The idea of representation has been used in various fields of study from data analysis to political science. In this paper, we define representativeness and describe a method to isolate data points that can represent the entire data set. Also, we show how the minimum set of representative data points can be generated. We use data from GLOBE (a project to study the effects on Land Change based on a set of parameters that include temperature, forest cover, human population, atmospheric parameters and many other variables) to test & validate the algorithm. Principal Component Analysis (PCA) is used to reduce the dimensions of the multivariate data set, so that the representative points can be generated efficiently and its Representativeness has been compared against Random Sampling of points from the data set.
A dimension reduction technique is a transformation which reduces number of dimensions required to represent a sample. The reduced set of dimensions may be a subset of the original set of dimensions (for example, using information gain) or could be a completely new set of dimensions. Some of standard dimension reduction techniques that can be used to transform a high dimensional data set are Principal Component Analysis (PCA), and Self Organizing Maps (SOM) @cite_11 . Neural Networks with GIS have also been used for constructing a which tries to forecast how usage changes @cite_5 . Self Organizing Maps have been used to perform environmental assessment of regions, grouping based on environmental conditions, and finding out which areas might deteriorate in the future @cite_0 . Once a SOM is trained, the nodes from the weight vector can be used as centroids representing their respective clusters. As the nodes may not be actual data points, the point closest to each node will be used as a representative point. Training a SOM may require updating the weight vector over several iterations of the data set. The time complexity of a SOM is @math @cite_12 .
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Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a sparse version of the fundamental tool in dimension reduction -- the Johnson-Lindenstrauss transform. Using hashing and local densification, we construct a sparse projection matrix with just O(1 e) non-zero entries per column. We also show a matching lower bound on the sparsity for a large class of projection matrices. Our bounds are somewhat surprising, given the known lower bounds of Ω(1 e2) both on the number of rows of any projection matrix and on the sparsity of projection matrices generated by natural constructions. Using this, we achieve an O(1 e) update time per non-zero element for a (1 e)-approximate projection, thereby substantially outperforming the O(1 e2) update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its O(d) worst case running time matches the best approach of Ailon and Liberty. The problem of dimension reduction is introduced as a way to overcome the curse of the dimensionality when dealing with vector data in high-dimensional spaces and as a modelling tool for such data. It is defined as the search for a low-dimensional manifold that embeds the high-dimensional data. A classification of dimension reduction problems is proposed. A survey of several techniques for dimension reduction is given, including principal component analysis, projection pursuit and projection pursuit regression, principal curves and methods based on topologically continuous maps, such as Kohonen’s maps or the generalised topographic mapping. Neural network implementations for several of these techniques are also reviewed, such as the projection pursuit learning network and the BCM neuron with an objective function. Several appendices complement the mathematical treatment of the main text. Dimensionality reduction techniques play an essential role in data analytics, signal processing, and machine learning. Dimensionality reduction is usually performed in a preprocessing stage that is separate from subsequent data analysis, such as clustering or classification. Finding reduced-dimension representations that are well-suited for the intended task is more appealing. This paper proposes a joint factor analysis and latent clustering framework, which aims at learning cluster-aware low-dimensional representations of matrix and tensor data. The proposed approach leverages matrix and tensor factorization models that produce essentially unique latent representations of the data to unravel latent cluster structure—which is otherwise obscured because of the freedom to apply an oblique transformation in latent space. At the same time, latent cluster structure is used as prior information to enhance the performance of factorization. Specific contributions include several custom-built problem formulations, corresponding algorithms, and discussion of associated convergence properties. Besides extensive simulations, real-world datasets such as Reuters document data and MNIST image data are also employed to showcase the effectiveness of the proposed approaches. Reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high-dimensional data analysis. Fisher discriminant analysis (FDA) is a traditional technique for supervised dimensionality reduction, but it tends to give undesired results if samples in a class are multimodal. An unsupervised dimensionality reduction method called locality-preserving projection (LPP) can work well with multimodal data due to its locality preserving property. However, since LPP does not take the label information into account, it is not necessarily useful in supervised learning scenarios. In this paper, we propose a new linear supervised dimensionality reduction method called local Fisher discriminant analysis (LFDA), which effectively combines the ideas of FDA and LPP. LFDA has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem. We demonstrate the practical usefulness and high scalability of the LFDA method in data visualization and classification tasks through extensive simulation studies. We also show that LFDA can be extended to non-linear dimensionality reduction scenarios by applying the kernel trick.