非线性PCA

Nonlinear PCA Nonlinear PCA toolbox for MATLAB by Matthias Scholz

Auto-associative neural network (Autoencoder)

Nonlinear principal component analysis (NLPCA) is commonly seen as a nonlinear generalization of standard principal component analysis (PCA). It generalizes the principal components from straight lines to curves (nonlinear). Thus, the subspace in the original data space which is described by all nonlinear components is also curved.
Nonlinear PCA can be achieved by using a neural network with an autoassociative architecture also known as autoencoder, replicator network, bottleneck or sandglass type network. Such autoassociative neural network is a multi-layer perceptron that performs an identity mapping, meaning that the output of the network is required to be identical to the input. However, in the middle of the network is a layer that works as a bottleneck in which a reduction of the dimension of the data is enforced. This bottleneck-layer provides the desired component values (scores).

linear PCA	The left plot shows standard PCA applied to a simple two-dimensional data set. The two resulting components are plotted as a grid which illustrates the linear PCA transformation. The plot on the right shows nonlinear PCA (autoencoder neural network) applied to a 3/4 circle with noise. Again, the two components are plotted as a grid, but the components are curved which illustrates the nonlinear transformation of NLPCA.	nonlinear PCA

Here, NLPCA is applied to 19-dimensional spectral data representing equivalent widths of 19 absorption lines of 487 stars, available atwww.cida.ve. The figure in the middle shows a visualisation of the data by using the first three components of standard PCA. Data of different colors belong to different spectral groups of stars. The first three components of linear PCA and of NLPCA are represented by grids in the left and right figure, respectively. Each grid represents the two-dimensional subspace given by two components while the third one is set to zero. Thus, the grids represent the new coordinate system of the transformation. In contrast to linear PCA (left) which does not describe the nonlinear characteristics of the data, NLPCA gives a nonlinear (curved) description of the data, shown on the right.

Publications by Matthias Scholz

• Nonlinear principal component analysis: neural network models and applications.
  Matthias Scholz, Martin Fraunholz, and Joachim Selbig.
  In Principal Manifolds for Data Visualization and Dimension Reduction, edited by Alexander N. Gorban, Balázs Kégl, Donald C. Wunsch, and Andrei Zinovyev. Volume 58 of LNCSE, pages 44-67. Springer Berlin Heidelberg, 2007.
  [ pdf (all book chapters) | pdf (Springer) | entire book (Springer) ]
  
  
• Analysing periodic phenomena by circular PCA.
  Matthias Scholz.
  In S. Hochreiter and R. Wagner, editors, Proceedings of the Conference on Bioinformatics Research and Development BIRD'07, LNCS/LNBI Vol. 4414, pages 38-47. Springer-Verlag Berlin Heidelberg, 2007.
  [ pdf (Springer) | pdf (author version) | bibtex ]
  
  
• Approaches to analyse and interpret biological profile data.
  Matthias Scholz.
  University of Potsdam, Germany. Ph.D. thesis. 2006.
  URN: urn:nbn:de:kobv:517-opus-7839
  URL: http://opus.kobv.de/ubp/volltexte/2006/783/
  [ pdf (library) | pdf (copy) | figures ]
  
  
• Non-linear PCA: a missing data approach.
  Matthias Scholz, Fatma Kaplan, Charles L. Guy, Joachim Kopka, and Joachim Selbig.
  Bioinformatics 21(20):3887-3895. 2005.
  [ pdf |Advance Access manuscript ]
  
  
• Nonlinear PCA based on neural networks.
  Matthias Scholz.
  Dep. of Computer Science, Humboldt-University Berlin. Diploma Thesis. 2002. In German.
  URN: urn:nbn:de:kobv:11-10086728
  [ pdf (library) |pdf (pre-print version)
  
  
• Nonlinear PCA: a new hierarchical approach.
  Matthias Scholz and Ricardo Vigário.
  In M. Verleysen, editor, Proceedings ESANN. 2002.
  [ pdf (pre-print version) |pdf (ESANN) ]

see all publications: [Matthias Scholz: publications]

Related algorithms

• LLE - Locally Linear Embedding (Sam T. Roweis and Lawrence K. Saul, 2000)
• Isomap (Josh Tenenbaum et al., 2000)
• SOM - Self-Organizing Map (Teuvo Kohonen, 1982)
• Principal Curves (Trevor Hastie and Werner Stuetzle, 1989),Matlab code by Jakob J. Verbeek
• Kernel PCA (Schölkopf et al., 1998)
• NFA - Nonlinear Factor Analysis (Harri Valpola and Antti Honkela, 2000)

See also:

• Nonlinear PCA toolbox for MATLAB
• Principal Component Analysis (PCA)
• Independent Component Analysis (ICA)
• Complex Networks and Systems

来源：

Nonlinear PCA: www.nlpca.org 代码下载：  Nonlinear PCA toolbox for MATLAB by Matthias Scholz  nlpca Nonlinear principal component analysis (NLPCA) Syntax [pc, net] = nlpca(data, k) pc = nlpca_get_components(net, data) data_reconstruction = nlpca_get_data(net, pc) Description The nonlinear PCA is based on an auto-associative neural network (autoencoder), see also: www.nlpca.org . pc = nlpca(data,k) extracts k nonlinear components from the data set. pc represents the estimated component values (scores). net is a data structure explaining the neural network parameters for the nonlinear transformation from data space to component space and reverse. net can be used in nlpca_get_components andnlpca_get_data to obtain component values (scores) for new data or reconstructed data for any component value. Example In this example nonlinear PCA (circular PCA) is applied to artificial data of a noisy circle. % generate circular data t=linspace(-pi , +pi , 100); % angular value t=-pi,...,+pi data = [sin(t);cos(t)]; % circle data = data + 0.2*randn(size(data)); % add noise % nonlinear PCA (circular PCA, inverse network architecture) [c,net]=nlpca(data, 1, 'type','inverse', 'circular','yes' ); % plot components nlpca_plot(net) See also the demos of the toolbox below. Download The NLPCA toolbox is distributed under the GNU General Public License. NLPCA can be downloaded as single package or individual files: NLPCA-0.88.zip NLPCA package (all files), version 0.88   package contains:       nlpca.m main program - component extraction     nlpca_get_data.m to reconstruct data from new component values     nlpca_get_components.m to estimate component values from new data     nlpca_plot.m to plot the components       demo_hierarchical_NLPCA_StarData.m    demo of hierarchical nonlinear PCA     demo_circular_PCA.m demo of circular units (Circular PCA)     demo_inverse_NLPCA.m demo of inverse network architecture     demo_missing_data.m demo of missing data estimation References Validation: Validation of nonlinear PCA Matthias Scholz Neural Processing Letters, 2012 [ pdf (pre-print) | pdf (Neural Process Lett) |poster RECOMB 2012 |Matlab code] review (book chapter): Nonlinear principal component analysis: neural network models and applications. Matthias Scholz, Martin Fraunholz, and Joachim Selbig. In Principal Manifolds for Data Visualization and Dimension Reduction, edited by Alexander N. Gorban, Balázs Kégl, Donald C. Wunsch, and Andrei Zinovyev. Volume 58 of LNCSE, pages 44-67. Springer Berlin Heidelberg, 2007. [ pdf (all book chapters) | pdf (Springer) | entire book (Springer)] Circular PCA: Analysing periodic phenomena by circular PCA. Matthias Scholz. In S. Hochreiter and R. Wagner, editors, Proceedings of the Conference on Bioinformatics Research and Development BIRD'07, LNCS/LNBI Vol. 4414, pages 38-47. Springer-Verlag Berlin Heidelberg, 2007. [ pdf (final version at Springer) |pdf (author's pre-version)] Inverse model, missing data: Non-linear PCA: a missing data approach. Matthias Scholz, Fatma Kaplan, Charles L. Guy, Joachim Kopka, and Joachim Selbig. Bioinformatics 21(20):3887-3895. 2005. [ pdf (final version) |pdf (pre-version in colour) ] Hierarchical NLPCA: Nonlinear PCA: a new hierarchical approach. Matthias Scholz and Ricardo Vigário. In M. Verleysen, editor, Proceedings ESANN. 2002. [ pdf (pre-print version) |pdf (ESANN) ]