Model for spectral and chromatographic data

A method of detelaring whether a sample matches a reference species, the method comprising:

• selecting N indices l1, l2, . . . lN of peaks in an indexed data set characterizing the reference species;
• selecting a first set of probabilities p1, p2, . . . pN that peaks will occur at indices l1, l2, . . . lN of an indexed data set that characterizes the sample when the sample matches the reference species;
• selecting a first set of probability density functions gi(yi;
• θi) that characterize a measurable feature yi of the peak at index li given the presence of a peak at index li of a data set that characterizes the sample when the sample matches the reference species;
• selecting a second set of probabilities q1, q2, . . . qN that peaks will occur at indices l1, l2, . . . lN of an indexed data set that characterizes the sample when the sample does not match the reference species;
• selecting a second set of probability density functions gi(yi;
• Ωi) that characterize the measurable feature yi of the peak at index li given the presence of a peak at index li of a data set that characterizes the sample when the sample does not match the reference species;
• selecting a threshold Kc;
• obtaining an indexed observation data set x1, x2, . . . xN where xi∈{0, 1} and xi=1 if and only if a peak is present in the sample at li;
• obtaining a feature data set yi, y2, . . . yN;
• and
• deciding that the sample matches the reference species if λ≦Kc where λ = ∑ i = 1 N ⁢ ⁢ [ log ⁢ ⁢ 1 - p i 1 - q i + x i ⁢ { log ⁢ ⁢ p i ⁢ ( 1 - q i ) q i ⁢ ( 1 - p i ) + log ⁢ ⁢ g i ⁢ ( y i ;
• θ i ) g i ⁢ ( y i ;
• Ω i ) } ] ;
• and
• deciding that the sample does not match the reference species if λ>Kc.

A method, wherein the status of a process at any point t in time is characterized by an indexed observation data set Xt={x1,t, x2,t, . . . xN,t}, where xj,t ∈{0, 1} and xj,t=1 if and only if a peak is present at time t in the sample at index lj, the method comprising:

• selecting a first set of probabilities p1, p2, . . . pN that peaks will occur at x1,t, x2,t, . . . xN,t, respectively, when the process is operating normally;
• selecting a second set of probabilities q1, q2, . . . qN that peaks will occur at x1,t, x2,t, . . . xN,t, respectively, when the process is not operating normally;
• acquiring a sequence X1, X2, . . . XT of indexed observation data sets;
• intervening in the process when it is determined that Cn equals or exceeds a predetermined value A, where
• C0=0;
• Cn=Sn−min1≦j≦n {Sj} for n≧1;
• and
• S n = ∑ 1 ≤ j ≤ n ⁢ ⁢ log ⁡ ( 1 - p j 1 - q j ) + ∑ 1 ≤ j ≤ n ⁢ ⁢ x j , t ⁢ log ⁡ [ p j ⁡ ( 1 - q j ) q j ⁡ ( 1 - p j ) ] .

A method,

• wherein the status of a process at any point t in time is characterized by
• an indexed observation data set Xt={x1,t, x2,t, . . . xN,t}, where xj,t ∈{0, 1} and xj,t=1 if and only if a peak is present at time t in the sample at index lj, and
• a feature data set Yt={y1,t, y2,t, . . . yN,t}, where if xj,t=0, yj,t=0, and if xj,t=1, yj,t quantifies a feature of the peak at time t in the sample at index lj,
• the method comprising:
• selecting a first set of probabilities p1, p2, . . . pN that peaks will occur at x1,t, x2,t, . . . xN,t, respectively, when the process is operating normally;
• selecting a first set of probability density functions gi(yi;
• θi) that characterize a measurable feature yi of the peak at index li given the presence of a peak at index li of a data set that characterizes the process when it is operating normally;
• selecting a second set of probabilities q1, q2, . . . qN that peaks will occur at x1,t, x2,t, . . . xN,t, respectively, when the process is not operating normally;
• selecting a second set of probability density functions gi(yi;
• Ωi) that characterize the measurable feature yi of the peak at index li given the presence of a peak at index li of a data set that characterizes the process when it is operating normally;
• acquiring a sequence X1, X2, . . . XT of indexed observation data sets;
• acquiring a sequence Y1, Y2, . . . YT of feature data sets;
• intervening in the process when it is determined that Cn equals or exceeds a predetermined value A, where
• C0=0;
• Cn=Sn−min1≦j≦n{Sj} for n≧1;
• and
• S n = ∑ i = 1 N ⁢ ⁢ [ log ⁢ ⁢ 1 - p i 1 - q i + x i ⁢ { log ⁢ ⁢ p i ⁡ ( 1 - q i ) q i ⁡ ( 1 - p i ) + log ⁢ ⁢ g i ⁡ ( y i ;
• θ i ) g i ⁡ ( y i ;
• Ω i ) } ] .

A system for analyzing a sample in comparison with a reference species, comprising:

• a processor;
• a memory storing data indicative of:
• probabilities p1, p2, . . . pN that peaks will occur at indices l1, l2, . . . lN of an indexed data set that characterizes the sample when the sample matches the reference species;
• probabilities q1, q2, . . . qN that peaks will occur at indices l1, l2, . . . lN of an indexed data set that characterizes the sample when the sample does not match the reference species;
• a threshold value;
• and
• an indexed sample data set x1, x2, . . . xN characterizing the sample, wherein each xi is a binary value that indicates whether or not a peak is present at index li;
• and
• a computer-readable medium encoded with programming instructions executable by said processor to:
• calculate a log-likelihood ratio λ, where λ = ∑ 1 ≤ j ≤ N ⁢ log ⁡ ( 1 - p 1 - q j ) + ∑ 1 ≤ j ≤ N ⁢ x j ⁢ log ⁡ [ p j ⁡ ( 1 - q j ) q j ⁡ ( 1 - p j ) ] ;
• generate a first signal when λ is less than said threshold value;
• and
• generate a second signal when λ is greater than said threshold value.

A method of performing discriminant analysis, the method comprising:

• selecting N indices l1, l2, . . . lN of peaks in an indexed data set characterizing a first reference species or a second reference species;
• selecting a first set of probabilities p1,1, p2,1, . . . pN,1 that peaks will occur at indices l1, l2, . . . lN, respectively, of an indexed data set that characterizes the sample when the sample matches the first reference species;
• selecting a second set of probabilities p1,2, p2,2, . . . pN,2 that peaks will occur at indices l1, l2, . . . lN, respectively, of an indexed data set that characterizes the sample when the sample matches the second reference species;
• selecting a third set of probabilities q1,1, q2,1, . . . qN,1 that peaks will occur at indices l1, l2, . . . lN, respectively, of an indexed data set that characterizes the sample when the sample matches a second reference species;
• selecting a fourth set of probabilities q1,2, q2,2, . . . qN,2 that peaks will occur at indices l1, l2, . . . lN, respectively, of an indexed data set that characterizes the sample when the sample matches a second reference species;
• obtaining an indexed observation data set x1, x2, . . . xN, where xj∈{0, 1} and xj=1 if and only if a peak is present in the sample at lj;
• calculating λ 1 = ∑ 1 ≤ j ≤ N ⁢ log ⁡ ( 1 - p j , 1 1 - q j , 1 ) + ∑ 1 ≤ j ≤ N ⁢ x j ⁢ log ⁡ [ p j , 1 ⁡ ( 1 - q j , 1 ) q j , 1 ⁡ ( 1 - p j , 1 ) ] ⁢ ⁢ and λ 2 = ∑ 1 ≤ j ≤ N ⁢ log ⁡ ( 1 - p j , 2 1 - q j , 2 ) + ∑ 1 ≤ j ≤ N ⁢ x j ⁢ log ⁡ [ p j , 2 ⁡ ( 1 - q j , 2 ) q j , 2 ⁡ ( 1 - p j , 2 ) ] ;
• and
• deciding that
• the sample matches the first reference species if λ1≦λ2;
• and
• the sample matches the second reference species if λ1>λ2.

A method of performing a cluster analysis of M samples, comprising:

• selecting N indices l1, l2, .. lN of possible peak locations in indexed data sets characterizing the M samples;
• obtaining indexed data sets Xi={x1,i, x2,i, . . . xN,i}:i=1, 2, . . . M, each data set corresponding to a different sample, wherein xj,i={0, 1} and xj,i=1 if and only if a peak exists in the data set for sample i at index lj;
• and
• defining P groups of samples by
• selecting a first array of probabilities pk,i:
• k=1, 2, . . . P;
• i=1, 2, . . . N that peaks will occur at indices l1, l2, . . . lN, respectively, of an indexed data set that characterizes a sample when the sample is in group k;
• selecting a second array of probabilities qk,i:
• k=1, 2, . . . P;
• i=1, 2, . . . N that peaks will occur at indices l1, l2, . . . lN, respectively, of an indexed data set that characterizes the sample when the sample is not in group k;
• and
• selecting gj ∈{1, 2, . . . P}:
• j=1, 2, . . . M, where sample j is in group gj;
• wherein pk,i, qk,i, and gj are selected to maximize λ = ∑ 1 ≤ j ≤ M ⁢ { ∑ 1 ≤ j ≤ N ⁢ [ log ⁡ ( 1 - p i 1 - q i ) + x i , j ⁢ log ⁡ [ p k , i ⁡ ( 1 - q k , i ) q k , i ⁡ ( 1 - p k , i ) ] ] | k = g j } .