CN104462187B - Gunz Validation of Data method based on maximum likelihood ratio - Google Patents

Abstract

The present invention provides a method for verifying the validity of crowd intelligence data based on maximum likelihood ratio. The data is classified according to the observed value; for all the data of the same measured value, the probability density function is calculated using kernel density estimation, and the confidence probability is calculated; the server waits for the user to upload new data; the measurer uses his mobile terminal to perform multiple measurements to obtain a set of The data, together with the observed components observed by the measurer, are uploaded to the server; the server compares the data provided by the user with the database, and calculates the likelihood of this set of data using a group intelligence data validity verification method based on the maximum likelihood ratio Reliability; the server decides whether to accept this set of data, pays remuneration according to reliability, updates the database of this measured value, and recalculates the probability density function and confidence probability.

Description

Crowd Intelligence Data Validation Verification Method Based on Maximum Likelihood Ratio

technical field

The present invention relates to the field of communication technology, in particular to a maximum likelihood ratio-based method for verifying the validity of crowd intelligence data.

Background technique

Crowdsourcing has a very broad prospect in the application of smart phones. With the rapid development of Internet technology, the number of individuals in the network is increasing rapidly, and the connection between individuals is getting closer and closer. In such a big environment, Swarm Intelligence Service came into being. How to effectively build a group intelligence service platform and promote resource sharing in society is an important issue that needs to be solved in the next generation Internet research.

Nowadays, information providers often adopt the Crowdsourcing Incentive Mechanism (Crowdsourcing Incentive Mechanism), assigning the work of collecting information to scattered users, and giving them certain returns for the information or services they provide. For example, if someone wants to know the congestion situation of a certain road, the information provided by the user on this road is not only faster but also more accurate than the information obtained by the provider sending people to investigate. Nowadays, mobile phone sensing technology (Mobile PhoneSensing) is developing vigorously, and a variety of sensing devices are being installed on smart phones, such as acceleration sensors, GPS, distance sensors, cameras, etc. It is an increasingly popular method to use the sensor technology of these decentralized users' smart phones to obtain the required information and upload it to the provider.

Although swarm intelligence has many advantages, its disadvantages are also inevitable. Since the measurers of the data are not professionally trained, the observation error of the measured data will be relatively large in general, and, because the measurers are not trained, the difference in the validity of different data will be greater than that obtained by traditional methods . In extreme cases, if the measurer is very unfamiliar with the test object, or even misuses it, causing the data to seriously deviate from the normal level, the use of this data will cause certain damage to the validity of the sample.

This is a unique error in the crowd intelligence scene, hereinafter referred to as observation error; the rest are called measurement errors. These two kinds of errors can usually be compensated by a larger sample size, but our purpose is to quantitatively evaluate and compare crowdsmart data through the method of probability theory. Further, the purpose is to select a part with higher relative validity, that is, a part with smaller observation error.

After searching the existing technical literature, it was found that M.Ramadan et al. proposed a video sharing mechanism based on cooperative downloading in "Implementation and evaluation of cooperative video streaming for mobile devices" published by International Symposium on Personal, Indoor and Mobile Radio Communications in 2008. However, this mechanism requires all participating users to know each other and actively form a wireless LAN, so the application scenarios are greatly limited. In "MicroCast: cooperative video streaming on smartphones" published by L.Keller et al. at the International Conference on Mobile Systems, Applications, and Services in 2012, a collaborative video download acceleration mechanism was proposed using wireless communication between mobile phones. However, this mechanism requires that all participating users want to download the same video, which is not satisfied in most cases, so it has great limitations.

Contents of the invention

Aiming at the defects in the prior art, the purpose of the present invention is to provide a method for verifying the validity of group intelligence data based on maximum likelihood ratio, which can better screen effective data by using the accumulated large amount of data content in the server database, and reduce Judgment bias caused by incorrect data entry.

According to a method for verifying the validity of crowd intelligence data based on maximum likelihood ratio provided by the present invention, it comprises the following steps:

Step 1: Experimentally obtain the prior probability p _lj , where p _lj represents the probability that an untrained measurer judges the observation component j as l for a certain observation component j;

Step 2: The server classifies all the accumulated data according to the observed value; for all the data of the same measured value j, use kernel density estimation to calculate the probability density function, and calculate the confidence probability α _j ;

Step 3: The server waits for the user to upload new data;

Step 4: The measurer i uses his mobile terminal to conduct multiple measurements to obtain a set of data, which is uploaded to the server together with the observed components observed by the measurer himself;

Step 5: The server compares the data provided by the user with the database, and calculates the likelihood reliability of this set of data;

Step 6: The server decides whether to accept this set of data, and pays according to the reliability; if the server accepts this set of data, return to step 2, update the database of the measured value j, and use the method in step 2 to calculate the probability density function and confidence probability α _j .

Preferably, said step 1 includes the following steps:

Step 1.1: During the training process of indoor positioning based on Wi-Fi signal strength, the measurer needs to determine his or her indoor position, resulting in an observation error; the measurer’s observation error is abstracted as being the closest to the room when it is at a point in the room The estimation error of the distance between the two walls of ;

Step 1.2: Determine the prior probability p _lj through a pre-experiment and apply the prior probability p _lj to all indoor positioning activities, specifically, let multiple measurers be fixed in a room without distance reference objects Point j judges its own position l, and collects the distribution of the judgment results of the multiple measurers as p _lj ;

Step 1.3: For p _lj that cannot be determined by an experiment in advance, the Kronecker function can be used:

Among them, δ _lj represents the Kronecker function.

Preferably, said step 2 includes the following steps:

Step 2.1: Each observation component in the database of the server corresponds to the accumulated data set D _j , j=1, 2, 3, ..., N, N represents the total number of observation components, each element D _j ^k in D _j , k=1, 2, 3,...T, subject to f ^j (x) distribution, T represents the total number of data of each observation component, f ^j (x) represents the probability density function obeyed by observation component j; T=| D _j |＞＞M, M represents the total number of data uploaded by the measurer once, then

Among them, K _h represents the kernel density function, and x represents the data variable;

Step 2.2: Set That is, n _s (x) represents the number of existing data in the database in [xh, x+h], and h represents the bandwidth of the kernel density function K _h ;

n _s (x) may have T+1 values and obey the distribution:

Among them, P( ) represents the probability mass function of n _s (x), n _s (x) represents the number of existing data in the database in [xh, x+h], n _s represents the possible value, which can be 0 , any value in 1,..., T, T+1, Indicates the number of combinations of n _s taken from T different elements, and h indicates the bandwidth of the kernel density function K _h ;

Step 2.3: Determine the expectation of r _il through the size of the database, and use this expectation as the confidence probability α, where r _il represents the probability density that the data uploaded by observer i belongs to the observation component l; obviously, the accumulated data amount corresponding to different observation values are different, so there are different confidence probabilities α _j for different observations.

Preferably, said step 4 includes the following steps:

Step 4.1: The measurer obtains a set of M data and writes it down as the following formula

in, Indicates a set of data obtained by the measurer i for multiple measurements on the same observed component, j indicates the true value of a component that needs to be observed in this set of M data, j∈{1, 2, 3, ..., N} , N represents the total number of observed components; x ^t _i obeys the distribution f ^j (x) corresponding to component j, and x ^t _i represents the tth data uploaded by measurer i;

Step 4.2: Observation error is reflected in that the measurer judges j as j′ and reports it to the server, that is,

Preferably, said step 5 includes the steps of:

Step 5.1: The server gets the data After calculating all {r _il }:

Among them, M represents the total number of data uploaded by the measurer at one time, f(·) represents the probability density function that the observed component obeys, l represents the number of the possible observed component, x ^t _ij′ represents the tth data uploaded by the observer i, and It is judged as the observation component j′, N represents the total number of observation components, and the physical meaning of r _il is The probability density belonging to the observed component l; obviously, when l=j is the largest;

Step 5.2: Define parameters

Among them, α _j is called the confidence probability, and p _lj' represents the probability that the measurer judges the observed component j' as the observed component l for the observed component j'; when α _j =1 The meaning of is the logarithm of the maximum possible probability density of the measured data; obviously for a set of data of the same length, The larger is more credible;

Step 5.3: Pass It can sort the effectiveness of all group intelligence data, and select the first few of them according to needs.

Preferably,

In step 2.1, the kernel density function is taken as the uniform kernel function: h is small enough to make the data approximately uniformly distributed within the bandwidth range, and the probability of falling into this area P _s =P(|xD _j ^k |<h)=f(x)2h;

In step 2.3, all The data of all have adopted value, the following is a method to calculate the expected E{r _il } of r _il :

Among them, f ^l (x ^t ) represents the probability density of the observed component l taking the value of x ^t , l represents the lth observed component, t represents the tth data uploaded by the observer, M represents the total number of data uploaded by the measurer once, ! Represents factorial, e represents natural base, P _s =P(|xD _j ^k |<h)=f( _xi )2h, f( _xi ) is estimated by kernel density; there are no variables other than T in the above formula , so the relationship between the confidence probability α _j and the database size T is determined.

Compared with the prior art, the present invention has the following beneficial effects:

1. The present invention can correct the judgment error of the swarm intelligence data observer through pre-experimentation;

2. The present invention can evaluate the effectiveness of new swarm intelligence data based on the existing reliable data sets, so as to reasonably make effective choices for the new swarm intelligence data.

Description of drawings

Other characteristics, objects and advantages of the present invention will become more apparent by reading the detailed description of non-limiting embodiments made with reference to the following drawings:

Fig. 1 is a flow chart of steps of the present invention.

detailed description

The present invention will be described in detail below in conjunction with specific embodiments. The following examples will help those skilled in the art to further understand the present invention, but do not limit the present invention in any form. It should be noted that those skilled in the art can make several modifications and improvements without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

The present invention provides a method for verifying the validity of crowd intelligence data based on maximum likelihood ratio. The data is classified according to the observed value; for all the data of the same measured value, the probability density function is calculated using kernel density estimation, and the confidence probability is calculated; the server waits for the user to upload new data; the measurer uses his mobile terminal to perform multiple measurements to obtain a set of The data, together with the observed components observed by the measurer, are uploaded to the server; the server compares the data provided by the user with the database, and calculates the likelihood of this set of data using a group intelligence data validity verification method based on the maximum likelihood ratio Reliability; the server decides whether to accept this set of data, pays remuneration according to reliability, updates the database of this measured value, and recalculates the probability density function and confidence probability.

Specifically, the present invention provides a maximum likelihood ratio-based crowd intelligence data validity verification method, which can better screen valid data by using a large amount of data content accumulated in the server database, and reduce judgment bias caused by wrong data entry.

Referring to accompanying drawing 1, the present invention is realized through the following technical solutions, and the present invention comprises the steps:

Step 1: Experimentally obtain the prior probability p _lj , which means that for a certain observed component j, an untrained ordinary person judges it as the probability of l.

Step 2: The server classifies all the accumulated data according to the observed value. For all data of the same measurement value j, use kernel density estimation to calculate the probability density function, and calculate the confidence probability α _j .

Step 3: The server waits for the user to upload new data.

Step 4: Measurer i uses his mobile terminal to conduct multiple measurements, obtain a set of data, and upload them to the server together with the observed components observed by the measurer himself.

Step 5: The server compares the data provided by the user with the database, and calculates the likelihood reliability of this set of data using a method of verifying the validity of group intelligence data based on the maximum likelihood ratio.

Step 6: The server decides whether to accept this set of data, and pays according to the reliability; if the server accepts this set of data, return to step 2, update the database of the measured value j, and use the method in step 2 to calculate the probability density function and confidence Probability α _j .

The implementation process of the present invention will be described in more detail below.

Step 1, assuming that the server needs to measure a certain measurement value through swarm intelligence data, and the measurement value includes several observation components. Affected by the observation error, the measurer misjudges a certain observed component j as another observed component l with probability p _lj . The experiment first obtains the prior probability p _lj .

For example, during the training process of indoor positioning based on Wi-Fi signal strength, the measurer needs to determine his or her indoor position, resulting in observation errors. The measurement error of the measurer can be abstracted as the estimation error of the distance between the two nearest walls of the room at a point in the room. This distribution p _lj can be determined and applied to all indoor positioning activities through a previous experiment. Recruit a large number of volunteers to judge their own position l at some fixed point j in a room without a significant distance from the reference object, and collect the distribution of their judgment results, which can be regarded as p _lj .

If p _lj cannot be determined through a previous experiment, the Kronecker Delta function can be used.

Step 2, each observation component in the database of the server corresponds to the accumulated data set D _j , j=1, 2, 3, ..., N, where each element D _j ^k , k = 1, 2, 3, .. .T, subject to f ^j (x) distribution, T=|D _j | is the size of the data set. It is assumed that f ^j (x) can be recovered with sufficient accuracy by kernel density estimation. but

The kernel density function can take other arbitrary forms, and those skilled in the art can make various deformations or modifications within the scope of the claims, which do not affect the essence of the present invention. For example, take the kernel density function as the uniform kernel function: h is small enough to make the data approximately evenly distributed in the bandwidth range, and the probability of falling in this area P _s =P(|xD _j ^k |<h)=f ^j (x)2h.

Assume That is, the number of existing data in the database in [xh, x+h]. n _s (x) may have T values whose distribution satisfies

Since different observation components accumulate different amounts of data, different observation components have different confidence probabilities α _j . Confidence probability α _j is used to measure the adoption value of user uploaded data in Indicates that user i uploads data, and the user judges it as observation component j′. If expressed by r _il belongs to the probability density of the observed component l, then the expectation of r _il can be used as the confidence probability α. The following is a method to calculate E{r _il }.

Wherein P _s =P(|xD _j ^k |<h)=f( _xi )2h, f( _xi ) is estimated by kernel density. There are no variables other than T in the formula, so the relationship between the confidence probability α _j and the database size T is determined.

Step 3, the server waits for the user to upload new data.

Step 4, measurer i obtains a set of M data for a certain measurement component and writes it down as the following formula

j represents the real value of the measurement component of this set of data, j∈{1, 2, 3, ..., N}. x ^t _i obeys the distribution f ^j (x) corresponding to component j. The observation error is reflected in the fact that the measurer judges the observed component j as j′ and reports it to the server, that is,

Step 5, the server obtains the data After calculating all {r _il }:

Obviously, it is the largest when l=j. define parameters

pass The system can sort the effectiveness of all group intelligence data, and select the first few of them according to needs.

The environmental parameters of this embodiment are:

Mobile terminal equipment: six Android smart phones, all Nexus 4, each smart phone is equipped with 1.5GHz Snapdragon APQ8064 CPU and 2 G RAM, and the operating system of the six smart phones is Android JellyBean (4.2). These six smartphones were used side by side as test phones for indoor positioning.

Server: Acer 4930G notebook computer, Core Duo processor, 2G memory, 2G main frequency.

Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art may make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention.

Claims (6)

1. A method for verifying the validity of crowd intelligence data based on maximum likelihood ratio, is characterized in that, comprises the steps: Step 1: Experimentally obtain the prior probability p _lj , where p _lj represents the probability that an untrained measurer judges the observation component j as the observation component l for a certain observation component j; Step 2: The server classifies all the accumulated data according to the observation value; for all the data of the same observation component j, use kernel density estimation to calculate the probability density function, and calculate the confidence probability α _j ; Step 3: The server waits for the user to upload new data; Step 4: The measurer i uses his mobile terminal to conduct multiple measurements to obtain a set of data, which is uploaded to the server together with the observed components observed by the measurer himself; Step 5: The server compares the data provided by the user with the database, and calculates the likelihood reliability of this set of data; Step 6: The server decides whether to accept this set of data, and pays according to the reliability; if the server accepts this set of data, return to step 2, update the database of this observation component j, and re-calculate the probability density function and confidence probability using the method in step 2 α _j . 2. the group intelligence data validation method based on maximum likelihood ratio according to claim 1, is characterized in that, described step 1 comprises the steps: Step 1.1: In the training process of indoor positioning based on Wi-Fi signal strength, the measurer needs to determine his indoor position, resulting in an observation error; the measurer's observation error is abstracted as the closest point to the room when he is at a point in the room The estimation error of the distance between the two walls of ; Step 1.2: Determine the prior probability p _lj through a previous experiment and apply the prior probability p _lj to all indoor positioning activities; specifically, let multiple measurers observe some The component j judges the observed component l, and collects the distribution of the judgment results of the multiple measurers as p _lj ; Step 1.3: For p _lj that cannot be determined by a previous experiment, take the Kronecker function: <mrow> <msub> <mi>p</mi> <mrow> <mi>l</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>&amp;delta;</mi> <mrow> <mi>l</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>i</mi> <mi>f</mi> <mi> </mi> <mi>l</mi> <mo>&amp;NotEqual;</mo> <mi>j</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mi>i</mi> <mi>f</mi> <mi> </mi> <mi>l</mi> <mo>=</mo> <mi>j</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> Among them, δ _lj represents the Kronecker function. 3. the group intelligence data validation method based on maximum likelihood ratio according to claim 1, is characterized in that, described step 2 comprises the steps: Step 2.1: Each observation component in the database of the server corresponds to the accumulated data set D _j , j=1, 2, 3,..., N, N represents the total number of observation components, each element D _j ^k in D _j , k= 1,2,3,...T, obey f ^j (x) distribution, T represents the total number of data of each observation component, f ^j (x) represents the probability density function that observation component j obeys; T=|D _j |＞ >M, M represents the total number of data uploaded by the measurer at one time, then <mrow> <msup> <mi>f</mi> <mi>j</mi> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>T</mi> </munderover> <msub> <mi>K</mi> <mi>h</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <msup> <msub> <mi>D</mi> <mi>j</mi> </msub> <mi>k</mi> </msup> <mo>)</mo> </mrow> </mrow> Among them, K _h represents the kernel density function, and x represents the data variable; Step 2.2: Set That is, n _s (x) represents the number of existing data in the database in [xh,x+h], and h represents the bandwidth of the kernel density function K _h ; n _s (x) may have T+1 values and obey the distribution: <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mi>s</mi> </msub> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <msub> <mi>n</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mi>T</mi> <msub> <mi>n</mi> <mi>s</mi> </msub> </msubsup> <mi>P</mi> <msup> <mrow> <mo>(</mo> <mo>|</mo> <mi>x</mi> <mo>-</mo> <msup> <msub> <mi>D</mi> <mi>j</mi> </msub> <mi>k</mi> </msup> <mo>|</mo> <mo>&lt;</mo> <mi>h</mi> <mo>)</mo> </mrow> <msub> <mi>n</mi> <mi>s</mi> </msub> </msup> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>P</mi> <mo>(</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>-</mo> <msup> <msub> <mi>D</mi> <mi>j</mi> </msub> <mi>k</mi> </msup> <mo>|</mo> <mo>&lt;</mo> <mi>h</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mi>T</mi> <mo>-</mo> <msub> <mi>n</mi> <mi>s</mi> </msub> </mrow> </msup> </mrow> 1 Among them, P(·) represents the probability mass function of n _s (x), n _s (x) represents the number of existing data in the database in [xh,x+h], and n _s takes 0,1,...,T , a value in T+1, Indicates the number of combinations of n _s taken from T different elements, and h indicates the bandwidth of the kernel density function K _h ; Step 2.3: Determine the expectation of r _il through the size of the database, and use this expectation as the confidence probability α, where r _il represents the probability density that the data uploaded by observer i belongs to the observation component l; obviously, the accumulated data amount corresponding to different observation values are different, so there are different confidence probabilities α _j for different observations. 4. the group intelligence data validation method based on maximum likelihood ratio according to claim 3, is characterized in that, described step 4 comprises the steps: Step 4.1: The measurer obtains a set of M data and writes it down as the following formula in, Indicates a set of data obtained by the measurer i for multiple measurements of the same observed component, j indicates the true value of a component that needs to be observed in this set of M data, j∈{1,2,3,...,N}, N Indicates the total number of observed components; x ^t _i obeys the distribution f ^j (x) corresponding to component j, and x ^t _i represents the tth data uploaded by measurer i; Step 4.2: Observation error is reflected in that the measurer judges j as j′ and reports it to the server, that is, 5. the group intelligence data validation method based on maximum likelihood ratio according to claim 4, is characterized in that, described step 5 comprises the steps: Step 5.1: The server gets the data After calculating all {r _il }: <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <munderover> <mi>&amp;Pi;</mi> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msup> <mi>f</mi> <mi>l</mi> </msup> <mrow> <mo>(</mo> <msub> <msup> <mi>x</mi> <mi>t</mi> </msup> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>N</mi> </mrow> Among them, M represents the total number of data uploaded by the measurer at one time, f(·) represents the probability density function that the observed component obeys, l represents the number of the observed component, x ^t _ij′ represents the tth data uploaded by the observer i, and its It is judged as the observation component j′, N represents the total number of observation components, and the physical meaning of r _il is The probability density belonging to the observed component l; obviously, when l=j is the largest; Step 5.2: Define parameters Among them, α _j is called the confidence probability, and p _lj' represents the probability that the measurer judges the observed component j' as the observed component l for the observed component j'; when α _j =1 The meaning of is the logarithm of the maximum possible probability density of the measured data; obviously for a set of data of the same length, The larger is more credible; Step 5.3: Pass It is possible to sort the effectiveness of all crowd intelligence data and take the first few of them. 6. the group intelligence data validation method based on maximum likelihood ratio according to claim 5, is characterized in that, In step 2.1, the kernel density function is taken as the uniform kernel function: h is small enough to make the data approximately uniformly distributed within the bandwidth range, and the probability of falling into this area P _s =P(|xD _j ^k |<h)=f(x)2h; In step 2.3, all The data of all have adopted value, the following is a method to calculate the expected E{r _il } of r _il : <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;alpha;</mi> <mi>j</mi> </msub> <mo>=</mo> <mi>E</mi> <mo>{</mo> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mi>f</mi> <mi>l</mi> </msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mi>t</mi> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mi>M</mi> </msup> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>x</mi> <mo>=</mo> <mo>-</mo> <mi>&amp;infin;</mi> </mrow> <mi>&amp;infin;</mi> </munderover> <mo>{</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>n</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>T</mi> </munderover> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>n</mi> <mi>s</mi> </msub> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> <mi>M</mi> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <msub> <mi>P</mi> <mi>s</mi> </msub> <msub> <mi>n</mi> <mi>s</mi> </msub> </msup> </mrow> <mrow> <msub> <mi>n</mi> <mi>s</mi> </msub> <mo>!</mo> </mrow> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>P</mi> <mi>s</mi> </msub> </mrow> </msup> <mo>}</mo> <msub> <mi>P</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> Among them, f ^l (x ^t ) represents the probability density of the observed component l taking the value of x ^t , l represents the lth observed component, t represents the tth data uploaded by the observer, M represents the total number of data uploaded by the measurer once, ! Represents factorial, e represents natural base, P _s =P(|xD _j ^k |<h)=f( _xi )2h, f( _xi ) is estimated by kernel density; there are no variables other than T in the above formula , so the relationship between the confidence probability α _j and the total number T of data of each observation component is determined.