CP decomposition
Encyclopedia
In multilinear algebra
, the canonical polyadic decomposition (CPD), historically known as PARAFAC and later CANDECOMP, is a generalization of the matrix singular value decomposition
(SVD) to tensor
s, with many applications in in statistics
, signal processing
, psychometrics
, linguistics
and chemometrics
. It originates from psychometrics
though going back to Hitchcock in 1927.
Algebraic algorithms:
Optimization algorithms:
Direct methods:
s for several samples, fluorescence
lifetime measured at several excitation and emission wavelengths or any kind of spectrum measured chromatographically for several samples. Determining such variables will give rise to three-way data; i.e., the data can be arranged in a cube instead of a matrix as in standard multivariate data sets.
s for multi-way data. The two main competitors are the Tucker3 method , and simply unfolding of the multi-way array to a matrix and then performing standard two-way methods as principal component analysis (PCA). The Tucker3 method should rightfully be called three-mode principal component analysis (or N-mode principal component analysis), but here the term Tucker3 or just Tucker decomposition
will be used instead. PARAFAC, Tucker and two-way PCA are all multi- or bi-linear decomposition methods, which decompose the array into sets of "scores" and "loadings", that hopefully describe the data in a more condensed form than the original data array. There are advantages and disadvantages with all the methods, and often several methods must be tried to find the most appropriate.
In the field of chemometrics, a number of diagnostic tools and techniques exist to help a PARAFAC user determine the best fitting model. These include the core consistency diagnostic (CORCONDIA), split-half analyses, examination of the loadings, and residual analysis.
Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...
, the canonical polyadic decomposition (CPD), historically known as PARAFAC and later CANDECOMP, is a generalization of the matrix singular value decomposition
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
(SVD) to tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s, with many applications in in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, psychometrics
Psychometrics
Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...
, linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
and chemometrics
Chemometrics
Chemometrics is the science of extracting information from chemical systems by data-driven means. It is a highly interfacial discipline, using methods frequently employed in core data-analytic disciplines such as multivariate statistics, applied mathematics, and computer science, in order to...
. It originates from psychometrics
Psychometrics
Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...
though going back to Hitchcock in 1927.
Calculating the CPD
Alternating algorithms:- alternating least squares (ALS)
- alternating slice-wise diagonalisation (ASD)
Algebraic algorithms:
- simultaneous diagonalization (SD)
- simultaneous generalized Schur decomposition (SGSD)
Optimization algorithms:
- Levenberg–Marquardt (LM)
- nonlinear conjugate gradient (NCG)
- limited memory BFGSL-BFGSThe limited-memory BFGS algorithm is a member of the broad family of quasi-Newton optimization methods that uses a limited memory variation of the Broyden–Fletcher–Goldfarb–Shanno update to approximate the inverse Hessian matrix...
(L-BFGS)
Direct methods:
- Direct multilinear decomposition (DMLD)
Chemometrics
Multi-way data are characterized by several sets of categorical variables that are measured in a crossed fashion. Chemical examples could be fluorescence emission spectra measured at several excitation wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s for several samples, fluorescence
Fluorescence
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation of a different wavelength. It is a form of luminescence. In most cases, emitted light has a longer wavelength, and therefore lower energy, than the absorbed radiation...
lifetime measured at several excitation and emission wavelengths or any kind of spectrum measured chromatographically for several samples. Determining such variables will give rise to three-way data; i.e., the data can be arranged in a cube instead of a matrix as in standard multivariate data sets.
Other decompositions
PARAFAC is one of several decomposition methodDecomposition method
In constraint satisfaction, a decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem that is binary and acyclic. Decomposition methods work by grouping variables into sets, and solving a subproblem for each set...
s for multi-way data. The two main competitors are the Tucker3 method , and simply unfolding of the multi-way array to a matrix and then performing standard two-way methods as principal component analysis (PCA). The Tucker3 method should rightfully be called three-mode principal component analysis (or N-mode principal component analysis), but here the term Tucker3 or just Tucker decomposition
Tucker decomposition
In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tuckeralthough it goes back to Hitchcock in 1927....
will be used instead. PARAFAC, Tucker and two-way PCA are all multi- or bi-linear decomposition methods, which decompose the array into sets of "scores" and "loadings", that hopefully describe the data in a more condensed form than the original data array. There are advantages and disadvantages with all the methods, and often several methods must be tried to find the most appropriate.
In the field of chemometrics, a number of diagnostic tools and techniques exist to help a PARAFAC user determine the best fitting model. These include the core consistency diagnostic (CORCONDIA), split-half analyses, examination of the loadings, and residual analysis.
Further reading
- Kolda, Tamara G.; Bader, Brett W. (2009) "Tensor Decompositions and Applications", SIAM Rev. 51, 455-500
External links
- PARAFAC Tutorial
- Parallel Factor Analysis (PARAFAC)
- FactoMineR (free exploratory multivariate data analysis software linked to R)