A solution to the wavelet transform optimization problem in multicomponent analysis

Coelho, Clarimar José; Galvão, Roberto Kawakami Harrop; Araújo, Mário César Ugulino de; Pimentel, Maria Fernanda; Silva, Edvan Cirino da

doi:10.1016/s0169-7439(03)00050-9

Cited by 27 publications

(8 citation statements)

References 21 publications

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“…Similar assumptions with Equation (16) result in Equation (18), which is the usual equation when error in X dominates [28,29]. If s 2 y dominates in Equation (13), then Vŷ y unk ð Þ%s 2…”

Section: Unknown Sample Prediction Variancementioning

confidence: 77%

Ridge regression optimization using a harmonious approach

Forrester

Kalivas

2004

Journal of Chemometrics

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A critical component of ridge regression (RR) is determining the optimal ridge parameter value, , where ! 0. Improper selection of not only generates an under-or overfitted model but also leads to incorrect conclusions in inter-model comparison studies such as between RR, PLS, PCR and other modeling methods. Several methods for determining the optimal RR model are evaluated in this paper. For example, the commonly used ridge trace is identified as subjective and impractical. A direct calculation method from the literature yields over-or underfitted RR models with either too small or to large respectively. Methods for determining based on a harmonious approach are discussed. The harmonious approach optimizes by inspecting the bias/variance tradeoff. Of the methods investigated, plotting a variance indicator against a bias measure to yield an L-curve appears not only to simplify selection of but also to reduce the chance of obtaining an under-or overfitted RR model. It is shown with four data sets that the L-curve harmonious approach consistently provides good models. The effective rank of models is also discussed in conjunction with the harmony/parsimony tradeoff.

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Section: Unknown Sample Prediction Variancementioning

confidence: 77%

Ridge regression optimization using a harmonious approach

Forrester

Kalivas

2004

Journal of Chemometrics

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“…The upsampled approximation and detail sequences are then filtered by lowpass (h r ) and highpass (g r ) reconstruction filters, respectively. In an orthogonal filter bank, these filters correspond to their decomposition counterparts in reverse order [6,22].…”

Section: The Wavelet Transform Implementationmentioning

confidence: 99%

“…In the present work, the constant extension method (also termed smooth padding of order 0) is adopted [19]. Such a method leads to less pronounced border effects, as compared to periodic extension [6,22] and zero padding [19]. However, the resulting transformation from the input signal to the wavelet coefficients is not orthogonal, even if an orthogonal filter bank is employed.…”

Section: The Wavelet Transform Implementationmentioning

confidence: 99%

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Combination of Models Obtained by Regression in the Wavelet Domain

Pinto¹,

Galvão²

2010

TEMA - Tendências Em Matemática Aplicada E Computacional

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Abstract. The wavelet transform is a useful tool to preprocess and compress datasets for linear regression modelling. However, the prediction performance of the resulting model depends on the choice of wavelet filter and number of decomposition levels, which may not be a straightforward task. This paper proposes an alternative approach, which consists of combining models obtained from different wavelet decompositions of the dataset. For this purpose, a method is developed to convert wavelet regression models back to the original domain. The proposed approach is illustrated in a case study involving the determination of density in gasoline samples by using infrared spectroscopy. The results are favourably compared to those obtained by using individual wavelet decompositions.Keywords. Wavelet Transform, Combination of Models, Multiple Linear Regression. IntrodutionThe wavelet transform (WT) has been used in a variety of signal processing applications, such as filtering [10], compression [24] and classification [8]. In particular, the WT has become a popular tool in a field of Analytical Chemistry known as Multivariate Calibration (MC) [7,15]. The MC problem consists of building a mathematical model for estimation of physical or chemical properties of a sample from indirect measurements. Such measurements could be, for instance, emission, absorption or reflection intensities acquired over different wavelengths [20]. Applications of MC include determination of metals in steel alloys [16], analysis of composition of pharmaceutical formulae [5] and prediction of fuel properties [3], among many others.

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“…Examples of the latter phenomena include the random amplitude of a deterministic machine signal, particle size-dependent light scattering and random optical path length, to name just a few. Input-output relations of the form (1) have been considered in the literature [3,11,18] and used as a benchmark in various simulation studies and tests of new algorithms [10,12,13,[25][26][27]. In Section 4 we present a theoretical analysis of PLS on error-free inputs of the form (1), while the case of inputs corrupted by noise according to (2) is considered in Section 5.…”

Section: A Probabilistic Model Of the Input Datamentioning

confidence: 99%

Partial least squares, Beer's law and the net analyte signal: statistical modeling and analysis

Nadler

Coifman

2005

Journal of Chemometrics

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Partial least squares (PLS) is one of the most common regression algorithms in chemistry, relating input-output samples (x i , y i ) by a linear multivariate model. In this paper we analyze the PLS algorithm under a specific probabilistic model for the relation between x and y. Following Beer's law, we assume a linear mixture model in which each data sample (x, y) is a random realization from a joint probability distribution where x is the sum of k components multiplied by their respective characteristic responses, and each of these components is a random variable. We analyze PLS on this model under two idealized settings: one is the ideal case of noise-free samples and the other is the case of an infinite number of noisy training samples. In the noise-free case we prove that, as expected, the regression vector computed by PLS is, up to normalization, the net analyte signal. We prove that PLS computes this vector after at most k iterations, where k is the total number of components. In the case of an infinite training set corrupted by unstructured noise, we show that PLS computes a final regression vector which is not in general purely proportional to the net analyte signal vector, but has the important property of being optimal under a mean squared error of prediction criterion. This result can be viewed as an asymptotic optimality of PLS in the limit of a very large but finite training set.

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A solution to the wavelet transform optimization problem in multicomponent analysis

Cited by 27 publications

References 21 publications

Ridge regression optimization using a harmonious approach

Ridge regression optimization using a harmonious approach

Combination of Models Obtained by Regression in the Wavelet Domain

Partial least squares, Beer's law and the net analyte signal: statistical modeling and analysis

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