Quantized Urysohn FilterThis is a nontraditional HOME page of large research. Instead of common phrases about nonlinear models, data filtering, artificial intelligence and system identification, we prefer to show a concept, which lays in foundation of new form of artificial intelligence.Observing of deterministic system and having indexed inputs $x[i]$ and outputs $y[i]$ recorded or processed by an automatic system, we can identify the following model by a few computational operations: $$y(t) = \int_{0}^T U[x(ts),s]ds. $$Input $x$ depends on time or observation index $i$ for discrete case. It is an argument of a kernel $U[\cdot, \cdot]$, which needs to be identified by $x$ and $y$ sequences without prior assumptions of how this kernel may look like. We can do that by 10 lines of code.If the signals are registered by digital equipment, they are provided as arrays with common notations $x[i]$, $y[i]$. The kernel is also an array $U[i, j]$ and here is the code fragment that do the trick: for (int i = T  1; i < N; ++i) { double predicted = 0.0; for (int j = 0; j < T; ++j) { predicted += U[(int)((x[ij]  xmin) / deltaX), j]; } double error = (y[i]  predicted) / T * learning_rate; for (int j = 0; j < T; ++j) { U[(int)((x[ij]  xmin) / deltaX), j] += error; } }Obviously, $N$ is size of input, output arrays, $T$ second size of kernel, $deltaX$ is quantization step for $x$ and $learningrate$ is computational filter for inaccurate data (must be within (0,1] interval). The identified array $U[\cdot, \cdot]$ may be shown as a grid structure with the values in nodes. All that may sound like an elegant solution for a particular mathematical problem of identification of nonlinear dynamic system, but, actually, it can be used as elementary brick in a large complicated structure functioning like artificial intelligence. And identified operator is a small part of it similar to a single neuron in a network. The site developers dedicated this resource to provide an experimental and theoretical proof for the following conceptual statements.
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