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2. Time series prediction."

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Table of contents

Key takeawaysCollaboration platforms are essential to the new way of workingEmployees prefer engati over emailEmployees play a growing part in software purchasing decisionsThe future of work is collaborativeMethodologyIn the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. They’re implemented to resolve domain problems that have a partial sample or a training data set that is too small.

In computational applications, functions of many variables often need to be approximated by other functions that are better understood or more readily evaluated. This may be for the purpose of displaying them frequently on a computer screen, for instance, so computer graphics is a field of practical use.

Radial basis functions are used to approximate functions, much as neural networks act as function approximators. The following sum:

represents a radial basis function network. The radial basis functions act as activation functions.

Radial basis functions are an efficient, frequently used way to do this. They also have additional uses, including function approximation, time series prediction, classification, and system control.

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

where t is a time index. The value of x at time t+1 is a parabolic function of x at time t. This equation represents the underlying geometry of the chaotic time series generated by the logistic map.

The 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 for f.

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. A small initial error is amplified with time. A measure of the divergence of time series with nearly identical initial conditions is known as the Lyapunov exponent.

Radial basis functions (RBFs) are a series of exact interpolation techniques; that is, the surface must pass through each measured sample value. There are five different basis functions:

- Thin-plate spline
- Spline with tension
- Completely regularized spline
- Multiquadric function
- Inverse multiquadric function

Each basis function has a different shape and results in a different interpolation surface. RBF methods are a special case of splines.

RBFs are conceptually similar to fitting a rubber membrane through the measured sample values while minimizing the total curvature of the surface. The basis function you select determines how the rubber membrane will fit between the values. The following diagram illustrates conceptually how an RBF surface fits through a series of elevation sample values. Notice in the cross section, the surface passes through the data values.

Being exact interpolators, the RBF methods differ from the global and local polynomial interpolators, which are both inexact interpolators that do not require the surface to pass through the measured points. When comparing an RBF to IDW (which is also an exact interpolator), IDW will never predict values above the maximum measured value or below the minimum measured value. However, the RBFs can predict values above the maximum and below the minimum measured values.

Radial basis function networks were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment.

RBFs are used to produce smooth surfaces from a large number of data points. The functions produce good results for gently varying surfaces such as elevation.

However, the techniques are inappropriate when large changes in the surface values occur within short distances and/or when you suspect the sample data is prone to measurement error or uncertainty.

Applications are manifold, they include

- Finite element or spectral methods for the solution of partial differential equations
- Neural networks with radial basis functions, and machine learning
- Approximations on spheres
- Statistical approximations, where positive definite kernels are very important,
- Geophysical research
- and many engineering applications

Further applications include the important fields of neural networks and learning theory. Since they are radially symmetric functions that are shifted by points in multidimensional Euclidean space and then linearly combined, they form data-dependent approximation spaces. This data-dependence makes the spaces so formed suitable for providing approximations to large classes of given functions. It also opens the door to existence and unique results for interpolating scattered data by radial basis functions in very general settings (in particular in many dimensions).

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