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Of the several algorithms described and detailed using ", FontFamily->"Geneva", FontSize->10], StyleBox["Mathematica", FontFamily->"Geneva", FontSize->10, FontSlant->"Italic"], StyleBox[ ", the back-propagated neural network (BPNN) is a well behaved algorithm \ which can be used in a variety of applications by students and researchers in \ different disciplines. The general purpose for using the BPNN is to classify \ as correctly as possible a set of exemplars based upon their varying \ attributes. In some instances, interest is only in whether the network \ carries out a successful classification, but in other instances common in the \ behavioral sciences, it is considered important to understand what the \ network has \"learned\", e.g., what distinguishing attributes of the \ exemplars are identified by the network for a successful classification \ [Elman1993]. By expanding upon the BPNN algorithm provided by Freeman, and \ developing further graphical representations of the dynamic and final state \ characteristics of the network using ", FontFamily->"Geneva", FontSize->10], StyleBox["Mathematica", FontFamily->"Geneva", FontSize->10, FontSlant->"Italic"], StyleBox[ ", a better understanding of what a network has learned can be obtained.", FontFamily->"Geneva", FontSize->10] }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["THE BACK-PROPAGATED NEURAL NETWORK", FontFamily->"Geneva", FontSize->10, FontWeight->"Bold"]], "Subsection"], Cell[TextData[{ StyleBox[ " A simple BPNN can be described as having a set of \"input cells\" which \ receive ", FontFamily->"Geneva", FontSize->10], StyleBox[ "information about the attributes of the exemplars, a layer of \"hidden \ cells\" which ", FontFamily->"Geneva", FontSize->10], StyleBox[ "receive information from each input cell, and one or more \"output cells\" \ which receive ", FontFamily->"Geneva", FontSize->10], StyleBox[ "information from all the hidden cells. 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A simple neural network consisting of 3 input cells,", FontFamily->"Geneva", FontSize->10], StyleBox["\n", FontSize->10], StyleBox["\t2 hidden cells, and 1 output cell.\n", FontFamily->"Geneva", FontSize->10] }], "Text"], Cell[TextData[{ StyleBox[ "At each input cell, one numerically encoded attribute of an exemplar is \ passed on directly to each cell in the hidden layer. Each cell of the hidden \ layer takes a linearly weighted sum (s) of all its inputs from the cells of \ the input layer, transforms the result by the logistic function y=1/(1+e", FontFamily->"Geneva", FontSize->10], StyleBox["-s", FontFamily->"Geneva", FontSize->9], StyleBox[ "), and passes the result (a response) to all the cells of the output \ layer. Each output cell behaves in a manner similar to a hidden cell by \ transforming a linearly weighted sum of all its inputs using the logistic \ function to provide the network's response. Random weights are used to start \ the process, and continuously adjusted (back-propagated to the output and \ hidden layer cells) according to a least squares criterion based on the \ difference between the desired network response and the obtained response \ each time an exemplar is presented. The adjustment is usually multiplied by a \ constant called the \"learning rate parameter\". Each presentation of an \ exemplar is called an iteration, and the presentation of the complete set of \ exemplars an epoch. The number of epochs required to reach an acceptable \ level of classification accuracy of the exemplars for some problems can be \ several thousands [Istvan et al 1995], and for some input data and \ configuration of cells a local minimum may prevent an adequate solution. ", FontFamily->"Geneva", FontSize->10] }], "Text"], Cell[BoxData[ StyleBox[\(THE\ BPNN\ ALGORITHM\ AND\ FREEMAN' S\ T\ AND\ C\ PROBLEM\), FontFamily->"Geneva", FontSize->10, FontWeight->"Bold"]], "Input"], Cell[TextData[ "In order to provide the student with a more exploratory simulation, the \ basic algorithm given by Freeman is expanded to include the following user \ options:\n\n1. Generation of random starting weights within defined \ boundaries, or to use weights from a previous run of the network.\n2. At the \ end of a prescribed number of epochs to increase or decrease the span of the \ exemplars presented to the network. This option allows exploration of \ network behavior when previously \"unseen\" exemplars are introduced together \ with those already presented. \n3. Increase the frequency of presentation of \ selected exemplars, i.e., provide more \"practice\" for the network in \ classifying specific exemplars.\n4. Setting the number of epochs, after \ which the results of the network are output together with the desired answers \ for the user to judge whether stability or a desired level of classification \ accuracy has been reached, and whether additional epochs are required.\n5. \ Change the learning rate parameter for the adjustment of the weights for \ cells at the output and hidden layers. \n6. Whether the presentation order of \ the exemplars is fixed or random. \n7. Writing to separate files the weights \ used and the output from the hidden and output cells.\n\nFreeman's T-and-C \ problem requires the BPNN to distinguish between the characters \"T\" and \"C\ \" given four different orientations of each character. Each character\ \[CloseCurlyQuote]s orientation is encoded into a 3 by 3 matrix with each \ matrix (exemplar) being a rotation of 0, 90, 180, or 270 degrees. 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The 3x3 character matrices forming the input patterns for the \ T-and-C problem.\n", FontFamily->"Geneva", FontSize->10]], "Text"], Cell[TextData[{ StyleBox[ " Each shaded and unshaded element, i.e., an attribute, of the character \ matrix is encoded as 0.9 and 0.1 respectively, and the 9 elements of each \ matrix defines an exemplar of the character for input to the network. The \ desired output from the network is encoded as 0.9 for T exemplars and 0.1 for \ C exemplars. The network Freeman uses for this problem contains 9 input \ cells, 3 hidden cells, and 1 output cell. Replicating Freeman's network, and \ running it for 75 epochs (600 iterations) produces five files or data sets: \ (1) hidden cell weights of order (75,3,9); (2) hidden cell response of order \ (75,8,3); (3) output cell weights of order (75,3,1); (4) output cell \ response of order (75,8,1); and an error output of order (600,1). 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hidden cell responses are the outputs from each hidden cell for \ each exemplar taken at the end of each epoch and form a matrix of order \ (75,8,3). Interest is in observing the discrimination made by hidden cells \ between the four T-and-C exemplars over the 75 epochs. To provide this plot \ the function ", FontFamily->"Geneva", FontSize->10], StyleBox["hdrAnlz1,", FontFamily->"Courier", FontSize->10], StyleBox[ " included in the accompanying notebook BPNNAppendix.nb, obtains the data \ from the appropriate file as specified by the user and calls ", FontFamily->"Geneva", FontSize->10], StyleBox["splt3d", FontFamily->"Courier", FontSize->10], StyleBox[" to carry out the plot. The function ", FontFamily->"Geneva", FontSize->10], StyleBox["splt3d[d_,ttl_]", FontFamily->"Courier", FontSize->10], StyleBox[" takes as arguments the data to be plotted (", FontFamily->"Geneva", FontSize->10], StyleBox["d_", FontFamily->"Courier", FontSize->10], StyleBox[") and a title (", FontFamily->"Geneva", FontSize->10], StyleBox["ttl_", FontFamily->"Courier", FontSize->10], StyleBox[ "), and allows the user to specify interactively which hidden cells are to \ be used, whether the start or end point should be marked for each \ exemplar-epoch sequence plotted, whether the plot points should be joined, \ the point size (if plot points are not joined), and whether single plots in \ addition to the combined plot of each exemplar-epoch sequence is to be \ displayed. The numbering of each exemplar-epoch sequence is made by a \ modification to the graphics object (", FontFamily->"Geneva", FontSize->10], StyleBox["pobj", FontFamily->"Courier", FontSize->10], StyleBox[") created by ", FontFamily->"Geneva", FontSize->10], StyleBox["ScatterPlot3D", FontFamily->"Courier", FontSize->10], StyleBox[" which is available in the Package ", FontFamily->"Geneva", FontSize->10], StyleBox["Graphics`Graphics3D`", FontFamily->"Courier", FontSize->10], StyleBox[". 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Each dot \ represents an epoch, ", FontFamily->"Geneva", FontSize->10], StyleBox[ "with the last dot replaced by the number of the exemplar to which the \ sequence belongs. ", FontFamily->"Geneva", FontSize->10], StyleBox[ "This 3D plot provides four items of information, i.e., three hidden \ weights and epoch. ", FontFamily->"Geneva", FontSize->10], StyleBox[ "The four exemplars of T are identified at the end of each individual plot \ by the numbers 1 ", FontFamily->"Geneva", FontSize->10], StyleBox[ "to 4 , and the exemplars of C by 5 to 8. After about 20 epochs the paths \ of the T ", FontFamily->"Geneva", FontSize->10], StyleBox[ "exemplars move in a direction opposite to those of C, while the C \ exemplars are ", FontFamily->"Geneva", FontSize->10], StyleBox[ "approaching near convergence. Hidden cells 1 and 3 provide the most marked \ ", FontFamily->"Geneva", FontSize->10], StyleBox[ "discrimination among the exemplars. With such a plot the migration of the \ hidden cell ", FontFamily->"Geneva", FontSize->10], StyleBox[ "responses to each exemplar can be observed in terms of their paths and \ rate of ", FontFamily->"Geneva", FontSize->10], StyleBox[ "convergence over the epoch sequence. By changing the network\ \[CloseCurlyQuote]s user options, e.g., the ", FontFamily->"Geneva", FontSize->10], StyleBox[ "learning rate parameter at the end of a set of epochs, one can observe the \ effect on the ", FontFamily->"Geneva", FontSize->10], StyleBox[ "migration paths of each exemplar providing an effective method for \ evaluating ", FontFamily->"Geneva", FontSize->10], StyleBox["modifications to the BPNN algorithm. ", FontFamily->"Geneva", FontSize->10] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Plotting the Hidden Cell Weights", FontFamily->"Geneva", FontSize->10, FontWeight->"Bold"]], "Subsection"], Cell[TextData[{ StyleBox[ "The data set of the hidden cell weights is of order (75,3,9) containing \ matrices of order ", FontFamily->"Geneva", FontSize->10], StyleBox[ "(3,9) for each of the 75 epochs. The (3,9) matrix holds the weights \ applied to each of ", FontFamily->"Geneva", FontSize->10], StyleBox[ "the 9 attributes by each of the 3 hidden cells. Using the calling \ function ", FontFamily->"Geneva", FontSize->10], StyleBox["hdwAnlz", FontFamily->"Courier", FontSize->10], StyleBox[" to ", FontFamily->"Geneva", FontSize->10], StyleBox["obtain the data file ", FontFamily->"Geneva", FontSize->10], StyleBox["hdwfile", FontFamily->"Courier", FontSize->10], StyleBox[" and to rearrange the levels of the hidden cell weights ", FontFamily->"Geneva", FontSize->10], StyleBox["matrix, the plotting function ", FontFamily->"Geneva", FontSize->10], StyleBox["splt3d ", FontFamily->"Courier", FontSize->10], StyleBox["is used to plot the weights assigned by the network ", FontFamily->"Geneva", FontSize->10], StyleBox["to each attribute. 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Thus, \ the network has identified the corner cells and assigned them considerably \ smaller weights than those for the center element. Finally, attributes 2, 4, \ 6, and 8 representing elements between the corner cells are identified but \ change little from their random starting weights indicating their relative \ unimportance to the classification process. ", FontFamily->"Geneva", FontSize->10], StyleBox["\n\n", FontSize->10], StyleBox[ " If the number of hidden cells is not equal to 3, then the 3-dimensional \ plot can be used only by selecting subsets of three hidden cells. 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StyleBox[ "weights for the corner elements, i.e., the plots in positions (1,1), \ (1,3), (3,1), and ", FontFamily->"Geneva", FontSize->10], StyleBox[ "(3,3), and between-corner elements as the simulation progresses. \ Clearly, the 3-", FontFamily->"Geneva", FontSize->10], StyleBox[ "dimensional and array plots provide much more insight into the behavior of \ the BPNN in ", FontFamily->"Geneva", FontSize->10], StyleBox[ "case of Freeman's example than that provided by the plot of the squared \ error. ", FontFamily->"Geneva", FontSize->10] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Discussion", FontFamily->"Geneva", FontSize->10, FontWeight->"Bold"]], "Subsection"], Cell[TextData[{ StyleBox["Using ", FontFamily->"Geneva", FontSize->10], StyleBox["Mathematica's", FontFamily->"Geneva", FontSize->10, FontSlant->"Italic"], StyleBox[" rich collection of graphical procedures can enhance one's ", FontFamily->"Geneva", FontSize->10], StyleBox[ "understanding of the dynamic and final state characteristics of the BPNN \ beyond that ", FontFamily->"Geneva", FontSize->10], StyleBox[ "available from a simple plot of the squared error function. Because the \ data produced by ", FontFamily->"Geneva", FontSize->10], StyleBox[ "the BPNN can be very extensive, graphic objects require some modification \ such as ", FontFamily->"Geneva", FontSize->10], StyleBox[ "removal of the plotting symbol. Because the network configuration used \ here had ", FontFamily->"Geneva", FontSize->10], StyleBox[ "relatively few cells some of the problems encountered when a very large \ number of cells ", FontFamily->"Geneva", FontSize->10], StyleBox[ "is used are not illustrated. For example, in problems which require the \ network to ", FontFamily->"Geneva", FontSize->10], StyleBox[ "identify characters through Morse code, or the diagnosis of cardiac \ conditions, 20 to 30 ", FontFamily->"Geneva", FontSize->10], StyleBox[ "hidden cells might be used. In such problems the user would require the \ option to select ", FontFamily->"Geneva", FontSize->10], StyleBox[ "the combination of three cells for a 3-dimensional plot. Furthermore, \ because the ", FontFamily->"Geneva", FontSize->10], StyleBox[ "attributes of the exemplars do not always lend themselves to a meaningful \ spatial ", FontFamily->"Geneva", FontSize->10], StyleBox[ "arrangement such as the 3x3 character matrix illustrated here, the display \ of plots in a ", FontFamily->"Geneva", FontSize->10], StyleBox[ "graphics array may not always be meaningful. A set of five Notebooks and \ a utility ", FontFamily->"Geneva", FontSize->10], StyleBox[ "Package containing the functions for the BPNN and plotting functions, \ which include ", FontFamily->"Geneva", FontSize->10], StyleBox[ "density plots and plots resulting from singular value decomposition, is in \ preparation ", FontFamily->"Geneva", FontSize->10], StyleBox["for MathSource.", FontFamily->"Geneva", FontSize->10] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Subsection"], Cell[TextData[{ "Elman, J.L., \"Learning and development in neural networks: the importance \ of starting small\", ", StyleBox["Cognition", FontSlant->"Italic"], ", ", StyleBox["48", FontWeight->"Bold"], ", (1993), 71-99. \n\nFreeman, J., A. ", StyleBox["Simulating Neural Networks with Mathematica", FontSlant->"Italic"], ", Addison-Wesley Publishing Co., Menlo Park (1994)\n\nWickham-Jones, T. ", StyleBox["Mathematica Graphic", FontSlant->"Italic"], "s, TELOS/Springer-Verlag, Santa Clara (1994).\n\nIstvan, S.N., Berkely, \ M.R.W., Dawson, D.A., Medler, D.P., Schopflocher, Hornsby, L. \"Density Plots \ of Hidden Value Unit Activations Reveal Interpretable Bands\", ", StyleBox["Connection Science", FontSlant->"Italic"], ", ", StyleBox["7", FontWeight->"Bold"], ",2, (1995).\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["About the Authors", "Subsection"], Cell["\<\ Steve Hunka is Mike Carbonaro is\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Electronic Subscriptions", "Subsection"], Cell[TextData[{ "Included in the distribution for each electronic subscription are the \ files ", StyleBox["BackProp.nb", "Input", FontFamily->"Courier"], " and ", StyleBox["BPNNAppendix.nb", "Input", FontFamily->"Courier"], " containing ", StyleBox["Mathematica", FontSlant->"Italic"], " code for the material described in this article." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"NeXT 3.0", ScreenRectangle->{{0, 1053}, {0, 832}}, WindowSize->{695, 699}, WindowMargins->{{Automatic, 15}, {Automatic, 19}}, Magnification->1.25 ] (*********************************************************************** Cached data follows. 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