A method for target identification includes receiving, at one or more processors, a number of measurements of a target, each measurement from the number of measurements being observed at a predetermined time (zk), a number of target types (T), each one of the number of measurements, each one the number of target type and each one of one or more hidden states, each hidden state (xk) being characterized at the predetermined time, being correlated to one another, providing, using the one or more processors, a first conditional probability distribution, a conditional probability of a target type given a number of measurements, defined inductively, and obtaining an estimate of the target type from the first conditional probability. Systems that implement the method are also disclosed.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A target identification system comprising: a memory for storing data comprising: a data structure stored in the memory, the data structure including: a number of measurements of a target, each measurement from the number of measurements being observed at a predetermined time (zk); a number of target types (T); each one of the number of measurements, each one the number of target type and each one of one or more hidden states, each hidden state (xk) being characterized at the predetermined time, being correlated to one another; one or more processors operatively connected to the memory for storing data; the one or more processors being configured to: provide a first conditional probability distribution, a conditional probability of a target type given a number of measurements, defined inductively by p  ( T | z 1 , 2 , … , k ) = p  ( z k , T | z 1 , 2 , … , k - 1 ) ∑ T ^   p  ( z k , T ^ | z 1 , 2 , … , k - 1 ) ,  and p  ( z k , T | z 1 , 2 , … , k - 1 ) = p  ( T | z 1 , 2 , … , k - 1 )  ∫ p  ( z k | T , x k )  p  ( x k | T , z 1 , 2 , … , k - 1 )   x k where p(zk,i|T,xk) is a conditional probability of an ith measurement at a kth instance giving a target type T and hidden states at the kth instance, and p(T|z1, 2, . . . k−1) is a conditional probability distribution f of a target type, T, given the plurality of measurements at the k−1th time, z1, . . . k−1; and obtain an estimate of the target type from the first conditional probability. 2. The target identification system of claim 1 further comprising a non-transitory computer usable media having computer readable code embodied therein, wherein the computer readable code, when executed by the one or more processors, causes the one or more processors to: provide a first conditional probability distribution, a conditional probability of a target type given a number of measurements, defined inductively by p  ( T | z 1 , 2 , … , k ) = p  ( z k , T | z 1 , 2 , … , k - 1 ) ∑ T ^   p  ( z k , T ^ | z 1 , 2 , … , k - 1 ) ,  and p  ( z k , T | z 1 , 2 , … , k - 1 ) = p  ( T | z 1 , 2 , … , k - 1 )  ∫ p  ( z k | T , x k )  p  ( x k | T , z 1 , 2 , … , k - 1 )   x k where p(zk,i|T,xk) is a conditional probability of an ith measurement at a kth instance giving a target type T and hidden states at the kth instance, and p(T|z1, 2, . . . k−1) is a conditional probability distribution f of a target type, T, given the plurality of measurements at the k−1th time, z1, . . . k−1; and obtain an estimate of the target type from the first conditional probability; thereby configuring the one or more processors. 3. The target identification system of claim 1 wherein p(xk|xk−1,T) is a multivariate Gaussian distribution and wherein p(xk|T,zk) is another multivariate Gaussian distribution, where p(xk|xk−1,T) is a conditional probability distribution of hidden states at instance k given hidden states at instance k−1 and target type T. 4. The target identification system of claim 3 wherein transition between one hidden state at one instance and the one hidden state at another instance is given by a predetermined dynamic model (referred to as ƒk(xk−1, T)) and wherein an expectation of p(xk|T, z1,2, . . . k−1) is given by μk,k−1x=∫ƒk(xk−1)N(xk;μk−1,k−1x,Pk−1,k−1xx)dxk−1 and a covariance matrix is given inductively by Pk,k−1xx=Qk+∫ƒk(xk−1)ƒkT(xk−1)N(xk;μk−1,k−1x,Pk−1,k−1xx)dxk−1−[μk,k−1x]Tμk,k−1x. 5. The target identification system of claim 1 wherein p(T|z1,2, . . . ,k) is approximated by adaptive quadrature; and wherein  p  ( T | z 1 , 2 , …   k ) = p  ( T , z k | z 1 , 2 , …   k - 1 ) / Σ T , p  ( T ′ , z k | z 1 , 2 , …   k - 1 ) ,   Where p  ( T , z k | z 1 , 2 , …   k - 1 ) = p  ( T | z 1 , 2 , …   k - 1 ) × ∑ i + 1 , …    w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i Where p(T,zk|z1,2, . . . k−1) is a second conditional probability, p(zk|T,xk) is a third conditional probability, a Fisher information is transformed by F=USVT, U & V are unitary, S is diagonal, Sii≧Sjj for i>j, s  = i max  S ii ≥ ɛ for some small ε, an i-th column of U is denoted by Ui and Û=[U1, . . . , Us], w=Ûx, μw is an expectation of w at step i, Pww is an expectation of wwT, Ũ is lower triangular and ŨTŨ=Pww, a new variable v is defined by v = 1 2  U ~ - 1  ( w - μ w ) , A transformed Fisher information is {circumflex over (F)}=[ÛŨ−1]TFÛŨ−1, q1={i: {circumflex over (F)}ii≦α}, q2={i: {circumflex over (F)}ii>α} where α is a constant, For each i in q1 select Hermite order as |β{circumflex over (F)}ii| where β is a constant yielding weights wi and knots ki, For each i in q2 select quadrature order as ⌈ F ^ ii χ ⌉ where χ is a constant yielding weights wi and knots ki, denote the column-wise Khatri-Rao product and Îi a sxi column vector of ones w=w1w2 . . . ws k=[Î0k1Îs−1][Î1k2Îs−2] . . . [Îs−1ksÎ0]. 6. The target identification system of claim 2 wherein p(T|z1,2, . . . ,k) is approximated by adaptive quadrature; and wherein  p  ( T | z 1 , 2 , …   k ) = p  ( T , z k | z 1 , 2 , …   k - 1 ) / Σ T , p  ( T ′ , z k | z 1 , 2 , …   k - 1 ) ,   Where p  ( T , z k | z 1 , 2 , …   k - 1 ) = p  ( T | z 1 , 2 , …   k - 1 ) × ∑ i + 1 , …    w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i Where p(T,zk|z1,2, . . . k−1) is a second conditional probability, p(zk|T,xk) is a third conditional probability, a Fisher information is transformed by F=USVT, U & V are unitary, S is diagonal, Sii≧Sjj for i>j, s  = i max  S ii ≥ ɛ for some small ε, an i-th column of U is denoted by Ui and Û=[U1, . . . , Us], w=Ûx, μw is an expectation of w at step i, Pww is an expectation of wwT, Ũ is lower triangular and ŨTŨ=Pww, a new variable v is defined by v = 1 2  U ~ - 1  ( w - μ w ) , A transformed Fisher information is {circumflex over (F)}=[ÛŨ−1]TFÛŨ−1, q1={i: {circumflex over (F)}ii≦α}, q2={i: {circumflex over (F)}ii>α} where α is a constant, For each i in q1 select Hermite order as |β{circumflex over (F)}ii| where β is a constant yielding weights wi and knots ki, For each i in q2 select quadrature order as ⌈ F ^ ii χ ⌉ where χ is a constant yielding weights wi and knots ki, denote the column-wise Khatri-Rao product and Îi a sxi column vector of ones w=w1w2 . . . ws k=[Î0k1Îs−1][Î1k2Îs−2] . . . [Îs−1ksÎ0]. 7. The target identification system of claim 1 wherein the estimate is obtained using a Bayes Classifier. 8. The target identification system of claim 1, wherein the measurements zi comprise one or more of a range to the target, a rate of change of a range to the target, an acceleration along a line of sight. 9. The target identification system of claim 1, wherein the hidden states xi comprise one or more of target geometry, target attitude, target velocity. 10. The target identification system of claim 1, wherein the plurality of target types comprise one or more of types of vehicles. 11. A method for target identification comprising: receiving, at one or more processors, a number of measurements of a target, each measurement from the number of measurements being observed at a predetermined time (zk), a number of target types (T); each one of the number of measurements, each one the number of target type and each one of one or more hidden states, each hidden state (xk) being characterized at the predetermined time, being correlated to one another; providing, using the one or more processors, a first conditional probability distribution, a conditional probability of a target type given a number of measurements, defined inductively by  p  ( T | z 1 , 2 , …  ,  k ) = p  ( z k , T | z 1 , 2 , …  , k - 1 ) ∑ T ^   p  ( z k , T ^ | z 1 , 2 , …  , k - 1 ) ,   and p  ( z k , T | z 1 , 2 , …  , k - 1 ) = p  ( T | z 1 , 2 , …  , k - 1 )  ∫ p  ( z k | T , x k )  p  ( x k | T , z 1 , 2 , …  , k - 1 )   x k where p(zk,i|T,xk) is a conditional probability of an ith measurement at a kth instance giving a target type T and hidden states at the kth instance, and p(T|z1, 2, . . . k−1) is a conditional probability distribution f of a target type, T, given the plurality of measurements at the k−1th time, z1, . . . k−1; and obtaining an estimate of the target type from the first conditional probability; wherein the providing a first conditional probability distribution and the obtaining an estimate of the target type are performed by the one or more processors executing computer readable code embodied in non-transitory computer usable media. 12. The method of claim 11 wherein p(xk|xk−1,T) is a multivariate Gaussian distribution and wherein p(xk|T, zk) is another multivariate Gaussian distribution, where p(xk|xk−1,T) is a conditional probability distribution of hidden states at instance k given hidden states at instance k−1 and target type T. 13. The method of claim 12 wherein transition between one hidden state at one instance and the one hidden state at another instance is given by a predetermined dynamic model (referred to as ƒk(xk−1, T)) and wherein an expectation of p(xk|T, z1,2, . . . k−1) is given by μk,k−1x=∫ƒk(xk−1)N(xk;μk−1,k−1x,Pk−1,k−1xx)dxk−1 and a covariance matrix is given inductively by Pk,k−1xx=Qk+∫ƒk(xk−1)ƒkT(xk−1)N(xk;μk−1,k−1x,Pk−1,k−1xx)dxk−1−[μk,k−1x]Tμk,k−1x. 14. The method of claim 13 wherein measurements are a predetermined function of target state and type (referred to as hk(xk, T)) corrupted by independent noise; and wherein p(xk|T, z1,2, . . . k−1) is a multivariate Gaussian distribution. 15. The method of claim 11 wherein p(T|z1,2, . . . ,k) is approximated by adaptive quadrature; and wherein  p  ( T | z 1 , 2 , …   k ) = p  ( T , z k | z 1 , 2 , …   k - 1 ) / Σ T , p  ( T ′ , z k | z 1 , 2 , …   k - 1 ) ,   Where p  ( T , z k | z 1 , 2 , …   k - 1 ) = p  ( T | z 1 , 2 , …   k - 1 ) × ∑ i + 1 , …    w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i Where p(T,zk|z1,2, . . . k−1) is a second conditional probability, p(zk|T,xk) is a third conditional probability, a Fisher information is transformed by F=USVT, U & V are unitary, S is diagonal, Sii≧Sjj for i>j, s  = i max  S ii ≥ ɛ for some small ε, an i-th column of U is denoted by Ui and Û=[U1, . . . , Us], w=Ûx, μw is an expectation of w at step i, Pww is an expectation of wwT, Ũ is lower triangular and ŨTŨ=Pww, a new variable v is defined by v = 1 2  U ~ - 1  ( w - μ w ) , A transformed Fisher information is {circumflex over (F)}=[ÛŨ−1]TFÛŨ−1, q1={i: {circumflex over (F)}ii≦α}, q2={i: {circumflex over (F)}ii>α} where α is a constant, ∀iεq1 select Hermite order as |β{circumflex over (F)}ii| where β is a constant yielding weights wi and knots ki, ∀iεq2 select quadrature order as ⌈ F ^ ii χ ⌉ where χ is a constant yielding weights wi and knots ki, denote the column-wise Khatri-Rao product and Îi a sxi column vector of ones w=w1w2 . . . ws k=[Î0k1,Îs−1][Î1k2Îs−2] . . . [Îs−1ksÎ0]. 16. The method of claim 11 wherein the estimate is obtained using a Bayes Classifier. 17. The method of claim 15 wherein providing a first conditional probability comprises: inductively obtaining hidden state information at a kth stage from hidden state information at a preceding (k−1)th stage; a obtaining a fourth conditional probability of a hidden state given a target type and a number of measurements, p(xk|T,z1,2, . . . ,k−1); and obtaining the first conditional probability, expressed inductively, as a function of the fourth conditional probability. 18. The method of claim 15 wherein in defining inductively the first conditional probability as:  p  ( T | z 1 , 2 , …  ,  k ) = p  ( z k , T | z 1 , 2 , …  , k - 1 ) ∑ T ^   p  ( z k , T ^ | z 1 , 2 , …  , k - 1 ) ,  p  ( z k , T | z 1 , 2 , …  , k - 1 ) = p  ( T | z 1 , 2 , …  , k - 1 )  ∫ p  ( z k | T , x k )  p  ( x k | T , z 1 , 2 , …  , k - 1 )   x k 19. The method of claim 18 wherein the third and fourth conditional probabilities are represented by multivariate Gaussian probability functions. 20. The method of claim 19 wherein providing the first conditional probability distribution further comprises: using subspace partitioning to reduce dimensions of integration variable space of integral ∫p(zk|T,xk)N(xk;μk,k−1x,Pk,k−1xx)dxk; transforming variables in the integral in order to apply whitening; and applying different approximation methods to regions of more importance and regions of less importance, importance measured by Fisher information, in order to approximate the integral by a sum, ∑ i + 1 , …    w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i . 21. The method of claim 20 wherein using subspace partitioning comprises: transforming the integration variable space by w=Ûx, where a Fisher information is transformed by F=USVT, U & V are unitary, S is diagonal, Sii≧Sjj for i>j, s = max i  S i   i ≥ ɛ for some small ε, and an i-th column of U is denoted by Ui and Û=[U1, . . . , Us]. 22. The method of claim 21 wherein transforming variables in the integral in order to apply whitening comprises: transforming from w to a new variable v defined by v = 1 2  U ~ - 1  ( w - μ w ) in order to apply whitening. 23. The method of claim 13 wherein: a transformed Fisher information is {circumflex over (F)}=[ÛŨ−1]TFÛŨ−1, q1={i: {circumflex over (F)}ii≦α}, q2={i: {circumflex over (F)}ii>α} where α is a constant, For each i in q1 select Hermite order as |β{circumflex over (F)}ii| where β is a constant yielding weights wi and knots ki, For each i in q2 select quadrature order as ⌈ F ^ ii χ ⌉ where χ is a constant yielding weights wi and knots ki, denote the column-wise Khatri-Rao product and Îi a sxi column vector of ones w=w1w2 . . . ws k=[Î0k1Îs−1][Î1k2Îs−2] . . . [Îs−1ksÎ0]; and wherein ∫ p  ( z | U ^  ( 2  U ~   v + μ w ) , T )   - v T  v   v ≈ ∑ i = 1 , …   w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i 24. A non-transitory computer readable medium having computer executable code embodied therein, said computer executable code, when executed in at least one processor, causing the at least one processor to: receive, at the at least one processor, a number of measurements of a target, each measurement from the number of measurements being observed at a predetermined time (zk), a number of target types (T); each one of the number of measurements, each one the number of target type and each one of one or more hidden states, each hidden state (xk) being characterized at the predetermined time, being correlated to one another; provide a first conditional probability distribution, a conditional probability of a target type given a number of measurements, defined inductively by  p  ( T | z 1 , 2 , …  , k ) = p  ( z k , T | z 1 , 2 , …  , k - 1 ) ∑ T ^   p  ( z k , T ^ | z 1 , 2 , …  , k - 1 )  and p  ( z k , T | z 1 , 2 , …  , k - l ) = p  ( T | z 1 , 2 , …  , k - 1 )  ∫ p  ( z k | T , x k )  p  ( x k | T , z 1 , 2 , …  , k - 1 )   x k where p(zk,i|T,xk) is a conditional probability of an ith measurement at a kth instance giving a target type T and hidden states at the kth instance, and p(T|z1, 2, . . . k−1) is a conditional probability distribution f of a target type, T, given the plurality of measurements at the k−1th time, z1, . . . k−1; and obtain an estimate of the target type from the first conditional probability. 25. The non-transitory computer readable medium of claim 24 wherein p(xk|xk−1,T) is a multivariate Gaussian distribution and wherein p(xk|T, zk) is another multivariate Gaussian distribution, where p(xk|xk−1,T) is a conditional probability distribution of hidden states at instance k given hidden states at instance k−1 and target type T. 26. The non-transitory computer readable medium of claim 25 wherein transition between one hidden state at one instance and the one hidden state at another instance is given by a predetermined dynamic model (referred to as ƒk(xk−1, T)) and wherein an expectation of p(xk|T, z1,2, . . . k−1) is given by μk,k−1x=∫ƒk(xk−1)N(xk;μk−1,k−1x,Pk−1,k−1xx)dxk−1 and a covariance matrix is given inductively by Pk,k−1xx=Qk+∫ƒk(xk−1)ƒkT(xk−1)N(xk;μk−1,k−1x,Pk−1,k−1xx)dxk−1−[μk,k−1x]Tμk,k−1x. 27. The non-transitory computer readable medium of claim 25 wherein measurements are a predetermined function of target state and type (referred to as hk(xk, T)) corrupted by independent noise; and wherein p(xk|T, z1,2, . . . k−1) is a multivariate Gaussian distribution. 28. The non-transitory computer readable medium of claim 24 wherein p(T|z1,2, . . . ,k) is approximated by adaptive quadrature; and wherein  p  ( T | z 1 , 2 , …   k ) = p  ( T , z k | z 1 , 2 , …   k - 1 ) / Σ T , p  ( T ′ , z k | z 1 , 2 , …   k - 1 ) ,   Where p  ( T , z k | z 1 , 2 , …   k - 1 ) = p  ( T | z 1 , 2 , …   k - 1 ) × ∑ i = 1 , …   w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i Where p(T,zk|z1,2, . . . k−1) is a second conditional probability, p(zk|T,xk) is a third conditional probability, a Fisher information is transformed by F=USVT, U & V are unitary, S is diagonal, Sii≧Sjj for i>j, s = max i  S i   i ≥ ɛ for some small ε, an i-th column of U is denoted by Ui and Û=[U1, . . . , Us], w=Ûx, μw is an expectation of w at step i, Pww is an expectation of wwT, Ũ is lower triangular and ŨTŨ=Pww, a new variable v is defined by v = 1 2  U ~ - 1  ( w - μ w ) , A transformed Fisher information is {circumflex over (F)}=[ÛŨ−1]TFÛŨ−1, q1={i: {circumflex over (F)}ii≦α}, q2={i: {circumflex over (F)}ii>α} where α is a constant, ∀iεq1 select Hermite order as |β{circumflex over (F)}ii| where β is a constant yielding weights wi and knots ki, ∀iεq2 select quadrature order as ⌈ F ^ ii χ ⌉ where χ is a constant yielding weights wi and knots ki, denote the column-wise Khatri-Rao product and Îia sxi column vector of ones w=w1w2 . . . ws k=[Î0k1Îs−1][Î1k2Îs−2] . . . [Îs−1ksÎ0]. 29. The non-transitory computer readable medium of claim 28 wherein providing a first conditional probability comprises: inductively obtaining hidden state information at a kth stage from hidden state information at a preceding (k−1)th stage; a obtaining a fourth conditional probability of a hidden state given a target type and a number of measurements, p(xk|T,z1,2, . . . ,k−1); and obtaining the first conditional probability, expressed inductively, as a function of the fourth conditional probability. 30. The non-transitory computer readable medium of claim 29 wherein in defining inductively the first conditional probability as  p  ( T | z 1 , 2 , …  , k ) = p  ( z k , T | z 1 , 2 , …  , k - 1 ) ∑  T   p  ( z k , T ^ | z 1 , 2 , …  , k - 1 ) ,  p  ( z k , T | z 1 , 2 , …  , k - 1 ) = p  ( T | z 1 , 2 , …  , k - 1 )  ∫ p  ( z k | T , x k )  p  ( x k | T , z 1 , 2 , …  ,  k - 1 )   x k . 31. The non-transitory computer readable medium of claim 30 wherein the third and fourth conditional probabilities are represented by multivariate Gaussian probability functions. 32. The non-transitory computer readable medium of 31 wherein providing the first conditional probability distribution further comprises: using subspace partitioning to reduce dimensions of integration variable space of integral ∫p(zk|T,xk)N(xk; μk,k−1x, Pk,k−1xx)dxk; transforming variables in the integral in order to apply whitening; and applying different approximation methods to regions of more importance and regions of less importance, importance measured by Fisher information, in order to approximate the integral by a sum, ∑ i = 1 , …   w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i . 33. The non-transitory computer readable medium of claim 33 wherein using subspace partitioning comprises: transforming the integration variable space by w=Ûx, where a Fisher information is transformed by F=USVT, U & V are unitary, S is diagonal, Sii≧Sjj for i>j, s = max i  S i   i ≥ ɛ for some small ε, and an i-th column of U is denoted by Ui and Û=[U1, . . . , Us]. 34. The non-transitory computer readable medium of claim 33 wherein transforming variables in the integral in order to apply whitening comprises: transforming from w to a new variable v defined by v = 1 2  U ~ - 1  ( w - μ w ) in order to apply whitening. 35. The non-transitory computer readable medium of claim 34 wherein a transformed Fisher information is {circumflex over (F)}=[ÛŨ−1]TFÛŨ−1, q1={i: {circumflex over (F)}ii≦α}, q2={i: {circumflex over (F)}ii>α} where α is a constant, For each i in q1 select Hermite order as |β{circumflex over (F)}ii| where β is a constant yielding weights wi and knots ki, For each i in q2 select quadrature order as ⌈ F ^ ii χ ⌉ where χ is a constant yielding weights wi and knots ki, denote the column-wise Khatri-Rao product and Îi a sxi column vector of ones w=w1w2 . . . ws k=[Î0k1Îs−1][Î1k2Îs−2] . . . [Îs−1ksÎ0]; and wherein ∫ p  ( z | U ^  ( 2  U ~   v + μ w ) , T )   - v T  v   v ≈ ∑ i = 1 , …   w    p  ( z | U ^  ( 2  U ~   k i + μ w ) , T )  w i .