CN108965178B - Intelligent self-adaptive equalizer based on machine learning and equalization demodulation method - Google Patents

Abstract

The invention discloses an intelligent self-adaptive equalizer based on machine learning and an equalization demodulation method, which are used for carrying out data preprocessing on an acquired signal, carrying out energy normalization on input data, then clustering clusters in the data by utilizing a Gaussian kernel function and a distance function without any other prior knowledge, labeling the clustered data to the clusters by utilizing a nearest neighbor algorithm, and realizing useful informatization of a modulated signal. And (4) the noise discrete points outside the cluster are not clustered, and the cluster halos without clustering labels are marked by using a weighted K nearest neighbor algorithm. And finally, integrating all the data to obtain an integral label, and comparing the integral label with a prestored label to estimate the system error rate. The method of the invention can identify the true cluster center without any other prior knowledge, regardless of the shape and size; the invention reduces the calculation complexity, obviously improves the accuracy of the classification result and can adapt to most modulation formats in the current communication system.

Description

Intelligent self-adaptive equalizer based on machine learning and equalization demodulation method

Technical Field

The invention relates to an intelligent coherent optical fiber communication method, in particular to an intelligent adaptive algorithm for a coherent optical communication system.

Background

With the increase of various bandwidth-intensive applications such as big data, cloud services, smart phones, network videos, and the like, network traffic is growing rapidly. The Cognitive Optical Network (CON) can provide automatic configuration, self optimization and autonomous network operation through self learning and cognitive decision, can coordinate a transmitter and a receiver, realizes real-time dynamic adjustment of a modulation format, a line rate, spectrum allocation and the like, and has better expandability and interoperability. By carrying out coordination management on the transceiver, the link functions of intelligent channel management, bandwidth allocation and the like of the network node can be realized, so that the service quality and the transmission quality are improved. Therefore, it is significant to develop a cognitive optical network in order to provide flexible and reasonably priced on-demand network services for elephant flow data.

In cognitive heterogeneous optical networks, the physical layer has various architectures and hardware and includes various types of optical signals with various modulation formats and symbol rates. Phase recovery in coherent optical receivers is highly dependent on the received signal modulation format, and it is challenging how to construct an optical receiver that is immune to modulation to automatically detect all types of optical signals.

In order to implement intelligent optical network management, it is considered to identify modulation format information and compensate for channel distortion in optical fiber transmission through a Digital Signal Processing (DSP) algorithm, and for this purpose, it is necessary to install different DSP algorithms for optical signals of various modulation formats, wherein the modulation format is identified before phase recovery, and then select an appropriate DSP algorithm for phase recovery according to the acquired modulation information. However, different signal modulation formats require different equalization and demodulation algorithms, which wastes hardware resources to some extent.

Therefore, how to implement adaptive equalization and demodulation of modulated signals with different formats in a cognitive optical network to provide efficient equalization performance under relatively low complexity conditions, thereby reducing the computational cost and realizing commercial application with low data redundancy is a technical problem to be solved in the art.

Disclosure of Invention

The invention aims to provide an intelligent self-adaptive equalization demodulation method based on machine learning, which is applied to signal reception in a cognitive optical network and used for equalizing the damage of signals in transmission by reducing the computational complexity and providing low data redundancy, such as: the signal is damaged by the nonlinearity of the optical fiber, the line width of the laser and the I/Q deflection of the modulator, and the like, so that the modulation signals of most formats are demodulated, and hardware resources are saved. It is another object of the invention to provide an equalizer implementing the method.

In order to achieve the above-mentioned objects, the present invention has the following general concepts: data preprocessing is carried out on the acquired signals, energy normalization is carried out on the input data, then clustering is carried out on clusters in the data by utilizing a Gaussian kernel function and a distance function under the condition of no other prior knowledge, and the clustered data are labeled by a Nearest Neighbor (NN) algorithm, so that useful informatization of the modulation signals is realized. Noise discrete points outside the cluster are not clustered, and cluster halos without clustering labels are marked by a weighted K Nearest Neighbor (KNN) algorithm. And finally, integrating all the data to obtain an integral label, and comparing the integral label with a prestored label to estimate the system Bit Error Rate (BER).

Specifically, the technical scheme adopted by the invention is as follows: an intelligent adaptive equalization demodulation method based on machine learning comprises the following steps:

(1) obtaining signal data to be processed from an optical receiving device, preprocessing the data, normalizing the carrier phase recovery signal based on average energy, expressed as

Wherein the content of the first and second substances,x _i is a data point that is,kis the number of data points that are,p _i is the data point after the average energy normalization of the first-level data point, and forms a data setP；

(2) In a data setPCalculating the Euclidean distanced _ij Wherein, in the step (A),iandjis the first in the data setiA data point andjthe number of data points is, for example,d _ij = d _ji ， i<jcalculating distances in ascending orderd _ij The value of the point at 2% is selected as the truncation distanced _c ；

(3) Acquiring a clustering center:

suppose that a data set of N classes consists of N data

Presentation, label thereofI _S = {1,2, …, N }, for each data point in the data set SX _i Separately calculating the densityρ _i And distanceδ _i With a Gaussian kernel function as a pointiDensity of (2)ρ _i The number of bits, written as,

whereind _ij Representing the distance between data point i and data point j,d _c is the truncation distance;

to be provided withq _i Representsρ _i Data, distance after descending orderδ _i Is defined as:

calculating distance parameters and density parameters of data points according to the definition to determine clustering centers, wherein the number of the clustering centers of the M-QAM signals is M, and obtaining the densityρ _i And distanceδ _i Product of (2)γ _i

Will be provided withγ _i Arranged in descending order, with the first M points selected as clustering centers;

(4) sticking a data label:

calculating the distance between the clustering center and a conventional QAM decision point, marking the clustering center by using the closest distance principle of a K-means algorithm, and using a label of the clustering center as a label of corresponding clustering data;

(5) labeling the cluster halos:

cluster halos are marked based on a weighted voting KNN algorithm,

using the marked clusters in the step (4) as a training set of weighted KNN, and enabling T = { ({ (x ₁ , y ₁ ), (x ₂ , y ₂ ), …, (x _N , y _N ) Contains known tags thereiny _i ∈{C ₁ , C ₂ ,C ₃ , … , C _m N training data ofx _i (i=1,2, …, N), m is the number of categories;

clustering halos without labels as test set of weighted KNN, finding K points closest to data point x in training set T by calculating Euclidean distance, these K points being written as N points in the x range_k(x) According to N_k(x) Obtaining K labels of the latest training data in the formula, obtaining a decision label y according to the K labels,

whereinω _i =1/D(x, x _i ), i = 1,2, …, NAndj = 1,2, …, mi is a function of index, wheny _i = C _j When I is 1, otherwise I is 0,D(x, x _i ) Is a pointxTo pointx _i The Euclidean distance of (a) is,

thereby achieving equalized demodulation of the signal data.

In a further technical scheme, the error rate of the system is calculated by comparing the realized labels of the constellation clusters including the cluster core and the cluster halo with the pre-stored labels.

The invention also provides an intelligent self-adaptive equalization demodulator based on machine learning, which comprises the following modules:

(1) a data preprocessing module:

obtaining signal data to be processed from an optical receiving device, preprocessing the data, normalizing the carrier phase recovery signal based on average energy, expressed as

Wherein the content of the first and second substances,x _i is a data point that is,kis the number of data points that are,p _i is the data point after the average energy normalization of the first-level data point, and forms a data setP；

(2) A distance calculation module:

in a data setPCalculating the Euclidean distanced _ij Wherein, in the step (A),iandjis the first in the data setiA data point andjthe number of data points is, for example,d _ij = d _ji ， i<jcalculating distances in ascending orderd _ij The value of the point at 2% is selected as the truncation distanced _c ；

(3) A clustering center obtaining module:

suppose that a data set of N classes consists of N data

Presentation, label thereofI _S = {1,2, …, N }, for each data point in the data set SX _i Separately calculating the densityρ _i And distanceδ _i With a Gaussian kernel function as a pointiDensity of (2)ρ _i The number of bits, written as,

whereind _ij Representing the distance between data point i and data point j,d _c is the truncation distance;

to be provided withq _i Representsρ _i Data, distance after descending orderδ _i Is defined as:

calculating distance parameters and density parameters of data points according to the definition to determine clustering centers, wherein the number of the clustering centers of the M-QAM signals is M, and obtaining the densityρ _i And distanceδ _i Product of (2)γ _i

Will be provided withγ _i Arranged in descending order, with the first M points selected as clustering centers;

(4) clustering cluster data label marking module:

calculating the distance between the clustering center and a conventional QAM decision point, marking the clustering center by using the closest distance principle of a K-means algorithm, and using a label of the clustering center as a label of corresponding clustering data;

(5) a cluster halo data tag marking module:

cluster halos are marked based on a weighted voting KNN algorithm,

using the labeled clusters as a training set of weighted KNNs, T = { ({ (C) } { (C) } C)x ₁ , y ₁ ), (x ₂ , y ₂ ), …, (x _N , y _N ) Contains known tags thereiny _i ∈{C ₁ , C ₂ ,C ₃ , … , C _m N training data ofx _i (i=1,2, …, N), m is the number of categories;

clustering halos without labels as test set of weighted KNN, finding K points closest to data point x in training set T by calculating Euclidean distance, these K points being written as N points in the x range_k(x) According to N_k(x) Obtaining K labels of the latest training data in the formula, obtaining a decision label y according to the K labels,

whereinω _i =1/D(x, x _i ), i = 1,2, …, NAndj = 1,2, …, mi is a function of index, wheny _i = C _j When I is 1, otherwise I is 0,D(x, x _i ) Is a pointxTo pointx _i The euclidean distance of (c).

Due to the application of the technical scheme, compared with the prior art, the invention has the following advantages:

1. in the process of processing coherent optical communication data, the invention adopts a brand-new algorithm of 'natural' processing based on machine learning, breaks through the defects of multiple iterations of a K-means clustering algorithm in the traditional machine learning and easy falling into local optimum, and the proposed method can identify a real clustering center without any other prior knowledge regardless of the shape and the size of the clustering center.

2. The balanced demodulator based on machine learning provided by the invention has the following advantages: (1) extra training parameters are not needed, any iterative calculation is not needed, and the calculation complexity can be greatly reduced; (2) because the noise of the system is mainly due to the external noise points, the method classifies the external noise points by using the weighted KNN, has stronger nonlinear classification capability, breaks through the defect of weak K-means nonlinear classification performance, and remarkably improves the classification result; (3) the algorithm is suitable for most modulation formats in the current communication system, and self-adaptive equalization is provided for the modulation formats, so that the future hardware cost can be saved.

Drawings

In fig. 1, (i) is the constellation cluster of 16-QAM signals classified by IACA, (ii) is the decision graphs ρ and δ, (iii) is γ calculated in descending order of 50 data in (i);

FIG. 2 shows (i) a blind K-means tagging method and (ii) a training sequence assisted K-means tagging method;

fig. 3 is a constellation diagram of a 64-QAM signal, in which (i) the noise is more (ii) the noise is less;

FIG. 4 is a schematic diagram of a structure of an intelligent adaptive equalizer based on a machine learning algorithm in an embodiment;

FIG. 5 is a flow chart of a method for intelligent adaptive equalization demodulation in an embodiment;

FIG. 6 is a comparison of Bit Error Rate (BER) versus signal to noise ratio (SNR) for 4/16/64/128/256-QAM with a slight phase rotation, with the corresponding decision diagram for the constellation diagram and cluster center for the insertion of 256-QAM signals, where IACA is an intelligent adaptive cluster-like algorithm, and the system is collectively referred to as an intelligent adaptive equalizer;

FIG. 7 is a diagram of the Bit Error Rate (BER) versus laser linewidth for a 16-QAM signal, with the corresponding constellation diagram inserted;

FIG. 8 is a graph of measured BER curves for 75-Gb/s 64-QAM signals versus transmit signal power in 130-km SSMF, inset by constellations for transmit signal power of-7.17 dBm and 4.82 dBm;

fig. 9 is a graph of BER versus I/Q phase offset for the measured modulator, and the inset is the constellation diagram for the corresponding skew.

Detailed Description

The invention is further described with reference to the following figures and examples:

the first embodiment is as follows: referring to fig. 4, at a receiving end of an optical network, after Chromatic Dispersion (CD) compensation, clock recovery, and carrier phase recovery are performed on a received signal, an equalization demodulator is constructed by using the method of this embodiment to demodulate the signal.

In this embodiment, an intelligent adaptive equalization demodulation method based on machine learning is shown in fig. 5, and includes the following steps:

(1) and (4) preprocessing data. Normalizing the carrier phase recovery signal based on the average energy, denoted as

Here, x_iIs a data point, k is the number of data points, p_iThe average energy that is the primary data point can be normalized.

(2) Distance between two adjacent platesd _ij And (4) calculating. In a data setP _i （d _ij = d _ji , i<j) Calculating the Euclidean distanced _ij And calculating the distances in ascending orderd _ij . Distance between two adjacent platesd _c Represents a truncation distance, whereind _c Is robust to the results of the data analysis, expressed as:

where f (-) denotes a rounded integer and n is the number of data points. Distance of truncationd _c Is to select d arranged from small to large_ijThen d is arranged_ijThe value of the point at 2% as the truncation distance, the weighting coefficient ω =0.1 in the present embodiment.

(3) Density ofρ _i And distanceδ _i And (4) calculating.

Suppose that a data set of N classes consists of N data

Presentation, label thereofI _S =1,2, …, N. For each data point in the data set SX _i It is necessary to calculate two quantities, the density of whichρ _i And distanceδ _i . Using a Gaussian kernel function as the density of points iρ _i The number of bits, written as,

whereind _ij Representing the distance between data point i and data point j,d _c is the truncation distance. Obviously, distanceX _i Is less thand _c The more data points that are present,ρ _i the larger the value of (c).

Suppose that

Represents

In descending order, distanceδ _i Can be defined as:

δ _i are data pointsX _i With other dots having a higher densityX _j The minimum distance between. However, whenX _i With the highest local density of the particles,δ _i give aX _i And in SX _i The distance between the data points with the largest distance. Thus, it can be found to have a larger sizeδ _i The clustering center of the values is fast.

Calculating data points according to the above density formula and distance formulaX _i Determines the cluster center, i.e. its density p and its distance δ. The number of the clustering centers of the M-QAM signal is M, and the density is obtainedρ _i And distanceδ _i Product of (2)γ _i

These parameters are illustrated for example in a 16-QAM signal, as shown in figure 1. Fig. 1 (i) shows a constellation diagram for a 16-QAM signal, where different blocks represent different clusters and the circle is the center of the cluster. Fig. 1 (ii) plots the decision graph for the calculated density ρ versus distance δ, where 16 points falling within the rectangle are determined as cluster centers. The calculated γ is plotted in fig. 1 (iii) in descending order, with the first 16 points selected as cluster centers. The results in fig. 1 are the same as described in the previous section.

(4) And sticking a data label.

The constellation cluster needs to label data correctly to function on the basis of classification.

And calculating the distance between the clustering center and the conventional QAM decision point, and marking the clustering center by using the closest distance principle of the K-means algorithm.

Fig. 2(i) and 2(ii) show labeling results based on the blind K-means algorithm and the training-assisted K-means algorithm, respectively, where the star points are conventional QAM decision points and the dots are cluster centers without labeling. Then, according to the clustering result in the step (3), the label of the clustering center is used as the label of the corresponding clustering data. For distortion signals and high-order modulation signals with low noise tolerance, the training auxiliary labeling method is simple in structure, high in calculation speed and high in complexity of O (n), wherein n is the number of test samples (namely the order of a modulation format, such as for M-QAM, n = M). Moreover, only 1% -2% of the training sequence is sufficient for accurate symbol labeling.

(5) And (5) labeling by cluster halo.

Each constellation cluster is composed of a cluster core and a cluster halo. Fig. 3 (i) and (ii) show the constellation diagrams of the 64-QAM signal with more noise and the 64-QAM signal with less noise, respectively, wherein the color blocks represent cluster nuclei, the surrounding black dots represent cluster halos, and the middle circle is the cluster center implemented based on the FSFDP algorithm. The cluster halo is far from the cluster center and only appears in the noise signal. For noise signals, the marker cluster halo is also important. In this embodiment, cluster halos are marked based on a weighted voting KNN algorithm.

Using the obtained known cluster as a training set of weighted KNN, wherein the training data set T = { ({ (R) } is obtainedx ₁ , y ₁ ), (x ₂ , y ₂ ), …, (x _N , y _N ) Contains known tags thereiny _i ∈{C ₁ , C ₂ ,C ₃ , … , C _m N training data ofx _i (i=1,2, …, N), m is the number of categories.

The cluster halos without labels serve as a test set of weighted KNN, and then K points closest to X can be found in the training set T by calculating Euclidean distances. In the X range, these K points are written as N_k(x) In that respect According to N_k(x) The K most recent training data labels in the formula are available, and most of these K labels are used as decision labels.

Whereinω _i =1/D(x, x _i ), i = 1,2, …, NAndj = 1,2, …, m.. I is an index function. When in usey _i = C _j When I is 1, otherwise I is 0,D(x, x _i ) Is a pointxTo pointx _i The euclidean distance of (c).

(6) And (6) performance estimation.

The BER performance of the system is calculated by comparing the implemented labels of the constellation clusters, including the cluster core and the cluster halo, to pre-stored labels.

The receiving system is constructed by adopting the method for testing, and the result is shown in the attached drawing.

Fig. 6 is a comparison of Bit Error Rate (BER) versus signal-to-noise ratio (SNR) for 4/16/64/128/256-QAM with a slight phase rotation. The insertion is a constellation diagram of the 256-QAM signal and a corresponding decision diagram of the cluster center. The IACA is an intelligent self-adaptive cluster algorithm, and the system is called an intelligent self-adaptive equalizer.

When the signal is worse, the estimated phase reference is inaccurate. This results in a rotation of the signal shown in fig. 6. For lower modulation level QPSK and 16-QAM signals, the effect of this rotation on BER is not severe. However, for the 64/128/256-QAM signal at higher modulation levels, phase rotation has a significant impact on BER performance because conventional linear symbol decisions cannot track signal variations, which leads to larger errors. The improvement obtained by using the proposed IACA is significant.

Figure 7 is the Bit Error Rate (BER) versus laser linewidth for a 16-QAM signal. The insertion is the corresponding constellation.

Higher order modulation signals are susceptible to laser linewidth, which can introduce severe phase noise. In general, kalman filtering is an effective method to compensate for signal impairments caused by laser linewidth. Figure 7 shows the degraded BER performance as the laser linewidth increases. The proposed IACA is superior to conventional kalman filtering, as shown by the curves marked with triangles and squares.

FIG. 8 is a graph of measured BER curves for 75-Gb/s 64-QAM signals versus transmit signal power versus insertion in a 130-km SSMF for constellations with transmit signal powers of-7.17 dBm and 4.82 dBm.

The BER performance was evaluated with phase noise. Fig. 8 shows the relationship between the measured BER and the signal power transmitted to 130 km SSMF. Higher signal power results in more severe phase noise due to fiber nonlinearities and lower signal power results in more severe ASE noise. This creates an arcuate curve, as shown in FIG. 8. A Support Vector Machine (SVM) with high computational complexity is an effective method for solving the problem of nonlinear classification, and can effectively improve the Bit Error Rate (BER) performance. The proposed IACA method with 2% training sequence outperforms the conventional SVM method with 10% training sequence as shown by the square and triangular labeling curves. The inset in fig. 8 is the constellation of signals with transmit signal powers of-7.17 dBm and 4.82 dBm.

Fig. 9 is a graph of BER versus I/Q phase offset of the measurement modulator. The inset is the constellation that is correspondingly skewed.

The method provided by the invention is an adaptive equalization algorithm, and can be used for well improving in-phase (I) and quadrature (Q) offset noises of a modulator. As shown in fig. 9.

Claims (3)

1. An intelligent adaptive equalization demodulation method based on machine learning comprises the following steps:

(1) obtaining signal data to be processed from an optical receiving device, preprocessing the data, normalizing the carrier phase recovery signal based on average energy, expressed as

Wherein the content of the first and second substances,x _i is a data point that is,kis the number of data points that are,p _i is the data point after the average energy normalization of the first-level data point, and forms a data setP；

(2) In a data setPCalculating the Euclidean distanced _ij Wherein, in the step (A),iandjis the first in the data setiA data point andjthe number of data points is, for example,d _ij = d _ji ， i<jcalculating distances in ascending orderd _ij The value of the point at 2% is selected as the truncation distanced _c ；

(3) Acquiring a clustering center:

suppose that a data set of N classes consists of N data

Presentation, label thereofI _S = {1,2, …, N }, for each data point in the data set SX _i Separately calculating the densityρ _i And distanceδ _i With a Gaussian kernel function as a pointiDensity of (2)ρ _i The number of bits, written as,

whereind _ij Representing the distance between data point i and data point j,d _c is the truncation distance;

to be provided withq _i Representsρ _i Data, distance after descending orderδ _i Is defined as:

calculating distance parameters and density parameters of data points according to the definition to determine clustering centers, wherein the number of the clustering centers of the M-QAM signals is M, and obtaining the densityρ _i And distanceδ _i Product of (2)γ _i

Will be provided withγ _i Arranged in descending order, with the first M points selected as clustering centers;

(4) sticking a data label:

calculating the distance between the clustering center and a conventional QAM decision point, marking the clustering center by using the closest distance principle of a K-means algorithm, and using a label of the clustering center as a label of corresponding clustering data;

(5) labeling the cluster halos:

cluster halos are marked based on a weighted voting KNN algorithm,

using the marked clusters in the step (4) as a training set of weighted KNN, and enabling T = { ({ (x ₁ , y ₁ ), (x ₂ , y ₂ ), …, (x _N , y _N ) Contains known tags thereiny _i ∈{C ₁ , C ₂ ,C ₃ , … , C _m N training data ofx _i (i=1,2, …, N), m is the number of categories;

clustering halos without labels as test set of weighted KNN, finding K points closest to data point x in training set T by calculating Euclidean distance, these K points being written as N points in the x range_k(x) According to N_k(x) Obtaining K labels of the latest training data in the formula, obtaining a decision label y according to the K labels,

whereinω _i =1/D(x, x _i ), i = 1,2, …, NAndj = 1,2, …, mi is a function of index, wheny _i = C _j When I is 1, otherwise I is 0,D(x, x _i ) Is a pointxTo pointx _i The Euclidean distance of (a) is,

thereby achieving equalized demodulation of the signal data.

2. The intelligent adaptive equalization demodulation method based on machine learning of claim 1, characterized in that: the error rate of the system is calculated by comparing the implemented tags of the constellation clusters, including the cluster core and the cluster halo, with pre-stored tags.

3. An intelligent adaptive equalization demodulator based on machine learning, comprising the following modules:

(1) a data preprocessing module:

obtaining signal data to be processed from an optical receiving device, preprocessing the data, normalizing the carrier phase recovery signal based on average energy, expressed as

Wherein the content of the first and second substances,x _i is a data point that is,kis the number of data points that are,p _i is the data point after the average energy normalization of the first-level data point, and forms a data setP；

(2) A distance calculation module:

in a data setPCalculating the Euclidean distanced _ij Wherein, in the step (A),iandjis the first in the data setiA data point andjthe number of data points is, for example,d _ij = d _ji ， i<jcalculating distances in ascending orderd _ij The value of the point at 2% is selected as the truncation distanced _c ；

(3) A clustering center obtaining module:

suppose that a data set of N classes consists of N data

Presentation, label thereofI _S = {1,2, …, N }, for each data point in the data set SX _i Separately calculating the densityρ _i And distanceδ _i With a Gaussian kernel function as a pointiDensity of (2)ρ _i The number of bits, written as,

whereind _ij Representing the distance between data point i and data point j,d _c is the truncation distance;

to be provided withq _i Representsρ _i Data, distance after descending orderδ _i Is defined as:

calculating distance parameters and density parameters of data points according to the definition to determine clustering centers, wherein the number of the clustering centers of the M-QAM signals is M, and obtaining the densityρ _i And distanceδ _i Product of (2)γ _i

Will be provided withγ _i Arranged in descending order, with the first M points selected as clustering centers;

(4) clustering cluster data label marking module:

calculating the distance between the clustering center and a conventional QAM decision point, marking the clustering center by using the closest distance principle of a K-means algorithm, and using a label of the clustering center as a label of corresponding clustering data;

(5) a cluster halo data tag marking module:

cluster halos are marked based on a weighted voting KNN algorithm,

using the labeled clusters as a training set of weighted KNNs, T = { ({ (C) } { (C) } C)x ₁ , y ₁ ), (x ₂ , y ₂ ), …, (x _N , y _N ) Contains known tags thereiny _i ∈{C ₁ , C ₂ ,C ₃ , … , C _m N training data ofx _i (i=1,2, …, N), m is the number of categories;

clustering halos without labels as test set of weighted KNN, finding K points closest to data point x in training set T by calculating Euclidean distance, these K points being written as N points in the x range_k(x) According to N_k(x) Obtaining K labels of the latest training data in the formula, obtaining a decision label y according to the K labels,

whereinω _i =1/D(x, x _i ), i = 1,2, …, NAndj = 1,2, …, mi is a function of index, wheny _i = C _j When I is 1, otherwise I is 0,D(x, x _i ) Is a pointxTo pointx _i The euclidean distance of (c).

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