Computing interval parameter bounds from fallible measurements using systems of nonlinear equations

1. A computer program product for computing interval parameter bounds from fallible measurements, comprising instructions for:

receiving a set of measurements z 1 , . . . , z n , wherein an observation model describes each z i as a function of a p-element vector parameter x=(x 1 , . . . , x p ),

wherein receiving the set of measurements involves receiving values for a set of conditions c 1 , . . . , c n under which the corresponding observations z i made,

wherein equations in the system of nonlinear equations account for the conditions c i , and are of the form z i −h(x|c i )=0 (i=1, . . . , n), and

wherein each condition c 1 is not known precisely but is contained within an interval c I i ;

storing the set of measurements z 1 , . . . , z n in a memory in a computer system;

forming a system of nonlinear equations z i −h(x)=0 (i=1, . . . , n) based on the observation model; and

solving the system of nonlinear equations to determine interval parameter bounds on x.

2. The computer program product of claim 1 , wherein the system of nonlinear equations is an “overdetermined system” in which there are more equations than unknowns.

3. The computer program product of claim 1 , wherein each measurement z i is actually a q-element vector of measurements z i =(z il , . . . , z iq ) T , and h is actually a q-element vector of functions h=(h 1 , . . . , h q ) T .

4. The computer program product of claim 1 , wherein each condition c i is actually an r-element vector of conditions c i =(c il , . . . , c ir ) T .

5. The computer program product of claim 1 , wherein equations in the system of nonlinear equations are of the form z i −h(x|c i )+ε I (x, c i )=0 (i=1, . . . , n), which includes an error model ε I (x, c i ) that provides interval bounds on measurement errors for z i .

6. The computer program product of claim 5 , wherein if z i is actually a q-element vector of measurements z i =(z il , . . . , z iq ) T , then ε I is actually a q-element vector ε I =(ε 1 , . . . , ε q ) T .

7. The computer program product of claim 5 , wherein if there exists no solution to the system of nonlinear equations, the method further comprises determining that at least one of the following is true:

at least one of the set of measurements z i , . . . , z n is faulty;

the observation model h(x|c i ) is false;

the error model ε I (x, c i ) is false; and

the computational system used to compute interval bounds on elements of x is flawed.

8. The computer program product of claim 1 , wherein solving the system of nonlinear equations involves:

linearizing the system of nonlinear equations to form a corresponding system of linear equations; and

solving the system of linear equations.

9. The computer program product of claim 8 , wherein solving the system of nonlinear equations involves using Gaussian Elimination.

10. A physical computer-readable storage medium storing instructions that when executed by a computer cause the computer to perform a method for computing interval parameter bounds from fallible measurements, wherein the computer-readable storage medium includes magnetic storage devices, optical storage devices, disk drives, magnetic tape. CDs (compact discs), and DVDs (digital versatile discs or digital video discs), the method comprising:

receiving a set of measurements z 1 , . . . , z n , wherein an observation model describes each z i as a function of a p-element vector parameter x=(x 1 , . . . , x p ),

wherein receiving the set of measurements involves receiving values for a set of conditions c 1 , . . . , c n under which the corresponding observations z i were made,

wherein equations in the system of nonlinear equations account for the conditions c i and are of the form z i −h(x|c i )=0 (i=1, . . . , n), and

wherein each condition c i is not known precisely but is contained within an interval c I i ;

storing the set of measurements z 1 , . . . , z n in a memory in a computer system;

forming a system of nonlinear equations z i −h(x)=0 (i=1, . . . , n) based on the observation model; and

solving the system of nonlinear equations to determine interval parameter bounds on x.

11. The computer-readable storage medium of claim 10 , wherein the system of nonlinear equations is an “overdetermined system” in which there are more equations than unknowns.

12. The computer-readable storage medium of claim 10 , wherein each measurement z i is actually a q-element vector of measurements z i =(z il , . . . , z iq ) T , and h is actually a q-element vector of functions h=(h 1 , . . . , h q ) T .

13. The computer-readable storage medium of claim 10 , wherein each condition c i is actually an r-element vector of conditions c i =(c il , . . . , c ir ) T .

14. The computer-readable storage medium of claim 10 , wherein equations in the system of nonlinear equations are of the form, z i −h(x|c i )+ε I (x, c i )=0 (i=1, . . . , n), which includes an error model ε I (x, c i ) that provides interval bounds on measurement errors for z i .

15. The computer-readable storage medium of claim 14 , wherein if z i is actually a q-element vector of measurements z i =(z il , . . . , z iq ) T , then ε I is actually a q-element vector ε I =(ε l , . . . , ε q ) T .

16. The computer-readable storage medium of claim 14 , wherein if there exists no solution to the system of nonlinear equations, the method further comprises determining that at least one of the following is true:

at least one of the set of measurements z i , . . . , z n is faulty;

the observation model h(x|c i ) is false;

the error model ε I (x, c i ) is false; and

the computational system used to compute interval bounds on elements of x is flawed.

17. The computer-readable storage medium of claim 10 , wherein solving the system of nonlinear equations involves:

linearizing the system of nonlinear equations to form a corresponding system of linear equations; and

solving the system of linear equations.

18. The computer-readable storage medium of claim 17 , wherein solving the system of nonlinear equations involves using Gaussian Elimination.

19. An apparatus that computes interval parameter bounds from fallible measurements, comprising:

a receiving mechanism configured to receive a set of measurements z 1 , . . . , z n , wherein an observation model describes each z i as a function of a

p-element vector parameter x=(x 1 , . . . , x p ),

wherein receiving the set of measurements involves receiving values for a set of conditions c 1 , . . . , c n under which the corresponding observations z 1 were made,

wherein equations in the system of nonlinear equations account for the conditions c i and are of the form z i −h(x|c i )=0 (i=1, . . . , n), and

wherein each condition c 1 is not known precisely but is contained within an interval c I i ;

a memory in a computer system for storing the set of measurements z 1 , . . . , z n ;

an equation forming mechanism configured to form a system of nonlinear equations z i −h(x)=0 (i=1, . . . , n) based on the observation model; and

a solver configured to solve the system of nonlinear equations to determine interval parameter bounds on x.

20. The apparatus of claim 19 , wherein the system of nonlinear equations is an “overdetermined system” in which there are more equations than unknowns.

21. The apparatus of claim 19 , wherein each measurement z i is actually a q-element vector of measurements z i =(z il , . . . , z iq ) T , and h is actually a q-element vector of functions h=(h 1 , . . . , h q ) T .

22. The apparatus of claim 19 , wherein each condition c i is actually an r-element vector of conditions c i =(c il , . . . , c ir ) T .

23. The apparatus of claim 19 , wherein equations in the system of nonlinear equations are of the form z i −h(x|c i )+ε I (x, c i )=0 (i=1, . . . , n), which includes an error model ε I (x, c i ) that provides interval bounds on measurement errors for z i .

24. The apparatus of claim 23 , wherein if z i is actually a q-element vector of measurements z i =(z il , . . . , z iq ) T , then ε I is actually a q-element vector ε I =(ε 1 , . . . , ε q ) T .

25. The apparatus of claim 23 , wherein if there exists no solution to the system of nonlinear equations, the solver is configured to determine that at least one of the following is true:

at least one of the set of measurements z i , . . . , z n is faulty;

the observation model h(x|c i ) is false;

the error model ε I (x, c i ) is false; and

the computational system used to compute interval bounds on elements of x is flawed.

26. The apparatus of claim 19 , wherein the solver is configured to:

linearize the system of nonlinear equations to form a corresponding system of linear equations; and to

solve the system of linear equations.

27. The apparatus of claim 26 , wherein the solver is configured to solve the system of nonlinear equations using Gaussian Elimination.