Multicomponent Determination of Chlorinated Hydrocarbons Using a Reaction-Based Chemical Sensor. 3. Medium-Rank Second-Order Calibration with Restricted Tucker Models

Smilde, Age K.; Tauler, Romà; Henshaw, John M.; Burgess, Lloyd W.; Kowalski, Bruce R.

doi:10.1021/ac00092a010

Cited by 86 publications

(40 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…To be able to compare with a Tucker3 core, the corresponding restricted Tucker3 core, the elements of which are given in Equation (14), must be used, just like the superdiagonal array T is used for the PARAFAC model. The unconstrained Tucker3 G is then compared with the expected core T, again using Equation (8).…”

Section: Appendix Imentioning

confidence: 99%

“…in the field of chemometrics, especially owing to its highly attractive uniqueness property. In certain applications of curve resolution [5,6] and secondorder calibration [7,8], this uniqueness of the PARAFAC model is essential for solving the problems. A major practical obstacle in the use of the PARAFAC model is how to determine the appropriate number of components.…”

Section: Introductionmentioning

confidence: 99%

“…The operator is the so-called Khatri-Rao product [14], which is equivalent to a column-wise Kronecker product [15]. The PARAFAC model may equivalently be stated in the form of a restricted Tucker3 model [8,9,16,17] as…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A new efficient method for determining the number of components in PARAFAC models

Bro¹,

Kiers

2003

Journal of Chemometrics

1,121

718

View full text Add to dashboard Cite

A new diagnostic called the core consistency diagnostic (CORCONDIA) is suggested for determining the proper number of components for multiway models. It applies especially to the parallel factor analysis (PARAFAC) model, but also to other models that can be considered as restricted Tucker3 models. It is based on scrutinizing the`appropriateness' of the structural model based on the data and the estimated parameters of gradually augmented models. A PARAFAC model (employing dimension-wise combinations of components for all modes) is called appropriate if adding other combinations of the same components does not improve the fit considerably. It is proposed to choose the largest model that is still sufficiently appropriate. Using examples from a range of different types of data, it is shown that the core consistency diagnostic is an effective tool for determining the appropriate number of components in e.g. PARAFAC models. However, it is also shown, using simulated data, that the theoretical understanding of CORCONDIA is not yet complete.

show abstract

Section: Appendix Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new efficient method for determining the number of components in PARAFAC models

Bro¹,

Kiers

2003

Journal of Chemometrics

1,121

718

View full text Add to dashboard Cite

show abstract

“…This illustrates how a given set of underlying phenomena can produce variation in the spectral domain and in the concentration domains that have different ranks. Some authors view this as showing that there is only one phenomenon evolving in the concentration/time domain-the reaction of M and N [11,15,16]. However, this perspective forces one to consider the spectral domain and time domain at different 'levels of description'-one for chemicals, the other for phenomena.…”

Section: Rank Deficiencymentioning

confidence: 99%

Modeling multi‐way data with linearly dependent loadings

Bro

Harshman

Sidiropoulos

et al. 2009

Journal of Chemometrics

104

159

View full text Add to dashboard Cite

A generalization/specialization of the PARAFAC model is developed that improves its properties when applied to multi-way problems involving linearly dependent factors. This model is called PARALIND (PARAllel profiles with LINear Dependences). Linear dependences can arise when the empirical sources of variation being modeled by factors are causally or logically linked during data generation, or circumstantially linked during data collection. For example, this can occur in a chemical context when end products are related to the precursor or in a psychological context when a single stimulus generates two incompatible feelings at once. For such cases, the most theoretically appropriate PARAFAC model has loading vectors that are linearly dependent in at least one mode, and when collinear, are nonunique in the others. However, standard PARAFAC analysis of fallible data will have neither of these features. Instead, latent linear dependences become high surface correlations and any latent nonuniqueness is replaced by a meaningless surface-level 'unique orientation' that optimally fits the particular random noise in that sample. To avoid these problems, any set of components that in theory should be rank deficient are re-expressed in PARALIND as a product of two matrices, one that explicitly represents their dependency relationships and another, with fewer columns, that captures their patterns of variation. To demonstrate the approach, we apply it first to fluorescence spectroscopy (excitation-emission matrices, EEM) data in which concentration values for two analytes covary exactly, and then to flow injection analysis (FIA) data in which subsets of columns are logically constrained to sum to a constant, but differently in each of two modes. In the PARAFAC solutions of the EEM data, all factors are 'unique' but this is only meaningful for two of the factors that are also unique at the latent level. In contrast, the PARALIND solutions directly display the extent and nature of partial nonuniqueness present at the latent level by exhibiting a corresponding partial uniqueness in their recovered loadings. For the FIA data, PARALIND constraints restore latent uniqueness to the concentration estimates. Comparison of the solutions shows that PARALIND more accurately recovers latent structure, presumably because it uses fewer parameters and hence fits less error.

show abstract

“…Most of the applications of these chemometric algorithms are focused on calibration which entails the standards of analytes of interest [18,20,21]. In analytical practice, unfortunately, there are many important cases where the calibration can not be performed because there is no previous information available about the system [22], which is a so-called black one [23].…”

Section: Introductionmentioning

confidence: 99%

Resolution of kinetic system of simultaneous degradations of chlorophyll a and b by PARAFAC

Tan

Jiang

Cui

et al. 2000

Analytica Chimica Acta

View full text Add to dashboard Cite

Multicomponent Determination of Chlorinated Hydrocarbons Using a Reaction-Based Chemical Sensor. 3. Medium-Rank Second-Order Calibration with Restricted Tucker Models

Cited by 86 publications

References 8 publications

A new efficient method for determining the number of components in PARAFAC models

A new efficient method for determining the number of components in PARAFAC models

Modeling multi‐way data with linearly dependent loadings

Resolution of kinetic system of simultaneous degradations of chlorophyll a and b by PARAFAC

Contact Info

Product

Resources

About