Oplecorr - Revision history

imported>Neal: /* Algorithm */

2013-12-17T20:08:31Z

Algorithm

← Older revision		Revision as of 13:08, 17 December 2013
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	<math>{{\mathbf{a}}^{T}}=\left[ \begin{matrix}		<math>{{\mathbf{a}}^{T}}=\left[ \begin{matrix}
	\mathbf{a}_{b}^{T} & \mathbf{a}_{r}^{T} \\		\mathbf{a}_{b}^{T} & \mathbf{a}_{r}^{T} \\
	\end{matrix} \right]</math>		\end{matrix} \right]\text{ (16)}</math>.
	.
			Next, a regression model is obtained to allow estimation of correction factors for future test samples using the following
			:<math>\mathbf{a}=\mathbf{Zb} +\mathbf{e}_{mathit{p}} \text{ (17)}</math>
			where the regression vector, <math> \mathbf{b} </math>, is estimated using PLS. Change ''options.center'' to ''true'' to use mean-centering for the PLS model. The correction factors for test samples are calculated using the following steps
			:<math>{{\mathbf{z}}_{test}}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right){{\mathbf{x}}_{test}}\text{ (18)}</math>, and
			:<math>{{a}_{test}}=\mathbf{z}_{test}^{T}\mathbf{b} \text{ (19)} </math>
			where the corrected spectrum is then given by
			:<math>{{\mathbf{z}}_{corr}}=\frac{1}{{{a}_{test}}}{{\mathbf{z}}_{test}}.\text{ (20)}</math>

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T19:52:24Z

Algorithm

← Older revision		Revision as of 12:52, 17 December 2013
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	& \left[ {{\mathbf{\Gamma} }_{:,2:J}}\odot \left( \mathbf{1c}_{b,\left( 2:J,1 \right)}^{T} \right)-\left( {{\mathbf{c}}_{r,\left( :,1 \right)}}{{\mathbf{1}}^{T}} \right)\odot {{\mathbf{\Gamma} }_{:,2:J}} \right]{{\mathbf{a}}_{b,\left( 2:end,2 \right)}}=\left( {{\mathbf{c}}_{r,\left( :,1 \right)}}-\mathbf{1}{{c}_{b,\left( 1,1 \right)}} \right)\odot {{\mathbf{\Gamma} }_{:,1}} \\		& \left[ {{\mathbf{\Gamma} }_{:,2:J}}\odot \left( \mathbf{1c}_{b,\left( 2:J,1 \right)}^{T} \right)-\left( {{\mathbf{c}}_{r,\left( :,1 \right)}}{{\mathbf{1}}^{T}} \right)\odot {{\mathbf{\Gamma} }_{:,2:J}} \right]{{\mathbf{a}}_{b,\left( 2:end,2 \right)}}=\left( {{\mathbf{c}}_{r,\left( :,1 \right)}}-\mathbf{1}{{c}_{b,\left( 1,1 \right)}} \right)\odot {{\mathbf{\Gamma} }_{:,1}} \\
	\end{align}.\text{ (15)}</math>		\end{align}.\text{ (15)}</math>
			Recognizing that the corrections, <math>\mathbf{a}</math>, must be non-negative implies that the remaining correction factors <math>{{\mathbf{a}}_{b2:end}}</math> should be obtained by solving Equation (15) using non-negative least squares. The result is correction factors for all the basis vectors
			<math>\mathbf{a}_{b}^{T}=\left[ \begin{matrix}
			1 & \mathbf{a}_{b,\left( 2:end \right)}^{T} \\
			\end{matrix} \right]</math>. that can be substituted into the sum of Equations (12) to give
			:<math>\begin{align}
			& \mathbf{\Gamma} \left( diag\left( {{\mathbf{c}}_{b,1}} \right)+\mathbf{I} \right){{\mathbf{a}}_{b}}=\left( diag\left( {{\mathbf{c}}_{r,1}} \right)+\mathbf{I} \right){{\mathbf{a}}_{r}} \\
			& {{\mathbf{a}}_{r}}={{\left( diag\left( {{\mathbf{c}}_{r,1}} \right)+\mathbf{I} \right)}^{-1}}\mathbf{\Gamma} \left( diag\left( {{\mathbf{c}}_{b,1}} \right)+\mathbf{I} \right){{\mathbf{a}}_{b}} \\
			\end{align}</math>
			The correction factors can be collected into a single vector given by
			<math>{{\mathbf{a}}^{T}}=\left[ \begin{matrix}
			\mathbf{a}_{b}^{T} & \mathbf{a}_{r}^{T} \\
			\end{matrix} \right]</math>
			.

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T19:47:06Z

Algorithm

← Older revision		Revision as of 12:47, 17 December 2013
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	Equating terms in Equation (11) gives two additional relationships:		Equating terms in Equation (11) gives two additional relationships:
	:<math>\mathbf{\Gamma} diag\left( {{\mathbf{c}}_{b,1}} \right){{\mathbf{a}}_{b}}=diag\left( {{\mathbf{c}}_{r,1}} \right){{\mathbf{a}}_{r}}\text{ (12)}</math>, and		:<math>\mathbf{\Gamma} diag\left( {{\mathbf{c}}_{b,1}} \right){{\mathbf{a}}_{b}}=diag\left( {{\mathbf{c}}_{r,1}} \right){{\mathbf{a}}_{r}}\text{ (12)}</math>, and
	:<math>{{\mathbf{a}}_{r}}=\Gamma {{\mathbf{a}}_{b}}.\text{ (13)}</math>		:<math>{{\mathbf{a}}_{r}}=\mathbf{\Gamma} {{\mathbf{a}}_{b}}.\text{ (13)}</math>
			Substitution of Equation (13) into (12) gives
			:<math>\begin{align}
			& \mathbf{\Gamma} diag\left( {{\mathbf{c}}_{b,1}} \right){{\mathbf{a}}_{b}}=diag\left( {{\mathbf{c}}_{r,1}} \right)\mathbf{\Gamma} {{\mathbf{a}}_{b}} \\
			& {{\left[ \mathbf{\Gamma} \odot \left( \mathbf{1c}_{b,1}^{T} \right) \right]}_{{{M}_{r}}\times J}}{{\left( {{\mathbf{a}}_{b}} \right)}_{J\times 1}}={{\left[ \left( {{\mathbf{c}}_{r,1}}{{\mathbf{1}}^{T}} \right)\odot \Gamma \right]}_{{{M}_{r}}\times J}}{{\left( {{\mathbf{a}}_{b}} \right)}_{J\times 1}} \\
			\end{align}.\text{ (14)}</math>
			Recall that <math>\mathbf{\Gamma} </math>, <math>{{\mathbf{c}}_{r,1}}</math> and <math>{{\mathbf{c}}_{b,1}}</math> are known but <math>{{\mathbf{a}}_{b}}</math> is unknown. However, as with MSC where the reference used for correction is arbitrary (e.g., the mean of the calibration set is often used as the spectrum to “correct to”), any element of <math>{{\mathbf{a}}_{b}}</math> can be set to one. Setting the first element of <math>{{\mathbf{a}}_{b}}</math> to one and rearranging Equation (14) yields
			:<math>\begin{align} & {{c}_{b,\left( 1,1 \right)}}{{\mathbf{\Gamma} }_{:,1}}+\left[ {{\mathbf{\Gamma} }_{:,2:J}}\odot \left( \mathbf{1c}_{b,\left( 2:J,1 \right)}^{T} \right) \right]{{\mathbf{a}}_{b,\left( 2:end,2 \right)}}={{\mathbf{c}}_{r,\left( :,1 \right)}}\odot {{\mathbf{\Gamma} }_{:,1}}+\left[ \left( {{\mathbf{c}}_{r,\left( :,1 \right)}}{{\mathbf{1}}^{T}} \right)\odot {{\mathbf{\Gamma} }_{:,2:J}} \right]{{\mathbf{a}}_{b,\left( 2:end,2 \right)}} \\
			& \left[ {{\mathbf{\Gamma} }_{:,2:J}}\odot \left( \mathbf{1c}_{b,\left( 2:J,1 \right)}^{T} \right)-\left( {{\mathbf{c}}_{r,\left( :,1 \right)}}{{\mathbf{1}}^{T}} \right)\odot {{\mathbf{\Gamma} }_{:,2:J}} \right]{{\mathbf{a}}_{b,\left( 2:end,2 \right)}}=\left( {{\mathbf{c}}_{r,\left( :,1 \right)}}-\mathbf{1}{{c}_{b,\left( 1,1 \right)}} \right)\odot {{\mathbf{\Gamma} }_{:,1}} \\
			\end{align}.\text{ (15)}</math>

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T19:37:53Z

Algorithm

@@ Line 79: / Line 79: @@
 :<math>{{\mathbf{Z}}_{r}}=\mathbf{\Gamma} {{\mathbf{Z}}_{b}}\text{     (6)}</math>
 where
-:<math>\Gamma ={{\mathbf{Z}}_{r}}\mathbf{Z}_{b}^{T}{{\left( {{\mathbf{Z}}_{b}}\mathbf{Z}_{b}^{T} \right)}^{-1}}.\text{     (7)}</math>
+:<math>\mathbf{\Gamma} ={{\mathbf{Z}}_{r}}\mathbf{Z}_{b}^{T}{{\left( {{\mathbf{Z}}_{b}}\mathbf{Z}_{b}^{T} \right)}^{-1}}.\text{     (7)}</math>
 Expanding a single measurement in <math>\mathbf{Z}_{\mathit{r}}</math> gives
 :<math>{{\mathbf{z}}_{m}}={{a}_{m}}{{c}_{m,1}}{{\mathbf{k}}_{1}}+{{a}_{m}}\sum\limits_{j=2}^{J-1}{{{c}_{m,j}}{{\mathbf{k}}_{j}}}+{{a}_{m}}{{c}_{m,J}}{{\mathbf{k}}_{J}}+\mathbf{e}_{m}^{*}.\text{     (8)}</math>
 ===See Also===
 [[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T19:28:15Z

Algorithm

← Older revision		Revision as of 12:28, 17 December 2013
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	where <math>{{\mathbf{P}}^{\dagger }}=\mathbf{P}{{\left( {{\mathbf{P}}^{T}}\mathbf{P} \right)}^{-1}}</math>, <math>\mathbf{K}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right)\mathbf{S}</math>, and <math>{{\mathbf{e}}^{*}}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right)\mathbf{e}</math>.		where <math>{{\mathbf{P}}^{\dagger }}=\mathbf{P}{{\left( {{\mathbf{P}}^{T}}\mathbf{P} \right)}^{-1}}</math>, <math>\mathbf{K}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right)\mathbf{S}</math>, and <math>{{\mathbf{e}}^{*}}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right)\mathbf{e}</math>.
	The measurements can be collected into a matrix <math>\mathbf{Z}</math> and it is recognized that a basis for the <math>\mathit{M}</math> measurements, <math>\mathbf{Z}_{\mathit{b}}</math>, can be obtained from a subset of linearly independent measurements. Partitioning <math>\mathbf{Z}</math> into the basis and remaining measurements, <math>\mathbf{Z}_{\mathit{r}}</math>, gives		The measurements can be collected into a matrix <math>\mathbf{Z}</math> and it is recognized that a basis for the <math>\mathit{M}</math> measurements, <math>\mathbf{Z}_{\mathit{b}}</math>, can be obtained from a subset of linearly independent measurements. Partitioning <math>\mathbf{Z}</math> into the basis and remaining measurements, <math>\mathbf{Z}_{\mathit{r}}</math>, gives
			:<math>\mathbf{Z}=\left[ \begin{matrix}
			{{\mathbf{Z}}_{b}} \\
			{{\mathbf{Z}}_{r}} \\
			\end{matrix} \right]=\left[ \begin{matrix}
			diag\left( {{\mathbf{a}}_{b}} \right){{\mathbf{C}}_{b}}{{\mathbf{K}}^{T}} \\
			diag\left( {{\mathbf{a}}_{r}} \right){{\mathbf{C}}_{r}}{{\mathbf{K}}^{T}} \\
			\end{matrix} \right]+{{\mathbf{E}}^{*}}\text{ (5)}</math>
			This partitioning implies that the remaining measurements, <math>\mathbf{Z}_{\mathit{r}}</math>, are linear combinations of <math>\mathbf{Z}_{\mathit{b}}</math> such that
			:<math>{{\mathbf{Z}}_{r}}=\mathbf{\Gamma} {{\mathbf{Z}}_{b}}\text{ (6)}</math>
			where
			:<math>\Gamma ={{\mathbf{Z}}_{r}}\mathbf{Z}_{b}^{T}{{\left( {{\mathbf{Z}}_{b}}\mathbf{Z}_{b}^{T} \right)}^{-1}}.\text{ (7)}</math>
			Expanding a single measurement in <math>\mathbf{Z}_{\mathit{r}}</math> gives
			:<math>{{\mathbf{z}}_{m}}={{a}_{m}}{{c}_{m,1}}{{\mathbf{k}}_{1}}+{{a}_{m}}\sum\limits_{j=2}^{J-1}{{{c}_{m,j}}{{\mathbf{k}}_{j}}}+{{a}_{m}}{{c}_{m,J}}{{\mathbf{k}}_{J}}+\mathbf{e}_{m}^{*}.\text{ (8)}</math>

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T19:17:45Z

Algorithm

← Older revision		Revision as of 12:17, 17 December 2013
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	\mathbf{1} & \lambda & {{\lambda }^{2}} \\		\mathbf{1} & \lambda & {{\lambda }^{2}} \\
	\end{matrix} \right]\text{ (3)}</math>		\end{matrix} \right]\text{ (3)}</math>
	where <math>\lambda</math> is the wavelength (or frequency) axis. However, it should be clear that <math>\mathbf{P}</math> is a matrix with columns that span physical artifacts not subject to closure.		where <math>\lambda</math> is the wavelength (or frequency) axis. However, it should be clear that <math>\mathbf{P}</math> is a matrix with columns that span physical artifacts not subject to closure. The <math>\mathbf{m}^{th}</math> measured signal, <math>\mathbf{x}_{\mathit{m}}</math>, <math>\mathit{m}=1,...,\mathit{M}</math>, orthogonal to <math>\mathbf{P}</math> is
			:<math>{{\mathbf{z}}_{m}}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right){{\mathbf{x}}_{m}}={{a}_{m}}\mathbf{K}{{\mathbf{c}}_{m}}+\mathbf{e}_{m}^{*}\text{ (4)}</math>
			where <math>{{\mathbf{P}}^{\dagger }}=\mathbf{P}{{\left( {{\mathbf{P}}^{T}}\mathbf{P} \right)}^{-1}}</math>, <math>\mathbf{K}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right)\mathbf{S}</math>, and <math>{{\mathbf{e}}^{*}}=\left( \mathbf{I}-{{\mathbf{P}}^{\dagger }}{{\mathbf{P}}^{T}} \right)\mathbf{e}</math>.
			The measurements can be collected into a matrix <math>\mathbf{Z}</math> and it is recognized that a basis for the <math>\mathit{M}</math> measurements, <math>\mathbf{Z}_{\mathit{b}}</math>, can be obtained from a subset of linearly independent measurements. Partitioning <math>\mathbf{Z}</math> into the basis and remaining measurements, <math>\mathbf{Z}_{\mathit{r}}</math>, gives

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T18:30:49Z

Algorithm

← Older revision		Revision as of 11:30, 17 December 2013
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	It is assumed that the measured signal, <math>\mathbf{x}</math> can be modeled as		It is assumed that the measured signal, <math>\mathbf{x}</math> can be modeled as
	:<math>\mathbf{x}=a\left( \mathbf{Sc}+\mathbf{Pt} \right)+\mathbf{e}\text{ (1)}</math>		:<math>\mathbf{x}=a\left( \mathbf{Sc}+\mathbf{Pt} \right)+\mathbf{e}\text{ (1)}</math>
			where <math>\mathbf{x}</math> is a <math>N\times 1</math> column vector, <math>\mathbf{S}</math> is a <math>N\times J</math> matrix with columns corresponding to analyte spectra, <math>\mathbf{c}</math> is a <math>J\times 1</math> vector of contributions, <math>\mathbf{P}</math> is a matrix with columns corresponding to physical artifacts in the spectra and <math>\mathbf{t}</math> is a vector corresponding scores (or contributions for the artifacts). The factor <math>\mathit{a}</math> is a multiplicative factor (e.g. due to changes in path-length) identified by the OPLEC algorithm. The <math>J</math> analyte contributions are subject to closure such that
			:<math>\sum\limits_{j=1}^{J}{{{c}_{j}}}=1\text{ ; }{{c}_{1}}=1-\sum\limits_{j=2}^{J}{{{c}_{j}}}.\text{ (2)}</math>
			Closure also implies that the contributions are non-negative. It is assumed that the contributions to the first analyte are known (i.e., the <math>M\times 1</math> column vector <math>\mathbf{c}_1</math> is known). It is also assumed that the matrix <math>\mathbf{P}</math> can be modeled ''a priori''. Examples for physical artifacts include an offset, slope and curvature of the baseline that can be accounted for by the <math>N\times 3</math> basis
			:<math>\mathbf{P}=\left[ \begin{matrix}
			\mathbf{1} & \lambda & {{\lambda }^{2}} \\
			\end{matrix} \right]\text{ (3)}</math>
			where <math>\lambda</math> is the wavelength (or frequency) axis. However, it should be clear that <math>\mathbf{P}</math> is a matrix with columns that span physical artifacts not subject to closure.

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T18:19:49Z

Algorithm

← Older revision		Revision as of 11:19, 17 December 2013
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	It is assumed that the measured signal, <math>\mathbf{x}</math> can be modeled as		It is assumed that the measured signal, <math>\mathbf{x}</math> can be modeled as
	:~~<math>\mathbf{x} = \mathit{a} ( \mathbf{Sc} + \mathbf{Pt} ) </math>~~		:<math>\mathbf{x}=a\left( \mathbf{Sc}+\mathbf{Pt} \right)+\mathbf{e}\text{ (1)}</math>



	<math>\mathbf{x}=a\left( \mathbf{Sc}+\mathbf{Pt} \right)+\mathbf{e}</math>

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Algorithm */

2013-12-17T18:08:01Z

Algorithm

← Older revision		Revision as of 11:08, 17 December 2013
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	===Algorithm===		===Algorithm===
			The OPLEC algorithm is based on the work Z-P Chen, J Morris, E Martin, “Extracting Chemical Information from Spectral Data with Multiplicative Light Scattering Effects by Optical Path-Length Estimation and Correction,” ''Anal. Chem.'', '''78''', 7674-7681 (2006). OPLEC is similar to extended multiplicative scatter correction (EMSC) except that it incorporates closure in the signal due to chemical analytes.

			It is assumed that the measured signal, <math>\mathbf{x}</math> can be modeled as
			:<math>\mathbf{x} = \mathit{a} ( \mathbf{Sc} + \mathbf{Pt} ) </math>



			<math>\mathbf{x}=a\left( \mathbf{Sc}+\mathbf{Pt} \right)+\mathbf{e}</math>

	===See Also===		===See Also===

	[[emscorr]], [[mscorr]], [[stdfir]]		[[emscorr]], [[mscorr]], [[stdfir]]

imported>Neal: /* Outputs */

2013-12-17T00:03:49Z

Outputs

← Older revision		Revision as of 17:03, 16 December 2013
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	* '''time''': time of creation, ...		* '''time''': time of creation, ...

	and		:and

	* '''sx''' = a M by N matrix of filtered ("corrected") spectra.		* '''sx''' = a M by N matrix of filtered ("corrected") spectra.