1. A method for analyzing measurement errors in a set of measured signals {x(k)} measured in an experiment, wherein k=1, 2, . . . , N, N being the number of signals in said set, said method comprising
(a) transforming said set of measured signals using a transformation, said transformation transforming said set of measured signals into a set of transformed signals {y(k)} such that said transformed signals have a near constant error; and
(b) determining error in each of said transformed signals by normalizing measurement error in said measured signal by a modeled error in said measured signal, said modeled error being calculated using an error model of said experiment; and
(c) outputting said set of transformed signals andor said errors determined in step (b) to a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying said set of transformed signals andor said errors determined in step (b).
2. The method of claim 1, wherein said error model is a three-term error model according to the equation
\u03c3x(k)=\u221a{square root over (c2+b2\xb7x(k)+a2\xb7x(k)2)}{square root over (c2+b2\xb7x(k)+a2\xb7x(k)2)}
wherein a is the fractional error coefficient of said experiment, b is the Poisson error coefficient of said experiment, and c is the standard deviation of background noise of said experiment.
3. The method of claim 2, wherein for each measured signal x(k) said transformation is carried out according to the equation
y
\u2061
(
k
)
=
ln
\u2061
(
b
2
+
2
\xb7
a
2
\xb7
x
\u2061
(
k
)
a
+
2
\xb7
c
2
+
b
2
\xb7
x
\u2061
(
k
)
+
a
2
\xb7
x
\u2061
(
k
)
2
)
a
+
d
,
wherein
d
=
–
ln
\u2061
(
b
2
a
+
2
\xb7
c
)
a
.
4. The method of claim 3, wherein for each measured signal x(k) said error of said transformed signal y(k) is determined according to the equation
\u0394
\u2062
\u2062
y
\u2061
(
k
)
=
\u0394
\u2062
\u2062
x
\u2061
(
k
)
c
2
+
b
2
\xb7
x
\u2061
(
k
)
+
a
2
\xb7
x
\u2061
(
k
)
2
wherein \u0394y(k) is said error in said transformed signal y(k), and wherein \u0394x(k) is an error in said measured signal x(k).
5. The method of claim 4, wherein for each measured signal x(k) said error in said measured signal x(k) is an error in said measured signal x(k) determined in said experiment.
6. The method of claim 5, wherein for each measured signal x(k) said error in said measured signal x(k) is the larger of an error in said measured signal x(k) determined in said experiment or said modeled error in said measured signal x(k).
7. The method of claim 1, further comprising before said step (a) the steps of
(i) determining said error model for said experiment;
(ii) determining said modeled error in each of said measured signals; and
(iii) determining said transformation for said experiment.
8. The method of claim 7, wherein said error model is a three-term error model according to the equation
\u03c3x(k)=\u221a{square root over (c2+b2\xb7x(k)+a2\xb7x(k)2)}{square root over (c2+b2\xb7x(k)+a2\xb7x(k)2)}
wherein a is the fractional error coefficient of said experiment, b is the Poisson error coefficient of said experiment, and c is the standard deviation of background noise of said experiment.
9. The method of claim 8, wherein for each measured signal x(k) said transformation is carried out according to the equation
y
\u2061
(
k
)
=
ln
\u2061
(
b
2
+
2
\xb7
a
2
\xb7
x
\u2061
(
k
)
a
+
2
\xb7
c
2
+
b
2
\xb7
x
\u2061
(
k
)
+
a
2
\xb7
x
\u2061
(
k
)
2
)
a
+
d
,
wherein
d
=
–
ln
\u2061
(
b
2
a
+
2
\xb7
c
)
a
.
10. The method of claim 9, wherein for each measured signal x(k) said error of said transformed signal y(k) is determined according to the equation
\u0394
\u2062
\u2062
y
\u2061
(
k
)
=
\u0394
\u2062
\u2062
x
\u2061
(
k
)
c
2
+
b
2
\xb7
x
\u2061
(
k
)
+
a
2
\xb7
x
\u2061
(
k
)
2
wherein \u0394y(k) is said error in said transformed signal y(k), and wherein \u0394x(k) is an error in said measured signal x(k).
11. The method of claim 10, wherein for each measured signal x(k) said error in said measured signal x(k) is an error in said measured signal x(k) determined in said experiment.
12. The method of claim 10, wherein for each measured signal x(k) said error in said measured signal x(k) is the larger of an error in said measured signal x(k) determined in said experiment or said modeled error in said measured signal x(k).
13. A method for determining a residue error in a measured signal x measured in an experiment, said method comprising
(a) transforming said measured signal into a transformed signal y by a method comprising using a transformation according to the equation
y
=
ln
\u2061
(
b
2
+
2
\xb7
a
2
\xb7
x
a
+
2
\xb7
c
2
+
b
2
\xb7
x
+
a
2
\xb7
x
2
)
a
+
d
,
wherein a is the fractional error coefficient of said experiment, b is the Poisson error coefficient of said experiment, and c is the standard deviation of background noise of said experiment, and
d
=
–
ln
\u2061
(
b
2
a
+
2
\xb7
c
)
a
(b) determining an error in said transformed signal y;
(c) determining said residue error by subtracting 1 from said error of said transformed signal y; and
(d) outputting said residue error to a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying said residue error.
14. The method of claim 13, wherein said error of said transformed signal y is determined according to the equation
\u0394
\u2062
\u2062
y
=
\u0394
\u2062
\u2062
x
c
2
+
b
2
\xb7
x
+
a
2
\xb7
x
2
;
wherein \u0394y is said error in said transformed signal y, and wherein \u0394x is an error in said measured signal x.
15. The method of claim 14, wherein said error in said measured signal x is an error in said measured signal x determined in said experiment.
16. The method of claim 14, wherein said error in said measured signal x is the larger of an error determined in said experiment or an error calculated according to the equation
\u0394x=\u221a{square root over (c2+b2\xb7x+a2\xb7x2)}.
17. A method for determining an error-weighted average of m signals xi measured in m replicate experiments, wherein i=1, 2, . . . m, said method comprising
(a) transforming each said signal xi into a transformed signal yi by a method comprising using a transformation
y
i
=
ln
\u2061
(
b
2
+
2
\xb7
a
2
\xb7
x
i
a
+
2
\xb7
c
2
+
b
2
\xb7
x
i
+
a
2
\xb7
x
i
2
)
a
+
d
,
wherein a is the fractional error coefficient of said experiments, b is the Poisson error coefficient of said experiments, and c is the averaged standard deviation of background noise of said experiments, and
d
=
–
ln
\u2061
(
b
2
a
+
2
\xb7
c
)
a
;
(b) determining an error \u0394yi in each said transformed signal y1;
(c) determining an error-weighted transformed signal according to the equation
y
_
=
\u2211
i
\u2062
(
y
i
\u0394
\u2062
\u2062
y
i
2
)
\u2211
i
\u2062
(
1
\u0394
\u2062
\u2062
y
i
2
)
;
(d) transforming said error-weighted transformed signal to produce said error-weighted average by a method comprising using a transformation according to the equation
x
_
=
–
(
4
\xb7
a
2
\xb7
c
2
–
a
2
\xb7
\u2147
2
\u2062
a
\xb7
(
y
_
–
d
)
+
2
\xb7
a
\xb7
b
2
\xb7
\u2147
a
\xb7
(
y
_
–
d
)
–
b
4
)
4
\xb7
a
3
\xb7
\u2147
a
\xb7
(
y
_
–
d
)
;
and
(e) outputting said error-weighted average x to a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying said error-weighted average x.
18. The method of claim 17, wherein said error \u0394yi of each said transformed signal yi is obtained according to the equation
\u0394
\u2062
\u2062
y
i
=
\u0394
\u2062
\u2062
x
i
c
2
+
b
2
\xb7
x
i
+
a
2
\xb7
x
i
2
wherein \u0394xi is an error in said measured signal xi.
19. The method of claim 18, wherein each said error \u0394xi is an error in said measured signal xi determined in said ith experiment.
20. The method of claim 18, wherein each said error \u0394xi is the larger of an error in said measured signal xi determined in said ith experiment or an error in said measured signal xi calculated according to the equation
\u0394xi=\u221a{square root over (c2+b2\xb7xi+a2\xb7xi2)}.
21. A method for analyzing m signals {x}i measured in m experiments, wherein i=1, 2, . . . m, said method comprising
(a) transforming each said signal xi into a transformed signal yi by a method comprising using a transformation
y
i
=
ln
\u2061
(
b
2
+
2
\xb7
a
2
\xb7
x
i
a
+
2
\xb7
c
2
+
b
2
\xb7
x
i
+
a
2
\xb7
x
i
2
)
a
+
d
,
wherein a is the fractional error coefficient of said experiments, b is the Poisson error coefficient of said experiments, c is the averaged standard deviation of background noise of said experiments, and
d
=
–
ln
\u2061
(
b
2
a
+
2
\xb7
c
)
a
;
(b) determining an error \u0394yi in each said transformed signal yi;
(c) analyzing said transformed signals and their errors, thereby analyzing said m signals; and
(d) outputting said transformed signal yi andor said error \u0394yi to a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying said transformed signal yi andor said error \u0394yi.
22. The method of claim 21, wherein for each measured signal xi said error in said transformed signal yi is obtained according to the equation
\u0394
\u2062
\u2062
y
i
=
\u0394
\u2062
\u2062
x
i
c
2
+
b
2
\xb7
x
i
+
a
2
\xb7
x
i
2
wherein \u0394yi is said error in said transformed signal yi, and wherein \u0394xi is an error in said measured signal xi.
23. The method of claim 22, wherein said error in said measured signal xi is an error in said measured signal xi determined in said ith experiment.
24. The method of claim 22, wherein said error in measured signal xi is the larger of an error in said measured signal xi determined in said ith experiment or an error in said measured signal xi calculated according to the equation
\u0394xi=\u221a{square root over (c2+b2\xb7xi+a2\xb7xi2)}.
25. The method of claim 24, wherein said step (c) is carried out by a method comprising performing ANOVA on said transformed signals and their errors.
26. The method of claim 25, wherein said step (c) is carried out by a method comprising performing a regression analysis on said transformed signals and their errors.
27. The method of claim 25, wherein said step (c) is carried out by a method comprising determining a variance of said transformed signals.
28. A method for determining if a measured signal x measured in an experiment is an outlier, comprising
(a) transforming said measured signal into a transformed signal y by a method comprising using a transformation according to the equation
y
=
ln
\u2061
(
b
2
+
2
\xb7
a
2
\xb7
x
a
+
2
\xb7
c
2
+
b
2
\xb7
x
+
a
2
\xb7
x
2
)
a
+
d
\u2062
,
wherein a is the fractional error coefficient of said experiment, b is the Poisson error coefficient of said experiment, and c is the standard deviation of background noise of said experiment, and
d
=
–
ln
\u2061
(
b
2
a
+
2
\xb7
c
)
a
\u2062
;
(b) determining an error in said transformed signal y;
(c) comparing said error in said transformed signal y with a predetermined threshold value, wherein said measured signal is identified as an outlier if said error in said transformed signal y is greater than said threshold value; and
(d) outputting a result of said comparing step (c) to a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying a result of said comparing step (c).
29. The method of claim 28, wherein said error of said transformed signal y is determined according to the equation
\u0394
\u2062
\u2062
y
=
\u0394
\u2062
\u2062
x
c
2
+
b
2
\xb7
x
+
a
2
\xb7
x
2
wherein \u0394y is said error in said transformed signal y, and wherein \u0394x is an error in said measured signal x.
30. The method of claim 28, wherein said error in measured signal x(k) is the larger of an error in said measured signal x determined in said experiment or an error in said measured signal x calculated according to the equation
\u0394x=\u221a{square root over (c2+b2\xb7x+a2\xb7x2)}.
31. A method of obtaining a difference of measured signals {x(k)} measured in an experiment, wherein k=1, 2, said method comprising
(a) transforming said signals into transformed signals {y(k)} by transforming each of said measured signals by a method comprising using a transformation according to
y
\u2061
(
k
)
=
ln
\u2061
(
b
2
+
2
\xb7
a
2
\xb7
x
\u2061
(
k
)
a
+
2
\xb7
c
2
+
b
2
\xb7
x
\u2061
(
k
)
+
a
2
\xb7
x
\u2061
(
k
)
2
)
a
+
d
\u2062
,
wherein a is the fractional coefficient of said experiment, b is the Poisson coefficient of said experiment, and c is the standard deviation of background noise of said experiment, and
d
=
–
ln
\u2061
(
b
2
a
+
2
\xb7
c
)
a
\u2062
;
(b) determining a difference between said transformed signals; and
(c) outputting said difference to a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying said difference.
32. The method of any one of claims 1, 7, 13, 17, 21, 28 and 31, wherein said experiment is an experiment in which measurements of a plurality of cellular constituents are obtained.
33. The method of claim 32, wherein said experiment is a microarray experiment and said cellular constituents are mRNAs.
34. The method of claim 32, wherein said cellular constituents are proteins.
35. The method of claim 32, wherein said measured signal is measured fluorescence intensity.
36. A computer program product for use in conjunction with a computer having a processor and a memory connected to the processor, said computer program product comprising a computer readable storage medium having a computer program mechanism encoded thereon, wherein said computer program mechanism may be loaded into the memory of said computer and cause said computer to carry out the method of claim 35.
37. A computer system comprising
a processor, and
a memory coupled to said processor and encoding one or more programs, wherein said one or more programs cause the processor to carry out the method of claim 35.
38. The method of claim 32, wherein said measured signal is a difference between a measured fluorescence intensity of a probe and a measured fluorescence intensity of a reference probe.
39. A computer system comprising
a processor, and
a memory coupled to said processor and encoding one or more programs, wherein said one or more programs cause the processor to carry out the method of claim 38.
40. A computer program product for use in conjunction with a computer having a processor and a memory connected to the processor, said computer program product comprising a computer readable storage medium having a computer program mechanism encoded thereon, wherein said computer program mechanism may be loaded into the memory of said computer and cause said computer to carry out the method of claim 38.
41. A computer system comprising
a processor, and
a memory coupled to said processor and encoding one or more programs, wherein said one or more programs cause the processor to carry out the method of any one of claims 1, 7, 13, 17, 21, 28 and 31.
42. A computer program product for use in conjunction with a computer having a processor and a memory connected to the processor, said computer program product comprising a computer readable storage medium having a computer program mechanism encoded thereon, wherein said computer program mechanism may be loaded into the memory of said computer and cause said computer to carry out the method of any one of claims 1, 7, 13, 17, 21, 28 and 31.
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A method for detecting breakage of a diesel particulate filter (DPF), the method comprising:
obtaining a learned value of a flow resistance in the DPF when a regeneration of the DPF is terminated;
calculating a predicted value of the flow resistance;
determining whether the learned value is within a permissible error range of the predicted value; and
determining that the DPF is broken if the learned value is not within the permissible error range.
2. The method of claim 1, further comprising:
storing an arithmetic average of the learned value according to a mileage of a vehicle; and
transforming a relation between the mileage and the learned value to a polynomial expression.
3. The method of claim 2, wherein the predicted value is calculated by using the polynomial expression.
4. The method of claim 1, which is reiterated if the learned value is within the permissible error range.