CN104715154B - Core K average Data Associations based on KMDL criterion criterions - Google Patents
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Abstract
The invention discloses a kind of core K average Data Associations based on KMDL criterion criterions, comprise the following steps：Step 1：Build typical track association scene；Step 2：Targetpath number is determined using KMDL criterion criterions；Step 3：Flight path observation scene is associated with core K mean algorithms.The core K average Data Associations based on KMDL criterion criterions of disclosure of the invention, based on target status information, KMDL criterions criterion and core K mean algorithms combine under solving complexity environment to (clutter is intensive, gtoal setting, target numbers are unknown) multi-target traces related question.This method makes full use of the movement state information of target, is effectively improved association accuracy rate, and association criterion is simple and easy, amount of calculation is small, association accuracy is high, it is insensitive to target intersection, it is adapted to carry out track association in heavy dense targets and Cross-environment, suitable for Project Realization.
Description
Technical field
The invention belongs to multi-sensor multi-target tracking technical field, and in particular to a kind of core based on KMDL criterion criterions
K- average Data Associations.
Background technology
Multi-sensor multi-target tracking system mainly receives the local tracks that each sensing system transmits by Data-Link
Information, then these local tracks information are associated, registration, fusion etc. key problem calculating, formed collaboration detection melt
Targetpath after conjunction.
Multi-sensor cooperation target following can realize the accurate tracking to target, and in practical application, target to be tracked is present
It is multiple, now need correctly to determine the measurement information received by sensor and the corresponding relation between target interested.So
And due to clutter caused by the radiation source of falseness, the reason such as interference noise and decoy, it can cause to measure the uncertain of origin
Property, i.e. target and the corresponding relation measured is uncertain, and it is to apply track association technology to eliminate this probabilistic method.
In distributed system architecture, fusion center receives the local tracks information that single sensor handles to obtain, then to these parts
Flight path carries out fusion formation and dbjective state is more accurately estimated, it is necessary first to knows the corresponding pass between local tracks and target
System.Track association problem seeks to judge whether come from same target in the local tracks that each sensor obtains.
True Research on Target has the characteristics that intensive, complexity is high, influencing each other between multiple target, the height of target flight
Mobility, the interference of clutter, the presence of sensor error in measurement, all track association problem is caused to become extremely difficult.
The content of the invention
The defects of in order to overcome above-mentioned prior art to exist, it is an object of the invention to provide one kind to be based on KMDL criterions
The kernel c-means Data Association of criterion, the multi-target traces that this method can be solved fast and accurately under complex environment close
Connection problem.
The present invention is to be achieved through the following technical solutions：
Based on the kernel c-means Data Association of KMDL criterion criterions, comprise the following steps：
Step 1：Build typical track association scene
First, the typical model close to target true motion pattern is selected, generation target measures；Secondly, using Poisson point
Cloth generates clutter；Then, the high motor-driven cross flying of target group, and the track association scene that heavy dense targets degree is high are generated；
Step 2：Targetpath number is determined using KMDL criterion criterions
The flight path observation scene at a certain moment, chooses the time data and is closed as flight path in any extraction track association scene
Join sample data, kernel function is incorporated into minimum description length MDL criterions, utilizes Minimum description length criterion under kernel function
KMDL criterions establish the model of accurate description object, carry out KMDL values and calculate, and the difference that KMDL value is chosen with K values changes, when
During KMDL minimums, it is optimal between model complexity and model data matching degree, the K values for choosing the moment are flight path optimization
Number；
Step 3：Flight path observation scene is associated with kernel c-means algorithm
Flight path observation scene is associated using kernel c-means algorithm, the number K of clustering cluster is the optimal boat that step 2 obtains
Mark number, K- mean clusters are then carried out in nuclear space, Kernel-based K-means clustering result is respectively the K flight path at a certain moment.
In step 2, if model set is M, the model that MDL criterions are selected is Mmdl, model MmdlStandard be minimize
Two sums below：
|Lm(Mi)|：Descriptive model MiRequired digit；
|Lc(D|Mi)|：Setting models Mi, the digit needed for description object D, | Lc(D|Mi) | it is based on model M i descriptions pair
As D language；
Then MDL is expressed as follows：
The MDL forms under kernel function are extended, covariance matrixes of the MDL only with descriptive model error | Σj| it is relevant, error by
Transformed space after former space and consideration convey change determines that MDL form is：
njIt is the number of samples of j-th of cluster, Dist is by sample XkThe error to j-th of cluster is included into, P (J, D, I) is
Penalty, D are the dimensions of sample, and I is the number of cluster；
Error function is the Euclidean distance calculated using kernel formula, distance of each point to its center in j-th of clustering cluster
Quadratic sum is the majorized function of kernel c-means：
Assuming that K is linear kernel, S is equal to variance, i.e. K (X in luv space Xk,Xl)=XkXl；
Obtain complete MDL formula, it is assumed that the variance of each dimension cluster is equal, replaces covariance matrix determinant ΣjFor：
It is assumed that each dimension indifference is clustered, thenSjObtain, then have in arbitrary kernel function：
Kernel c-means algorithm described in step 3, comprises the following steps：
1) K sample in flight path scene is arbitrarily selected in nuclear space as initial cluster center
2) in nuclear space, by each sample φ (xi) be assigned to according to nearest neighbouring rule in each classification, lead to distance meter
Calculate, φ (xi) nearest from which cluster centre, just belong to the category；
3) cluster centre is recalculatedAnd JφValue；
4) repeat step 2) and 3), until continuous n iteration JφBe worth it is constant untill.
J in step 3)φThe computational methods of value are：
First, the classification according to obtained by step 2), a sample is found in each category as such center；So
Afterwards, in each category, respectively using each sample as class center, calculate class in other each sample points to class center distance, and
Calculate apart from sum, the class center minimum apart from sum is exactly such center；Finally, minimum range sum is exactly such error
Quadratic sumWill be all kinds ofAddition obtains JφValue.
K- mean clusters described in step 3, it is that the flight path sample to be sorted of nuclear space is passed through into non-linearity mapping phi:
Rn→ F, x → φ (x), original sample (x1,x2,…xn) it is mapped as (φ (x1),…,φ(xN)), it is based on distance in nuclear space
Division, similarity measure between sample it is big be classified as one kind, until the central point of cluster is restrained.
K- mean cluster concrete operations are：
First, the object function in following formula is minimized：
Wherein, average：I=1 ..., N, k=1 ..., K, NkIt is the number of samples of kth class；
Then, in nuclear space, any one sample φ (x) and certain class averageThe distance between calculated with following formula：
Wherein, k () is exactly kernel function, uses radial direction kernel function gaussian kernel function：
Compared with prior art, the present invention has technique effect beneficial below：
The kernel c-means Data Association based on KMDL criterion criterions of disclosure of the invention, believed based on dbjective state
Breath, KMDL criterions criterion and kernel c-means algorithm is combined under solving complexity environment (clutter is intensive, gtoal setting, target
Number is unknown) multi-target traces related question.First, typical track association scene is built, is selected close to target true motion
The typical model of pattern, generation target measure, can be preferably close to the real motion pattern of target；Clutter is generated, will be true
Target is correctly matched, and is that association scene is more life-like, is effectively investigated algorithm interrelating effect；In track association Scenario Design
In, the high motor-driven cross flying of generation target group, the higher simulating scenes of heavy dense targets degree.Secondly, kernel function is incorporated into most
In small description length MDL criterions, the number of accurate description object is established using Minimum description length criterion KMDL criterions under kernel function
Model is learned, handling a certain moment flight path with KMDL criterions criterion observes scene data point.Finally, scene core K- is observed to flight path
Mean algorithm associates, and is extended on traditional K- means clustering algorithms based on division, in clustering algorithm association process
Kernel function is introduced, K- mean clusters are then carried out in nuclear space.The inventive method makes full use of the movement state information of target,
Association accuracy rate is effectively improved, association criterion is simple and easy, and amount of calculation is small, association accuracy is high, target is intersected unwise
Sense, is adapted to carry out track association in heavy dense targets and Cross-environment, suitable for Project Realization.
Brief description of the drawings
Fig. 1 is the true flight path panorama sketch of target under CT models；
Fig. 2 is flight path observation chart of 30 sensors to 5 targets；
Fig. 3 is a certain moment observation chart of flight path；
Fig. 4 is schematic diagram after a certain moment denoising of flight path；
Fig. 5 is KMDL curve synoptic diagrams；
Fig. 6 is that flight path observation scene is based on kernel c-means algorithm associated diagram.
Embodiment
With reference to specific embodiment, the present invention is described in further detail, it is described be explanation of the invention and
It is not to limit.
The kernel c-means Data Association based on KMDL criterion criterions of the present invention, comprises the following steps：
Step 1: build typical track association scene
1st, target measures generation
For maneuvering target, target motor pattern is uncertain, and kinetic characteristic is unforeseen, is difficult to machine
Moving-target establishes single accurate model.The target movement model conventional from three kinds：Uniform motion model (CV), uniformly accelerated motion
Conventional typical model is chosen in model (CA), at the uniform velocity turning motion model (CT), generation target measures, preferably close to target
Real motion pattern.
2nd, clutter generates
In actual applications under true operation scene, often by clutter and false-alarm to target when sensor detects target
The interference of detection.The presence of clutter and false-alarm may cause measuring point to be more than real goal number, it is also possible to have influence on sensor
To the effect of real goal detection.Clutter is generated, whether checking association algorithm effectively can exclude clutter and noise spot, will be true
Real target is correctly matched, and makes association scene more life-like, effectively investigates algorithm interrelating effect.
3rd, architectural characteristic generates between target group
Under normal conditions, often there is special architectural characteristic between target group.High machine be present in the motion of target group
Dynamic property, target there may be parallel flight or cross flying, and extraterrestrial target closeness has the several scenes such as height difference.Navigating
In mark association Scenario Design, the high motor-driven cross flying of generation target group, the higher simulating scenes of heavy dense targets degree；Closed in flight path
Join in algorithm application, sufficiently using its architectural characteristic, investigate the validity of association algorithm.
Here at the uniform velocity turning motion model is chosen as a kind of typical association scene.Assuming that 30 sensor observation clutters
In 5 targets.Before emulation starts, there are 5 targets to obey and be evenly distributed on coordinate as [- 3km, 3km] × [- 3km, 3km's]
In region, the motion models of all targets obeys approximate at the uniform velocity turning motion model, initial velocity direction be in [- 2 π, 2 π] with
Machine is distributed, speed 20m/s.Sampling time interval is 1s, and the flight path cumulative time is 60s.
Dbjective stateMotion model：
xk=Fxk-1+wk-1
Wherein xkIt is the state value of k moment targets, px,kIt is k moment x position,It is k moment x speed, py,kIt is k
Moment y position,It is k moment y speed, wkIt is k moment corresponding process noise, F is state-transition matrix, and ω is
Turning rate, T are the sampling times.
Process noise is white Gaussian noise, average 0, and variance is：
δu=0.005 (π/180) rad/s.
Measurement model is nonlinear equation：
Measurement noise is white Gaussian noise, average 0, and variance is：
vk~N (；[0.001,0]T,Rk)；
σθ=0.1 (π/180) rad；σr=5m.
Fig. 1 is illustrated at the uniform velocity turning motion model, flight path of the target within all sampling times.Mesh is detected in the scene
It is 5 to mark number, sensor 30.
Step 2: calculate flight path observation scene KMDL values and choose optimal targetpath number
The data sharing information of flight path needs the correct matching between target, and transmitting terminal sends the observation letter of a series of complex
Breath, receiving terminal are matched it with local observation data with a kind of matching algorithm.Sensor is carried out by observing data
During association, it is contemplated that system deviation, random error, false-alarm, the presence for leaking situations such as detection, and by sensing system itself
The constraint of condition, in order to solve complicated related question, in CT motion models, extraction multiple target is within a certain sampling time
Scene is observed, chooses 30 sensors to a certain moment measuring point of 5 targets as track association sample point.
Kernel function is incorporated into minimum description length MDL criterions, utilizes Minimum description length criterion KMDL under kernel function
Criterion establishes the mathematical modeling of accurate description object.A certain moment flight path observation scene data point is handled with KMDL criterions criterion：
When the undermatching flight path observation contextual data of descriptive model curve, model is not enough to capture the rule of multi-target traces data reflection
Property, though at this moment model complexity is low, model and track data matching degree are poor, and description length is bigger than normal；When model curve is crossed
With data point, while capture and fluctuated caused by noise, it is impossible to the rule of data reflection is captured, in such case model sum
Though high according to matching degree, model complexity is high, causes description length bigger than normal；When model complexity and model data matching degree it
Between when being optimal, regularity that the data that model capture are reflected, model complexity is relatively low, model and data matching degree compared with
Height, now corresponding KMDL values are minimum, and corresponding K values are flight path optimization number.
A certain moment point flight path observation scene is arbitrarily extracted, chooses the time data as track association sample data.Fig. 2
It is flight path observation chart of 30 sensors to 5 targets.Fig. 3 is a certain moment observation chart of flight path.Fig. 4 is a certain moment denoising of flight path
Schematic diagram afterwards.
If model set is M, the model that MDL criterions are selected is Mmdl, model MmdlStandard be to minimize following two
(describing length) sum：|Lm(Mi)|：Descriptive model MiRequired digit (can be regarded as the code for needing all parameter codings
Long summation, also referred to as describes length).|Lc(D|Mi)|：Setting models Mi, the digit needed for description object D.Here | Lc(D|Mi)|
For the language based on model M i description objects D.MDL is expressed as follows：
Extend kernel function under MDL forms, MDL only and descriptive model error covariance matrix | Σj| it is relevant.Error by
Transformed space after former space and consideration convey change determines that the more general forms of MDL are：
njIt is the number of samples of j-th of cluster, Dist is by sample XkThe error to j-th of cluster is included into, P (J, D, I) is
Penalty, D are the dimensions of sample, and I is the number of cluster.
Simplest error function is the Euclidean distance calculated using kernel formula, and each point is to wherein in j-th of clustering cluster
The square distance of the heart and the majorized function for being kernel c-means
Assuming that K is linear kernel, S is equal to variance, i.e. K (X in luv space Xk,Xl)=XkXl.It is public to obtain complete MDL
Formula, it is assumed that the variance of each dimension cluster is equal, can replace covariance matrix determinant ΣjFor：
It is assumed that each dimension of cluster is not as differenceIn view of SjIt can be obtained in arbitrary kernel function, can
Scene is observed to a certain moment point flight path, chooses the value that the time data calculates KMDL, KMDL value is selected with K values
The different changes taken, it is optimal between model complexity and model data matching degree during KMDL minimums, chooses K now as most
Good cluster number is target number.Fig. 5 is KMDL curve synoptic diagrams, and KMDL minimum values are -732.2927, and now K values are
5。
Step 3: scene clustering is observed to flight path with kernel c-means algorithm
Flight path observation scene is associated using kernel c-means algorithm, the wherein number K of clustering cluster is obtained by KMDL criterions
Target number, it is extended on traditional K- means clustering algorithms based on division, is introduced in clustering algorithm association process
Kernel function, K- mean clusters are then carried out in nuclear space, K- averages are carried out inside a new feature space (nuclear space)
Cluster, the flight path sample to be sorted of nuclear space pass through non-linearity mapping phi:Rn→ F, x → φ (x) are original sample (x1,x2,…
xn) it is mapped as (φ (x1),…,φ(xN)), in division of the nuclear space based on distance, similarity measure between sample it is big be classified as one
Class, until the close convergence of the central point of cluster.Kernel-based K-means clustering result is respectively the K flight path at a certain moment.
Specifically, scene clustering is observed to flight path using Kernel-based K-means clustering algorithm, first with a Nonlinear Mapping
φ:Rn→ F, x → φ (x), by original space RnSample x to be sorted be mapped in a high-dimensional feature space F (nuclear space),
Purpose is can to protrude the feature difference between different classes of sample so that sample becomes linear separability, and (or approximately linear can
Point), then carry out K- mean clusters in higher-dimension nuclear space.In nuclear space, sample to be sorted is changed into (φ (x1),…,φ
(xN)), carry out Kernel-based K-means clustering and seek to minimize the object function in following formula：
Wherein, average：I=1 ..., N, k=1 ..., K, NkIt is the number of samples of kth class.
In nuclear space, any one sample φ (x) and certain class averageThe distance between calculated with following formula：
Wherein, k () is exactly kernel function, used here as radial direction kernel function gaussian kernel function
Feature space corresponding to gaussian kernel function is infinite dimensional, can not be limited the higher dimensional space that data map by dimension.
There are following steps using Kernel-based K-means clustering algorithm：
(1) initial cluster center is determinedI.e.：K in flight path scene are arbitrarily selected in nuclear space
Sample is as initial cluster center.
(2) in nuclear space, by each sample φ (xi) be assigned to according to nearest neighbouring rule in each classification.Pass through above formula
Distance calculates, φ (xi) nearest from which cluster centre, just belong to the category.
(3) cluster centre is recalculatedAnd JφValue.Due in nuclear space, it is impossible in clearly calculating
The heart, therefore a sample can only be selected to replace class center in each category, specific method is：According to obtained by step (2)
Classification, a sample is found in each category as such center.In each category, respectively using each sample as in class
The heart, calculates the distance that other each sample points in class arrive class center, and calculates apart from sum, and the class center apart from sum minimum is exactly
Such center.Minimum range sum is exactly such error sum of squaresWill be all kinds ofAdd up and just obtain JφValue.
(4) repeat step (2) and (3), until continuous n iteration JφUntill being worth constant (or varying less).
Simulation result such as Fig. 6.By above-mentioned steps, a certain moment flight path scene extracted in CT motion models is used
KMDL criterion criterions determine targetpath number K, carry out track association cluster using kernel c-means algorithm, every one kind as a result is
To come from the local tracks of same target.
Claims (6)
1. the kernel c-means Data Association based on KMDL criterion criterions, it is characterised in that comprise the following steps：
Step 1：Build typical track association scene
First, the typical model close to target true motion pattern is selected, generation target measures；Secondly, given birth to using Poisson distribution
Into clutter；Then, the high motor-driven cross flying of target group, and the track association scene that heavy dense targets degree is high are generated；
Step 2：Targetpath number is determined using KMDL criterion criterions
The flight path observation scene at a certain moment, chooses the time data as track association sample in any extraction track association scene
Notebook data, kernel function is incorporated into minimum description length MDL criterions, sentenced using Minimum description length criterion KMDL under kernel function
According to the model for establishing accurate description object, carry out KMDL values and calculate, the difference change that KMDL value is chosen with K values, when KMDL most
Hour, it is optimal between model complexity and model data matching degree, the K values for choosing the moment are flight path optimization number；
Step 3：Flight path observation scene is associated with kernel c-means algorithm
Flight path observation scene is associated using kernel c-means algorithm, the number K of clustering cluster is the flight path optimization that step 2 obtains
Number, K- mean clusters are then carried out in nuclear space, Kernel-based K-means clustering result is respectively the K flight path at a certain moment.
2. the kernel c-means Data Association according to claim 1 based on KMDL criterion criterions, it is characterised in that step
In rapid two, if model set is M, the model that MDL criterions are selected is Mmdl, model MmdlStandard be minimize following two it
With：
|Lm(Mi)|：Descriptive model MiRequired digit；
|Lc(D|Mi)|：Setting models Mi, the digit needed for description object D, | Lc(D|Mi) | it is based on model M i description objects D's
Language；
Then MDL is expressed as follows：
The MDL forms under kernel function are extended, covariance matrixes of the MDL only with descriptive model error | ∑j| relevant, error is by former empty
Between and consideration convey change after transformed space determine that MDL form is：
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3. the kernel c-means Data Association according to claim 1 based on KMDL criterion criterions, it is characterised in that step
Kernel c-means algorithm described in rapid three, comprises the following steps：
1) K sample in flight path scene is arbitrarily selected in nuclear space as initial cluster center
2) in nuclear space, by each sample φ (xi) be assigned to according to nearest neighbouring rule in each classification, calculated by distance,
φ(xi) nearest from which cluster centre, just belong to the category；
3) cluster centre is recalculatedWith object function JφValue；
4) repeat step 2) and 3), until continuous n iterative target function JφBe worth it is constant untill.
4. the kernel c-means Data Association according to claim 3 based on KMDL criterion criterions, it is characterised in that step
It is rapid 3) in object function JφThe computational methods of value are：
First, the classification according to obtained by step 2), a sample is found in each category as such center；Then, exist
In each classification, respectively using each sample as class center, in calculating class other each sample points to class center distance, and calculate away from
From sum, the class center minimum apart from sum is exactly such center；Finally, minimum range sum is exactly such error sum of squaresWill be all kinds ofAddition obtains JφValue.
5. the kernel c-means Data Association according to claim 1 based on KMDL criterion criterions, it is characterised in that step
K- mean clusters described in rapid three, it is that the flight path sample to be sorted of nuclear space is passed through into non-linearity mapping phi:Rn→F,x→φ
(x), from original space RnMapped to nuclear space F, original sample (x1,x2,…xn) it is mapped as (φ (x1),…,φ(xN)),
Division of the nuclear space based on distance, similarity measure between sample it is big be classified as one kind, until the central point of cluster is restrained.
6. the kernel c-means Data Association according to claim 5 based on KMDL criterion criterions, it is characterised in that K-
Mean cluster concrete operations are：
First, the object function in following formula is minimized：
<mrow>
<msup>
<mi>J</mi>
<mi>&phi;</mi>
</msup>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>K</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</munderover>
<mo>|</mo>
<mo>|</mo>
<mi>&phi;</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msubsup>
<mi>m</mi>
<mi>k</mi>
<mi>&phi;</mi>
</msubsup>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
</mrow>
Wherein, average：I=1 ..., N, k=1 ..., K, NkIt is the number of samples of kth class；
Then, in nuclear space, any one sample φ (x) and certain class averageThe distance between calculated with following formula：
<mrow>
<mo>|</mo>
<mo>|</mo>
<mi>&phi;</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msubsup>
<mi>m</mi>
<mi>k</mi>
<mi>&phi;</mi>
</msubsup>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>=</mo>
<mo>|</mo>
<mo>|</mo>
<mi>&phi;</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</munderover>
<mi>&phi;</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>=</mo>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mn>2</mn>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</munderover>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</munderover>
<mi>k</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
Wherein, k () is exactly kernel function, uses radial direction kernel function gaussian kernel function：
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