Self-organizing map
A self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher dimensional data set while preserving the topological structure of the data. For example, a data set with "p" variables measured in "n" observations could be represented as clusters of observations with similar values for the variables. These clusters then could be visualized as a two-dimensional "map" such that observations in proximal clusters have more similar values than observations in distal clusters. This can make high-dimensional  data easier to visualize and analyze. An SOM is a type of artificial neural network but is trained using competitive learning rather than the error-correction learning (e.g., backpropagation with gradient descent) used by other artificial neural networks. The SOM was introduced by the Finnish professor Teuvo Kohonen in the 1980s and therefore is sometimes called a Kohonen map or Kohonen network. The Kohonen map or network is a computationally convenient abstraction building on biological models of neural systems from the 1970s and morphogenesis models dating back to Alan Turing in the 1950s. Overview. Self-organizing maps, like most artificial  neural networks, operate in two modes: training and mapping. First, training uses an input data set (the "input space") to generate a lower-dimensional representation of the input data (the "map space"). Second, mapping classifies additional input data using the generated map. In most cases, the goal of training is to represent an input space with "p" dimensions as a map space with two dimensions. Specifically, an input space with "p" variables is said to have "p" dimensions. A map space consists of components called "nodes" or "neurons," which are arranged as a hexagonal or rectangular grid with two dimensions . The number of nodes and their arrangement are specified beforehand based on the larger goals of the analysis and exploration of the data. Each node in the map space is associated with a "weight" vector, which is the position of the node in the input space. While nodes in the map space stay fixed, training consists in moving weight vectors toward the input data (reducing a distance metric such as Euclidean distance) without spoiling the topology induced from the map space. After training, the map can be used to classify additional observations for the input space by finding the node  with the closest weight vector (smallest distance metric) to the input space vector. Learning algorithm. The goal of learning in the self-organizing map is to cause different parts of the network to respond similarly to certain input patterns. This is partly motivated by how visual, auditory or other sensory information is handled in separate parts of the cerebral cortex in the human brain. The weights of the neurons are initialized either to small random values or sampled evenly from the subspace spanned by the two largest principal component eigenvectors. With the latter alternative, learning is much faster because the  initial weights already give a good approximation of SOM weights. The network must be fed a large number of example vectors that represent, as close as possible, the kinds of vectors expected during mapping. The examples are usually administered several times as iterations. The training utilizes competitive learning. When a training example is fed to the network, its Euclidean distance to all weight vectors is computed. The neuron whose weight vector is most similar to the input is called the best matching unit (BMU). The weights of the BMU and neurons close to it in the SOM grid are adjusted  towards the input vector. The magnitude of the change decreases with time and with the grid-distance from the BMU. The update formula for a neuron v with weight vector W(s) is where s is the step index, t an index into the training sample, u is the index of the BMU for the input vector D(t), α(s) is a monotonically decreasing learning coefficient; Θ(u, v, s) is the neighborhood function which gives the distance between the neuron u and the neuron v in step s. Depending on the implementations, t can scan the training data set systematically (t is  0, 1, 2...T-1, then repeat, T being the training sample's size), be randomly drawn from the data set (bootstrap sampling), or implement some other sampling method (such as jackknifing). The neighborhood function Θ(u, v, s) (also called "function of lateral interaction") depends on the grid-distance between the BMU (neuron "u") and neuron "v". In the simplest form, it is 1 for all neurons close enough to BMU and 0 for others, but the Gaussian and mexican-hat functions are common choices, too. Regardless of the functional form, the neighborhood function shrinks with time. At the beginning when  the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of neurons, the weights are converging to local estimates. In some implementations, the learning coefficient α and the neighborhood function Θ decrease steadily with increasing s, in others (in particular those where t scans the training data set) they decrease in step-wise fashion, once every T steps. This process is repeated for each input vector for a (usually large) number of cycles λ. The network winds up associating output nodes with groups or patterns in the input  data set. If these patterns can be named, the names can be attached to the associated nodes in the trained net. During mapping, there will be one single "winning" neuron: the neuron whose weight vector lies closest to the input vector. This can be simply determined by calculating the Euclidean distance between input vector and weight vector. While representing input data as vectors has been emphasized in this article, any kind of object which can be represented digitally, which has an appropriate distance measure associated with it, and in which the necessary operations for training are possible can be used  to construct a self-organizing map. This includes matrices, continuous functions or even other self-organizing maps. Variables. These are the variables needed, with vectors in bold, Initialization options. Selection of initial weights as good approximations of the final weights is a well-known problem for all iterative methods of artificial neural networks, including self-organizing maps. Kohonen originally proposed random initiation of weights. (This approach is reflected by the algorithms described above.) More recently, principal component initialization, in which initial map weights are chosen from the space of the first principal components, has become popular due to the exact  reproducibility of the results. A careful comparison of random initialization to principal component initialization for a one-dimensional map, however, found that the advantages of principal component initialization are not universal. The best initialization method depends on the geometry of the specific dataset. Principal component initialization was preferable (for a one-dimensional map) when the principal curve approximating the dataset could be univalently and linearly projected on the first principal component (quasilinear sets). For nonlinear datasets, however, random initiation performed better. Interpretation. There are two ways to interpret a SOM. Because in the training phase weights of the whole neighborhood  are moved in the same direction, similar items tend to excite adjacent neurons. Therefore, SOM forms a semantic map where similar samples are mapped close together and dissimilar ones apart. This may be visualized by a U-Matrix (Euclidean distance between weight vectors of neighboring cells) of the SOM. The other way is to think of neuronal weights as pointers to the input space. They form a discrete approximation of the distribution of training samples. More neurons point to regions with high training sample concentration and fewer where the samples are scarce. SOM may be considered a nonlinear generalization of  Principal components analysis (PCA). It has been shown, using both artificial and real geophysical data, that SOM has many advantages over the conventional feature extraction methods such as Empirical Orthogonal Functions (EOF) or PCA. Originally, SOM was not formulated as a solution to an optimisation problem. Nevertheless, there have been several attempts to modify the definition of SOM and to formulate an optimisation problem which gives similar results. For example, Elastic maps use the mechanical metaphor of elasticity to approximate principal manifolds: the analogy is an elastic membrane and plate. Examples. Fisher's iris flower data. Consider an array of  nodes, each of which contains a weight vector and is aware of its location in the array. Each weight vector is of the same dimension as the node's input vector. The weights may initially be set to random values. Now we need input to feed the map. Colors can be represented by their red, green, and blue components. Consequently, we will represent colors as vectors in the unit cube of the free vector space over generated by the basis: The diagram shown compares the results of training on the data sets and the original images. Note the striking resemblance  between the two. Similarly, after training a grid of neurons for 250 iterations with a learning rate of 0.1 on Fisher's Iris, the map can already detect the main differences between species.