Probabilistic principal component analysis

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss the advantages conveyed by the definition of a probability density function for PCA.

Probabilistic principal component analysis

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss the advantages conveyed by the definition of a probability density function for PCA.

Design of a flat-top fiber Bragg filter via quasi-random modulation of the refractive index

The statistics of the reflection spectrum of a short-correlated disordered fiber Bragg grating are studied. The averaged spectrum appears to be flat inside the bandgap and has significantly suppressed sidelobes compared to the uniform grating of the same bandwidth. This is due to the Anderson localization of the modes of a disordered grating. This observation prompts a new algorithm for designing passband reflection gratings. Using the stochastic invariant imbedding approach it is possible to obtain the probability distribution function for the random reflection coefficient inside the bandgap and obtain both the variance of the averaged reflectivity as well as the distribution of the time delay of the grating.

Revealing statistical properties of quasi-CW fibre lasers in bandwidth-limited measurements

We introduce a general technique how to reveal in experiments of limited electrical bandwidth which is lower than the optical bandwidth of the optical signal under study, whether the statistical properties of the light source obey Gaussian distribution or mode correlations do exist. To do that one needs to perform measurements by decreasing the measurement bandwidth. We develop a simple model of bandwidth-limited measurements and predict universal laws how intensity probability density function and intensity auto-correlation function of ideal completely stochastic source of Gaussian statistics depend on limited measurement bandwidth and measurement noise level. Results of experimental investigation are in good agreement with model predictions. In particular, we reveal partial mode correlations in the radiation of quasi-CW Raman fibre laser.

Influence of the generated power, measurement bandwidth, and noise level on intensity statistics of a quasi-CW Raman fiber laser

In the present paper we numerically study instrumental impact on statistical properties of quasi-CW Raman fiber laser using a simple model of multimode laser radiation. Effects, that have the most influence, are limited electrical bandwidth of measurement equipment and noise. To check this influence, we developed a simple model of the multimode quasi- CW generation with exponential statistics (i.e. uncorrelated modes). We found that the area near zero intensity in probability density function (PDF) is strongly affected by both factors, for example both lead to formation of a negative wing of intensity distribution. But far wing slope of PDF is not affected by noise and, for moderate mismatch between optical and electrical bandwidth, is only slightly affected by bandwidth limitation. The generation spectrum often becomes broader at higher power in experiments, so the spectral/electrical bandwidth mismatch factor increases over the power that can lead to artificial dependence of the PDF slope over the power. It was also found that both effects influence the ACF background level: noise impact decreases it, while limited bandwidth leads to its increase. © (2014) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.