PARAFAC: Parallel factor analysis

Harshman, Richard A.; Lundy, Margaret E.

doi:10.1016/0167-9473(94)90132-5

Cited by 463 publications

(285 citation statements)

References 34 publications

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“…The orthogonality constraint was implemented by replacing Step 3 of the ALS algorithm in Appendix A with 3. C = Z(Z ′ Z) −1/2 where Z = X c (B ⊙ A) (Harshman & Lundy, 1994). We chose to appy the constraint on C (instead of A or B) to ensure that each component provided non-redundant information concerning interrelationships between the joints, while allowing the components to be linearly related with respect to the subjects and time points.…”

Section: Parallel Factor Analysis Of Joint Angle Datamentioning

confidence: 99%

Parallel Factor Analysis of gait waveform data: A multimode extension of Principal Component Analysis

Helwig

Hong

Polk

2012

Human Movement Science

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Gait data are typically collected in multivariate form, so some multivariate analysis is often used to understand interrelationships between observed data. Principal Component Analysis (PCA), a data reduction technique for correlated multivariate data, has been widely applied by gait analysts to investigate patterns of association in gait waveform data (e.g., interrelationships between joint angle waveforms from different subjects and/or joints). Despite its widespread use in gait analysis, PCA is for two-mode data, whereas gait data are often collected in higher-mode form. In this paper, we present the benefits of analyzing gait data via Parallel Factor Analysis (Parafac), which is a component analysis model designed for three-or higher-mode data. Using three-mode joint angle waveform data (subjects × time × joints), we demonstrate Parafac's ability to (a) determine interpretable components revealing the primary interrelationships between lowerlimb joints in healthy gait and (b) identify interpretable components revealing the fundamental differences between normal and perturbed subjects' gait patterns across multiple joints. Our results offer evidence of the complex interconnections that exist between lower-limb joints and limb segments in both normal and abnormal gaits, confirming the need for the simultaneous analysis of multi-joint gait waveform data (especially when studying perturbed gait patterns).ii To my loving and supportive family.iii

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Section: Parallel Factor Analysis Of Joint Angle Datamentioning

confidence: 99%

Parallel Factor Analysis of gait waveform data: A multimode extension of Principal Component Analysis

Helwig

Hong

Polk

2012

Human Movement Science

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“…1). The first of these models is known as "Parallel Factor Analysis" or "PARAFAC" (Harshman and Lundy, 1994) and extends the bilinear umixing model (Eq. S1 in the Supplement) to a trilinear model (Fig.…”

Section: Three-dimensional Array Factorizationmentioning

confidence: 99%

Three-dimensional factorization of size-resolved organic aerosol mass spectra from Mexico City

et al. 2012

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Abstract.A size-resolved submicron organic aerosol composition dataset from a high-resolution time-of-flight mass spectrometer (HR-ToF-AMS) collected in Mexico City during the MILAGRO campaign in March 2006 is analyzed using 3-dimensional (3-D) factorization models. A method for estimating the precision of the size-resolved composition data for use with the factorization models is presented here for the first time. Two 3-D models are applied to the dataset. One model is a 3-vector decomposition (PARAFAC model), which assumes that each chemical component has a constant size distribution over all time steps. The second model is a vector-matrix decomposition (Tucker 1 model) that allows a chemical component to have a size distribution that varies in time. To our knowledge, this is the first report of an application of 3-D factorization models to data from fast aerosol instrumentation, and the first application of this vector-matrix model to any ambient aerosol dataset. A larger number of degrees of freedom in the vector-matrix model enable fitting real variations in factor size distributions, but also make the model susceptible to fitting noise in the dataset, giving some unphysical results. For this dataset and model, more physically meaningful results were obtained by partially constraining the factor mass spectra using a priori information and a new regularization method. We find four factors with each model: hydrocarbon-like organic aerosol (HOA), biomass-burning organic aerosol (BBOA), oxidized organic aerosol (OOA), and a locally occurring organic aerosol (LOA). These four factors have previously been reported from 2-dimensional factor analysis of the highresolution mass spectral dataset from this study. The size distributions of these four factors are consistent with previous reports for these particle types. Both 3-D models produce useful results, but the vector-matrix model captures real variability in the size distributions that cannot be captured by the 3-vector model. A tracer m/z-based method provides a useful approximation for the component size distributions in this study. Variation in the size distributions is demonstrated in a case study day with a large secondary aerosol formation event, in which there is evidence for the coating of HOA-containing particles with secondary species, shifting the HOA size distribution to larger particle sizes. These 3-D factorizations could be used to extract size-resolved aerosol composition data for correlation with aerosol hygroscopicity, cloud condensation nuclei (CCN), and other aerosol impacts. Furthermore, other fast and chemically complex 3-D datasets, including those from thermal desorption or chromatographic separation, could be analyzed with these 3-D factorization models. Applications of these models to new datasets requires careful construction of error estimates and appropriate choice of models that match the underlying structure of those data. Factorization studies with these 3-D datasets have the potential to provide further insights into organic aero...

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“…For instance, the singular value decomposition can be expressed as a Tucker decomposition of a second order tensor with orthogonal factor matrices. Furthermore, Candecomp / Parafac (CP) [10,7] can be described as a Tucker decomposition with the additional constraints that the core tensor W is superdiagonal and r 1 = r 2 = · · · = r m . Similarly, the Block-Term decomposition (BTD) [8] can be viewed as imposing the constraint that the core tensor W is block-diagonal.…”

Section: Tensor Factorizationsmentioning

confidence: 99%

An Analysis of Tensor Models for Learning on Structured Data

Nickel

Tresp

2013

Advanced Information Systems Engineering

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Abstract. While tensor factorizations have become increasingly popular for learning on various forms of structured data, only very few theoretical results exist on the generalization abilities of these methods. Here, we discuss the tensor product as a principled way to represent structured data in vector spaces for machine learning tasks. By extending known bounds for matrix factorizations, we are able to derive generalization error bounds for the tensor case. Furthermore, we analyze analytically and experimentally how tensor factorization behaves when applied to overand understructured representations, for instance, when two-way tensor factorization, i.e. matrix factorization, is applied to three-way tensor data.

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PARAFAC: Parallel factor analysis

Cited by 463 publications

References 34 publications

Parallel Factor Analysis of gait waveform data: A multimode extension of Principal Component Analysis

Parallel Factor Analysis of gait waveform data: A multimode extension of Principal Component Analysis

Three-dimensional factorization of size-resolved organic aerosol mass spectra from Mexico City

An Analysis of Tensor Models for Learning on Structured Data

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