702 determines the number p of output signals b_{jk }that each expected output pattern B_{k }includes, and the number n of input signals a_{hk }that each input pattern A_{k }includes. From this information, the network evolver 702 defines an initial topology for the neural network 100 that includes (i) an input layer F_{x }having a bias processing element PEx_{0 }and n input processing elements PEx_{h}, (ii) an output layer F_{z }having p output processing elements PEz_{j}, and (iii) a hidden layer F_{y }having a bias processing element PEy_{0 }and q hidden processing elements PEy_{i}.
[0099] A suitable number q of hidden processing elements can vary widely according to the application and bears a relationship to the number of statistically significant factors that exist in the input data. If there are too few hidden processing elements PEy_{i}, the network evolver 702 will probably fail to train the neural network 100. If there are just barely enough, the network evolver 702 may successfully train the neural network 100, but the resulting neural network 100 may fail to generate appropriate output patterns Z_{k }for input patterns A_{k }that were not part of the training pattern set TRAIN_{SET}. Moreover, the resulting neural network 100 will probably not handle noisy data well. Conversely, if there are too many hidden processing elements PEy_{i}, the resulting neural network 100 probably will not generalize very well. In other words, the resulting neural network 100 probably will not generate appropriate output patterns Z_{k }for input patterns A_{k }that were not part of the training pattern set TRAIN_{SET}.
[0100] Accordingly, the neural network may need to be trained several different times with different values of q until a suitable number q of hidden processing elements PEy_{i }is found. A suitable number q of hidden processing elements PEy_{i}, however, may often be obtained by taking the square root of the number n of input processing elements PEx_{h }squared plus the number p of output processing elements squared PEz_{j }plus a few additional processing elements. This relationship for q is represented mathematically by the following equation (7):
5$\begin{array}{cc}q=\mathrm{ceil}\ue8a0\left(\sqrt{{n}^{2}+{p}^{2}}\right)+o& \left(7\right)\end{array}$
[0101] where n represents the number of input processing elements PEx_{h}, p represents the number of output processing elements PEz_{j}, ceil( ) represents a ceiling function which rounds a noninteger number up to the next integer number, and o represents a small integer with respect to
6$\mathrm{ceil}\ue8a0\left(\sqrt{{n}^{2}+{p}^{2}}\right).$
[0102] In an exemplary embodiment of the present invention, the values n, q, and p are supplied by a user via a configuration file that contains appropriate values for n, q, and p as well as the above discussed parameters. However, it should be appreciated that a user may alternatively supply the values n, q, and p as well as the above discussed parameters via an input device such as a mouse or a keyboard. Accordingly, the network evolver 702 in step 804 may determine appropriate values for n, q and p based upon (i) the input patterns A_{k }and output pattern B_{k }of the training pattern set TRAIN_{SET}, and/or (ii) user supplied parameters received via an input device and/or configuration file.
[0103] After defining an initial network topology for the neural network 100, the network evolver 702 in step 806 initializes a swarm S of particles P_{0}, P_{1}, . . . P_{P#} that represent P# possible definitions for the neural network 100. More specifically, the network evolver 702 defines for each particle P_{x }of the swarm S, (i) a position POS_{x }in Ddimensional hyperspace, and a velocity vector V_{x }through the Ddimensional hyperspace. More specifically, the network evolver 702 defines the Ddimensional hyperspace for the neural network 100 such that each dimension of the Ddimensional hyperspace represents a weighted connection W_{hi }of the weighted connections matrix W, a weighted connection u_{ij }of the weighted connections matrix U, a slope factor α_{i }of a slope vector A, or a slope factor β_{j }of a slope vector B.
[0104] For example, if the neural network 100 is initially defined to include an (i) input layer F_{x }having a bias processing element PEx_{0 }and two input processing elements PEx_{1 }and PEx_{2}, a hidden layer F_{y }having a bias processing element PEy_{0 }and four hidden processing elements PEy_{1}, PEy_{2}, . . . and PEy_{4}, and an output layer F_{z }having a single output processing element PEz_{1}, then the neural network 100 would have a weighted connections matrix W consisting of 12 weighted connections w_{10}, w_{20}, . . . w_{40}, w_{11}, w_{21}, . . . w_{41}, w_{12}, w_{22}, . . . w_{42}, a weighted connections matrix U consisting of 5 weighted connections u_{10}, u_{11}, . . . u_{14}, a slope vector A consisting of 4 slope factors α_{1}, α_{2}, . . . α_{4}, and a slope vector B consisting of 1 slope factor β_{1}. Therefore, the network evolver 702 in this example, would define a 22dimensional hyperspace in which each position in the 22dimensional hyperspace represents a possible solution for the 12 weighted connections of the weighted connections matrix W, the 5 weighted connections of the weighted connection matrix U, the 4 slope factors of the slope vector A, and the 1 slope factor of the slope vector B.
[0105] In an exemplary embodiment, the network evolver 702 in step 808 randomly assigns each particle P_{x }of the particle swarm S an initial position POS_{x }and velocity vector V_{x}. More specifically, the network evolver 702 randomly assigns the initial positions POS_{1}, POS_{2}, . . . POS_{P#} of the particles P_{1}, P_{2}, . . . P_{P#} such that the weight connections W_{hi }and u_{ij }represented by the positions POS_{1}, POS_{2}, . . . POS_{P#} are between the initial lower limit LOWER_{0 }and initial upper limit UPPER_{0}. Moreover, the network evolver 702 assigns the initial positions POS_{1}, POS_{2}, . . . POS_{P#} of the particles P_{1}, P_{2}, . . . P_{P#} such that the slope factors α_{i }and β_{j }are initially equal to 1. Furthermore, the network evolver randomly assigns the Ddimensional velocity vectors V_{1}, V_{2}, . . . V_{P#} of the particles P_{1}, P_{2}, . . . P_{P#} such that each dimensional velocity component v_{x1}, V_{x2}, V_{xD }of a velocity vector V_{x }is between the initial lower limit LOWER_{0 }and the initial upper limit UPPER_{0}.
[0106] After initializing the positions POS_{1}, POS_{2}, . . . POS_{P#} of the particles P_{1}, P_{2}, . . . P_{P#}, the network evolver 702 in step 810 determines a personal best value PBEST_{x}, a personal best position PBESTX_{x}, a local best value LBEST_{x}, and a local best position LBESTX_{x }for each particle of the particles P_{1}, P_{2}, . . . P_{P#}. More specifically, each personal best value PBEST_{x }represents the corresponding best definition obtained by the particle P_{x }for the neural network 100, and each personal best position PBESTX_{x }represents the position in hyperspace where the particle P_{x }obtained its corresponding particle best value PBEST_{x}. Similarly, each local best value LBEST_{x }represents the corresponding best definition obtained by a particle group PG_{x }that includes particles P_{x−L}, . . . P_{x−1}, P_{x}, P_{x+1}, . . . P_{x+L}, and each local best position LBESTX_{x }represents the position in hyperspace where the particle group PG_{x }obtained its corresponding local best value LBEST_{x}.
[0107] In an exemplary embodiment, the particle groups PG_{1}, PG_{2}, . . . PG_{P#} are defined in a circular array fashion based upon a local neighbor parameter L. More specifically, if the local neighbor parameter L is equal to 2 and the number of particles is equal to 20, then the first particle group PG_{1 }would include particles P_{19}, P_{20}, P_{1}, P_{2}, and P_{3 }and the second particle group PG_{2 }would include the particles P_{20}, P_{1}, P_{2}, P_{3}, and P_{4}. It should be appreciated by those skilled in the art that if the local neighbor parameter L is equal to or greater than onehalf the number of particles P#, then a single local best value LBEST and corresponding local best position LBESTX may be used since all of the particle groups PG_{1}, PG_{2}, . . . PG_{P#} would include every particle P_{1}, P_{2}, . . . P_{P#} of the particle swarm S. This special case is referred to as a global version of the particle swarm optimizer implemented by the network evolver 702. Moreover, the single local best value LBEST and corresponding local best position LBESTX in this special case are referred to as the global best value GBEST and the global best position GBESTX, respectively.
[0108] It has been found that the larger the local neighbor parameter L becomes the quicker (i.e. less iterations) on average the particle swarm optimizer of the network evolver 702 converges to an optimum. However, as the local neighbor parameter L becomes larger, the particle swarm optimizer of the network evolver 702 is more likely to converge on a local optimum instead of a global optimum. In other words, as the local neighbor parameter L becomes larger, the more likely the particle swarm optimizer of the network evolver 702 will fail to obtain a definition for the neural network 100 that achieves a desired level of performance. Accordingly, in an exemplary embodiment, the network evolver 702 utilizes a local neighbor parameter L of 2 which has been found to cause the particle swarm optimizer to converge on a global optimum as opposed to a local optimum at highly successful rate.
[0109] In order to determine a particle best value PBEST_{x }and a particle best position PBESTX_{x }for each particle P_{x }of the particle swarm S, the network evolver 702 computes a fitness value FV_{x }for each definition of the neural network 100 as defined by the particle positions POS_{1}, POS_{2}, . . . POS_{P#}. More specifically, the network evolver 702 in an exemplary embodiment calculates a fitness value FV_{x }for each particle P_{x }based upon a fitness function FIT( ) of (i) the output patterns Z_{1}, Z_{2}, . . . Z_{PAT#} generated in response to propagating the input patterns A_{1}, A_{2}, . . . A_{PAT#} of the training pattern set TRAIN_{SET }through the neural network 100 as defined by the selected particle P_{x}, and (ii) the corresponding expected output patterns B_{1}, B_{2}, . . . B_{PAT#} of the training pattern set TRAIN_{SET}. In an exemplary embodiment of the present invention, the network evolver 702 calculates the fitness value FV_{x }of a particle P_{x }based upon the following fitness function FIT( ) which computes the average sumsquared error between the output patterns Z_{1}, Z_{2}, . . . Z_{PAT#} and the expected output patterns B_{1}, B_{2}, . . . B_{PAT#}:
7$\begin{array}{cc}{\mathrm{FV}}_{x}=\mathrm{FIT}\ue8a0\left(B,Z\right)=\frac{0.5\ue89e\sum _{k=1}^{\mathrm{PAT}\ue89e\#}\ue89e\text{}\ue89e\sum _{j=1}^{q}\ue89e\text{}\ue89e{\left({b}_{\mathrm{kj}}{z}_{\mathrm{kj}}\right)}^{2}}{\mathrm{PAT}\ue89e\#}& \left(8\right)\end{array}$
[0110] where q represents the number of output processing elements of the neural network output layer F_{z}, Z_{kj }represents the output signal of the output processing element PEz_{j }in response to the input pattern A_{k }being applied to the neural network input layer F_{x}, b_{kj }represents the corresponding expected output signal of the output processing element PEz_{j}, and PAT# represents the number of patterns of the training pattern set TRAIN_{SET}.
[0111] To this end of generating fitness values FV_{1}, FV_{2}, . . . FV_{P#} for the particles P_{1}, P_{2}, . . . P_{P#}, the network evolver 702 in step 812 initializes a particle index N to a value of 1 in order to cause the particle index N to identify the first particle P_{1 }of the particle swarm S. The network evolver 702 then in step 814 selects the particle P_{N }identified by the particle index N.
[0112] After selecting the particle P_{N }identified by the particle index N, the network evolver 702 in step 816 calculates a fitness value FV_{N }for the particle definition of the neural network 100 via the above fitness function FIT( ). In particular, the network evolver 702 in an exemplary embodiment generates output patterns Z_{1}, Z_{2}, . . . Z_{PAT#} in response to applying the input patterns A_{1}, A_{2}, . . . A_{PAT#} of the training pattern set TRAIN_{SET }to a definition of the neural network 100 as defined by the position POS_{N }of the selected particle P_{N}. More specifically, the network evolver 702 generates the output patterns Z_{1}, Z_{2}, . . . Z_{PAT#} based upon a definition of the neural network 100 in which the neural network 100 has a weighted connections matrix W, a weighted connections matrix U, a slope vector A, and a slope vector B as defined by the position POS_{N }of the selected particle P_{N}. Moreover, the network evolver 702, in the exemplary embodiment, generates a fitness value FV_{N }for the selected particle P_{N }based upon above equation (8) which causes the network evolver 702 to calculate the average sumsquared error between the generated output patterns Z_{1}, Z_{2}, . . . Z_{PAT#} and the expected output patterns B_{1}, B_{2}, . . . B_{PAT#} of the training pattern set TRAIN_{SET}.
[0113] After obtaining the fitness value FV_{N }for the selected particle P_{N}, the network evolver 702 in step 818 determines whether the personal best value PBEST_{N }and personal best position PBESTX_{N }for the selected particle P_{N }need to be updated. To this end, the network evolver 702 determines whether the obtained fitness value FV_{N }for the selected particle P_{N }is better than the current personal best value PBEST_{N }for the selected particle P_{N}. If the obtained fitness value FV_{N }for the selected particle P_{N }is better than the current personal best value PBEST_{N }for the selected particle P_{N}, then the network evolver 702 proceeds to step 820 in order to update the personal best value PBEST_{N }and personal best position PBESTX_{N }for the selected particle P_{N}. Otherwise, if the obtained fitness value FV_{N }for the selected particle P_{N }is not better than the current personal best value PBEST_{N }for the selected particle P_{N}, then the network evolver 702 proceeds to step 822 in order to determine whether a fitness value FV_{x }has been obtained for each particle P_{x}.
[0114] In an exemplary embodiment, the network evolver 702 attempts to minimize the fitness value FV_{x }(i.e. average sumsquared error) for the neural network 100. Accordingly, in the exemplary embodiment, the network evolver 702 determines that an obtained fitness value FV_{N }is better than a personal best value PBEST_{N }if the fitness value FV_{N }is less than the personal best value PBEST_{N}. However, it should be appreciated that the fitness function FIT( ) may be defined in such a way that the network evolver 702 needs to maximize the fitness values FV_{x }in order to properly train the neural network 100. Accordingly, in such a maximization environment, the network evolver 702 would determine that the a fitness value FV_{N }is better than a personal best value PBEST_{N }if the fitness value FV_{N }is greater than the personal best value PBEST_{N}. Moreover, it should be appreciated that the fitness function of equation (8) is merely exemplary and that other fitness functions may be used.
[0115] After determining that the fitness value FV_{N }is better than the personal best value PBEST_{N }for the selected particle P_{N}, the network evolver 702 in step 820 updates the personal best value PBEST_{N }and the personal best position PBESTX_{N }for the selected particle P_{N}. More specifically, the network evolver 702 sets the personal best value PBEST_{N }equal to the calculated fitness value FV_{N }for the selected particle P_{N}. Moreover, the network evolver 702 sets the personal best position PBEST_{N }equal to the position POS_{N }of the selected particle P_{N}.
[0116] It should be noted that in an exemplary embodiment, the personal best values PBEST_{x }are initially set to zero so that network evolver 702 during the first iteration through the network evolution method 800 (i.e. iteration counter ITER equal to 1) will update the personal best values PBEST_{N }with the calculated fitness value FV_{N }for the selected particle P_{N}.
[0117] In step 822, the network evolver 702 determines whether a fitness value FV_{x }has been obtained for each particle P_{1}, P_{2}, . . . P_{P#} of the particle swarm S. If the network evolver 702 determines that a fitness value FV_{x }has been obtained for each particle P_{1}, P_{2}, . . . P_{P#} of the particle swarm S, then the network evolver 702 proceeds to step 826 of the network evolution method 800. However, if the network evolver 702 determines that a fitness value FV_{x }has not been obtained for each particle P_{1}, P_{2}, . . . P_{P#} of the particle swarm S, then the network evolver 702 proceeds to step 824 of the network evolution method 800. In an exemplary embodiment of the present invention, the network evolver 702 determines that a fitness value FV_{x }has been obtained for each particle P_{1}, P_{2}, . . . P_{P#} of the particle swarm S if the particle index N is greater than the number of particles P# included in the particle swarm S.
[0118] After determining that a fitness value FV_{x }has not been obtained for each particle P_{1}, P_{2}, . . . P_{P#}, the network evolver 702 in step 824 updates the particle index N and returns to step 814 in order to select the particle P_{N }identified by the updated particle index N and obtain a fitness value FV_{N }for the newly selected particle P_{N}. In an exemplary embodiment, the network evolver 702 updates the particle index N by incrementing the particle index N by a value of 1.
[0119] After determining that a fitness value FV_{x }has been obtained for each particle P_{1}, P_{2}, . . . P_{P#}, the network evolver 702 updates the local best value LBEST_{x }and local best position LBESTX_{x }for each particle P_{1}, P_{2}, . . . P_{P#}. To this end, the network evolver 702 in step 826 initializes a group index M to a value of 1 in order to obtain a group index M that identifies a first particle group PG_{1}. Then, the network evolver 702 in step 828 determines whether any particle P_{M−L}, . . . P_{M−1}, P_{M}, P_{M+1}, . . . P_{M+L }of the identified group PG_{M }has a personal best value PBEST_{x }that is better than the current local best value LBEST_{M }for the identified particle group PG_{M}. If any of the personal best values PBEST_{M−L}, . . . PBEST_{M−1}, PBEST_{M}, PBEST_{M+1}, . . . PBEST_{M+L }of the identified group PG_{M }is better than the current local best value LBEST_{M }for the identified particle group PG_{M}, then the network evolver proceeds to step 830 in order to update the local best value LBEST_{M }for the identified particle group PG_{M}. Otherwise, if none of the personal best values PBEST_{M−L}, . . . PBEST_{M−1}, PBEST_{M}, PBEST_{M+1}, . . . PBEST_{M+L }of the identified group PG_{M }are better than the current local best value LBEST_{M }for the identified particle group PG_{M}, then the computer system proceeds to step 832 in order to determine whether all particle groups PG_{1}, PG_{2}, . . . PG_{P#} have been processed.
[0120] After determining that at least one personal best value PBEST_{M−L}, . . . PBEST_{M−1}, PBEST_{M}, PBEST_{M+1}, . . . PBEST_{M+L }of the identified group PG_{M }is better than the current local best value LBEST_{M}, the network evolver 702 in step 830 updates the local best value LBEST_{M }and the local best position LBESTX_{M }for the identified particle group PG_{M}. More specifically, the network evolver 702 sets the local best value LBEST_{M }equal to the best, personal best value PBEST_{B }of the identified group PG_{M}. Moreover, the network evolver 702 sets the local best position LBESTX_{M }equal to the personal best position PBESTX_{B }associated with the best, personal best value PBEST_{B }of the identified group PG_{M}.
[0121] It should be noted that in an exemplary embodiment, the local best values LBEST_{x }are initially set to zero so that the network evolver 702 during the first iteration through the network evolution method 800 (i.e. iteration counter ITER equal to 1) will update the local best values LBEST_{x }with one of the personal best values PBEST_{M−L}, . . . PBEST_{M−1}, PBEST_{M}, PBEST_{M+1}, . . . PBEST_{M+L }of its respective particle group PG_{x}. In another exemplary embodiment, the network evolver 702 during the first iteration through the network evolution method 800 sets each LBEST_{x }of a particle P_{x }equal to its respective personal best value PBEST_{x }in step 820. Under either exemplary embodiment, each local best value LBEST_{x }should be equal to the best, personal best value PBEST_{X−L}, . . . PBEST_{X−1}, PBEST_{X}, PBEST_{X+1}, . . . PBEST_{X+L }of its respective particle group PG_{x }after the first iteration through the network evolution method 800.
[0122] In step 832, the network evolver 702 determines whether all of the particle groups PG_{1}, PG_{2}, . . . PG_{P#} have been processed. If all of the particle groups PG_{1}, PG_{2}, . . . PG_{P#} have been processed, then the network evolver 702 proceeds to step 836 in order to determine whether termination criteria have been satisfied. However, if all of the particle groups PG_{1}, PG_{2}, . . . PG_{P#} have not been processed, then the network evolver 702 proceeds to step 834. In an exemplary embodiment, the network evolver 702 determines that all of the particle groups PG_{1}, PG_{2}, . . . PG_{P#} have been processed if the particle group index M is greater than the number of particles P# of the swarm S.
[0123] In step 834, the network evolver 702 updates the particle group index M and returns to step 828 in order to process the next particle group PG_{M}. In an exemplary embodiment, the network evolver 702 updates the particle group index M by incrementing the particle group index M by a value of 1.
[0124] After processing all of the particle groups PG_{1}, PG_{2}, . . . PG_{P#}, the network evolver 702 in step 836 determines whether defined termination criteria have been satisfied. If the network evolver 702 determines that the defined termination criteria have been satisfied, then the network evolver 702 proceeds to step 844 in order define the weighted connections matrix W, the weighted connections matrix U, the slope vector A, and the slope vector B. However, if the network evolver 702 determines that the defined termination criteria have not been satisfied, then the network evolver 702 proceeds to step 838 in order to update the velocity vectors V_{1}, V_{2}, . . . V_{P#} associated with the particles P_{1}, P_{2}, . . . P_{P#}.
[0125] In an exemplary embodiment of the present invention, the termination criteria are defined by a maximum number of iterations ITER_{MAX }and an error cutoff E_{CUT}. More specifically, the network evolver 702 in an exemplary embodiment determines to terminate the network evolution method 800 in response to (i) the iteration counter ITER having a predetermined relationship to the maximum number of iterations ITER_{MAX}, or (ii) the best of the local best values LBEST_{1}, LBEST_{2}, . . . LBEST_{P#} having a predetermined relationship to the desired error cutoff E_{CUT}. For example, in an exemplary embodiment which attempts to minimize the fitness values FV_{x}, the network evolver 702 may be implemented to terminate the network evolution method 800 if either (i) the iteration counter ITER is equal to the maximum number of iterations ITER_{MAX}, or (ii) the best of the local best values LBEST_{1}, LBEST_{2}, . . . LBEST_{P#} is less than the desired error cutoff E_{CUT}.
[0126] In step 838, the network evolver 702 updates the velocity vector V_{x }for each particle P_{x }of the swarm S. More specifically, the network evolver 702 updates the velocity vector V_{x }of a particle P_{x }based upon (i) an inertia weight wi, (ii) the personal best position PBESTX_{x }of the particle P_{x}, and (iii) the local best position LBESTX_{x }for the particle group PG_{x }to which the particle P_{x }belongs. In an exemplary embodiment, the network evolver 702 updates each velocity component V_{x1}, V_{x2}, . . . V_{xD }of the velocity vector V_{x }for a particle P_{x }based upon the following velocity equation (9):
v_{xd}′=wi* v_{xd}+c_{1}*rand( )*(pbestx_{xd}−pos_{xd})+c_{2}*Rand( )*(lbestx_{xd}−pos_{xd}) (9)
[0127] In the above velocity equation (9), V_{xd}40 represents an updated velocity component of the velocity vector V_{x }in the d^{th }dimension, v_{xd }represents the current velocity component of the velocity vector V_{x }in the d^{th }dimension, wi represents the inertia weight parameter, and c_{1 }and c_{2 }represent acceleration constants. Moreover, rand( ) and Rand( ) represent random functions that each generate a random number in the range between 0 and 1. Furthermore, pos_{xd }represents the position of the particle P_{x }in the d^{th }dimension, pbestx_{xd }represents the position of the personal best value PBEST_{x }in the d^{th }dimension, and lbestx_{xd }represents the position of the local best value LBEST_{x }in the d^{th }dimension.
[0128] The inertia weight parameter wi and the acceleration constants c_{1 }and c_{2 }may be used to control the tension in the system. Lower values (less than 1) for c_{1 }and c_{2 }tend to increase the time it takes for particles P_{x }to arrive in the vicinity of their respective personal best values PBEST_{x }and local best values LBEST_{x}, whereas higher values (greater than 1) for the acceleration constants c_{1 }and c_{2 }tend to allow particles P_{x }to explore larger regions around their respective personal best values PBEST_{x }and local best values LBEST_{x}. Similarly, high values for the inertia weight parameter wi tend to allow particles P_{x }to roam farther from their respective personal best values PBEST_{x }and local best values LBEST_{x}, whereas low values for the inertia weight parameter wi tend to restrict particles P_{x }to regions nearer their respective personal best values PBEST_{x }and local best values LBEST_{x}.
[0129] In an exemplary embodiment of the present invention, the network evolver 702 sets the acceleration constants equal to a value of 2. Moreover, the network evolver 702 decreases the inertia weight wi linearly over the defined maximum iterations ITER_{MAX }from the initial inertia weight wi_{0 }to a final inertia weight such as 0.4. The advantage of adjusting the inertia weight wi from a high value of 0.9 to a low value of 0.4 is that the particles P_{1}, P_{2}, . . . P_{P#} initially perform a more global exploration and move gradually toward a more local exploration.
[0130] Moreover, in an exemplary embodiment, the network evolver 702 clips the updated velocity components v_{x1}′, v_{x2}′, . . . v_{xD}′ such that no updated velocity component v_{xd}′ is greater than the maximum velocity parameter V_{MAX}. For example, if the network evolver 702 obtains an updated velocity component v_{xd}′ equal to 11.3 and the maximum velocity parameter V_{MAX }is set to 10.0, then the network evolver 702 would set the updated velocity component v_{xd}′ equal to 10.0 instead of 11.3.
[0131] After updating the velocity vector V_{x }for each particle P_{x }of the particle swarm S, the network evolver 702 in step 840 updates each position POS_{x }of each particle P_{x}. In particular, the network evolver 702 updates each position component pos_{x1}, pos_{x2}, . . . pos_{xD }of each position POS_{x }based upon the updated velocity vector V_{x }for corresponding particle P_{x}. To this end, the network evolver 702 in an exemplary embodiment updates each position component pos_{x1}, pos_{x2}, . . . pox_{xD }of a particle position POS_{X }based upon the following position equation (10).
pos_{xd}′=pos_{xd}+v_{xd}′ (10)
[0132] where pos_{xd }represents the current position component of the particle position POS_{x }in the d^{th }dimension, pos_{xd}′ represents the updated position component of the particle position POS_{x }in the d^{th }dimension, and v_{xd}′ represents the updated velocity component for the particle P_{x }in the d^{th }dimension. Moreover, in an exemplary embodiment, the network evolver 702 clips the updated position components pos_{x1}′, pos_{x2}′, . . . pos_{xD}′ such that no updated position component pos_{xd}′ is greater than the maximum position parameter POS_{MAX}. For example, if the network evolver 702 obtains an updated position component pos_{xd}′ equal to −11.3 and the maximum position parameter POS_{MAX }is set to 10.0, then the network evolver 702 would set the updated position component pos_{xd}′ equal to −10.0 instead of −11.3. However, it should be appreciated that the network evolver 702 may also be implemented such that the updated positions components pos_{x1}′, pos_{x2}′, . . . pos_{xD}′ are not limited to a specified range.
[0133] In step 842, the network evolver 702 updates the iteration counter ITER and returns to step 810 in order to process the updated particles P_{1}, P_{2}, . . . P_{P#} of the particle swarm S. In particular, the network evolver 702 in an exemplary embodiment increments the iteration counter ITER by a value of 1 before returning to step 810.
[0134] After the termination criteria are satisfied, the network evolver 702 in step 844 obtains a neural network definition for the neural network 100 from the particle swarm S. More specifically, the network evolver 702 obtains the position in the Ddimensional hyperspace that achieved that best fitness value and uses this position to define the weighted connections matrix W, the weighted connections matrix U, the slope matrix A, and the slope matrix B of the neural network 100. To this end, the network evolver 702 in an exemplary embodiment (i) obtains the best, personal best position PBESTX_{B }associated with the best of the personal best values PBEST_{1}, PBEST_{2}, . . . PBEST_{P#}, (ii) sets each weighted connection w_{hi }of the weighted connections matrix W, each weighted connection u_{ij }of the weighted connections matrix U, each slope factor α of the slope factor A, and each slope factor β of the slope vector B equal to a respective position component pbestx_{B1}, pbestx_{B2}, . . . pbest_{BD }of the best, personal best position PBESTX_{B}.
[0135] Similarly, in another exemplary embodiment, the network evolver 702 (i) obtains the best, local best position LBESTX_{B }associated with the best of the local best values LBEST_{1}, LBEST_{2}, . . . LBEST_{P#}, (ii) sets each weighted connection w_{hi }of the weighted connections matrix W, each weighted connection u_{ij }of the weighted connection matrix U, each slope factor α of the slope factor A, and each slope factor β of the slope vector B equal to a respective position component lbestx_{B1}, lbestx_{B2}, . . . lbest_{BD }of the best, local best position PBESTX_{B}. It should be appreciated that either of the two exemplary embodiments should achieve the same results for the weighted connections matrix W, the weighted connections matrix U, the slope vectorA, and the slope vector B.
[0136] Exemplary Network Simplification Method
[0137] Referring now to FIG. 9, there is illustrated a flowchart of an network simplification method 900. In general, the network simplification method 900 when executed by the network simplifier 704 causes the network simplifier 704 to simplify a definition for the neural network 100 of the computation intelligence 50 in order to obtain a less complex definition of the neural network 100. More specifically, the network simplifier 704 in implementing the network simplification method 900 generally (i) removes unnecessary processing elements from the neural network 100, and/or (ii) replaces complex activation functions of certain processing elements with less computationally complex activation functions.
[0138] To this end, the network simplifier 704 in step 902 determines based upon the slope vector A and the slope vector B whether any of the activation functions of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }or the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p }may be simplified. In an exemplary embodiment, each hidden processing element PEy_{i }is initially implemented with the sigmoid threshold function of equation (3) and each output processing element PEz_{j }is initially implemented with the sigmoid threshold function of equation (6). As can be seen from equations (3) and (6), if the slope factors α and β are positive and have a sufficiently large magnitude, the sigmoid activation function essentially becomes the following step threshold function (11):
8$\begin{array}{cc}f\ue8a0\left(x\right)=\{\begin{array}{cc}1& \mathrm{if}\ue89e\text{}\ue89ex\ge 0\\ 0& \mathrm{if}\ue89e\text{}\ue89ex<0\end{array}& \left(11\right)\end{array}$
[0139] Similarly, if the slope factors α and β are negative and have a sufficiently large magnitude, the sigmoid threshold function essentially becomes the following step threshold function (12):
9$\begin{array}{cc}f\ue8a0\left(x\right)=\{\begin{array}{cc}1& \mathrm{if}\ue89e\text{}\ue89ex\le 0\\ 0& \mathrm{if}\ue89e\text{}\ue89ex>0\end{array}& \left(12\right)\end{array}$
[0140] As a result of the above properties of the sigmoid threshold function, the network simplifier 704 in an exemplary embodiment determines that activation functions of the neural network 100 may be simplified if any of the slope factors α and β of the slope vectors A and B has a magnitude greater than a slope upper limit SLOPE_{ULIM}. It has been found that a slope upper limit SLOPE_{ULIM }as small as 10 often results in a simplified neural network 100 with minimal effect on the accuracy of the output patterns Z_{k }generated by the neural network 100.
[0141] In step 904, the network simplifier 704 redefines the activation functions for those processing elements PE which may be implemented with a simpler activation function. In particular, for each hidden processing element PEy_{i }with a positive slope factor α_{i }having a magnitude greater than the slope upper limit SLOPE_{ULIM}, the network simplifier 704 in an exemplary embodiment replaces the sigmoid threshold function of the hidden processing element PEy_{i }with the step threshold function of equation (11). Moreover, for each hidden processing element PEy_{i }with a negative slope factor α_{i }having a magnitude greater than the slope upper limit SLOPE_{ULIM}, the network simplifier 704 replaces the sigmoid threshold function of the hidden processing element PEy_{i }with the step threshold function of equation (12). Similarly, for each output processing element PEz_{j }with a positive slope factor β_{j }having a magnitude greater than the slope upper limit SLOPE_{ULIM}, the network simplifier 704 in an exemplary embodiment replaces the sigmoid threshold function of the output processing element PEz_{j }with the step threshold function of equation (11). Moreover, for each output processing element PEz_{j }with a negative slope factor β_{j }having a magnitude greater than the slope upper limit SLOPE_{ULIM}, the network simplifier 704 replaces the sigmoid threshold function of the output processing element PEz_{j }with the step threshold function of equation (12).
[0142] The network simplifier 704 then in step 906 determines whether any of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }may be removed from the neural network 100. If the network simplifier 704 determines that at least one of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }may be removed from the neural network 100, then the network simplifier 704 proceeds to step 908. However, if the network simplifier 704 determines that none of the hidden processing elements PEy_{1}, PEy_{2}, . . . PEy_{q }may be removed from the neural network 100, then the network simplifier proceeds to step 910.
[0143] As stated above, each hidden processing elements PEy_{i}, of the neural network 100 in an exemplary embodiment is initially implemented with the sigmoid threshold function of equation (3). As can be seen from the sigmoid threshold function of equation (3), the output of the sigmoid threshold function is roughly equal to a constant value of 0.5 when the magnitude of the slope factor α is less than a slope lower limit SLOPE_{LLIM}. Accordingly, the network simplifier 704 in an exemplary embodiment determines that a hidden processing element PEy_{i }may be removed from the neural network 100 if the slope factor α_{i }associated with the hidden processing element PEy_{i }is less than the slope lower limit SLOPE_{LLIM}. Appropriate values for the slope lower limit SLOPE_{LLIM }are dependent upon the application; however, the network simplifier 704 in an exemplary embodiment uses a lower slope limit SLOPEL_{LIM }of 0.1 which may be sufficient for many different types of applications.
[0144] In step 908, the network simplifier 704 removes those hidden processing elements PEy_{i }identified in step 906 as being appropriate for removal. To this end, for each hidden processing element PEy_{i }with a slope factor α_{i }having a magnitude less than the slope lower limit SLOPE_{LLIM}, the network simplifier 704 (i) removes the identified hidden processing element PEy_{i}, (ii) removes the weighted connections vector W_{i }from the weighted connections matrix W associated with the removed hidden processing element PEy_{i}, (iii) updates the biasing weight connections u_{10}, u_{20}, . . . u_{p0 }in order to replicate the function of the removed hidden processing element PEy_{i}, and (iv) removes the weighted components u_{1i}, u_{2i}, . . . u_{pi }from the weighted connections matrix U associated with the removed hidden processing element PEy_{i}.
[0145] More specifically, since the removed hidden processing element PEy_{i }essentially generated a constant output signal y_{i }of 0.5, the removed hidden processing PEy_{i }essentially affected each of the output processing elements PEz_{1}, PEz_{2}, . . . PEz_{p }by an amount of 0.5 times the respective weighted connections u_{1i}, U_{2i}, . . . u_{pi}. Accordingly, the network simplifier 704 in an exemplary embodiment replicates the effect of the removed hidden processing element PEy_{i }by increasing the biasing weighted connections u_{10}, u_{20}, . . . u_{p0 }from the biasing processing element PEy_{0 }by 0.5 times the removed weight connections u_{1i}, u_{2i}, . . . u_{pi}. The following equation (13) represents this update of the biasing weight connections u_{10}, u_{20}, . . . u_{p0}:
u_{k0}′=u_{k0}+0.5*u_{ki} (13)
[0146] where u_{k0}′ represents the updated k^{th }weighted connection from the biasing processing element PEy_{0}, u_{k0 }represents the current k^{th }weighted connection from the biasing processing element PEy_{0}, and u_{ki }represents the k^{th }weighted connection associated with the removed hidden processing element PEy_{i}. For example, FIG. 10 illustrates a simplified neural network 100′ in which the hidden processing element PEy_{1 }has been removed from the neural network 100 of FIG. 5, and the biasing weighted connections u_{10}, u_{20}, . . . u_{p0 }have been updated to replicate the effect of the removed hidden processing element PEy_{1}.
[0147] In step 910, the network simplifier 704 transfers the simplified definition of the neural network 100 to the network verifier 706. In particular, the network simplifier 704 transfers the obtained weighted connections matrix W, the weighted connections matrix U, the slope vector A, and the slope vector B to the network verifier 706. The network verifier 706 may then test the obtained simplified definition for the neural network 100 by applying test input patterns A_{k }of the test pattern set TEST_{SET}, and calculating a fitness value for the simplified definition based upon generated output patterns Z_{k }and expected output patterns B_{k }of the test pattern set TEST_{SET}.
[0148] It should be appreciated by those skilled in the art that the above exemplary training mechanism 60 may be used to train neural networks having processing elements that utilize different activation functions. For example, the above exemplary training mechanism 60 may be used to evolve and simplify neural networks that utilize the threshold functions of below TABLE 1. More specifically, the network simplifier 706 may replace a processing element threshold function with the simpler threshold function if the slope factor α of the processing element meets the criteria of TABLE 1. Moreover, the network simplifier 706 may remove hidden layer processing elements of the neural network 100 if the simplified threshold function generates a constant output.
1
TABLE 1 


Function Name  Function  Simplification 

  
Hyperbolic tangent 
10$f\ue8a0\left(x\right)=\mathrm{tanh}\ue8a0\left(\mathrm{\alpha x}\right)=\frac{{e}^{\mathrm{\alpha x}}{e}^{\mathrm{\alpha x}}}{{e}^{\mathrm{\alpha x}}+{e}^{\mathrm{\alpha x}}}$

11$\begin{array}{c}\mathrm{For}\ue89e\text{}\ue89e\mathrm{small}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=0\\ \mathrm{for}\ue89e\text{}\ue89e\mathrm{large}\ue89e\text{}\ue89e\mathrm{positive}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=\{\begin{array}{cc}1& \mathrm{if}\ue89e\text{}\ue89ex\ge 0\\ 1& \mathrm{if}\ue89e\text{}\ue89ex<0\end{array}\\ \mathrm{for}\ue89e\text{}\ue89e\mathrm{large}\ue89e\text{}\ue89e\mathrm{positive}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=\{\begin{array}{cc}1& \mathrm{if}\ue89e\text{}\ue89ex\ge 0\\ 1& \mathrm{if}\ue89e\text{}\ue89ex<0\end{array}\end{array}$


Hyperbolic secant 
12$f\ue8a0\left(x\right)=\mathrm{sech}\ue8a0\left(\mathrm{\alpha x}\right)=\frac{2}{{e}^{\mathrm{\alpha x}}+{e}^{\mathrm{\alpha x}}}$

13$\begin{array}{c}\mathrm{For}\ue89e\text{}\ue89e\mathrm{small}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=1\\ \mathrm{for}\ue89e\text{}\ue89e\mathrm{large}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=0\end{array}$


Gaussian Function, or Radial Basis Function 
14$f\ue8a0\left(x\right)={e}^{{\left(\frac{x}{\alpha}\right)}^{2}}$

15$\begin{array}{c}\mathrm{For}\ue89e\text{}\ue89e\mathrm{small}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=0\\ \mathrm{for}\ue89e\text{}\ue89e\mathrm{large}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=1\end{array}$


Augmented Ratio of Squares 
16$f\ue8a0\left(x\right)=\{\begin{array}{cc}\frac{{\mathrm{\alpha x}}^{2}}{1+{\mathrm{\alpha x}}^{2}}& \mathrm{if}\ue89e\text{}\ue89ex>0\\ 0& \mathrm{if}\ue89e\text{}\ue89ex\le 0\end{array}$

17$\begin{array}{c}\mathrm{For}\ue89e\text{}\ue89e\mathrm{small}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=0\\ \mathrm{for}\ue89e\text{}\ue89e\mathrm{large}\ue89e\text{}\ue89e\alpha ,\\ \text{}\ue89ef\ue8a0\left(x\right)=\{\begin{array}{cc}1& \mathrm{if}\ue89e\text{}\ue89ex>0\\ 0& \mathrm{if}\ue89e\text{}\ue89ex\le 0\end{array}\end{array}$


[0149] Exemplary Implementations of Neural Networks and Network Evolution Systems
[0150] It should be appreciated by those skilled in the art that the movement monitoring device 20, the preprocessor 30, multiplexor 40, computational intelligence system 50, and the training mechanism 60 may be implemented with various hardware components such a digital signal processors, digital logic components, and analog components. Moreover, it should be appreciated that the preprocessor 30, multiplexor 40, computational intelligence system 50, and/or the training mechanism 60 may be implemented with properly programmed general purpose computer systems, multiprocessor computer systems, and distributed clusters of computer systems.
[0151] For example, FIG. 11 illustrates a general processing system 1100 which is suitable for implementing the preprocessor 30, multiplexor 40, computational intelligence system 50, and/or the training mechanism 60 of the analysis system 10. To this end, the general processing system 1100 includes a processor 1102, memory 1104, mass storage device 1106, video display 1108, and input device 1110. Moreover, the general processing system 1100 includes a mass storage controller 1112 for controlling the mass storage device 1106, a video controller 1114 for controlling the video display 1108, an I/O controller 1116 for controlling the input device 1110, and a system bus 1118. The system bus 1118 operably couples the processor 1102 to the memory 1104, the mass storage controller 1112, the video controller 1114, and the I/O controller 1116.
[0152] The memory 1104 includes random access memory (RAM) such as SRAM (static RAM), DRAM (dynamic RAM), and SDRAM (synchronous DRAM) which store software routines obtained from computer readable media such as a floppy disk, CDROM disc, DVD disc, and hard disks. The memory 1104 may also include nonvolatile computer readable media such as PROM (programmable read only memory), EPROM (erasable PROM), EEPROM (electrically erasable PROM), and flash memory that store firmware routines.
[0153] The processor 1102 is operable to execute the software and/or firmware routines stored in the memory 1104, and communicate with the mass storage device 1106, the video display 1108, and the input device 1110 via the mass storage controller 1112, the video controller 1114, and the I/O controller 1116 respectively. Most importantly, the processor 1102 is operable to execute software and/or firmware routines of the memory 1104 which cause the computer 1100 to implement the analysis system 10 of FIG. 1. Thus, for example, the processor 1102 may be operable to perform the functions of the preprocessor 30, the multiplex or 40, the computational intelligence system 50, the training mechanism 60 or any combination thereof.
[0154] It should be appreciated by those skilled in the art that, since the slope factors α and β may become arbitrarily large and the input signal a_{1k}, a_{2k}, and a_{nk }may be arbitrarily large, precautions must be taken with a computer system implementation of the computational intelligence system 50 and the exemplary training mechanism 60 to ensure against overflow and underflow errors. For example, in an exemplary computer system embodiment of the present invention, the computer system 1100 in calculating the sigmoid activation function of equations (3) and (6) first test to see if the product of the slope factor α or β and the resulting combinatory value c is greater than a threshold number such as 90. If the product is greater than 90, then the computer system 1100 generates a value of 0 for the result of the sigmoid activation function instead of performing the rest of the sigmoid activation calculation. This threshold test insures that the computer system 1100 will not encounter an overflow or underflow error due to computational limits inherent to digital computations.
[0155] While the invention has been illustrated and described in detail in the drawings and foregoing description, such illustration and description is to be considered as exemplary and not restrictive in character, it being understood that only the exemplary embodiments have been shown and described and that all changes and modifications that come within the spirit of the invention are desired to be protected.