Spectroscopic instrument calibration

In an indirect method for determining a property-p for a sample-u, using spectral data-k measured by an instrument-k, capable of determining such spectral data-k for said sample-u, in combination with a calibration equation-k, having at least one constant-k, a dependent variable-k, and at least one independent variable-k;

• wherein each value for said dependent variable-k is determined specifically by particular values for each said at least one independent variable-k, and wherein each said at least one independent variable-k has a value-k equal to a spectral feature-k, functionally determined by said spectral data-k, wherein the improvement comprises:
• (1) Defining said calibration equation-k by recourse to a calibration equation-i for instrument-i, wherein said calibration equation-i has at least one constant-i;
• at least one independent variable-i;
• and a dependent variable-i;
• wherein each value for said dependent variable-i is determined specifically by specific values for each said at least one independent variable-i, and wherein each said at least one independent variable-i has a value-i equal to a spectral feature-i, functionally determined from spectral data-i, and wherein calibration equation-i and -k determine, respectively, values for each said dependent variable-k and-i, that are functionally related to one another by a function-f that transforms dependent variable-i into dependent variable-k;
• for property-p;
• said method for defining said calibration equation-k by recourse to said calibration equation-i comprising:
• (a) obtaining a calibration equation for said instrument-i and assuming a calibration equation-k for said instrument-k has the following form:
• Y.sub.s.sup.k =A.sup.k +.SIGMA..sub.j {B.sub.j.sup.k X.sub.js.sup.k } and calibration equation-i has the same or different form:
• where:
• Y.sub.s.sup.i is a dependent variable-i in calibration equation-i, and Y.sub.s.sup.k is functionally related to Y.sub.s.sup.i by a function-f;
• A.sup.k and A.sup.i are each intercept constants-i and -k in calibration equations-i and -k, respectively;
• X.sub.js.sup.i and X.sub.js.sup.k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, having a value equal to at least one said spectral feature-i and -k, respectively, of a member s, at wavelength-j;
• and
• B.sub.j.sup.i and B.sub.j.sup.k are slope constants-i and -k, respectively, corresponding to coefficients for each of said at least one of independent variable-i and -k at wavelength-j, respectively;
• (b) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-c comprising at least one member c, thereby uniquely determining value-k and -i, respectively, equal to said at least one spectral feature-k and-i, respectively, for each member of sample-c used in step (c);
• (c) determining said intercept constant-k and said at least one slope constant-k corresponding respectively to A.sup.k and B.sub.j.sup.k which substantially minimizes the following expression:
• .SIGMA..sub.c (Y.sub.c.sup.i -A.sup.k -.SIGMA..sub.j B.sub.j.sup.k X.sub.jc.sup.k).sup.2 ;
• whereby a calibration equation-k for instrument-k is obtained by reference to said calibration equation for instrument-i;
• and
• (2) obtaining spectral data-u for said sample-u by means of instrument-k, and determining functionally a value-u for each spectral feature-u therein;
• wherein each said value-u is used in combination with said calibration equation-k to determine a value for dependent variable-k, Y.sub.u.sup.k, for sample-u which is functionally related by the inverse of function-f to said property-p of said sample-u.

In an indirect method for determining a property-p for a sample-u, using spectral data-k measured by an instrument-k, capable of determining such spectral data-k for said sample-u, in combination with a calibration equation-k, having at least one constant-k, a dependent variable-k, and at least one independent variable-k;

• wherein each value for said dependent variable-k is determined specifically by specific values for each said at least one independent variable-k, and wherein each said at least one independent variable-k has a value-k equal to a spectral feature-k, functionally determined by said spectral data-k;
• wherein the improvement comprises:
• (1) defining said calibration equation-k by recourse to a calibration equation-i for instrument-i, wherein said calibration equation-i has:
• at least one constant-i comprising an intercept constant-i and at least one slope constant-i;
• at least one independent variable-i;
• and a dependent variable-i;
• wherein each value for said dependent variable-i is determined specifically by specific values for each said at least one independent variable-i, and wherein each said at least one independent variable-i has a value-i equal to a spectral feature-i, functionally determined from spectral data-i, and wherein calibration equation-i and -k determine, respectively, values for each said dependent variable-k and -i, that are functionally related to one another by a function-f that transforms dependent variable-i into dependent variable-k;
• said method for defining said calibration equation-k by recourse to said calibration equation-i comprising:
• (a) obtaining a calibration equation for said instrument-i for property-p, and assuming a calibration equation-k for said instrument-k;
• wherein each have the following form:
• Y.sub.s.sup.i =A.sup.i +.SIGMA..sub.j {B.sub.j.sup.i X.sub.js.sup.i }
• and
• Y.sub.s.sup.k =A.sup.k +.SIGMA..sub.j {B.sub.j.sup.k X.sub.js.sup.k }
• where:
• Y.sub.s.sup.i is the dependent variable-i for member s in calibration equation-i and Y.sub.s.sup.k is functionally related to Y.sub.s.sup.i by a function-f;
• A.sup.k and A.sup.i are each intercept constants-i and -k in calibration equations-i and -k, respectively;
• X.sub.js.sup.i and X.sub.js.sup.k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, each having a value equal to at least one said spectral feature-i and -k, respectively, of each member s, of sample-s at wavelength-j;
• and
• B.sub.j.sup.i and B.sub.j.sup.k are slope constants-i and -k, respectively, corresponding to coefficients for each of said at least one of independent variables-i and -k at wavelength-j, respectively;
• (b) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-c comprising at least one member c, and functionally determining value-k and -i, respectively, equal to said at least one spectral feature-k and -i, respectively, for each member of sample-c used in step (c);
• (c) determining values of B.sub.j.sup.k and A.sup.k so that the following two equations are simultaneously substantially satisfied:
• B.sub.j.sup.k =B.sub.j.sup.i .SIGMA..sub.c X.sub.jc.sup.i X.sub.jc.sup.k /.SIGMA..sub.c (X.sub.jc.sup.k).sup.2 and A.sup.k =.SIGMA..sub.c (Y.sub.c.sup.i -.SIGMA..sub.j B.sub.j.sup.k X.sub.jc.sup.k)/n;
• where n=number of members to sample-c;
• whereby a calibration equation-k for instrument-k is obtained by reference to said calibration equation for instrument-i;
• and
• (2) obtaining spectral data-u by means of instrument-k for said sample-u, and functionally determining a value-u for each spectral features-u therein;
• wherein each said value-u is used in combination with said calibration equation-k to determine a value for dependent variable-k, Y.sub.u.sup.k, for sample-u which is functionally related by the inverse of function-f to said property-p of said sample-u.

In an indirect method for determining a property-p for a sample-u, using spectral data-k measured by an instrument-k, capable of determining such spectral data-k for said sample-u in combination with a calibration equation-k, having at least one constant-k, a dependent variable-k, and at least one independent variable-k;

• wherein each value for said dependent variable-k is determined specifically by specific values for each said at least one independent variable-k, and wherein each said at least one independent variable-k has a value-k equal to a spectral feature-k, uniquely determined by reference to said spectral data-k;
• wherein the improvement comprises:
• (1) defining said calibration equation-k by recourse to a calibration equation-i for instrument-i, wherein said calibration equation-i has:
• at least one constant-i;
• at least one independent variable-i;
• and a dependent variable-i;
• wherein each value for said dependent variable-i is determined specifically by specific values for each said at least one independent variable-i, and wherein each said at least one independent variable-i has a value-i equal to a spectral feature-i, functionally determined from spectral data-i, and wherein calibration equation-i and -k determine, respectively, values for each said dependent variable-k and -i, that are functionally related to one another by a function-f that transforms dependent variable-i into dependent variable-k;
• said method for defining said calibration equation-k by recourse to said calibration equation-i comprising:
• (a) obtaining a calibration equation for said instrument-i for property-p, and assuming a calibration equation-k for said instrument-k wherein each have the following form:
• Y.sub.s.sup.i =A.sup.i +.SIGMA..sub.j {B.sub.j.sup.i X.sub.js.sup.i }
• and
• Y.sub.s.sup.k =A.sup.k +.SIGMA..sub.j {B.sub.j.sup.k X.sub.js.sup.k }
• where:
• Y.sub.s.sup.i is the dependent variable-i in calibration equation-i, and Y.sub.s.sup.k is functionally related to Y.sub.s.sup.i by a function-f;
• A.sup.k and A.sup.i are each intercepts constants-i and -k in calibration equations-i and -k, respectively;
• X.sub.js.sup.i and X.sub.js.sup.k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, each having a value equal to at least one said spectral feature-i and -k, respectively;
• and
• B.sub.j.sup.i and B.sub.j.sup.k are slope constants-i and -k, respectively, corresponding to coefficients for each of said at least one of independent variables-i and -k corresponding to spectral features-i and -k at least one wavelength-j, respectively;
• (b) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-s comprising at least one member c, used in step (c), thereby uniquely determining value-k and -i, respectively, equal to said at least one spectral feature-k and -i, respectively, for each member of sample-c having a total of n members;
• (c) determining said intercept constant-k and said at least one slope constant-k corresponding, respectively, to values of B.sub.j.sup.k and A.sup.k so that the following are substantially satisfied:
• B.sub.j.sup.k =B.sub.j.sup.i .SIGMA..sub.c X.sub.jc.sup.i /.SIGMA..sub.c (X.sub.jc.sup.k) and A.sup.k =.SIGMA..sub.c (Y.sub.c.sup.i -.SIGMA..sub.j B.sub.j.sup.k X.sub.jc.sup.k)/n;
• whereby a calibration equation-k for instrument-k is obtained by reference to said calibration equation for instrument-i;
• (2) obtaining spectral data-u for said sample-u by means of instrument-k, and determining functionally a value-u for each spectral feature-u therein;
• wherein each said value-u is used in combination with said calibration equation-k to determine a value for dependent variable-k, Y.sub.u.sup.k, for sample-u which is functionally related by the inverse of function-f to said property-p of said sample-u.

In a method for recalibrating an instrument-k in light of its former calibrated condition as instrument-i or for calibrating instrument-k, previously uncalibrated, in light of a calibrated instrument-i wherein a calibrated instrument is one having a particular calibration equation, here calibration equation-k, based upon which equation spectraldata of a sample can be used to predict a physical or chemical property of said sample, the improvement which comprises:

• determining said calibration equation-k by means of a calibration equation-i, each said calibration equation-k and -i, respectively, having:
• at least one constant-k and -i;
• a dependent variable-k and -i;
• and at least one independent variable-k and -i;
• wherein each value, respectively, for said dependent variable-k and -i is determined specifically by particular values for each said at least one independent variable-k and -i and wherein each said at least one independent variable-k and -i has respectively a value-k and -i equal to a spectral feature-k and -i, functionally determined, respectively, by spectral data-k and -i, determined respectively, by an instrument-k and -i capable of determining, respectively, said spectral data-k and -i for a sample, s;
• said method comprising:
• (a) obtaining respectively with instrument-i and instrument-k spectral data-i and spectral data-k, respectively, for each member of a sample-c, comprising at least one member c, thereby functionally determining value-k and value-i, respectively, equal to said at least one spectral feature-k and said at least one spectral feature-i, respectively, for each member of sample-c used in step (b);
• (b) determining in accordance with calibration equation-i, a value for said dependent variable-i for each member of sample-c and in place of each said value of said dependent variable-k in calibration equation-k inserting a value which is related by a function-f to each said value for said dependent variable-i and selecting an appropriate value for each said at least one constant-k in said calibration equation-k to produce a calibration equation-k;
• wherein said calibration equation-k in conjunction with said at least one spectral feature-k and the inverse of function-f functionally define each predicted value corresponding to each dependent variable-k for each member c, of sample-c, wherein a sum of absolute differences between each said predicted value and said dependent variable-i for each member c, of sample-c is minimized at least so that said calibration equation-k predicts a value for property-p of each unknown sample that has a value for property-p within a range of interest with a standard error of prediction not substantially greater than a standard error value selected from the group consisting of a standard error of estimate and a standard error of prediction wherein at least one of said standard errors is necessarily present in a direct method used to determine values for property-p to obtain directly or indirectly calibration equation-i.

In the method of 32, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

• determining values for m and b, which substantially satisfy the following expressions:
• m={(.SIGMA..sub.c Y.sub.c.sup.a .SIGMA.Y.sub.c.sup.k)/I.SIGMA..sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{.SIGMA..sub.c Y.sub.c.sup.k .SIGMA..sub.c Y.sub.c.sup.k }
• /I-.SIGMA..sub.c Y.sub.c.sup.k Y.sub.c.sup.k }
• b={.SIGMA..sub.c Y.sub.c.sup.a -m.SIGMA..sub.c Y.sub.c.sup.k }/I
• where:
• Y.sub.c.sup.a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;
• I is the number of members of sample-p;
• substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows:
• Y.sub.c.sup.k ={mA.sup.k +b}+.SIGMA..sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };
• whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

In an indirect method for determining a property-p for a sample-u by measuring and using spectral data-(ku) measured by an instrument-k capable of measuring such spectral data-(ku) for said sample-u in conjunction with a calibration equation-k,

• wherein said calibration equation-k defines a dependent-variable-(ku), in terms of at least one constant-k and at least one independent-variable-(kju),
• wherein said at least one independent-variable-(kju) has a value equal to at least one spectral-feature-(kju) determined from said spectral data-(ku) for said sample-u by a first functions that involves wavelength-j,
• and wherein said dependent-variable-(ku) has a value substantially equal to said property-p for said sample-u when values for each said at least one constant-k and each said at least one independent-variable-(kju) are substituted appropriately into said calibration equation-k;
• wherein the improvement comprises:
• determining a value for each said at least one constant-k by recourse to a calibration equation-i,
• wherein said calibration equation-i defines dependent-variable-(ic) as a function of an independent-variable-(ijc) which independent-variable-(ijc) is equal to a spectral feature-(ijc) functionally determined from spectral data-(ic) corresponding both to member c of a calibration sample-c having a total of n members and to each wavelength-j, and said dependent-variable-(ic) has a value substantially equal to said property-p of said member c, of said sample-c;
• said determining a value for each said at least one constant-k comprises the steps of:
• (1) using respectively spectral data-(ic) and spectral data-(kc) for each member c, of said calibration sample-c measured respectively by instrument-i and instrument-k, to determine each said at least one independent-variable-(kjc) corresponding to each member c, and to determine from said calibration equation-i a value for said dependent-variable-(ic) corresponding to each member c;
• (2) maintaining consistent a set of equations that results from substituting in place of the value for dependent-variable-(kc) that value for dependent-variable-(ic), while selecting a value for each said at least one constant-k so that a sum of absolute differences in value between each new-dependent-variable-(kc) and dependent-variable-(ic) corresponding to each said member c, is substantially minimized;
• wherein said new-dependent-variable-(kc) is determined by calibration equation-k after substituting each selected value for each said at least one constant-k along with each said at least one independent-variable-(kjc) in said calibration equation-k.

In a method for recalibrating an instrument-k in light of its former calibrated condition as instrument-i or for calibrating instrument-k, previously uncalibrated, in light of a calibrated instrument-i wherein a calibrated instrument is one having a particular calibration equation, here calibration equation-k, based upon which equation spectral data of a sample can be used to predict a physical or chemical property of said sample, the improvement which comprises:

• determining said calibration equation-k for instrument-k by means of said calibration equation-i for instrument-i, wherein said calibration equation-k defines a dependent-variable-(ku), in terms of at least one constant-k and at least one independent-variable-(kju), wherein said at least one independent-variable-(kju) has a value equal to at least one spectral feature-(kju) determined from said spectral data-(ku) measured by said instrument-k for said sample-u by a first function that involves wavelength-j, and wherein said dependent-variable-(ku) has a value substantially equal to a property-p for said sample-u when values for each said at least one constant-k and each said at least one independent-variable-(kju) are substituted appropriately into said calibration equation-k;
• wherein said calibration equation-i defines dependent-variable-(iu) as a function of an independent-variable-(iju) which independent-variable-(iju) is equal to a spectral feature-(iju) determined from spectral data-(iu) measured by said instrument-i for said sample-u corresponding to each wavelength-j, and has a value substantially equal to said property-p of said sample-u;
• said method for determining said calibration equation-k by means of said calibration equation-i comprising:
• (1) using respectively spectral data-(ic) and spectral data-(kc) for each member c, of a group of samples having n members that are each measured respectively by said instrument-i and instrument-k, to determine each said at least one independent-variable-(ijc) and independent-variable-(kjc), respectively, involving each wavelength-j corresponding to each member c, and to determine from calibration equation-i a value for said dependent-variable-(ic) corresponding to each member c;
• (2) maintaining consistent a set of equations that results from substituting in place of the value for dependent-variable-(kc) that value for dependent-variable-(ic), while selecting a value for each said at least one constant-k so that a sum of absolute differences between each new-dependent-variable-(kc) and dependent-variable-(ic) corresponding to each said member c, is minimized;
• wherein said each new-dependent-variable-(kc) is determined by calibration equation-k after substituting each selected value for each said at least one constant-k along with each said at least one independent-variable-(kjc) in said calibration equation-k.

A method for calibrating or recalibrating a first instrument, instrument-k, by reference to a second instrument, instrument-i, comprising in any order:

• (a) obtaining spectral data by means of instrument-k and instrument-i, corresponding respectively to data-k and data-i for each member of a calibration set;
• (b) determining by means of a calibration equation, calibration equation-i, of said instrument-i, a value, value-i, for a dependent variable, dependent variable-i, corresponding to each member of said calibration set;
• (c) substituting a value, value-f, related by function-f to said value-i for said dependent variable-i, for each member of the calibration set in place of that value for said dependent variable-k in a calibration equation-k which contains constants-k and a dependent variable, dependent variable-k;
• (d) selecting values for said constants-k in calibration equation-k so that the sum of absolute differences between:
• (1) each dependent variable-k calculated based on said calibration-k with said selected constants-k and data-k for each member of said calibration set;
• and
• (2) corresponding values for identical members of said calibration set determined by function-f and corresponding value-i for dependent variable-i of each said member;
• is substantially minimized;
• whereby said constants-k can be inversely transformed by an inverse to function-f to define calibration equation-k.

A method for determining constants in a calibration equation, calibration equation-1, for a first instrument by recourse to a second instrument having a calibration equation, calibration equation-2, comprising in any order:

• (a) determining from calibration equation-2 a value for each dependent variable-2 corresponding to each member of a calibration set based on spectral data-2 for each member of said calibration set measured by means of instrument-2;
• and
• (b) determining constants-1 in said calibration equation-1 so that the sum of absolute differences between:
• (1) said value for each said dependent variable-2 for each member of said calibration set;
• and
• (2) that corresponding value of dependent variable-1 calculated for each said member based upon calibration equation-1;
• is substantially minimized.