So far, you have not used the data values at all. The weights w 1 , w 2 , w 3 , and so on, are found by requiring that, when the prediction is moved to a location with a measured value, the data value is predicted exactly.

## How radial basis functions work

This forms N equations with N unknowns and can be solved uniquely. Thus, the surface passes through the data values, making predictions exact. The radial basis function in this example is a special case of the multiquadric RBF. Geostatistical Analyst also allows you to use other RBFs such as completely regularized splines, thin-plate splines, splines with tension, and inverse multiquadric. Sometimes, the difference between these is not great, but you may have reason to choose one, or you can try several and use cross-validation to select one.

Each of the RBFs has a parameter that controls the smoothness of the surface. For all methods except inverse multiquadric, the higher the parameter value, the smoother the map; the opposite is true for inverse multiquadric. Feedback on this topic? Skip To Content. Back to Top. Available with Geostatistical Analyst license.

## How radial basis functions work—ArcGIS Pro | ArcGIS Desktop

The basic properties of radial basis functions can be illustrated with a simple mathematical map, the logistic map , which maps the unit interval onto itself. It can be used to generate a convenient prototype data stream. The logistic map can be used to explore function approximation , time series prediction , and control theory.

The map originated from the field of population dynamics and became the prototype for chaotic time series. The map, in the fully chaotic regime, is given by. This equation represents the underlying geometry of the chaotic time series generated by the logistic map.

Generation of the time series from this equation is the forward problem. The examples here illustrate the inverse problem ; identification of the underlying dynamics, or fundamental equation, of the logistic map from exemplars of the time series. The goal is to find an estimate. Since the input is a scalar rather than a vector , the input dimension is one. The training is performed with one pass through the training points.

### Mini-Degrees

The rms error is 0. Again, we choose the number of basis functions as five and the size of the training set to be exemplars generated by the chaotic time series. The rms error on a test set of exemplars is 0.

Normalization yields accuracy improvement. Typically accuracy with normalized basis functions increases even more over unnormalized functions as input dimensionality increases. Once the underlying geometry of the time series is estimated as in the previous examples, a prediction for the time series can be made by iteration:.

A comparison of the actual and estimated time series is displayed in the figure. The estimated times series starts out at time zero with an exact knowledge of x 0. It then uses the estimate of the dynamics to update the time series estimate for several time steps. Note that the estimate is accurate for only a few time steps.

This is a general characteristic of chaotic time series. This is a property of the sensitive dependence on initial conditions common to chaotic time series.

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A small initial error is amplified with time. Crossref Google Scholar [16] Regis R.

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Crossref Google Scholar [21] Vasu A. Google Scholar [36] Liu H. Crossref Google Scholar [38] Fasshauer G. Crossref Google Scholar [39] Acar E. Previous article.

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Figures References Related Details Cited By A radial basis function-based multi-fidelity surrogate model: exploring correlation between high-fidelity and low-fidelity models. Efficient global optimization with ensemble and selection of kernel functions for engineering design.

## Using Neural Networks for Regression: Radial Basis Function Networks

Comparative studies of error metrics in variable fidelity model uncertainty quantification. A survey of adaptive sampling for global metamodeling in support of simulation-based complex engineering design. An adaptive sampling approach for Kriging metamodeling by maximizing expected prediction error. Volume 54, Number 10 October