US 20090030655A1 - Analysis Methods for unmixing the response of non-linear, cross-reactive sensors and related system to single and multiple stimulants

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Abstract

Disclosed herein are methods of analysis for unmixing non-linear, cross-reactive sensors and related system. Use of the disclosed methods, related systems and computer program product permits better analysis of the magnitudes of various stimulants including but not limited to chemical concentrations. One method may add one or more additional signal vectors to the sensor response before linearizing each channel. A second method may add one or more exponential terms to the response curve when using curve parameterization to unmix the sensor response. A third method may use non-linear iterative solutions that estimates an optical depth, linearizes the optical depth, solves for a correction to the estimated optical depth, and updates the optical depth. Also, the disclosed methods and related systems include combinations of the methods described herein.

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49 Claims

1. A method for analyzing signals from a plurality of cross reactive channels of a sensor of a system where the signals vary non-linearly with the quantities of stimulant that induce these signals, wherein the said method comprises:
- obtaining signals from a plurality of channels from said sensor at a variety of known levels for each stimulant;
  
  producing a net normalized result from each signal from the known levels;
  
  creating a library of signatures that includes the net normalized result at each of the known levels of stimulant;
  
  obtaining one or more signals from a plurality of channels from said sensor at unknown levels of stimulant;
  
  producing a net normalized result from each signal from the unknown levels; and
  
  unmixing the net normalized result from the unknown levels by projecting the sensor response onto the library of known signatures, with the various levels of each stimulant being treated as independent stimulants.
- View Dependent Claims (2, 3, 4, 20, 21, 22, 23)
- - 2. The method of claim 1, wherein said net normalized result is a signal of a given channel that is reference corrected by a predetermined mathematical formula defined as:
    - $r = \frac{s (0) - s}{s (0)} = \frac{\int [(ɛ W_{s} - W_{v}) (1 - e^{- \sum_{i} α_{i} (λ) C_{i} })] τ_{f} (λ) \partial λ}{\int [ɛ_{s} W_{s} (λ)] τ_{f} (λ) \partial λ}$ wherein;
      
      r is the net normalized result;
      
      s is the signal for a given channel;
      
      s(0) is a reference signal for the given channel;
      
      τ
      
      (λ
      
      ) is the transmission by a radiation filter attached to said channel;
      
      α
      
      _i(λ
      
      ) is the absorptivity of chemical species i;
      
      λ
      
      is the wavelength of the radiation transmitted by the filter attached to said channel;
      
      C_iis the concentration of chemical species i;
      
      l is the path length of radiative absorption;
      
      W_sand W_vrepresent the directional spectral radiant emission from the heat source and the detected chemical, respectively, and are defined by the Planck function at their respective temperatures; and
      
      ε
      
      _sis the spectral emissivity of the source.
  - 3. The method of claim 1, wherein said net normalized result is a signal that is reference corrected by subtracting it from a known reference signal and dividing the net result by the reference signal.
  - 4. The method of claim 1, wherein said unmixing by projecting finds the concentration of the chemicals or the magnitudes of the stimulants by a predetermined mathematical formula defined as:
    - $r_{j} \approx \sum_{i} a_{ij} x_{i}$ wherein;
      
      r_jis the net normalized result of channel j;
      
      x_iis a convenient short hand for the concentration of chemical i, or the optical depth of chemical i, or the magnitude of the i^thstimulant; and
      
      coefficients a_ijrepresent the contribution of the absorptivity of chemical i or the response created by the i^thstimulant to the net normalized result of channel j.
  - 20. The method of any one of claims 1, 5, and 17, wherein said sensor comprises at least one of a remote sensor and/or in-situ sensor.
  - 21. The method of claim 20, wherein said at least one remote sensor comprises an optical sensor.
  - 22. The method of claim 21, wherein said optical sensor comprises at least one of TOTALLY OPTICAL VAPOR ANALYZER (TOVA) type sensor, a differential radiometer absorption type sensor, a Fourier transform type spectrometer or radiometer, a tunable etalon type sensor, a grating based spectrometer type sensor, or a lidar type sensor, a differential absorption lidar (DIAL) type sensor, or any combination thereof.
  - 23. The method of claim 20, wherein said at least one in-situ sensor comprises at least one of the following types of sensors:
    - surface acoustic wave (SAW), micro-cantilever (MC), ELECTRONIC NOSE (EN) type sensor, chemi-resitor type sensor, gas chromatograph type sensor, interferometric type waveguide sensor, chemical paper type sensor, TOTALLY OPTICAL VAPOR ANALYZER (TOVA) type sensor, a differential absorption type sensor, a Fourier transform type spectrometer or radiometer, a tunable etalon type sensor, a grating based spectrometer type sensor, a lidar type sensor, a differential absorption lidar (DIAL) type sensor, or Ion Mobility Spectrometer (IMS), or any combination thereof.

5. A method for analyzing signals from a plurality of cross reactive channels of a sensor of a system, where the signals vary non-linearly with the quantity of stimulants that induce these signals wherein the said method comprises:
- obtaining signals from a plurality of channels from said sensor at a variety of known levels for each stimulant;
  
  producing a net normalized result from each signal from the known levels;
  
  creating a library of signatures by developing a parameterized equation that matches the net normalized result at each of the known levels of each stimulant in terms of the level of stimulant;
  
  obtaining one or more signals from a plurality of channels from said sensor at unknown levels of stimulants;
  
  producing a net normalized result from each signal from the unknown levels; and
  
  unmixing the net normalized result from the unknown levels by comparing it to the library of parameterized signatures.
- View Dependent Claims (6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16)
- - 6. The method of claim 5, wherein the match is determined by a best match.
  - 7. The method of claim 5, wherein the parameterized equation approximates the net normalized result at each of the known levels of each stimulant in terms of the level of stimulant.
  - 8. The method of claim 5, wherein said net normalized result is a signal that is reference corrected by a predetermined mathematical formula defined as:
    - $r = \frac{s (0) - s}{s (0)} = \frac{\int [(ɛ W_{s} - W_{v}) (1 - e^{- \sum_{i} α_{i} (λ) C_{i} })] τ_{f} (λ) \partial λ}{\int [ɛ_{s} W_{s} (λ)] τ_{f} (λ) \partial λ}$ wherein;
      
      r is the net normalized result;
      
      s is the signal for a given channel;
      
      s(0) is a reference signal for the given channel;
      
      τ
      
      (λ
      
      ) is the transmission by a radiation filter attached to said channel;
      
      α
      
      _i(λ
      
      ) is the absorptivity of chemical species i;
      
      λ
      
      is the wavelength of the radiation transmitted by the filter attached to said channel;
      
      C_iis the concentration of chemical species i;
      
      l is the path length;
      
      W_sand W_vrepresent the directional spectral radiant emission from the heat source and gas, respectively, and are defined by the Planck function at their respective temperatures; and
      
      ε
      
      _sis the spectral emissivity of the source.
  - 9. The method of claim 5, wherein said net normalized result is a signal that is reference corrected by subtracting a known reference signal and dividing the net result by the measured signal.
  - 10. The method of claim 5, wherein said parameterized equation of said net normalized results is defined as:
    - $r = r_{sat} (1 - \exp^{- \frac{m_{lin}}{r_{sat}} C})$ wherein;
      
      r is the net normalized result,r_satis the net normalized result at saturation defined as the point where increasing the optical depth or the magnitude of the stimulant no longer affects the response for the given chemical or stimulant,m_linis its slope at low concentrations or low magnitude of the stimulant, andC_il is the optical depth of the i^thchemical or the response created by the i^thstimulant.
  - 11. The method of claim 5, wherein said parameterized equation of said net normalized results is defined as:
    - $r = r_{sat} (1 - (1 - ɛ) \exp^{- \frac{m_{lin}}{r_{sat}} C} - ɛ \exp^{- σ \frac{m_{lin}}{r_{sat}} C})$ wherein;
      
      r is the net normalized result;
      
      r_satis the net normalized result at optical saturation defined as the point where increasing the optical depth no longer affects the response for the given chemical;
      
      m_linis its slope at low concentrations or low magnitude of stimulant;
      
      C_il is the optical depth of i^thchemical or the response created by the i^thstimulant;
      
      σ
      
      >
      
      1 is a parameter that accounts for the effects of the higher exponentials; and
      
      0≦
      
      ε
      
      ≦
      
      1.
  - 12. The method of any one claims 5, 10 and 11, wherein said unmixing is performed using a least-squares projection defined by the formula:
    - {circumflex over (x)}=(A′
      
      S^−
      
      1A)^−
      
      1A′
      
      S^−
      
      1{tilde over (r)}wherein;
      
      {tilde over (r)} is a vector of the terms ln(1−
      
      r_j/r_sat) of the net normalized result, where j refers to each channel;
      
      A is a matrix containing the terms m_lin/r_satfor each channel and each candidate chemical;
      
      A′
      
      is the transpose of matrix A;
      
      S is the noise covariance matrix of the system; and
      
      {circumflex over (x)} is a vector for the concentration of the chemicals included in the library, or the optical depth of chemical included in the library, or the magnitude of the stimulants included in the library.
  - 13. The method of claim 12, wherein the noise covariance matrix of the system is omitted and replaced with an identity matrix.
  - 14. The method of any one of claims 12 and 13, wherein a stimulant is identified by finding the distances between the vectors of the predicted net normalized results for each stimulant as determined from the values of {circumflex over (x)} and the corresponding vectors of the net normalized result as obtained by the sensor and selecting among those distances the shortest distance.
  - 15. The method of claim 13, wherein the distance between the vectors is the Mahalanobis distance, which is
    d_stimulant²=(r−
    - {circumflex over (r)})′
      
      S^−
      
      1(r−
      
      {circumflex over (r)})wherein;
      
      r is the vector of net normalized responses as obtained by the sensor;
      
      {circumflex over (r)} is the vector of the predicted net normalized responses as obtained by using the calculated values of {circumflex over (x)}; and
      
      S is the noise covariance matrix of the system.
  - 16. The method of claim 15, wherein the noise covariance matrix of the system is omitted and replaced with an identity matrix.

17. A method for analyzing signals from a plurality of cross reactive channels of a sensor of a system where the signals vary non-linearly with the magnitude of stimulants that induce these signals, wherein the said method comprises:
- obtaining signals from a plurality of channels from said sensor;
  
  producing a net normalized result from each signal;
  
  estimating initially the magnitude or magnitudes of stimulant(s);
  
  predicting a net normalized result from a model of the physics of the system by using said estimated magnitude or magnitudes of stimulant(s);
  
  determining the difference between the actual net normalized result and the predicted net normalized result;
  
  using said difference to solve for a correction to the estimated magnitude(s);
  
  updating the magnitude or magnitudes of stimulant(s); and
  
  repeating the prediction, determination of the difference, determination of the correction, and updating the magnitude(s).
- View Dependent Claims (18, 19)
- - 18. The method of claim 17, wherein said updating of the stimulants is determined by estimating the correction, δ
    - , to the magnitude by using a predetermined mathematical formula defined as;
      
      Aδ
      
      =r−
      
      {circumflex over (r)}_kwherein;
      
      r is a vector containing the actual net normalized result for each channel,{circumflex over (r)}_kis a vector containing the predicted net normalized result,A is a matrix containing the theoretical, linearized response of the system at the predicted net normalized results, {circumflex over (r)}_k, andwherein the magnitudes of the stimulants are updated by predetermined mathematical formula defined as;
      
      x_k+1=x_k+δ
      
      wherein x_kis the current estimate of the magnitude of the stimulant at iteration k, and x_k+1is the new estimate.
  - 19. The method of claim 18, wherein said solving for a correction is determined mathematically by using any suitable linear techniques.

24. A system for analyzing signals from a plurality of cross reactive channels of a sensor of the system where the signals vary non-linearly with the quantities of stimulant that induce these signals, wherein the said system comprises at least one data processor adapted to:
- obtain signals from a plurality of channels from said sensor at a variety of known levels for each stimulant;
  
  produce a net normalized result from each signal from the known levels;
  
  create a library of signatures that includes the net normalized result at each of the known levels of stimulant;
  
  obtain one or more signals from a plurality of channels from said sensor at unknown levels of stimulant;
  
  produce a net normalized result from each signal from the unknown levels; and
  
  unmix the net normalized result from the unknown levels by projecting the sensor response onto the library of known signatures, with the various levels of each stimulant being treated as independent stimulants.
- View Dependent Claims (25, 26, 27, 43, 44, 45, 46)
- - 25. The system of claim 24, wherein said net normalized result is a signal of a given channel that is reference corrected by a predetermined mathematical formula defined as:
    - $r = \frac{s (0) - s}{s (0)} = \frac{\int [(ɛ W_{s} - W_{v}) (1 - e^{- \sum_{i} α_{i} (λ) C_{i} })] τ_{f} (λ) \partial λ}{\int [ɛ_{s} W_{s} (λ)] τ_{f} (λ) \partial λ}$ wherein;
      
      r is the net normalized result;
      
      s is the signal for a given channel;
      
      s(0) is a reference signal for the given channel;
      
      τ
      
      (λ
      
      ) is the transmission by a radiation filter attached to said channel;
      
      α
      
      _i(λ
      
      ) is the absorptivity of chemical species i;
      
      λ
      
      is the wavelength of the radiation transmitted by the filter attached to said channel;
      
      C_iis the concentration of chemical species i;
      
      l is the path length of radiative absorption;
      
      W_sand W_vrepresent the directional spectral radiant emission from the heat source and the detected chemical, respectively, and are defined by the Planck function at their respective temperatures; and
      
      ε
      
      _sis the spectral emissivity of the source.
  - 26. The system of claim 24, wherein said net normalized result is a signal that is reference corrected by subtracting it from a known reference signal and dividing the net result by the reference signal.
  - 27. The system of claim 24, wherein said unmixing by projecting finds the concentration of the chemicals or the magnitudes of the stimulants by a predetermined mathematical formula defined as:
    - $r_{j} \approx \sum_{i} a_{ij} x_{i}$ wherein;
      
      r_jis the net normalized result of channel j;
      
      x_iis a convenient short hand for the concentration of chemical i, or the optical depth of chemical i, or the magnitude of the i^thstimulant; and
      
      coefficients a_ijrepresent the contribution of the absorptivity of chemical i or the response created by the i^thstimulant to the net normalized result of channel j.
  - 43. The system of any one of claims 24, 28, and 40, wherein said sensor comprises at least one of a remote sensor and/or in-situ sensor.
  - 44. The system of claim 43, wherein said at least one remote sensor comprises an optical sensor.
  - 45. The system of claim 44, wherein said optical sensor comprises at least one of TOTALLY OPTICAL VAPOR ANALYZER (TOVA) type sensor, a differential radiometer absorption type sensor, a Fourier transform type spectrometer or radiometer, a tunable etalon type sensor, a grating based spectrometer type sensor, or a lidar type sensor, a differential absorption lidar (DIAL) type sensor, or any combination thereof.
  - 46. The system of claim 43, wherein said at least one in-situ sensor comprises at least one of the following types of sensors:
    - surface acoustic wave (SAW), micro-cantilever (MC), ELECTRONIC NOSE (EN) type sensor, chemi-resitor type sensor, gas chromatograph type sensor, interferometric type waveguide sensor, chemical paper type sensor, TOTALLY OPTICAL VAPOR ANALYZER (TOVA) type sensor, a differential absorption type sensor, a Fourier transform type spectrometer or radiometer, a tunable etalon type sensor, a grating based spectrometer type sensor, a lidar type sensor, a differential absorption lidar (DIAL) type sensor, or Ion Mobility Spectrometer (IMS), or any combination thereof.

28. A system for analyzing signals from a plurality of cross reactive channels of a sensor of the system, where the signals vary non-linearly with the quantity of stimulants that induce these signals wherein the said system comprises at least one data processor adapted to:
- obtain signals from a plurality of channels from said sensor at a variety of known levels for each stimulant;
  
  produce a net normalized result from each signal from the known levels;
  
  create a library of signatures by developing a parameterized equation that matches the net normalized result at each of the known levels of each stimulant in terms of the level of stimulant;
  
  obtain one or more signals from a plurality of channels from said sensor at unknown levels of stimulants;
  
  produce a net normalized result from each signal from the unknown levels; and
  
  unmix the net normalized result from the unknown levels by comparing it to the library of parameterized signatures.
- View Dependent Claims (29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39)
- - 29. The system of claim 28, wherein the match is determined by a best match.
  - 30. The system of claim 28, wherein the parameterized equation approximates the net normalized result at each of the known levels of each stimulant in terms of the level of stimulant.
  - 31. The system of claim 28, wherein said net normalized result is a signal that is reference corrected by a predetermined mathematical formula defined as:
    - $r = \frac{s (0) - s}{s (0)} = \frac{\int [(ɛ W_{s} - W_{v}) (1 - e^{- \sum_{i} α_{i} (λ) C_{i} })] τ_{f} (λ) \partial λ}{\int [ɛ_{s} W_{s} (λ)] τ_{f} (λ) \partial λ}$ wherein;
      
      r is the net normalized result;
      
      s is the signal for a given channel;
      
      s(0) is a reference signal for the given channel;
      
      τ
      
      (λ
      
      ) is the transmission by a radiation filter attached to said channel;
      
      α
      
      _i(λ
      
      ) is the absorptivity of chemical species i;
      
      λ
      
      is the wavelength of the radiation transmitted by the filter attached to said channel;
      
      C_iis the concentration of chemical species i;
      
      l is the path length;
      
      W_sand W_vrepresent the directional spectral radiant emission from the heat source and gas, respectively, and are defined by the Planck function at their respective temperatures; and
      
      ε
      
      _sis the spectral emissivity of the source.
  - 32. The system of claim 28, wherein said net normalized result is a signal that is reference corrected by subtracting a known reference signal and dividing the net result by the measured signal.
  - 33. The system of claim 28, wherein said parameterized equation of said net normalized results is defined as:
    - $r = r_{sat} (1 - \exp^{- \frac{m_{lin}}{r_{sat}} C})$ wherein;
      
      r is the net normalized result,r_satis the net normalized result at saturation defined as the point where increasing the optical depth or the magnitude of the stimulant no longer affects the response for the given chemical or stimulant,m_linis its slope at low concentrations or low magnitude of the stimulant, andC_il is the optical depth of the i^thchemical or the response created by the i^thstimulant.
  - 34. The system of claim 28, wherein said parameterized equation of said net normalized results is defined as:
    - $r = r_{sat} (1 - (1 - ɛ) \exp^{- \frac{m_{lin}}{r_{sat}} C} - ɛ \exp^{- σ \frac{m_{lin}}{r_{sat}} C})$ wherein;
      
      r is the net normalized result;
      
      r_satis the net normalized result at optical saturation defined as the point where increasing the optical depth no longer affects the response for the given chemical;
      
      m_linis its slope at low concentrations or low magnitude of stimulant;
      
      C_il is the optical depth of i^thchemical or the response created by the i^thstimulant;
      
      σ
      
      >
      
      1 is a parameter that accounts for the effects of the higher exponentials; and
      
      0≦
      
      ε
      
      ≦
      
      1.
  - 35. The system of any one claims 28, 33 and 34, wherein said unmixing is performed using a least-squares projection defined by the formula:
    - {circumflex over (x)}=(A′
      
      S^−
      
      1A)^−
      
      1A′
      
      S^−
      
      1{tilde over (r)}wherein;
      
      {tilde over (r)} is a vector of the terms ln(1−
      
      r_j/r_sat) of the net normalized result, where j refers to each channel;
      
      A is a matrix containing the terms m_lin/r_satfor each channel and each candidate chemical;
      
      A′
      
      is the transpose of matrix A;
      
      S is the noise covariance matrix of the system; and
      
      {circumflex over (x)} is a vector for the concentration of the chemicals included in the library, or the optical depth of chemical included in the library, or the magnitude of the stimulants included in the library.
  - 36. The system of claim 35, wherein the noise covariance matrix of the system is omitted and replaced with an identity matrix.
  - 37. The system of any one of claims 35 and 36, wherein a stimulant is identified by finding the distances between the vectors of the predicted net normalized results for each stimulant as determined from the values of {circumflex over (x)} and the corresponding vectors of the net normalized result as obtained by the sensor and selecting among those distances the shortest distance.
  - 38. The system of claim 36, wherein the distance between the vectors is the Mahalanobis distance, which is
    d_stimulant²=(r−
    - {circumflex over (r)})′
      
      S^−
      
      1(r−
      
      {circumflex over (r)})wherein;
      
      r is the vector of net normalized responses as obtained by the sensor;
      
      {circumflex over (r)} is the vector of the predicted net normalized responses as obtained by using the calculated values of {circumflex over (x)}; and
      
      S is the noise covariance matrix of the system.
  - 39. The system of claim 38, wherein the noise covariance matrix of the system is omitted and replaced with an identity matrix.

40. A system for analyzing signals from a plurality of cross reactive channels of a sensor of the system where the signals vary non-linearly with the magnitude of stimulants that induce these signals, wherein the said system comprises at least one data processor adapted to:
- obtain signals from a plurality of channels from said sensor;
  
  produce a net normalized result from each signal;
  
  estimate initially the magnitude or magnitudes of stimulant(s);
  
  predict a net normalized result from a model of the physics of the system by using said estimated magnitude or magnitudes of stimulant(s);
  
  determine the difference between the actual net normalized result and the predicted net normalized result;
  
  use said difference to solve for a correction to the estimated magnitude(s);
  
  update the magnitude or magnitudes of stimulant(s); and
  
  repeat the prediction, determination of the difference, determination of the correction, and updating the magnitude(s).
- View Dependent Claims (41, 42)
- - 41. The system of claim 40, wherein said updating of the stimulants is determined by estimating the correction, δ
    - , to the magnitude by using a predetermined mathematical formula defined as;
      
      Aδ
      
      =r−
      
      {circumflex over (r)}_kwherein;
      
      r is a vector containing the actual net normalized result for each channel,{circumflex over (r)}_kis a vector containing the predicted net normalized result,A is a matrix containing the theoretical, linearized response of the system at the predicted net normalized results, {circumflex over (r)}_k, andwherein the magnitudes of the stimulants are updated by predetermined mathematical formula defined as;
      
      x_k+1=x_k+δ
      
      wherein x_kis the current estimate of the magnitude of the stimulant at iteration k, and x_k+1is the new estimate.
  - 42. The system of claim 41, wherein said solving for a correction is determined mathematically by using any suitable linear techniques.

47. A computer program product comprising a computer useable medium having computer program logic for enabling one processor in a computer system to analyze signals from a plurality of cross reactive channels of a sensor of a system where the signals vary non-linearly with the quantities of stimulant that induce these signals, said computer program logic comprises:
- obtaining signals from a plurality of channels from said sensor at a variety of known levels for each stimulant;
  
  producing a net normalized result from each signal from the known levels;
  
  creating a library of signatures that includes the net normalized result at each of the known levels of stimulant;
  
  obtaining one or more signals from a plurality of channels from said sensor at unknown levels of stimulant;
  
  producing a net normalized result from each signal from the unknown levels; and
  
  unmixing the net normalized result from the unknown levels by projecting the sensor response onto the library of known signatures, with the various levels of each stimulant being treated as independent stimulants.

48. A computer program product comprising a computer useable medium having computer program logic for enabling one processor in a computer system to analyze signals from a plurality of cross reactive channels of a sensor of a system, where the signals vary non-linearly with the quantity of stimulants that induce these signals, said computer program logic comprises:
- obtaining signals from a plurality of channels from said sensor at a variety of known levels for each stimulant;
  
  producing a net normalized result from each signal from the known levels;
  
  creating a library of signatures by developing a parameterized equation that matches the net normalized result at each of the known levels of each stimulant in terms of the level of stimulant;
  
  obtaining one or more signals from a plurality of channels from said sensor at unknown levels of stimulants;
  
  producing a net normalized result from each signal from the unknown levels; and
  
  unmixing the net normalized result from the unknown levels by comparing it to the library of parameterized signatures.

49. A computer program product comprising a computer useable medium having computer program logic for enabling one processor in a computer system to analyze signals from a plurality of cross reactive channels of a sensor of a system where the signals vary non-linearly with the magnitude of stimulants that induce these signals, said computer program logic comprises:
- obtaining signals from a plurality of channels from said sensor;
  
  producing a net normalized result from each signal;
  
  estimating initially the magnitude or magnitudes of stimulant(s);
  
  predicting a net normalized result from a model of the physics of the system by using said estimated magnitude or magnitudes of stimulant(s);
  
  determining the difference between the actual net normalized result and the predicted net normalized result;
  
  using said difference to solve for a correction to the estimated magnitude(s);
  
  updating the magnitude or magnitudes of stimulant(s); and
  
  repeating the prediction, determination of the difference, determination of the correction, and updating the magnitude(s).

Specification

Analysis Methods for unmixing the response of non-linear, cross-reactive sensors and related system to single and multiple stimulants

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Analysis Methods for unmixing the response of non-linear, cross-reactive sensors and related system to single and multiple stimulants

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