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The rst developments of MDS were metric [360,227]; the values used had to be quantitative, complete, and symmetric. The inability of this class of MDS to handle asymmetric and incomplete data proves too restrictive for most real datasets. This led to the development of nonmetric MDS to allow for these possibilities and additionally enabling the use of ordinal data [188,326]. Other extensions to the approach include replicated MDS (RMDS) [404] where the simultaneous analysis of several matrices of similarity data is allowed, and weighted MDS (WMDS) [404]. WMDS generalizes the distance model so that several similarity matrices can be assumed to differ from each other in systematically nonlinear or nonmonotonic ways. Whereas RMDS only accounts for individual differences in the ways subjects use the response scale (in psychological terms), WMDS incorporates a model to account for individual differences in the fundamental perceptual or cognitive processes that generate the responses. For this reason WMDS is often called individual differences scaling.
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4.1.2 Nonlinear Methods
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As useful as the previous methods are for reducing dimensionality, their utility fails for data exhibiting nonlinearity. Given a dataset containing nonlinear relationships, these methods detect only the Euclidean structure. This brought about the need for methods that can effectively handle nonlinearity. The rst attempts were extensions to the original PCA process, either by clustering data initially and performing PCA within clusters [43] or by greedy optimization processes [187]. Both suffer from problems brought on as a result of simply attempting to extend linear PCA [111]. This motivates the development of techniques designed to suitably and successfully handle nonlinearity. Isomap [358] is an extension of MDS in which embeddings are optimized to preserve geodesic distances between pairs of data points, estimated by calculating the shortest paths through large sublattices of data. The algorithm can discover nonlinear degrees of freedom as these geodesic distances represent the true low-dimensional geometry of the manifold. The success of Isomap depends on being able to choose a neighborhood size (either or K) that is neither so large that it introduces short-circuit edges into the neighborhood graph nor so small that the graph becomes too sparse to approximate geodesic paths accurately. Short-circuit edges occur where there are links between data points that are not near each other geodesically and can lead to low-dimensional embeddings that do not preserve a manifold s true topology. Locally linear embedding (LLE) [293] is an eigenvector method for the problem of nonlinear DR. It calculates low-dimensional neighborhood-preserving reconstructions (embeddings) of data of high dimensionality. LLE achieves this by exploiting the local symmetries of linear reconstructions. To conceptualize this, consider the following informal analogy. The initial data is three-dimensional and forms the topology of a two-dimensional rectangular manifold bent into a
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three-dimensional S-curve. Scissors are then used to cut this manifold into small squares representing locally linear patches of the nonlinear surface. These squares can then be arranged onto a at tabletop while angular relationships between neighboring squares are maintained. This linear mapping is due to the fact that all transformations involve translation, scaling, or rotation only. This way the algorithm identi es the nonlinear structure through a series of linear steps. LLE avoids the need to solve large dynamic programming problems. It also tends to accumulate very sparse matrices whose structure can be exploited to save time and space. However, in [293] there is no indication as to how a test data point may be mapped from the input space to the manifold space, or how a data point may be reconstructed from its low-dimensional representation. Additionally, LLE suffers from the problem of short-circuit edges as described previously for Isomap. Multivariate adaptive regression splines (MARS) [105] is an implementation of techniques for solving regression-type problems, with the main purpose of predicting the values of a continuous decision feature from a set of conditional features. It is a nonparametric regression procedure that makes no assumptions about the underlying functional relationships. MARS instead constructs this relation from a set of coef cients and basis functions, de ned by control points, that are selected based on the data only. However, this is a relatively complex process, and if suffers from the curse of dimensionality. Each dimension of the hyperplane requires one dimension for the approximation model, and an increase in the time and space required to compute and store the splines. The time required to perform predictions increases exponentially with the number of dimensions. Noise can also distort the model by causing MARS to generate a much more complex model as it tries to incorporate the noisy data into its approximation. Also worth mentioning are methods based on arti cial neural networks (ANNs). ANNs are mathematical models that are inspired by the connections and the functioning of neurons in biological systems, based on the topology of nodes and connections between them, and transfer functions that relate the input and output of each node. ANNs are often used as a way of optimizing a classi cation (or pattern recognition) procedure. They also usually have more input than output nodes; they may thus also be viewed as performing a dimensionality reduction on input data, in a way more general than principal component analysis and multidimensional scaling [36,187,192].
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