FIGURE 6-11 Complete program for the nonlinear predictor in .NET

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FIGURE 6-11 Complete program for the nonlinear predictor
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Neural Networks with Memory
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Vector signal = signal{-1} | signal[1] = sTRUE Vector x = x{-1} | x[1] = signal[m] ------------Vector vv = tanh(WW1 * xx) | -note bias Vector y = WW2 * vv | -no limiter needed on output! -Vector error = sTRUE - y | -backpropagation Vector vvdelta = WW2% * error * (1 - vv^2) DELTA WW1 = WW1gain * vvdelta * xx DELTA WW2 = WW2gain * error * vv ----------------------------------------------------------ERRORx5 = 5 * error[1] - 05 * scale dispt y[1], ERRORx5, sTRUE
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FIGURE 6-11 (Continued)
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to produce the current predictor input s(t) = signal[m] The simulated predictor then tries to predict sTRUE(t) by minimizing the sample average of g = (y sTRUE)2 with the backpropagation algorithm of Section 6-12a Prediction results necessarily depend on the frequency content of the input signal To provide a fairly difficult prediction task, sTRUE = sTRUE(t) is the Mackay Glass chaotic time series [17] defined by17
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Sd(t) = sTRUE(t tau) (d/dt) sTRUE = a Sd/(1 + Sdc) b sTRUE
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Figure 6-12 shows the future signal value sTRUE, the predictor output y, and the prediction error sTRUE y during training and recall We used 20 training runs with a total of 250,000 training steps to learn prediction m = 20 steps ahead
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In Figure 6-11, this is programmed with
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tdelay Sd = DD, sTRUE, tau sTRUEdot = a * Sd/(1 + Sd^c) - b * sTRUE d/dt sTRUE = sTRUEdot
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where tdelay is a library time-delay routine that implements Sd(t) = sTRUE(t tau) by storing samples of its input sTRUE in an array DD declared in the experiment protocol; one sample for each DT step of the simple Euler integration routine (Section 1-7a) used here The example mglasslst in the book CD lets you experiment with the generator
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sTRUE, y
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error x 5 error x 5 1e+03 scale = 3 125e+03 15e+03 y[1],ERROR 5,sTRUE vs t 105e+04 scale = 3 108e+04 11e+04 y[1],ERROR 5,sTRUE vs t
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FIGURE 6-12 Time histories of the future signal sTRUE, the predictor output y, and the scaled predictor error 5(sTRUE y) during training and recall
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6-23 The Gamma Delay Line Layer J Principe and his associates [7] replaced the tapped-delay-line definition (6-45), or
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x[i] = x[i 1] (i = 2,3, , nx) x[1] = s(t)
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(6-48)
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with a cascade of simple difference equations
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x[i] = x[i] + mu(x[i 1] x[i]) (i = 2, 3, , nx) x[1] = s(t)
(6-49a)
where mu is a positive parameter Our compact vector notation models this gamma delay line with a single vector difference equation (Section 3-4)
Vectr delta x = mu * (x{ 1} x) | x[1] = s(t)
(6-49b)
Each tapped-delay-line neuron [Eq (6-46)] remembers just one past input value But each neuron output x[i] in the gamma delay line is affected by all past input values This extra information about the past history of s(t) may allow a reduction in the number nx of delay-line sections in the block diagram of Figure 6-9 compared to that needed with a simple tapped delay line Figure 6-13 displays the tap-value responses to the initial condition x[1] = 1 for nx = 8 and two different values of mu The memory effect decreases with time The maximum time interval analyzed by a delay-line-fed neural network is nx COMINT for a simple tapped delay line For a gamma delay line, the effective memory period (memory depth) still depends on nx but is mainly determined by the difference-equation parameter mu Suitable values
Pulsed-neuron Replication
mu = 0025
mu = 004
1 scale = 05
300 600 [1], [2], [3], [4], [5], [6], [7], [8] vs t
1 scale = 05
300 600 [1], [2], [3], [4], [5], [6], [7], [8] vs t
FIGURE 6-13 Response of the tap outputs of an 8-tap gamma delay line to the initial condition x[1] = 1 for mu = 0025 and mu = 004 Curves in the original display were in different colors; the small squares at the bottom are color keys
of this parameter are often found by trial and error; References [7] and [14] discuss automatic training The simplest static neural networks used with an input gamma delay line are again linear (weighted-sum) layers (6-46) or (6-47), which can be optimized with the LMS algorithm The tap activation functions in Figure 6-13 serve as a useful set of basis functions for regression, as in Eq (6-31) Reference [7] discusses more advanced networks and a number of applications
PULSED-NEURON REPLICATION 6-24 Pulsed-neuron Models Biological neurons propagate electrical signals, but their actual inputs and outputs are fluctuating release rates of chemical substances (transmitters) fed into synaptic clefts between neurons [17, 23] Neuron activations in the simplified neural networks discussed in Sections 6-1 to 6-23 model running averages of pulsed-neuron inputs and outputs In the receiving neuron, a transmitter substance reacts with receptor chemicals to change the neuron-membrane permeability to ions passing into and out of the neuron Multiple excitatory and/or inhibitory inputs roughly add with different individual gains and fire the neuron when their weighted sum exceeds a threshold value Firing or ion transition through the neuron membrane produces a positive 20 to 300-mV voltage pulse across the membrane at a specific location This pulse propagates down a neuron fiber (axon)