CN114384565B - Dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition - Google Patents

Dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition Download PDF

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CN114384565B
CN114384565B CN202210292045.5A CN202210292045A CN114384565B CN 114384565 B CN114384565 B CN 114384565B CN 202210292045 A CN202210292045 A CN 202210292045A CN 114384565 B CN114384565 B CN 114384565B
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张�杰
曹相
陈倩倩
徐磊
高旺
潘树国
刘宏
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Southeast University
Nanjing Institute of Measurement and Testing Technology
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Abstract

The invention discloses a dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition. Firstly, VMD decomposition is carried out on a dynamic positioning coordinate sequence to obtain a low-frequency trend component and a high-frequency noise component of the coordinate sequence, and the standard deviation of the high-frequency noise component is calculated; then, identifying abnormal values by using the coordinate sequence, the low-frequency trend component and the noise standard deviation, and interpolating abnormal points to obtain a new sequence; performing VMD decomposition, abnormal value identification and interpolation on the new sequence iteration, and calculating the change percentage between the noise standard deviation of the last iteration and the noise standard deviation of the last iteration; and comparing the change percentage with a threshold, if the change percentage is larger than the threshold, iterating again, and if the change percentage is smaller than the threshold, considering that the abnormal value in the sequence is identified and eliminated. By using the method provided by the invention, abnormal values with different amplitudes in the dynamic positioning coordinate sequence can be effectively identified and eliminated, and the precision and the reliability of the positioning sequence are improved.

Description

Dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition
Technical Field
The present invention relates to a Global Navigation Satellite System (GNSS) Satellite positioning method, and more particularly, to a dynamic positioning coordinate series abnormal value identification method based on a Variational Mode Decomposition (VMD).
Background
In dynamic positioning data processing based on Beidou/GNSS and combined positioning thereof, errors such as ionosphere errors, troposphere errors, satellite orbit errors, satellite clock errors and the like which affect positioning accuracy calculation can be effectively processed generally through a differential mode or a state domain correction mode, and positioning calculation results are mainly affected by observation noise and unmodeled errors. Under the environment of serious signal shielding of urban canyons and complex mountainous areas, the real-time dynamic positioning sequence calculated according to the satellite-based or ground-based enhancement contains more obvious observation noise influence, and even rough calculation may exist. For dynamic positioning sequences, the gross error identification methods widely adopted at home and abroad at present can be divided into two major types, namely a statistical gross error detection method and a non-statistical gross error detection method, the statistical gross error detection method mainly comprises a standard deviation-based inspection method and a quartile spacing method, and the two methods have good detection effects under the conditions of small observed gross error quantity and small data dispersion degree, but are difficult to deal with coordinate sequences with large observed gross error quantity and large data dispersion amplitude. The non-statistical gross error detection method mainly uses wavelet analysis, but the selection of wavelet basis mainly depends on manual experience, is lack of unified standard, and cannot bring ideal effect when the selection is improper. In order to realize reliable estimation of a dynamic positioning sequence in a complex environment, the problem of multiple gross errors possibly existing in a real-time resolving coordinate sequence needs to be fully considered on the basis of the existing method, and a more applicable positioning abnormal gross error identification and elimination algorithm is developed.
The Variable Mode Decomposition (VMD) algorithm is mainly based on wiener filtering, hilbert transform and heterodyne demodulation of frequency mixing, and compared with a common EMD method using circular screening, the VMD algorithm decomposes an original signal into a preset number of Intrinsic Mode Function (IMF) components by constructing and solving a constraint variable problem. The algorithm has good complex data decomposition precision and strong anti-interference capability, can effectively avoid problems of mode aliasing, boundary effect and the like, and can be used for determining the frequency center and the bandwidth of signal components such as observation noise, tendency and the like in a dynamic coordinate sequence.
Disclosure of Invention
In order to solve the problem that abnormal values exist in the dynamic positioning coordinate sequence, the invention provides the dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition, which can effectively identify and eliminate the abnormal values with different amplitudes in the dynamic positioning coordinate sequence and improve the accuracy and reliability of the positioning sequence.
In order to achieve the purpose, the invention is realized by adopting the following technical scheme:
a dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition comprises the following steps:
step 1, VMD decomposition is carried out on a dynamic positioning coordinate sequence of satellite navigation to obtain a low-frequency trend component and a high-frequency noise component of the coordinate sequence, and the standard deviation of the high-frequency noise component is calculated;
step 2, identifying abnormal values by using the coordinate sequence, the standard deviation of the low-frequency trend component and the high-frequency noise component, and interpolating abnormal points to obtain a new sequence;
step 3, performing VMD decomposition, abnormal value identification and interpolation on the new sequence iteration, and calculating the change percentage between the standard deviation of the high-frequency noise component of the last iteration and the standard deviation of the high-frequency noise component of the last iteration;
and 4, comparing the change percentage with a set threshold value, if the change percentage is larger than the threshold value, iterating again, and if the change percentage is smaller than the threshold value, determining that the abnormal value in the sequence is identified and eliminated.
In step 1, the VMD decomposes a dynamic positioning coordinate sequence of the satellite navigation into a preset number of low-frequency trend components and high-frequency noise components by constructing and solving a constraint variation problem, and a constraint variation equation is expressed as:
Figure 502992DEST_PATH_IMAGE001
（1）
wherein the content of the first and second substances,
Figure 750434DEST_PATH_IMAGE002
a sequence of dynamic positioning coordinates is represented,
Figure 509574DEST_PATH_IMAGE003
each of the components resulting from the decomposition is represented,
Figure 766243DEST_PATH_IMAGE004
a high-frequency noise component is represented,
Figure 996236DEST_PATH_IMAGE005
a low-frequency tendency component is represented,
Figure 680158DEST_PATH_IMAGE006
a center frequency representing a high frequency noise component, a low frequency trend component;
Figure 917805DEST_PATH_IMAGE007
it is shown that the partial derivative is calculated over time,
Figure 774902DEST_PATH_IMAGE008
the unit impulse function is expressed as a function of unit impulse,
Figure 672451DEST_PATH_IMAGE009
represents the unit imaginary number;
Figure 979805DEST_PATH_IMAGE010
，
Figure 697225DEST_PATH_IMAGE011
representing the identification of the signal components for the number of the signal components;
Figure 46429DEST_PATH_IMAGE013
is a natural index;
Figure 126380DEST_PATH_IMAGE014
represents an amount of signal time;
introducing a secondary penalty factor
Figure 152105DEST_PATH_IMAGE015
And lagrange multiplier
Figure 606089DEST_PATH_IMAGE016
Converting the constraint variable division problem into an unconstrained variable division problem; the augmented lagrange expression is expressed as:
Figure 945935DEST_PATH_IMAGE017
（2）
and (3) solving the formula (2) by using an alternating direction multiplier iterative algorithm, and optimizing to obtain each modal component and the center frequency.
After the high frequency noise component is decomposed based on the VMD, the corresponding standard deviation is calculated as follows:
Figure 565879DEST_PATH_IMAGE018
（3）
wherein, the first and the second end of the pipe are connected with each other,
Figure 824822DEST_PATH_IMAGE019
which represents the standard deviation of the noise sequence,
Figure 782413DEST_PATH_IMAGE020
representing high frequency noise components
Figure 706376DEST_PATH_IMAGE021
The number of the data is one,
Figure 370706DEST_PATH_IMAGE022
represents the mean of the high-frequency noise components,
Figure 754546DEST_PATH_IMAGE023
the number of data indicating the dynamic coordinate time series.
In step 2, the discrimination parameter of the abnormal value
Figure 261750DEST_PATH_IMAGE024
Expressed as:
Figure 740136DEST_PATH_IMAGE025
（4）
wherein the content of the first and second substances,
Figure 570558DEST_PATH_IMAGE024
is shown as
Figure 46670DEST_PATH_IMAGE026
The abnormal value of the individual data is used to determine the parameter,
Figure 732516DEST_PATH_IMAGE027
and
Figure 76909DEST_PATH_IMAGE028
respectively representing the dynamic positioning coordinate sequence and the second in the low-frequency trend component
Figure 512570DEST_PATH_IMAGE029
A piece of data; when in use
Figure 736747DEST_PATH_IMAGE030
Above a given threshold, then
Figure 625068DEST_PATH_IMAGE031
If the data is abnormal data, the abnormal data is removed, and the previous data is utilized for interpolation, wherein the calculation formula of the interpolation data is as follows:
Figure 399251DEST_PATH_IMAGE032
（5）
wherein the content of the first and second substances,
Figure 954998DEST_PATH_IMAGE033
and
Figure 163125DEST_PATH_IMAGE034
respectively represent
Figure 788010DEST_PATH_IMAGE035
The first two data, i.e. thei-1 and 2i-2 data.
In step 3, the percentage of change between the standard deviation of the high-frequency noise component of the last iteration and the standard deviation of the high-frequency noise component of the last iteration is represented as:
Figure 411890DEST_PATH_IMAGE036
（6）
wherein, the first and the second end of the pipe are connected with each other,
Figure 773208DEST_PATH_IMAGE037
represents the percentage change in the standard deviation of the noise,
Figure 89919DEST_PATH_IMAGE038
and
Figure 280729DEST_PATH_IMAGE039
denotes the first
Figure 957567DEST_PATH_IMAGE040
Second and third
Figure 160010DEST_PATH_IMAGE041
Noise standard deviation of the sub-iteration.
In step 4, the judgment condition that the abnormal value is identified and eliminated is as follows:
Figure 132776DEST_PATH_IMAGE042
（7）
wherein the content of the first and second substances,
Figure 748565DEST_PATH_IMAGE043
a threshold value representing a percentage of change.
The invention has the beneficial effects that: the method comprises the steps of firstly carrying out VMD decomposition on a dynamic positioning coordinate sequence to obtain a low-frequency trend component and a high-frequency noise component of the coordinate sequence, calculating a standard deviation of the high-frequency noise component, identifying an abnormal value, carrying out interpolation on an abnormal point to obtain a new sequence, and carrying out iteration until the abnormal value is completely eliminated. By using the method provided by the invention, abnormal values with different amplitudes in the dynamic positioning coordinate sequence can be effectively identified and eliminated, and the precision and the reliability of the positioning sequence are improved.
Drawings
FIG. 1 is a flow chart of a method of an embodiment of the present invention;
FIG. 2 is a time and frequency domain plot of the components at the first VMD decomposition;
FIG. 3 is a diagram showing the first recognized abnormal value and the coordinate sequence before and after interpolation;
FIG. 4 is a time domain and frequency domain plot of the components at the second VMD decomposition;
FIG. 5 is a diagram showing the second recognized abnormal value and the coordinate sequence before and after interpolation;
FIG. 6 is a time domain and frequency domain plot of the components at the third VMD decomposition;
FIG. 7 is a diagram showing the third recognized abnormal value and the coordinate sequence before and after interpolation;
FIG. 8 is a comparison of standard deviations of high frequency noise components for three VMD decompositions.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, so that those skilled in the art can implement the technical solutions in reference to the description text.
Referring to fig. 1, the invention is a dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition, comprising the following steps:
step 1, VMD decomposition is carried out on the satellite navigation dynamic positioning coordinate sequence to obtain a low-frequency trend component and a high-frequency noise component of the coordinate sequence, and the standard deviation of the high-frequency noise component is calculated. The VMD decomposes an original signal into a preset number of low-frequency trend components and high-frequency noise components by constructing and solving a constraint variation problem, and a constraint variation equation is expressed as follows: in step 1, the VMD decomposes a dynamic positioning coordinate sequence of the satellite navigation into a preset number of low-frequency trend components and high-frequency noise components by constructing and solving a constraint variation problem, and a constraint variation equation is expressed as:
Figure 42143DEST_PATH_IMAGE001
（1）
wherein, the first and the second end of the pipe are connected with each other,
Figure 676256DEST_PATH_IMAGE044
a sequence of dynamic positioning coordinates is represented,
Figure 6874DEST_PATH_IMAGE045
each of the components resulting from the decomposition is represented,
Figure 875340DEST_PATH_IMAGE046
a high-frequency noise component is represented,
Figure 379134DEST_PATH_IMAGE047
a low-frequency tendency component is represented,
Figure 946381DEST_PATH_IMAGE048
a center frequency representing a high frequency noise component, a low frequency trend component;
Figure 759485DEST_PATH_IMAGE049
it is shown that the partial derivative is calculated over time,
Figure 84287DEST_PATH_IMAGE050
the unit impulse function is expressed as a function of unit impulse,
Figure 345767DEST_PATH_IMAGE051
represents a unit imaginary number;
Figure 439625DEST_PATH_IMAGE010
，
Figure 502259DEST_PATH_IMAGE011
representing the identification of the signal components for the number of the signal components;
Figure 563624DEST_PATH_IMAGE052
is a natural index;
Figure 674800DEST_PATH_IMAGE053
represents an amount of signal time; in order to solve the variation problem of the formula (1), a secondary penalty factor and a Lagrange multiplier are introduced, and the constraint variation problem is converted into an unconstrained variation problem; the augmented lagrange expression is expressed as:
Figure 574229DEST_PATH_IMAGE017
（2）
solving the formula (2) by using an alternating direction multiplier iterative Algorithm (ADMM), and optimizing to obtain each modal component and the center frequency.
After the high-frequency noise component is decomposed based on the VMD, calculating the corresponding standard deviation:
Figure 745448DEST_PATH_IMAGE054
（3）
wherein the content of the first and second substances,
Figure 107159DEST_PATH_IMAGE055
which represents the standard deviation of the noise sequence,
Figure 271293DEST_PATH_IMAGE056
representing high frequency noise components
Figure 231421DEST_PATH_IMAGE021
The number of the data is one,
Figure 307961DEST_PATH_IMAGE057
represents the mean of the high-frequency noise components,
Figure 156969DEST_PATH_IMAGE058
the number of data representing the dynamic coordinate time series.
And 2, identifying abnormal values by using the coordinate sequence, the low-frequency trend component and the noise standard deviation, and interpolating abnormal points to obtain a new sequence. The discrimination parameters for the outliers are expressed as:
Figure 859214DEST_PATH_IMAGE025
（4）
wherein the content of the first and second substances,
Figure 985433DEST_PATH_IMAGE059
is shown as
Figure 252116DEST_PATH_IMAGE061
The abnormal value of each data is used to judge the parameter,
Figure 260524DEST_PATH_IMAGE062
and
Figure 579509DEST_PATH_IMAGE063
respectively representing the dynamic positioning coordinate sequence and the second in the low-frequency trend component
Figure 871819DEST_PATH_IMAGE029
A piece of data; when in use
Figure 493425DEST_PATH_IMAGE064
Above a given threshold (typically set to 3 with a corresponding confidence of 99.73%), the determination is made that
Figure 739861DEST_PATH_IMAGE065
If the data is abnormal data, the abnormal data is removed, and the previous data is utilized for interpolation, wherein the calculation formula of the interpolation data is as follows:
Figure 862537DEST_PATH_IMAGE032
（5）
wherein the content of the first and second substances,
Figure 494507DEST_PATH_IMAGE066
and
Figure 598598DEST_PATH_IMAGE067
respectively represent
Figure 784860DEST_PATH_IMAGE068
The first two data, i.e. thei-1 and 2i2 data.
Step 3, calculating the change percentage between the noise standard deviation of the last iteration and the noise standard deviation of the last iteration, and expressing as:
Figure 131134DEST_PATH_IMAGE036
（6）
wherein, the first and the second end of the pipe are connected with each other,
Figure 617610DEST_PATH_IMAGE069
represents the percentage change in the standard deviation of the noise,
Figure 971231DEST_PATH_IMAGE070
and
Figure 690795DEST_PATH_IMAGE071
is shown as
Figure 30640DEST_PATH_IMAGE040
Second and third
Figure 387934DEST_PATH_IMAGE072
Noise standard deviation of the sub-iteration.
Step 4, comparing the percentage of change with a threshold value, expressed as:
Figure 912457DEST_PATH_IMAGE073
（7）
wherein the content of the first and second substances,
Figure 604469DEST_PATH_IMAGE074
a threshold value representing a percentage of change. When the condition of the formula (7) is satisfied, the abnormal value in the coordinate sequence is considered to be identified and removed; and when the condition of the formula (7) is not satisfied, indicating that part of abnormal values still remain in the coordinate sequence, returning to the step 3, and performing VMD iterative decomposition and judgment until the condition of the formula (7) is satisfied.
Experimental verification was performed on the basis of the measured data as follows: the adopted data is a group of satellite navigation dynamic positioning data which is actually acquired, the frequency is 1Hz, 10878 epochs in total from 14:50 in one day are selected for carrying out an abnormal value identification experiment, the change percentage threshold of the standard deviation of the noise component is 2%, and the algorithm meets the precision requirement after three times of iterative identification and interpolation.
The experimental results are shown in fig. 2 to 7, wherein fig. 2, 4, and 6 are time domain and frequency domain graphs of low-frequency trend components and high-frequency noise components obtained by VMD decomposition in three iterations, it can be seen from the graphs that the low-frequency trend components are matched with the trend of the original sequence, and the frequency domain analysis results are mainly ultra-low frequency components; the high frequency noise component is substantially white noise and contains no trend component.
Fig. 3, 5, and 7 show the abnormal points identified in the triple iteration process and the sequences before and after interpolation, and fig. 8 shows the standard deviation of the high-frequency noise component calculated in the triple iteration. As can be seen from fig. 3, 5 and 7, the outlier identification method used in the present invention mainly identifies outliers with larger amplitude in the first iteration, and some outliers with smaller amplitude are not identified in the first iteration; because the first interpolation weakens the influence of the abnormal value with larger amplitude, the standard deviation of the high-frequency noise component in the sequence is reduced, and the abnormal value with smaller amplitude can be effectively identified and interpolated in the subsequent second iteration and the third iteration. As can be seen from FIG. 8, the standard deviation of the high frequency noise component was reduced from 2.266mm to 2.020 mm. Therefore, experiments prove that the abnormal values with different amplitudes in the dynamic positioning coordinate sequence can be effectively identified and weakened by adopting the VMD iterative decomposition-based dynamic positioning coordinate sequence abnormal value identification method, and the accuracy and the reliability of the positioning sequence are improved.
Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention and not for limiting the same, and although the present invention is described in detail with reference to the above embodiments, those of ordinary skill in the art should understand that: modifications and equivalents may be made to the embodiments of the invention without departing from the spirit and scope of the invention, which is to be covered by the claims.

Claims (6)

1. A dynamic positioning coordinate sequence abnormal value identification method based on VMD iterative decomposition is characterized by comprising the following steps:
step 1, VMD decomposition is carried out on a dynamic positioning coordinate sequence of satellite navigation to obtain a low-frequency trend component and a high-frequency noise component of the coordinate sequence, and the standard deviation of the high-frequency noise component is calculated;
step 2, identifying abnormal values by using the coordinate sequence, the standard deviation of the low-frequency trend component and the high-frequency noise component, and interpolating abnormal points to obtain a new sequence;
step 3, performing VMD decomposition, abnormal value identification and interpolation on the new sequence iteration, and calculating the change percentage between the standard deviation of the high-frequency noise component of the last iteration and the standard deviation of the high-frequency noise component of the last iteration;
and 4, comparing the change percentage with a set threshold value, if the change percentage is larger than the threshold value, iterating again, and if the change percentage is smaller than the threshold value, determining that the abnormal value in the sequence is identified and eliminated.
2. The method for identifying the abnormal value of the dynamic positioning coordinate sequence based on the VMD iterative decomposition as claimed in claim 1, wherein: in step 1, the VMD decomposes a dynamic positioning coordinate sequence of the satellite navigation into a preset number of low-frequency trend components and high-frequency noise components by constructing and solving a constraint variation problem, and a constraint variation equation is expressed as:
Figure DEST_PATH_IMAGE001
（1）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE002
a sequence of dynamic positioning coordinates is represented,
Figure DEST_PATH_IMAGE003
each of the components resulting from the decomposition is represented,
Figure DEST_PATH_IMAGE004
a high-frequency noise component is represented,
Figure DEST_PATH_IMAGE005
a low-frequency tendency component is represented,
Figure DEST_PATH_IMAGE006
a center frequency representing a high frequency noise component, a low frequency trend component;
Figure DEST_PATH_IMAGE007
it is shown that the partial derivative is calculated over time,
Figure DEST_PATH_IMAGE008
the unit impulse function is expressed as a function of unit impulse,
Figure DEST_PATH_IMAGE009
represents the unit imaginary number;
Figure DEST_PATH_IMAGE010
，
Figure DEST_PATH_IMAGE011
representing the identification of the signal components for the number of the signal components;
Figure DEST_PATH_IMAGE013
is a natural index;
Figure DEST_PATH_IMAGE014
represents an amount of signal time;
introducing a secondary penalty factor
Figure DEST_PATH_IMAGE015
And lagrange multiplier
Figure DEST_PATH_IMAGE016
Converting the constrained variable problem into an unconstrained variable problem; the augmented lagrange expression is expressed as:
Figure DEST_PATH_IMAGE018
（2）
and (3) solving the formula (2) by using an alternating direction multiplier iterative algorithm, and optimizing to obtain each modal component and the center frequency.
3. The VMD iterative decomposition-based dynamic positioning coordinate sequence outlier identification method of claim 2, wherein: after the high frequency noise component is decomposed based on the VMD, the corresponding standard deviation is calculated as follows:
Figure DEST_PATH_IMAGE019
（3）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE020
which represents the standard deviation of the noise sequence,
Figure DEST_PATH_IMAGE021
second to express high frequency noise component
Figure DEST_PATH_IMAGE022
The number of the data is one,
Figure DEST_PATH_IMAGE023
represents the mean of the high-frequency noise components,
Figure DEST_PATH_IMAGE024
the number of data representing the dynamic coordinate time series.
4. The VMD iterative decomposition-based dynamic positioning coordinate sequence outlier identification method of claim 3, wherein: in step 2, the discrimination parameter of the abnormal value
Figure DEST_PATH_IMAGE025
Expressed as:
Figure DEST_PATH_IMAGE026
（4）
wherein the content of the first and second substances,
Figure 49511DEST_PATH_IMAGE025
is shown as
Figure DEST_PATH_IMAGE027
The abnormal value of the individual data is used to determine the parameter,
Figure DEST_PATH_IMAGE028
and
Figure DEST_PATH_IMAGE029
respectively representing the dynamic positioning coordinate sequence and the second in the low-frequency trend component
Figure DEST_PATH_IMAGE030
A piece of data; when in use
Figure DEST_PATH_IMAGE031
Above a given threshold, then
Figure DEST_PATH_IMAGE032
As abnormal data, thisAnd eliminating the abnormal data, and performing interpolation by using the previous data, wherein the calculation formula of the interpolation data is as follows:
Figure DEST_PATH_IMAGE033
（5）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE034
and
Figure DEST_PATH_IMAGE035
respectively represent
Figure DEST_PATH_IMAGE036
The first two data, i.e. thei-1 and ai2 data.
5. The method for identifying the abnormal value of the dynamic positioning coordinate sequence based on the VMD iterative decomposition as claimed in claim 1, wherein: in step 3, the percentage of change between the standard deviation of the high-frequency noise component of the last iteration and the standard deviation of the high-frequency noise component of the last iteration is expressed as:
Figure DEST_PATH_IMAGE037
（6）
wherein, the first and the second end of the pipe are connected with each other,
Figure DEST_PATH_IMAGE038
represents the percentage change in the standard deviation of the noise,
Figure DEST_PATH_IMAGE039
and with
Figure DEST_PATH_IMAGE040
Is shown as
Figure DEST_PATH_IMAGE041
Second and third
Figure DEST_PATH_IMAGE042
Noise standard deviation of the sub-iteration.
6. The method for identifying the abnormal value of the dynamic positioning coordinate sequence based on the VMD iterative decomposition as claimed in claim 1, wherein: in step 4, the judgment condition that the abnormal value has been identified and removed is as follows:
Figure DEST_PATH_IMAGE043
（7）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE044
a threshold value representing a percentage of change.
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