P. Demartines and J. Hrault. Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transaction on Neural Networks, 8(1):148--154, January 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....faithfully as possible. Thus, the high dimensional data set can be visualized while still preserving its essential topological properties. Examples of such nonlinear projection methods include multidimensional scaling techniques [39] 40] Sammon s mapping [41] and curvilinear component analysis [42]. A special technique is to project the prototype vectors into a color space so that similar map units are assigned similar colors [43] 44] Of course, based on the visualization, one can select clusters manually. However, this is a tedious process and nothing guarantees that the manual ....

P. Demartines and J. Hrault, "Curvilinear component analysis: A selforganizing neural network for nonlinear mapping of data sets," IEEE Trans. Neural Networks, vol. 8, pp. 148--154, Jan. 1997.

....dimensions. This makes it possible to evaluate how much individual documents contribute in terms of the total stress. To overcome the discretization of SOMs, a relatively new algorithm for performing topology preserving nonlinear dimension reduction, Curvilinear Components Analysis (CCA) [8, 6], could be explored in such situations where there are large number of text documents. Additional dimensionality reduction techniques found in information retrieval and text classification, such as LSI [10, 9] could be incorporated into future studies. 6 Acknowledgments We gratefully ....

P. Demartines and J. Herault. Curvilinear component analysis: A selforganizing neural network for nonlinear mapping of data sets. IEEE Transactions on Neural Networks, 8(1):148--154, January 1997.

....in more detail [36] The highdimensional analysis was pointed out in [11] using phase diagrams. To overcome the topological mismatch problem growing variants of SOM have been developed [12] or input pruning was tried [21] Structure adaptation is closely related to the connection of SOM and PCA [35] and its non linear extension of principal curves [63] Ritter has shown that (in case of topology preserving mapping) SOM can be taken as an approximation of principal curves [109] Probability density matching and magnification: As mentioned above, the SOM is not an optimal vector quantizer in ....

P. Demartines and J. H6rault. Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Trans. on Neural Networks, 8(1):148 154, January 1997.

....non linear autoregressive matrix that could be used for further prediction. 599 Many non linear projection methods exist. Kohonen s self organizing map is probably the most widely known example. Yet in our experiments we will use another method, the Curvilign Component Analysis (CCA) [3]; unlike the Kohonen maps, this method doe not make any assumption on the shape of the projection space, and was found to give better results in our application. 2.3 Non linear forecasting After this projection, we obtain the required non linear autoregressive matrix. Its rows will be used as ....

Demartines P., Hrault J.: Curvilinear Component Analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Trans. on Neural Networks 8(1) (1997) 148-154

....di#er in the way they use the measured distances. In Isomap, the distance reproduction from the d dimensional space to the p dimensional space is achieved algebraically by the traditional MDS, while CDA works with neural methods. Actually, CDA derives from the Curvilinear Component Analysis (CCA, [3, 7]) and from Sammon s Nonlinear Mapping (NLM, 12] Those two techniques act by preserving distances, like the MDS, but they proceed with an energy function which is minimized by gradient descent (NLM) or by stochastic gradient descent (CCA) Formally, the error function of CDA is written as: ....

P. Demartines and J. Herault. Curvilinear Component Analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transaction on Neural Networks, 8(1):148--154, January 1997.

....With this single parameter, PCA automatically determines the dimension p of the space where to project and gives the best LMS projection of the database. It would be nice to hve such a user friendly projection method, but with nonlinear capabilities. T o reach this goal, the CCA algorithm [1, 2, 3], described below, is a promising approach. This work w as realized with the support of the Minisfere de la Rgion wallonne , under the Programme de Formation et d Impulsion la Rec herche Scien tifique et T echnologique iN.D. is working to wards the Ph.D. degree under FRIA fellship. M.V. ....

....reproducing global topology only when it is possible. A t first glance, CCA looks lile Sammon s [4] mapping or nonlinear MDS [5] How ever, some differences in the objecti function makes CCA more powerful in real cases. More details and a comparison between the three methods can be found in [1]. 3 How to implement CCA CCA is implemented b y a modified stohastic gradient descent applied to the objective function E,CA: Vi j, Ax din j diP J ( n,x(d,j) ii,j) x x) 2) where , t) learning factor) and h(t) neighborhood factor) are time decreasing parameters, with values betw ....

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P. Demartines, J. trault, Curvilinear Component Analysis: A self-or ganizing neural network for nonlinear mapping of data sets. IEEE Transaction on Neural Networks 8: (1) 148-154, Jan uary 1997.

....Fig. 1. The two steps of the methodology. To perform the transformation between the initial inputs and the new variables, we may choose to use a linear method or a non linear one. In this paper, we will use Principal Component Analysis as a linear transformation and Curvilinear Component Analysis [5] as a non linear one. 2. Dimension reduction 2.1. Intrinsic Dimension First, it is important to evaluate the projection dimension, i.e. the dimension of the space of new variables. If the estimation of this dimension is too small, information will be lost in the projection. If it is too large, ....

....live in R . A possible way of computing this intrinsic dimension is explained in [7] but its determination remains very difficult to apply, not to say approximate, for high dimensional data sets. 2.2. Curvilinear Component Analysis This nonlinear extension of the Principal Component Analysis [5] spreads out the manifold that contains the data and projects it from a high dimensional space to a smaller dimensional one. The projection of the horseshoe distribution carried out by Curvilinear Component Analysis (CCA) is illustrated in Fig. 3. Preprocessing Prediction Algorithm Dimension ....

Demartines P., Hrault J., "Curvilinear Component Analysis: A selforganizing neural network for nonlinear mapping of data sets", IEEE Trans. on Neural Networks, 8(1), pp. 148-154, 1997.

....spaces. Our opinion is that the use of these methods is often preferred because of the traditional limitation of Kohonen maps to 2dimensional grids, rather than the distance preservation property. Multi dimensional scaling [12 13] Sammon s mapping [14] and Curvilinear Component Analysis [15] are methods based on the same principle: if we have n data points in a d dimensional space, they try to place n points in the m dimensional projection space, keeping the mutual distances between any pair of points unchanged between the input space and the corresponding pair in the projection ....

....similar to MDS. However, the objective function is now a mean square error of the differences between distances (between pairs of samples) in the input and output spaces. Contrary to MDS, weighting is done with respect to input distances in Sammon s mapping. Curvilinear Component Analysis (CCA) [15] also measures the mean square error between distances. But contrary to Sammon s mapping, weighting is done with respect to distances in the projection space. Dematines claims that this modification enhances the quality of the projection in many situations; our own experience with these ....

P. Demartines, J. Hrault, Curvilinear Component Analysis: a self-organizing neural network for nonlinear mapping of data sets, IEEE Trans. on Neural Networks,. 8-1 (1997) 148-154.

....created to address this issue. For example, the nonmetric Multidimensional Scaling (MDS, 12] and Sammon s nonlinear mapping (NLM [11] are based on the preservation of either pairwise dissimilarities or Euclidean distances. Neural versions of the NLM, like Curvilinear Component Analysis (CCA, [3, 4]) generally show better performance, particularly when they do not use the traditional Euclidean metrics [9, 13] Finally, nonlinear projection can be achieved by the Self Organizing Map (SOM, 8, 14, 10] that works with true topology preservation rather than the more constraining distance ....

P. Demartines and J. Herault. Curvilinear Component Analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transaction on Neural Networks, 8(1):148--154, January 1997.

....encoder decoder may be found in [6] 4 However, while some convergence results do not exist, so far, the SOM has shown to give excellent results in many practical applications. Chapter 4. Curvilinear Components Analysis 83 Curvilinear Components Analysis. This algorithm, proposed by Demartines [4], basically combines vector quantization and projection to perform a continuous mapping of some data set while preserving topology as best as possible. The CCA is still at an elementary stage of development, but seems to be promizing for our concern. The remaining of this chapter will therefore ....

....of development, but seems to be promizing for our concern. The remaining of this chapter will therefore be devoted to a presentation of this interesting algorithm. 4. 3 The CCA algorithm This section presents the basic principles underlying Demartines Curvilinear Components Analysis algorithm [4]. We then expose the (straightforward) extension of the dy dx representation to CCA data structures. The method is finally tested on some classical data sets, and thereby evaluated. 4.3.1 CCA learning scheme CCA was formerly referred to as Vector Quantization and Projection (VQP) 5] after the ....

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P. Demartines, J. H'erault, Curvilinear Component Analysis: a SelfOrganizing Neural Network for Nonlinear Mapping of Data sets, In: IEEE Transactions on Neural Networks, 8(1), (IEEE, New York, USA, 1997) p. 148--54.

....not effect the old labels of sTo: they are left as they were. label map based on the data using add mode sM = somautolabel(sM,sD) 37 label data based on map sD = somautolabel(sD,sM) label map using the 5th column of labels in the data and vote mode sM = somautolabel(sM,sD, vote ,[5]) 4.2.7 Visualization functions The visualization functions in the Toolbox can basically be divided to three groups: ffl The som show family (som show, som show add, som show clear and som recolorbar and some functions in the contributed code) which are high level tools for making cellstyle ....

....the same coordinate system for the positions of map units (unless the coordinates are explicitly set as is possible in the case of generic functions) The coordinate system insures that if two visualizations of the map of same size are drawn on the same figure they will match. sombarplane( hexa ,[10 5],rand(50,4) hold on; somcplane( hexa , 10 5] none ) somgrid( hexa , 10 5] For example, som grid can be used for plotting labels on top a visualization made using som cplane. The coordinates are specified by function som vis coords. Most visualizations have been implemented so that after ....

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P. Demartines and J. H'erault. Curvilinear Component Analysis: A Self-Organizing Neural Network for Nonlinear Mapping of Data Sets. IEEE Transactions on Neural Networks, 8(1):148--154, January 1997.

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P. Demartines and J. Hrault. Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transaction on Neural Networks, 8(1):148--154, January 1997.

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Pierre Demartines and Jeanny Herault, "Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets," IEEE Transactions on Neural Networks, vol. 8, no. 1, pp. 148--154, January 1997.

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P. Demartines, and J. Herault, "Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets", IEEE Transactions on Neural Networks, 8 (1), 1997, pp. 148-154.

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P. Demartines and J. Herault, Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets, IEEE Transactions on Neural Networks 8 (1997) 148--154.

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Demartines, P. and Hrault, J. (1997). Curvilinear Component Analysis: a Self-Organizing Neural Network for Nonlinear Mapping of Data Sets. IEEE Transactions on Neural Networks, 8:148154.

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P. Demartines and J. Herault. Curvilinear Component Analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transactions on Neural Networks, 8(1):148--154, January 1997.

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P. Demartines and J. Herault, "Curvilinear Component Analysis: A SelfOrganizing Neural Network for Nonlinear Mapping of Data Sets," IEEE Trans. on Neural Networks, vol. 8, no. 1, pp. 148--154, January 1997.

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Demartines P, He rault J. Curvilinear component analysis: a selforganizing neural network for nonlinear mapping of data sets. IEEE Transaction on Neural Networks 1997; 8: 148--154.

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P. Demartines and J. Herault. Curvilinear Component Analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transaction on Neural Networks, 8(1):148--154, January 1997.

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Demartines, P., Hrault, J.: Curvilinear Component Analysis: a self-organizing neural network for nonlinear mapping of data sets, IEEE T. Neural Networks,. 8-1 (1997) 148-154

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P. Demartines and J. Herault, \Curvilinear component analysis : A self-organizing neural network for nonlinear mapping of data sets," IEEE Tr. Neur. Net. 8(1), pp. 148-154, 1997.

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P. Demartines, J. Hrault (1997) Curvilinear Component Analysis: A self-organizing neural network for nonlinear mapping of data sets, IEEE Trans. on Neural Networks, 8(1), pp. 148-154.

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Demartines, P., Herault, J., "Curvilinear Component Analysis: A self-Organizing Neural Network for Nonlinear Mapping of data Sets", IEEE Trans. on Neural Networks, vol. 8, no. 1, January 1997

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P. Demartines and J. H'eerault, Curvilinear Component Analysis: A Self-Organizing Neural Network for Nonlinear Mapping of Data Sets, in IEEE Transactions on Neural Networks (1997) 8(1):148-154.

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