Elastic map

Elastic maps provide a tool for nonlinear dimensionality reduction.

By their construction, they are a system of elastic springs embedded in the data space.

The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules).

With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds.

This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates.

be a data set in a finite-dimensional Euclidean space.

Elastic map is represented by a set of nodes

It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points

On the set of nodes an additional structure is defined.

is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy

is a linear problem with the sparse matrix of coefficients.

Therefore, similar to principal component analysis or k-means, a splitting method is used: This expectation-maximization algorithm guarantees a local minimum of

For improving the approximation various additional methods are proposed.

coefficients) and finishes with soft grids (small

The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images.

[10] This reconstruction is used for quantifying the geodesic distances between trichomes and their patterning, which is a marker of the capability of a plant to resist to pathogenes.

Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of financial portfolios.

[11] The method of elastic maps has been systematically tested and compared with several machine learning methods on the applied problem of identification of the flow regime of a gas-liquid flow in a pipe.

The simplest and most common method used to identify the flow regime is visual observation.

This approach is, however, subjective and unsuitable for relatively high gas and liquid flow rates.

Therefore, the machine learning methods are proposed by many authors.

The methods are applied to differential pressure data collected during a calibration process.

The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure.

Here, ANN stands for the backpropagation artificial neural networks, SVM stands for the support vector machine, SOM for the self-organizing maps.

The hybrid technology was developed for engineering applications.

[13] In this technology, elastic maps are used in combination with Principal Component Analysis (PCA), Independent Component Analysis (ICA) and backpropagation ANN.

The textbook[14] provides a systematic comparison of elastic maps and self-organizing maps (SOMs) in applications to economic and financial decision-making.