PCOP: Principal Curves of Oriented Points

PCOP: Principal Curves of Oriented Points

Here you can find our contributions on Principal Curves (papers and software) and some links to other related pages.

Papers: A list of our works on Principal curves:

Delicado, P. and Smrekar, M. (2008) Measuring non-linear dependence for two random variables distributed along a curve. Statistics and Computing, (to appear). doi: 10.1007/s11222-008-9090-y (First authors' version: PDF file).

Delicado, P. and Smrekar, M. (2007) Mixture of nonlinear models: A Bayesian fit for principal curves. In Proceedings of the International Joint Conference on Neural Networks, Orlando, Florida, USA. ISBN: 1-4244-1380-X. (PDF file).

Cedano, J., Huerta, M., Estrada, I., Ballllosera, F., Conchillo, O., Delicado, P. and Querol, E. (2007) A web server for automatic analysis and extraction of relevant biological knowledge. Computers in Biology and Medicine, 37 (11) 1672-1675 (Abstract, PDF).

Delicado, P. and Huerta, M. (2003) Principal Curves of Oriented Points: Theoretical and computational improvements. Computational Statistics, 18 (2) 293-315. (PS file) (Extended working paper version: Document de Recerca EIO (UPC) 2002/06. )

Delicado, P.(2001) Another look at principal curves and surfaces. Journal of Multivariate Analysis, 77, 84-116.

Software: We have developed an object oriented application implementing the PCOP methods. The main part is written in C++. Graphical outputs are obtained in Matlab. Here you can find the source code and some examples:

R-functions to compute CovGC and CorGC (Generalized covariance and correlation along a generating curve). See Delicado and Smrekar (2008).

PCOP Application (Exec. file for Win-98) (Present version: June 2003).

Delicado and Huerta (2003) (working paper version)

readme file.

The application source code is also available. It is free for non-commercial use, and you can apply it to your data, modify it, and integrate it into another program as long as you notify us beforehand. We are not responsible for any implications from the use of this software.

A Matlab function file used to obtain graphical outputs.

Some examples of graphical outputs:

Population Pyramids Data Set: Population pyramids of 223 countries are analyzed. The first four principal components are considered and the first principal curve is calculated on this 4-dimensional data set. The PCOP is projected on the first two principal components plane. (Population pyramids A to G in the lower row of the graphic correspond to Mauritania, Pakistan, Peru, The Bahamas, Taiwan, Canada, and Japan, respectively).
Ellipse: Two dimensional artificial data randomly distributed around an ellipse . There are four graphics, that are the Matlab graphical routine standard output:

Figure 1: Principal curve over the data cloud.
Figure 2: Original coordinate values versus principal curve location.
Figure 3: Density, span, and residual variances along the principal curve.
Figure 4: Princiapl directions over the data cloud.

Helix: Three dimensional artificial data randomly distributed around a helix.

helix1.eps: Noise to signal rate: Low.
helix2.eps: Noise to signal rate: High.

Links: Some sites dedicates to principal curves and other interesting links:

Balázs Kégl dedicates a well documented web page to principal curves, including links to papers on principal curves, algorithms to find them, and some practical applications.

The web page of Jakob J. Verbeek contains its contributions to Principal Curves and some references.

Last modification: June 6, 2007.

Responsible for this page:
Pedro Delicado (Web page)