CN106991080A - A kind of quantile of data determines method and device - Google Patents

Abstract

Method and device is determined the embodiment of the invention discloses a kind of quantile of data, methods described includes：The training data chosen from target data is fitted, the corresponding probability density function p (x) of training data is obtained；Using the probability density function p (x), the corresponding fitting distribution function F (x) of training data and its inverse function F are calculated^‑1(x), wherein, the fitting distribution function F (x) be nonlinear function；For presetting each quantile that quantile sequence P is included, the inverse function F is utilized^‑1(x) the corresponding fitting quantile of the quantile, is calculated, and by the fitting quantile storage into fitting quantile sequence B；Obtain the target data sequence D of quantile to be calculated；For the target data sequence D, the fitting distribution function F (x), the inverse function F are utilized^‑1(x) and the fitting quantile sequence B, determine that each quantile distinguishes corresponding current quantile.Using the embodiment of the present invention, the error for the quantile determined is reduced.

Description

A kind of quantile of data determines method and device

Technical field

The present invention relates to technical field of data processing, more particularly to a kind of quantile of data determines method and device.

Background technology

In the current big data epoch, it is more and more the need for data are flowed into the further working process of row, to obtain More useful information.Quantile is carried out to data stream and determines it is the important processing mode of one of which, in computer and finance Had a wide range of applications Deng field.Quantile refers to the numerical value that the probability distribution scope of a stochastic variable is divided into several equal portions Point, conventional has median, quartile, percentile etc.., can intuitively observed number by determining a series of its quantile According to the cumulative distribution function of stream, and then analyze its statistical property.

Existing quantile determines method, it may not be necessary to which the probability Distribution Model to data flow is analyzed, but logical Cross and extract Partial Elements representative in data flow and calculated, so that it is determined that going out the quantile of data flow.Due to right The probability distribution of data flow without analyse, can determine quantile using linear function, for the data flow of distribution uniform, profit Determine that error is smaller during quantile with linear function.

But in actual applications, heavytailed distribution situation i.e. pockety is often presented in data flow.In this case, It is the equally distributed linear function of default data stream due to what is utilized, the quantile error determined can be caused larger, it is impossible to is accurate Really reflect the statistical nature of data flow.

The content of the invention

The purpose of the embodiment of the present invention is that providing a kind of quantile of data determines method and device, is determined with reducing Quantile error.

To reach above-mentioned purpose, method is determined the embodiment of the invention discloses a kind of quantile of data, method includes：Number According to training process and fractional-dimension calculus process；

The data training process, including：

The training data chosen from target data is fitted, the corresponding probability density function p of training data is obtained (x)；

Using the probability density function p (x), the corresponding fitting distribution function F (x) of training data and its inverse function are calculated F^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；

For presetting each quantile that quantile sequence P is included, the inverse function F is utilized^-1(x) this point of position, is calculated The corresponding fitting quantile of point, and by the fitting quantile storage into fitting quantile sequence B；

The fractional-dimension calculus process, including：

Obtain the target data sequence D of quantile to be calculated；

For the target data sequence D, the fitting distribution function F (x), the inverse function F are utilized^-1(x) and institute Fitting quantile sequence B is stated, determines that each quantile distinguishes corresponding current quantile.

Preferably, described utilize the probability density function p (x), the corresponding fitting distribution function F of training data is calculated And its inverse function F (x)^-1(x) the step of, including：

For the probability density function p (x), the p (x) is calculated from the negative infinite definite integral to variable x, obtains described It is fitted distribution function F (x)；

Calculate the inverse function F of the fitting distribution function F (x)^-1(x)。

Preferably, described each quantile included for default quantile sequence P, utilizes the inverse function F^-1(x), The step of quantile is corresponding to be fitted quantile is calculated, including：

For presetting each quantile that quantile sequence is included, the value of the quantile is substituted into the inverse function, The corresponding contrafunctional value of the quantile is calculated, the value of the corresponding fitting quantile of the quantile is used as.

Preferably, the step of target data sequence D of the acquisition quantile to be calculated, including：

1 data element d in target data is received, and d is stored to the target data sequence D that size is N, Zhi Daosuo Target data sequence D is stated to be filled；

It is described to be directed to the target data sequence D, utilize the fitting distribution function F (x), the inverse function F^-1(x) with And the fitting quantile sequence B, the step of each quantile distinguishes corresponding current quantile is determined, including：

S1：Judge whether the first design variables C currency is more than 0, wherein, the initial value of the C is 0；

If more than 0, performing S2；

If equal to 0, the second design variables T initialization value is set to 0, and by default corresponding point of quantile sequence P All quantiles in digit sequence Q are reset, and perform S3, wherein, P={ p₀, p₁..., p_m..., p_M, Q={ q₀, q₁..., q_m..., q_M, m ∈ [0, M]；

S2：For each element a in quantile sequence Q and the union X of data sequence D, if a is less than q₀, then really Determine the corresponding CDF of element a_QValue be 0, perform S3；

If a is more than or equal to q_M, it is determined that the corresponding CDF of element a_QValue be 1, perform S3；

If a is more than or equal to q₀And less than q_M, then quantile q is searched from quantile sequence Q_m, so that a is more than or waited In q_mAnd less than q_m+1, wherein, m is more than or equal to 0 and less than or equal to (M-1)；

Q is obtained from fitting quantile sequence B_mCorresponding b_m, and q_m+1Corresponding b_m+1, wherein, B={ b₀, b₁..., b_m..., b_M, m ∈ [0, M]；

Determine the corresponding CDF of element a_QValue CDF_Q(a)=F (b_m+(b_m+1-b_m)*(a-q_m)/(q_m+1-q_m)), perform S3；

S3：Order by the data in data sequence D according to data value from small to large, is ranked up, after being sorted Data sequence D₁, S4 is performed, wherein, D₁={ d₁, d₂..., d_n..., d_N, n ∈ [0, N]；

S4：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b is less than or equal to d₁, it is determined that the corresponding ECDF of element b_D ^-Value be 0, perform S5；

If b is more than d_N, it is determined that the corresponding ECDF of element b_D ^-Value be 1, perform S5；

If b is more than d₁And less than or equal to d_N, then from data sequence D₁Middle lookup d_n, so that b is more than d_nAnd be less than or wait In d_n+1；

Determine the corresponding ECDF of element b_D ^-Value ECDF_D ^-(b)=n/ (n+M), performs S5；

S5：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than d₁, then really Determine the corresponding ECDF of element b_D ⁺Value be 0, perform S6；

If b is more than or equal to d_N, it is determined that the corresponding ECDF of element b_D ⁺Value be 1, perform S6；

If b is more than or equal to d₁And less than d_N, then using binary chop, from data sequence D₁Middle lookup d_n, so that b More than or equal to d_nAnd less than d_n+1；

Determine the corresponding ECDF of element b_D ⁺Value ECDF_D ⁺(b)=n/ (n+M), performs S6；

S6：For quantile sequence Q and data sequence D₁Union X₁In each element b, calculate the element b's ACDF^-Value ACDF^-(b)=(T*CDF_Q(b)+N*ECDF_D ^-(b))/(T+N), and the element b ACDF⁺Value ACDF⁺ (b)=(T*CDF_Q(b)+N*ECDF_D ⁺(b))/(T+N), performs S7；

S7：For presetting each quantile p in quantile sequence P_m, calculate the quantile p_mCorresponding q_m ^-And q_m ⁺ Value, perform S8, wherein, q_m ^-=max { b ∣ ACDF^-(b)≤p_m, b ∈ X₁, q_m ⁺=max { b ∣ ACDF⁺(b)≥p_m, b ∈ X₁}；

S8：For presetting each quantile p in quantile sequence P_m, judge quantile p_mCorresponding q_m ^-And q_m ⁺'s Whether value is identical；

If identical, by q_m ^-Value be defined as the quantile p_mCorresponding current quantile q_mValue；

If it is not the same, then by (k*q_m ^-+(1-k)*q_m ⁺), it is defined as the quantile p_mCorresponding current quantile q_m's Value, wherein, k=F^-(ACDF⁺(q_m ⁺)-F^-(p_m))/F^-(ACDF⁺(q_m ⁺)-ACDF^-(q_m ^-))。

Preferably, after step s8, methods described also includes：

S9：The currency of the second design variables T is added into N, and the currency of the first design variables C is added 1, is held Row S10；

S10：Judge Jia 1 after the first design variables C currency whether the value for being the 3rd design variables cycle Integral multiple；

If not being the integral multiple of cycle value, return and perform the target data sequence D for obtaining quantile to be calculated The step of；

If the integral multiple of cycle value, the fitting quantile sequence is updated, the execution acquisition is returned to be calculated The step of target data sequence D of quantile, wherein, each quantile that the fitting quantile sequence B after renewal is included It is worth for (T*b_m+M*q_m)/(T+M)。

To reach above-mentioned purpose, the embodiment of the present invention puies forward the quantile determining device for disclosing a kind of data, and device includes： Data training module and fractional-dimension calculus module；

The data training module, for being fitted the training data chosen from target data, obtains training number According to corresponding probability density function p (x)；Using the probability density function p (x), the corresponding fitting distribution of training data is calculated Function F (x) and its inverse function F^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；For presetting quantile Each quantile that sequence P is included, utilizes the inverse function F^-1(x) the corresponding fitting quantile of the quantile, is calculated, and will The fitting quantile storage is into fitting quantile sequence B；

The fractional-dimension calculus module, the target data sequence D for obtaining quantile to be calculated；For the number of targets According to sequence D, the fitting distribution function F (x), the inverse function F are utilized^-1(x) and it is described fitting quantile sequence B, it is determined that Each quantile distinguishes corresponding current quantile.

Preferably, the data training module includes：

Function Fitting submodule, for being fitted to the training data chosen from target data, obtains training data Corresponding probability density function p (x)；

First calculating sub module, for utilizing the probability density function p (x), calculates the corresponding fitting point of training data Cloth function F (x) and its inverse function F^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；

Second calculating sub module, for for presetting each quantile that quantile sequence P is included, utilizing the inverse letter Number F^-1(x) the corresponding fitting quantile of the quantile, is calculated, and fitting quantile sequence B is arrived into the fitting quantile storage In；

The fractional-dimension calculus module, including：

Target data sequence obtains submodule, the target data sequence D for obtaining quantile to be calculated；

Quantile determination sub-module, for for the target data sequence D, using the fitting distribution function F (x), The inverse function F^-1(x) and the fitting quantile sequence B, determine that each quantile distinguishes corresponding current quantile.

Preferably, first calculating sub module, specifically for：

For the probability density function p (x), the p (x) is calculated from the negative infinite definite integral to variable x, obtains described It is fitted distribution function F (x)；

Calculate the inverse function F of the fitting distribution function F (x)^-1(x)。

Preferably, second calculating sub module, specifically for：

For presetting each quantile that quantile sequence is included, the value of the quantile is substituted into the inverse function, The corresponding contrafunctional value of the quantile is calculated, the value of the corresponding fitting quantile of the quantile is used as.

Preferably, the target data sequence obtains submodule, specifically for：

1 data element d in target data is received, and d is stored to the target data sequence D that size is N, Zhi Daosuo Target data sequence D is stated to be filled；

The quantile determination sub-module, specifically for：

S1：Judge whether the first design variables C currency is more than 0, wherein, the initial value of the C is 0；

If more than 0, performing S2；

If equal to 0, the second design variables T initialization value is set to 0, and by default corresponding point of quantile sequence P All quantiles in digit sequence Q are reset, and perform S3, wherein, P={ p₀, p₁..., p_m..., p_M, Q={ q₀, q₁..., q_m..., q_M, m ∈ [0, M]；

S2：For each element a in quantile sequence Q and the union X of data sequence D, if a is less than q₀, then really Determine the corresponding CDF of element a_QValue be 0, perform S3；

If a is more than or equal to q_M, it is determined that the corresponding CDF of element a_QValue be 1, perform S3；

If a is more than or equal to q₀And less than q_M, then quantile q is searched from quantile sequence Q_m, so that a is more than or waited In q_mAnd less than q_m+1, wherein, m is more than or equal to 0 and less than or equal to (M-1)；

Q is obtained from fitting quantile sequence B_mCorresponding b_m, and q_m+1Corresponding b_m+1, wherein, B={ b₀, b₁..., b_m..., b_M, m ∈ [0, M]；

Determine the corresponding CDF of element a_QValue CDF_Q(a)=F (b_m+(b_m+1-b_m)*(a-q_m)/(q_m+1-q_m)), perform S3；

S3：Order by the data in data sequence D according to data value from small to large, is ranked up, after being sorted Data sequence D₁, S4 is performed, wherein, D₁={ d₁, d₂..., d_n..., d_N, n ∈ [0, N]；

S4：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b is less than or equal to d₁, it is determined that the corresponding ECDF of element b_D ^-Value be 0, perform S5；

If b is more than d_N, it is determined that the corresponding ECDF of element b_D ^-Value be 1, perform S5；

If b is more than d₁And less than or equal to d_N, then from data sequence D₁Middle lookup d_n, so that b is more than d_nAnd be less than or wait In d_n+1；

Determine the corresponding ECDF of element b_D ^-Value ECDF_D ^-(b)=n/ (n+M), performs S5；

S5：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than d₁, then really Determine the corresponding ECDF of element b_D ⁺Value be 0, perform S6；

If b is more than or equal to d_N, it is determined that the corresponding ECDF of element b_D ⁺Value be 1, perform S6；

If b is more than or equal to d₁And less than d_N, then using binary chop, from data sequence D₁Middle lookup d_n, so that b More than or equal to d_nAnd less than d_n+1；

Determine the corresponding ECDF of element b_D ⁺Value ECDF_D ⁺(b)=n/ (n+M), performs S6；

S6：For quantile sequence Q and data sequence D₁Union X₁In each element b, calculate the element b's ACDF^-Value ACDF^-(b)=(T*CDF_Q(b)+N*ECDF_D ^-(b))/(T+N), and the element b ACDF⁺Value ACDF⁺ (b)=(T*CDF_Q(b)+N*ECDF_D ⁺(b))/(T+N), performs S7；

S7：For presetting each quantile p in quantile sequence P_m, calculate the quantile p_mCorresponding q_m ^-And q_m ⁺ Value, perform S8, wherein, q_m ^-=max { b ∣ ACDF^-(b)≤p_m, b ∈ X₁, q_m ⁺=max { b ∣ ACDF⁺(b)≥p_m, b ∈ X₁}；

S8：For presetting each quantile p in quantile sequence P_m, judge quantile p_mCorresponding q_m ^-And q_m ⁺'s Whether value is identical；

If identical, by q_m ^-Value be defined as the quantile p_mCorresponding current quantile q_mValue；

If it is not the same, then by (k*q_m ^-+(1-k)*q_m ⁺), it is defined as the quantile p_mCorresponding current quantile q_m's Value, wherein, k=F^-(ACDF⁺(q_m ⁺)-F^-(p_m))/F^-(ACDF⁺(q_m ⁺)-ACDF^-(q_m ^-))。

Preferably, described device also includes：

Plus computing module, for the currency of the second design variables T to be added into N, and by the first design variables C's Currency adds 1；

Judge module, for judging whether the currency of the first design variables C after Jia 1 is the 3rd design variables The integral multiple of cycle value；In the case where the judge module judged result is no, triggering target data sequence obtains submodule Block；

Update module, in the case of being in the judge module judged result, updates the fitting quantile sequence Row, and target data sequence acquisition submodule is triggered, wherein, each point of position that the fitting quantile sequence B after renewal is included Several values is (T*b_m+M*q_m)/(T+M)。

As seen from the above technical solutions, method and dress are determined the embodiments of the invention provide a kind of quantile of data Put, including：Data training process and fractional-dimension calculus process；The data training process, including：To being chosen from target data Training data be fitted, obtain the corresponding probability density function p (x) of training data；Utilize the probability density function p (x) the corresponding fitting distribution function F (x) of training data and its inverse function F, are calculated^-1(x), wherein, the fitting distribution function F (x) it is nonlinear function；For presetting each quantile that quantile sequence P is included, the inverse function F is utilized^-1(x), count The corresponding fitting quantile of the quantile is calculated, and by the fitting quantile storage into fitting quantile sequence B；Described point of position Number estimation procedure, including：Obtain the target data sequence D of quantile to be calculated；For the target data sequence D, institute is utilized State fitting distribution function F (x), the inverse function F^-1(x) and the fitting quantile sequence B, each quantile point is determined Not corresponding current quantile.

It can be seen that, because fitting distribution function F (x) is nonlinear function, thus it is distribution that heavytailed distribution, which is presented, in data flow In the case of uneven, it is possible to use embody the fitting distribution function F (x) of data flow non-uniform Distribution, rather than default data Equally distributed linear function is flowed, with reference to the inverse function F calculated^-1(x) and it is described fitting quantile sequence B go determine point position Number, so as to reduce the error for the quantile determined.

Certainly, any product or method for implementing the present invention it is not absolutely required to while reaching all the above excellent Point.

Brief description of the drawings

In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing There is the accompanying drawing used required in technology description to be briefly described, it should be apparent that, drawings in the following description are only this Some embodiments of invention, for those of ordinary skill in the art, on the premise of not paying creative work, can be with Other accompanying drawings are obtained according to these accompanying drawings.

Fig. 1 determines a kind of schematic flow sheet of method for the quantile of data provided in an embodiment of the present invention；

Fig. 2 be embodiment illustrated in fig. 1 in, a kind of step S105 idiographic flow schematic diagram；

Fig. 3 is a kind of structural representation of the quantile determining device of data provided in an embodiment of the present invention；

Fig. 4 is another structural representation of the quantile determining device of data provided in an embodiment of the present invention.

Embodiment

Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete Site preparation is described, it is clear that described embodiment is only a part of embodiment of the invention, rather than whole embodiments.It is based on Embodiment in the present invention, it is every other that those of ordinary skill in the art are obtained under the premise of creative work is not made Embodiment, belongs to the scope of protection of the invention.

Method, which is described in detail, to be determined to a kind of quantile of data provided in an embodiment of the present invention first below.

Referring to Fig. 1, Fig. 1 determines a kind of schematic flow sheet of method for the quantile of data provided in an embodiment of the present invention, It can include：Data training process and fractional-dimension calculus process；

The data training process, may include steps of：

S101, is fitted to the training data chosen from target data, obtains the corresponding probability density of training data Function p (x)；

Specifically, target data can be target data stream, the high-speed data-flow for example constantly reached.

Wherein it is possible to choose a part of data as training data from target data stream, probability density is obtained by fitting Function p (x).Data flow is different, then the probability density function fitted is different.For example, assuming that data flow Normal Distribution On the premise of, it is respectively μ and σ to calculate and obtain the average and standard deviation of training data, then

Wherein it is possible to be realized using prior art：The average and standard deviation of training data are calculated, the embodiment of the present invention is herein It is not repeated.

In actual applications, it can be realized using prior art：The training data chosen from target data is intended Close, obtain the corresponding probability density function p (x) of training data.

S102, using the probability density function p (x), calculate the corresponding fitting distribution function F (x) of training data and its Inverse function F^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；

Specifically, described utilize the probability density function p (x), the corresponding fitting distribution function F of training data is calculated And its inverse function F (x)^-1(x) the step of, it can include：For the probability density function p (x), the p (x) is calculated from negative nothing The poor definite integral for arriving variable x, obtains the fitting distribution function F (x)；Calculate the inverse function F of the fitting distribution function F (x)^-1 (x)。

From the foregoing,It is public that the probability density function p (x) that fitting is obtained brings the integration into Formula, you can calculating obtains fitting distribution function F (x), found a function to F (x) inverse and then obtains inverse function F^-1(x)。

S103, for presetting each quantile that quantile sequence P is included, utilizes the inverse function F^-1(x), calculating should The corresponding fitting quantile of quantile, and by the fitting quantile storage into fitting quantile sequence B；

Specifically, described each quantile included for default quantile sequence P, utilizes the inverse function F^-1(x), The step of quantile is corresponding to be fitted quantile is calculated, can be included：For presetting each point that quantile sequence is included Site, the inverse function is substituted into by the value of the quantile, is calculated the corresponding contrafunctional value of the quantile, is used as this point of position The value of the corresponding fitting quantile of point.

For example, P={ p₀, p₁..., p_m..., p_M, p₀、p₁、…、p_MIt is the quantile that quantile sequence P is included, wherein, m ∈ [0, M].Each quantile p that P is included_mBring inverse function F into respectively^-1(x) F, is calculated^-1(p_m) value, be used as p_mCorrespondence Fitting quantile value, by value storage to fitting quantile sequence B={ b₀, b₁..., b_m..., b_MIn, i.e. b_m=F^-1 (p_m), wherein, b₀、b₁、…、b_MIt is to be fitted the fitting quantile that quantile sequence B is included.

The fractional-dimension calculus process, may include steps of：

S104, obtains the target data sequence D of quantile to be calculated；

Specifically, the step of target data sequence D of the acquisition quantile to be calculated, can include：Receive number of targets 1 data element d in, and d is stored to the target data sequence D that size is N, until the target data sequence D quilt Fill up.

In actual applications, a data element d can be received from target data stream, it is N's to be stored to size In data sequence D, then judge whether D is filled：

If D is not filled, continuation receives another data element e from target data stream, is then stored into D, then Judge whether D is filled ... and so circulate, untill data sequence D is filled, next step S105 is performed afterwards；

If D is already filled up, next step S105 is directly performed.

S105, for the target data sequence D, utilizes the fitting distribution function F (x), the inverse function F^-1(x) with And the fitting quantile sequence B, determine that each quantile distinguishes corresponding current quantile.

Specifically, referring to Fig. 2, Fig. 2 be embodiment illustrated in fig. 1 in, a kind of step S105 idiographic flow schematic diagram.It is described Step 105, for the target data sequence D, the fitting distribution function F (x), the inverse function F are utilized^-1(x) and institute Fitting quantile sequence B is stated, determines that each quantile distinguishes corresponding current quantile, may comprise steps of：

S105A：Judge whether the first design variables C currency is more than 0；Wherein, the initial value of the C is 0；If being more than 0, perform S105B；If equal to 0, performing S105C；

S105B：For each element a in quantile sequence Q and the union X of data sequence D, if a is less than q₀, then Determine the corresponding CDF of element a_QValue be 0, perform S105D；

If a is more than or equal to q_M, it is determined that the corresponding CDF of element a_QValue be 1, perform S105D；

If a is more than or equal to q₀And less than q_M, then quantile q is searched from quantile sequence Q_m, so that a is more than or waited In q_mAnd less than q_m+1, wherein, m is more than or equal to 0 and less than or equal to (M-1)；

Q is obtained from fitting quantile sequence B_mCorresponding b_m, and q_m+1Corresponding b_m+1, wherein, B={ b₀, b₁..., b_m..., b_M, m ∈ [0, M]；

Determine the corresponding CDF of element a_QValue CDF_Q(a)=F (b_m+(b_m+1-b_m)*(a-q_m)/(q_m+1-q_m)), perform S105D；

S105C：Second design variables T initialization value is set to 0, and by the corresponding quantiles of default quantile sequence P All quantiles in sequence Q are reset, and perform S105D, wherein, P={ p₀, p₁..., p_m..., p_M, Q={ q₀, q₁..., q_m..., q_M, m ∈ [0, M]；

S105D：Order by the data in data sequence D according to data value from small to large, is ranked up, and obtains after sequence Data sequence D₁, S105E is performed, wherein, D₁={ d₁, d₂..., d_n..., d_N, n ∈ [0, N]；

S105E：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b is less than or waited In d₁, it is determined that the corresponding ECDF of element b_D ^-Value be 0, perform S105F；

If b is more than d_N, it is determined that the corresponding ECDF of element b_D ^-Value be 1, perform S105F；

If b is more than d₁And less than or equal to d_N, then from data sequence D₁Middle lookup d_n, so that b is more than d_nAnd be less than or wait In d_n+1；

Determine the corresponding ECDF of element b_D ^-Value ECDF_D ^-(b)=n/ (n+M), performs S105F；

S105F：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than d₁, Then determine the corresponding ECDF of element b_D ⁺Value be 0, perform S105G；

If b is more than or equal to d_N, it is determined that the corresponding ECDF of element b_D ⁺Value be 1, perform S105G；

If b is more than or equal to d₁And less than d_N, then using binary chop, from data sequence D₁Middle lookup d_n, so that b More than or equal to d_nAnd less than d_n+1；

Determine the corresponding ECDF of element b_D ⁺Value ECDF_D ⁺(b)=n/ (n+M), performs S105G；

S105G：For quantile sequence Q and data sequence D₁Union X₁In each element b, calculate the element b ACDF^-Value ACDF^-(b)=(T*CDF_Q(b)+N*ECDF_D ^-(b))/(T+N), and the element b ACDF⁺Value ACDF⁺(b)=(T*CDF_Q(b)+N*ECDF_D ⁺(b))/(T+N), performs S105H；

S105H：For presetting each quantile p in quantile sequence P_m, calculate the quantile p_mCorresponding q_m ^- And q_m ⁺Value, perform S105I, wherein, q_m ^-=max { b ∣ ACDF^-(b)≤p_m, b ∈ X₁, q_m ⁺=max { b ∣ ACDF⁺(b)≥p_m, b ∈X₁}；

S105I：For presetting each quantile p in quantile sequence P_m, judge quantile p_mCorresponding q_m ^-And q_m ⁺Value it is whether identical；

If identical, by q_m ^-Value be defined as the quantile p_mCorresponding current quantile q_mValue；

If it is not the same, then by (k*q_m ^-+(1-k)*q_m ⁺), it is defined as the quantile p_mCorresponding current quantile q_m's Value, wherein, k=F^-(ACDF⁺(q_m ⁺)-F^-(p_m))/F^-(ACDF⁺(q_m ⁺)-ACDF^-(q_m ^-))。

It should be noted that when the currency for judging the C in step S105A is equal to 0, step S105B can be skipped, turned And step S105C is performed, then perform S105D.When going to step S105G, ACDF is calculated^-And ACDF (b)⁺(b) value In formula, now CDF_Q(b) value is also equal to 0.

Also, in step S105D, data sequence D is sorted from small to large, the data sequence D after sequence₁= {d₁, d₂..., d_n..., d_N, you can know d₁Value it is minimum, d₂Take second place, increase successively, until d_NValue it is maximum.

In addition, in step S105F, can be realized using prior art：Using binary chop, from data sequence D₁In look into Look for d_n, so that b is more than or equal to d_nAnd less than d_n+1。

Specifically, after step S105I, can also comprise the following steps：

The currency of the second design variables T is added into N, and the currency of the first design variables C is added 1；

Judge Jia 1 after the first design variables C currency whether the integer for the value for being the 3rd design variables cycle Times；

If not being the integral multiple of cycle value, return and perform the target data sequence D for obtaining quantile to be calculated The step of；

If the integral multiple of cycle value, the fitting quantile sequence is updated, the execution acquisition is returned to be calculated The step of target data sequence D of quantile, wherein, each quantile that the fitting quantile sequence B after renewal is included It is worth for (T*b_m+M*q_m)/(T+M)。

Wherein, in actual applications, design variables cycle value can be set to 50 or 100.

It can be seen that, because fitting distribution function F (x) is nonlinear function, thus it is distribution that heavytailed distribution, which is presented, in data flow In the case of uneven, it is possible to use embody the fitting distribution function F (x) of data flow non-uniform Distribution, rather than default data Equally distributed linear function is flowed, with reference to the inverse function F calculated^-1(x) and it is described fitting quantile sequence B go determine point position Number, so as to reduce the error for the quantile determined.

Referring to Fig. 3, Fig. 3 is a kind of structural representation of the quantile determining device of data provided in an embodiment of the present invention, Corresponding with the flow shown in Fig. 1, the determining device can include：Data training module 301 and fractional-dimension calculus module 302；

The data training module 301,

For being fitted to the training data chosen from target data, the corresponding probability density letter of training data is obtained Number p (x)；Using the probability density function p (x), the corresponding fitting distribution function F (x) of training data and its inverse function are calculated F^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；For presetting each point that quantile sequence P is included Site, utilizes the inverse function F^-1(x) the corresponding fitting quantile of the quantile, is calculated, and the fitting quantile is stored Into fitting quantile sequence B；

The fractional-dimension calculus module 302, the target data sequence D for obtaining quantile to be calculated；For the mesh Data sequence D is marked, the fitting distribution function F (x), the inverse function F is utilized^-1(x) and it is described fitting quantile sequence B, Determine that each quantile distinguishes corresponding current quantile.

Referring to Fig. 4, Fig. 4 is another structural representation of the quantile determining device of data provided in an embodiment of the present invention Figure, the determining device can include：Data training module 400 and fractional-dimension calculus module 410；

The data training module 400, can include：

Function Fitting submodule 401, for being fitted the training data chosen from target data, obtains training number According to corresponding probability density function p (x)；

First calculating sub module 402, for utilizing the probability density function p (x), calculates the corresponding fitting of training data Distribution function F (x) and its inverse function F^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；

Specifically, first calculating sub module 402, specifically can be used for：

For the probability density function p (x), the p (x) is calculated from the negative infinite definite integral to variable x, obtains described It is fitted distribution function F (x)；

Calculate the inverse function F of the fitting distribution function F (x)^-1(x)。

Second calculating sub module 403, for for presetting each quantile for including of quantile sequence P, using described Inverse function F^-1(x) the corresponding fitting quantile of the quantile, is calculated, and fitting quantile sequence is arrived into the fitting quantile storage Arrange in B；

Specifically, second calculating sub module 403, specifically can be used for：

For presetting each quantile that quantile sequence is included, the value of the quantile is substituted into the inverse function, The corresponding contrafunctional value of the quantile is calculated, the value of the corresponding fitting quantile of the quantile is used as.

The fractional-dimension calculus module 410, can include：

Target data sequence obtains submodule 411, the target data sequence D for obtaining quantile to be calculated；

Quantile determination sub-module 412, for for the target data sequence D, utilizing the fitting distribution function F (x), the inverse function F^-1(x) and the fitting quantile sequence B, determine that each quantile distinguishes corresponding current point of position Number.

Specifically, the target data sequence obtains submodule 411, specifically it can be used for：

1 data element d in target data is received, and d is stored to the target data sequence D that size is N, Zhi Daosuo Target data sequence D is stated to be filled；

The quantile determination sub-module 412, specifically can be used for：

S1：Judge whether the first design variables C currency is more than 0, wherein, the initial value of the C is 0；

If more than 0, performing S2；

If equal to 0, the second design variables T initialization value is set to 0, and by default corresponding point of quantile sequence P All quantiles in digit sequence Q are reset, and perform S3, wherein, P={ p₀, p₁..., p_m..., p_M, Q={ q₀, q₁..., q_m..., q_M, m ∈ [0, M]；

S2：For each element a in quantile sequence Q and the union X of data sequence D, if a is less than q₀, then really Determine the corresponding CDF of element a_QValue be 0, perform S3；

If a is more than or equal to q_M, it is determined that the corresponding CDF of element a_QValue be 1, perform S3；

If a is more than or equal to q₀And less than q_M, then quantile q is searched from quantile sequence Q_m, so that a is more than or waited In q_mAnd less than q_m+1, wherein, m is more than or equal to 0 and less than or equal to (M-1)；

Q is obtained from fitting quantile sequence B_mCorresponding b_m, and q_m+1Corresponding b_m+1, wherein, B={ b₀, b₁..., b_m..., b_M, m ∈ [0, M]；

Determine the corresponding CDF of element a_QValue CDF_Q(a)=F (b_m+(b_m+1-b_m)*(a-q_m)/(q_m+1-q_m)), perform S3；

S3：Order by the data in data sequence D according to data value from small to large, is ranked up, after being sorted Data sequence D₁, S4 is performed, wherein, D₁={ d₁, d₂..., d_n..., d_N, n ∈ [0, N]；

S4：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b is less than or equal to d₁, it is determined that the corresponding ECDF of element b_D ^-Value be 0, perform S5；

If b is more than d_N, it is determined that the corresponding ECDF of element b_D ^-Value be 1, perform S5；

If b is more than d₁And less than or equal to d_N, then from data sequence D₁Middle lookup d_n, so that b is more than d_nAnd be less than or wait In d_n+1；

Determine the corresponding ECDF of element b_D ^-Value ECDF_D ^-(b)=n/ (n+M), performs S5；

S5：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than d₁, then really Determine the corresponding ECDF of element b_D ⁺Value be 0, perform S6；

If b is more than or equal to d_N, it is determined that the corresponding ECDF of element b_D ⁺Value be 1, perform S6；

If b is more than or equal to d₁And less than d_N, then using binary chop, from data sequence D₁Middle lookup d_n, so that b More than or equal to d_nAnd less than d_n+1；

Determine the corresponding ECDF of element b_D ⁺Value ECDF_D ⁺(b)=n/ (n+M), performs S6；

S6：For quantile sequence Q and data sequence D₁Union X₁In each element b, calculate the element b's ACDF^-Value ACDF^-(b)=(T*CDF_Q(b)+N*ECDF_D ^-(b))/(T+N), and the element b ACDF⁺Value ACDF⁺ (b)=(T*CDF_Q(b)+N*ECDF_D ⁺(b))/(T+N), performs S7；

S7：For presetting each quantile p in quantile sequence P_m, calculate the quantile p_mCorresponding q_m ^-And q_m ⁺ Value, perform S8, wherein, q_m ^-=max { b ∣ ACDF^-(b)≤p_m, b ∈ X₁, q_m ⁺=max { b ∣ ACDF⁺(b)≥p_m, b ∈ X₁}；

S8：For presetting each quantile p in quantile sequence P_m, judge quantile p_mCorresponding q_m ^-And q_m ⁺'s Whether value is identical；

If identical, by q_m ^-Value be defined as the quantile p_mCorresponding current quantile q_mValue；

If it is not the same, then by (k*q_m ^-+(1-k)*q_m ⁺), it is defined as the quantile p_mCorresponding current quantile q_m's Value, wherein, k=F^-(ACDF⁺(q_m ⁺)-F^-(p_m))/F^-(ACDF⁺(q_m ⁺)-ACDF^-(q_m ^-))。

Specifically, described device can also include：

Plus computing module, for the currency of the second design variables T to be added into N, and by the first design variables C's Currency adds 1；

Judge module, for judging whether the currency of the first design variables C after Jia 1 is the 3rd design variables The integral multiple of cycle value；In the case where the judge module judged result is no, triggering target data sequence obtains submodule Block 304；

Update module, in the case of being in the judge module judged result, updates the fitting quantile sequence Row, and target data sequence acquisition submodule 304 is triggered, wherein, each point that the fitting quantile sequence B after renewal is included The value of digit is (T*b_m+M*q_m)/(T+M)。

It can be seen that, because fitting distribution function F (x) is nonlinear function, thus it is distribution that heavytailed distribution, which is presented, in data flow In the case of uneven, it is possible to use embody the fitting distribution function F (x) of data flow non-uniform Distribution, rather than default data Equally distributed linear function is flowed, with reference to the inverse function F calculated^-1(x) and it is described fitting quantile sequence B go determine point position Number, so as to reduce the error for the quantile determined.

It should be noted that herein, such as first and second or the like relational terms are used merely to a reality Body or operation make a distinction with another entity or operation, and not necessarily require or imply these entities or deposited between operating In any this actual relation or order.Moreover, term " comprising ", "comprising" or its any other variant are intended to Nonexcludability is included, so that process, method, article or equipment including a series of key elements not only will including those Element, but also other key elements including being not expressly set out, or also include being this process, method, article or equipment Intrinsic key element.In the absence of more restrictions, the key element limited by sentence "including a ...", it is not excluded that Also there is other identical element in process, method, article or equipment including the key element.

Each embodiment in this specification is described by the way of related, identical similar portion between each embodiment Divide mutually referring to what each embodiment was stressed is the difference with other embodiment.It is real especially for device Apply for example, because it is substantially similar to embodiment of the method, so description is fairly simple, related part is referring to embodiment of the method Part explanation.

Can one of ordinary skill in the art will appreciate that realizing that all or part of step in above method embodiment is To instruct the hardware of correlation to complete by program, described program can be stored in computer read/write memory medium, The storage medium designated herein obtained, such as：ROM/RAM, magnetic disc, CD etc..

The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the scope of the present invention.It is all Any modification, equivalent substitution and improvements made within the spirit and principles in the present invention etc., are all contained in protection scope of the present invention It is interior.

Claims (10)

1. a kind of quantile of data determines method, it is characterised in that methods described includes：Data training process and quantile are estimated Meter process；

The data training process, including：

The training data chosen from target data is fitted, the corresponding probability density function p (x) of training data is obtained；

Using the probability density function p (x), the corresponding fitting distribution function F (x) of training data and its inverse function F are calculated^-1 (x), wherein, the fitting distribution function F (x) be nonlinear function；

For presetting each quantile that quantile sequence P is included, the inverse function F is utilized^-1(x) quantile pair, is calculated The fitting quantile answered, and by the fitting quantile storage into fitting quantile sequence B；

The fractional-dimension calculus process, including：

Obtain the target data sequence D of quantile to be calculated；

For the target data sequence D, the fitting distribution function F (x), the inverse function F are utilized^-1(x) and it is described intend Quantile sequence B is closed, determines that each quantile distinguishes corresponding current quantile.

2. according to the method described in claim 1, it is characterised in that described to utilize the probability density function p (x), calculate instruction Practice the corresponding fitting distribution function F (x) of data and its inverse function F^-1(x) the step of, including：

For the probability density function p (x), the p (x) is calculated from the negative infinite definite integral to variable x, the fitting is obtained Distribution function F (x)；

Calculate the inverse function F of the fitting distribution function F (x)^-1(x)。

3. according to the method described in claim 1, it is characterised in that each included for default quantile sequence P Quantile, utilizes the inverse function F^-1(x) the step of quantile is corresponding to be fitted quantile, is calculated, including：

For presetting each quantile that quantile sequence is included, the value of the quantile is substituted into the inverse function, calculated The corresponding contrafunctional value of the quantile, is used as the value of the corresponding fitting quantile of the quantile.

4. method according to claim 2, it is characterised in that the target data sequence D of the acquisition quantile to be calculated The step of, including：

1 data element d in target data is received, and d is stored to the target data sequence D that size is N, until the mesh Mark data sequence D is filled；

It is described to be directed to the target data sequence D, utilize the fitting distribution function F (x), the inverse function F^-1(x) and institute Fitting quantile sequence B is stated, the step of each quantile distinguishes corresponding current quantile is determined, including：

S1：Judge whether the first design variables C currency is more than 0, wherein, the initial value of the C is 0；

If more than 0, performing S2；

If equal to 0, the second design variables T initialization value is set to 0, and by the corresponding quantiles of default quantile sequence P All quantiles in sequence Q are reset, and perform S3, wherein, P={ p₀, p₁..., p_m..., p_M, Q={ q₀, q₁..., q_m..., q_M, m ∈ [0, M]；

S2：For each element a in quantile sequence Q and the union X of data sequence D, if a is less than q₀, it is determined that this yuan The corresponding CDF of plain a_QValue be 0, perform S3；

If a is more than or equal to q_M, it is determined that the corresponding CDF of element a_QValue be 1, perform S3；

If a is more than or equal to q₀And less than q_M, then quantile q is searched from quantile sequence Q_m, so that a is more than or equal to q_mAnd Less than q_m+1, wherein, m is more than or equal to 0 and less than or equal to (M-1)；

Q is obtained from fitting quantile sequence B_mCorresponding b_m, and q_m+1Corresponding b_m+1, wherein, B={ b₀, b₁..., b_m..., b_M, m ∈ [0, M]；

Determine the corresponding CDF of element a_QValue CDF_Q(a)=F (b_m+(b_m+1-b_m)*(a-q_m)/(q_m+1-q_m)), perform S3；

S3：Order by the data in data sequence D according to data value from small to large, is ranked up, the data after being sorted Sequence D₁, S4 is performed, wherein, D₁={ d₁, d₂..., d_n..., d_N, n ∈ [0, N]；

S4：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than or equal to d₁, then Determine the corresponding ECDF of element b_D ^-Value be 0, perform S5；

If b is more than d_N, it is determined that the corresponding ECDF of element b_D ^-Value be 1, perform S5；

If b is more than d₁And less than or equal to d_N, then from data sequence D₁Middle lookup d_n, so that b is more than d_nAnd be less than or equal to d_n+1；

Determine the corresponding ECDF of element b_D ^-Value ECDF_D ^-(b)=n/ (n+M), performs S5；

S5：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than d₁, it is determined that should The corresponding ECDF of element b_D ⁺Value be 0, perform S6；

If b is more than or equal to d_N, it is determined that the corresponding ECDF of element b_D ⁺Value be 1, perform S6；

If b is more than or equal to d₁And less than d_N, then using binary chop, from data sequence D₁Middle lookup d_n, so that b be more than or Equal to d_nAnd less than d_n+1；

Determine the corresponding ECDF of element b_D ⁺Value ECDF_D ⁺(b)=n/ (n+M), performs S6；

S6：For quantile sequence Q and data sequence D₁Union X₁In each element b, calculate the ACDF of the element b^- Value ACDF^-(b)=(T*CDF_Q(b)+N*ECDF_D ^-(b))/(T+N), and the element b ACDF⁺Value ACDF⁺(b)= (T*CDF_Q(b)+N*ECDF_D ⁺(b))/(T+N), performs S7；

S7：For presetting each quantile p in quantile sequence P_m, calculate the quantile p_mCorresponding q_m ^-And q_m ⁺'s Value, performs S8, wherein, q_m ^-=max { b ∣ ACDF^-(b)≤p_m, b ∈ X₁, q_m ⁺=max { b ∣ ACDF⁺(b)≥p_m, b ∈ X₁}；

S8：For presetting each quantile p in quantile sequence P_m, judge quantile p_mCorresponding q_m ^-And q_m ⁺Value be It is no identical；

If identical, by q_m ^-Value be defined as the quantile p_mCorresponding current quantile q_mValue；

If it is not the same, then by (k*q_m ^-+(1-k)*q_m ⁺), it is defined as the quantile p_mCorresponding current quantile q_mValue, Wherein, k=F^-(ACDF⁺(q_m ⁺)-F^-(p_m))/F^-(ACDF⁺(q_m ⁺)-ACDF^-(q_m ^-))。

5. method according to claim 4, it is characterised in that after step s8, methods described also includes：

S9：The currency of the second design variables T is added into N, and the currency of the first design variables C is added 1, is performed S10；

S10：Judge Jia 1 after the first design variables C currency whether the integer for the value for being the 3rd design variables cycle Times；

If not being the integral multiple of cycle value, the step for performing the target data sequence D for obtaining quantile to be calculated is returned Suddenly；

If the integral multiple of cycle value, the fitting quantile sequence is updated, returns and performs to be calculated point of position of the acquisition The step of several target data sequence D, wherein, the value for each quantile that the fitting quantile sequence B after renewal is included is (T*b_m+M*q_m)/(T+M)。

6. the quantile determining device of a kind of data, it is characterised in that described device includes：Data training module and quantile are estimated Count module；

The data training module, for being fitted to the training data chosen from target data, obtains training data pair The probability density function p (x) answered；Using the probability density function p (x), the corresponding fitting distribution function F of training data is calculated And its inverse function F (x)^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；For presetting quantile sequence P Comprising each quantile, utilize the inverse function F^-1(x) the corresponding fitting quantile of the quantile, is calculated, and will be described Quantile storage is fitted into fitting quantile sequence B；

The fractional-dimension calculus module, the target data sequence D for obtaining quantile to be calculated；For the target data sequence D is arranged, the fitting distribution function F (x), the inverse function F is utilized^-1(x) and it is described fitting quantile sequence B, determine each Individual quantile distinguishes corresponding current quantile.

7. device according to claim 6, it is characterised in that the data training module includes：

Function Fitting submodule, for being fitted to the training data chosen from target data, obtains training data correspondence Probability density function p (x)；

First calculating sub module, for utilizing the probability density function p (x), calculates the corresponding fitting distribution letter of training data Number F (x) and its inverse function F^-1(x), wherein, the fitting distribution function F (x) be nonlinear function；

Second calculating sub module, for for presetting each quantile that quantile sequence P is included, utilizing the inverse function F^-1 (x) the corresponding fitting quantile of the quantile, is calculated, and by the fitting quantile storage into fitting quantile sequence B；

The fractional-dimension calculus module, including：

Target data sequence obtains submodule, the target data sequence D for obtaining quantile to be calculated；

Quantile determination sub-module, for for the target data sequence D, utilizes the fitting distribution function F (x), described Inverse function F^-1(x) and the fitting quantile sequence B, determine that each quantile distinguishes corresponding current quantile.

8. device according to claim 7, it is characterised in that first calculating sub module, specifically for：

For the probability density function p (x), the p (x) is calculated from the negative infinite definite integral to variable x, the fitting is obtained Distribution function F (x)；

Calculate the inverse function F of the fitting distribution function F (x)^-1(x)。

9. device according to claim 7, it is characterised in that second calculating sub module, specifically for：

For presetting each quantile that quantile sequence is included, the value of the quantile is substituted into the inverse function, calculated The corresponding contrafunctional value of the quantile, is used as the value of the corresponding fitting quantile of the quantile.

10. device according to claim 8, it is characterised in that the target data sequence obtains submodule, specific to use In：

1 data element d in target data is received, and d is stored to the target data sequence D that size is N, until the mesh Mark data sequence D is filled；

The quantile determination sub-module, specifically for：

S1：Judge whether the first design variables C currency is more than 0, wherein, the initial value of the C is 0；

If more than 0, performing S2；

If equal to 0, the second design variables T initialization value is set to 0, and by the corresponding quantiles of default quantile sequence P All quantiles in sequence Q are reset, and perform S3, wherein, P={ p₀, p₁..., p_m..., p_M, Q={ q₀, q₁..., q_m..., q_M, m ∈ [0, M]；

S2：For each element a in quantile sequence Q and the union X of data sequence D, if a is less than q₀, it is determined that this yuan The corresponding CDF of plain a_QValue be 0, perform S3；

If a is more than or equal to q_M, it is determined that the corresponding CDF of element a_QValue be 1, perform S3；

If a is more than or equal to q₀And less than q_M, then quantile q is searched from quantile sequence Q_m, so that a is more than or equal to q_mAnd Less than q_m+1, wherein, m is more than or equal to 0 and less than or equal to (M-1)；

Q is obtained from fitting quantile sequence B_mCorresponding b_m, and q_m+1Corresponding b_m+1, wherein, B={ b₀, b₁..., b_m..., b_M, m ∈ [0, M]；

Determine the corresponding CDF of element a_QValue CDF_Q(a)=F (b_m+(b_m+1-b_m)*(a-q_m)/(q_m+1-q_m)), perform S3；

S3：Order by the data in data sequence D according to data value from small to large, is ranked up, the data after being sorted Sequence D₁, S4 is performed, wherein, D₁={ d₁, d₂..., d_n..., d_N, n ∈ [0, N]；

S4：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than or equal to d₁, then Determine the corresponding ECDF of element b_D ^-Value be 0, perform S5；

If b is more than d_N, it is determined that the corresponding ECDF of element b_D ^-Value be 1, perform S5；

If b is more than d₁And less than or equal to d_N, then from data sequence D₁Middle lookup d_n, so that b is more than d_nAnd be less than or equal to d_n+1；

Determine the corresponding ECDF of element b_D ^-Value ECDF_D ^-(b)=n/ (n+M), performs S5；

S5：For quantile sequence Q and data sequence D₁Union X₁In each element b, if b be less than d₁, it is determined that should The corresponding ECDF of element b_D ⁺Value be 0, perform S6；

If b is more than or equal to d_N, it is determined that the corresponding ECDF of element b_D ⁺Value be 1, perform S6；

If b is more than or equal to d₁And less than d_N, then using binary chop, from data sequence D₁Middle lookup d_n, so that b be more than or Equal to d_nAnd less than d_n+1；

Determine the corresponding ECDF of element b_D ⁺Value ECDF_D ⁺(b)=n/ (n+M), performs S6；

S6：For quantile sequence Q and data sequence D₁Union X₁In each element b, calculate the ACDF of the element b^- Value ACDF^-(b)=(T*CDF_Q(b)+N*ECDF_D ^-(b))/(T+N), and the element b ACDF⁺Value ACDF⁺(b)= (T*CDF_Q(b)+N*ECDF_D ⁺(b))/(T+N), performs S7；

S7：For presetting each quantile p in quantile sequence P_m, calculate the quantile p_mCorresponding q_m ^-And q_m ⁺'s Value, performs S8, wherein, q_m ^-=max { b ∣ ACDF^-(b)≤p_m, b ∈ X₁, q_m ⁺=max { b ∣ ACDF⁺(b)≥p_m, b ∈ X₁}；

S8：For presetting each quantile p in quantile sequence P_m, judge quantile p_mCorresponding q_m ^-And q_m ⁺Value be It is no identical；

If identical, by q_m ^-Value be defined as the quantile p_mCorresponding current quantile q_mValue；

If it is not the same, then by (k*q_m ^-+(1-k)*q_m ⁺), it is defined as the quantile p_mCorresponding current quantile q_mValue, Wherein, k=F^-(ACDF⁺(q_m ⁺)-F^-(p_m))/F^-(ACDF⁺(q_m ⁺)-ACDF^-(q_m ^-))。