Elastic map

of breast cancer microarray data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and can not be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The “basal” breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the elastic maps algorithm. Data are available for public competition. Software is available for free non-commercial use. Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. 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.…

of breast cancer microarray data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and can not be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The “basal” breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the elastic maps algorithm. Data are available for public competition. Software is available for free non-commercial use. ]] Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. 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. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998. Let {\mathcal S} be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes {\bf w}_j in the same space. Each datapoint s \in {\mathcal S} has a host node, namely the closest node {\bf w}_j (if there are several closest nodes then one takes the node with the smallest number). The data set {\mathcal S} is divided into classes K_j=\{s \ | \ {\bf w}_j \mbox{ is a host of } s\} . The approximation energy D is the distortion D=\frac{1}{2}\sum_{j=1}^k \sum_{s \in K_j}\|s-{\bf w}_j\|^2 , which is the energy of the springs with unit elasticity which connect each data point with its host node. 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 \{s_i\} . On the set of nodes an additional structure is defined. Some pairs of nodes, ({\bf w}_i,{\bf w}_j) , are connected by elastic edges. Call this set of pairs E . Some triplets of nodes, ({\bf w}_i,{\bf w}_j,{\bf w}_k) , form bending ribs. Call this set of triplets G . The stretching energy is U_{E}=\frac{1}{2}\lambda \sum_{({\bf w}_i,{\bf w}_j) \in E} \|{\bf w}_i -{\bf w}_j\|^2 , The bending energy is U_G=\frac{1}{2}\mu \sum_{({\bf w}_i,{\bf w}_j,{\bf w}_k) \in G} \|{\bf w}_i -2{\bf w}_j+{\bf w}_k\|^2 , where \lambda and \mu are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. The total energy of the elastic map is thus U=D+U_E+U_G. The position of the nodes \{{\bf w}_j\} is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy U .

References

Text derived from https://en.wikipedia.org/w/index.php?oldid=1295567548 under CC BY-SA 4.0; modifications have been made.