The parametric methods have limited ability to take into account the complexity and heterogeneity of data distributions, and it is the problem of this category of methods. This paper presents a nonparametric method using local polynomial splines to estimate heterogeneous mixtures of distributions. By leveraging B-splines, the method adapts to various data distributions without strict assumptions about their shapes. This approach improves parameter estimation accuracy in complex data structures where traditional parametric methods are inadequate.
The Support Vector Machine (SVM) is a supervised learning method used for classification and
regression. The basic idea is to find the optimal hyperplane that best separates objects of different
classes on the feature plane. One of the advantages of SVM is its effectiveness in dealing with
highdimensional data and the ability to work with heterogeneous mixtures of distributions, but it can
be sensitive to large amounts of data and requires proper selection of hyperparameters, such as the
regularization parameter and kernel selection [
main advantages of the k-means method are its simplicity and speed, which allows you to perform clustering quickly even on large amounts of data. In addition, this method works well when clusters are spherical.
However, the k-means method has its drawbacks. It is sensitive to the choice of initial cluster
centers, which can lead to different results under different initial conditions. Paper [
The Expectation-Maximization (EM) algorithm is an iterative method used to estimate the
parameters of statistical models with hidden variables, such as a mixture of distributions. This
method combines an Expectation step and a Maximization step to iteratively update and find the
most likely model parameters. One of the advantages of the EM algorithm is the ability to work
with complex data distributions and find their parameters, but it can get stuck in local maxima and
require multiple runs from different initial conditions. Publications [
The use of polynomial splines in the context of recreating mixtures of distributions can be an
effective approach that allows you to adapt to the diversity of the data and approximate the
distribution density of each component of the mixture. The basic idea is to approximate complex
functions using piecewise polynomial functions that maintain smoothness and continuity over the
entire data interval. They can approximate even non-homogeneous distributions well, which can
be useful in cases where other methods may not give satisfactory results. In addition, polynomial
splines do not require any explicit assumptions about the shape of the data distribution, making
them a versatile tool for analyzing a variety of data sets [
Let an array Ω2, N ={X l ; l=1 , N }, X l∈ R2 consisting of N observations be given. Suppose that this data set consists of a mixture of K distributions. Let η1( X ) , η2( X ) , … , ηk ( X ) be the distributional densities of each mixed component that make up the mixture. The goal is to reproduce the estimated distributional density of the mixture f ( X ) without reducing the commonality f ( X )∈ Cr ,…,r, which is a linear combination of the densities of each component:
K f ( X )=∑ ρk ηk ( X ) , k=1
K where ρk is the weighting factor of the kth component of the mixture, ∑ ρk=1. k=1
corresponding weighting factors ρk.
splines close to the interpolation mean, a mathematical tool that allows you to approximate functions using smooth curves, adapting to the diversity of the data. The main idea is to build a flexible model that takes into account the heterogeneity of distributions and provides accurate parameter estimates. This approach allows us to take into account the complexity of the data structure.
{( x1,i , x2, j ) , ni , j , pi , j ; i , j∈ Z } based on a uniform partition of the set Δh1,h2 , where ( x1,i , x2, j ) is a variant that determines the center point of the ( i , j )th element of the partition Δh1,h2 ; h1 , h2 is the partition step; ni , j is the frequency (the number of elements that fall within the boundaries of the ( i , j )th element of the partition Δh1,h2); pi , j is the relative frequency of the variant.
S2,0 ( p , x1 , x2)=∑ ∑ B2,h1 ( x1−i h1) B2,h2 ( x2− j hσ ) pi , j ,
i∈ Z j∈ Z
where (with the notation accuracy up to the split step) [
B2,h ( x )={
0 , (3+2 x / h)2/ 8 , 3 / 4−(2 x / h)2/ 4 , (3−2 x / h)2/ 8 , x∉ [−3 h / 2 ; 3 h / 2] , x∈ [−3 h / 2 ;−h / 2] , x∈ [−h / 2 ; h / 2] , x∈ [ h / 2 ; 3 h / 2] .
Step 3. We search for local maxima of the spline (Fig. 1c) M q={( m iq , m jq ); q=1 , g }, where g is the number of local maxima. We consider in pairs { M v , M s }, v , s=1 , g , v ≠ s. For each pair of local maxima, we implement the following algorithm:
Draw a line f v ( x )=k α x +b passing through the points М v ( x1, v , x2, v ) and М s ( x1, s , x2, s ), where k α= x2, s− x2, v =tgα, b= x1, s x2, v− x1, v x2, s .
x1, s− x1, v x1, s− x1, v Draw lines d v ,ξ ( x )∥f v ( x ), d v ,ξ ( x )=k α x +b+ cξ, where cξ−const, ξ is the number of parallel lines drawn.
Draw lines av ,ψ ( x )=k γ x +b at an angle φ to f v ( x ), where γ =α ± φ, k γ= 1∓tgαtg±αt⋅gφtgφ =
kα ± tgφ 1∓ kα⋅ tgφ
, ψ is the number of lines drawn at an angle. 4. Form an array of lines U ={uτ , τ =1 , e }, where uτ={f v ( x ) , d v ,ξ ( x ) , av ,ψ ( x )}, e is the number of lines drawn, e=v⋅ ( 1+ξ +ψ ). Find the point on each line ( x1,τ , x2,τ ) for which S2,0( pτ , x1,τ , x2,τ )=min. Form an array of the found minimum values (Fig. 1d): U min= argmin U q= argmin uq ,τ ( x1 , x2 ) ,
S2,0( p , x1, x2) S2,0( p , x1, x2) where q=1 , g, τ =1 , e.
created. To describe the algorithm, let’s choose the simplest example and reproduce the linear regression (Fig. 1e) x2,q= βq ,0+ βq ,1 x1, where x2,q is the dependent variable, x1 is the independent variable and ⃗Β={βq ,0 ; βq ,1} is the vector of parameters that is unknown. We find the estimates of the parameters ^β using the least squares method
e B^=argmin ∑ ( x2,q ,τ− ^βq ,0− ^βq ,1 x1,τ )2 .
β τ=1 Similarly, we find the linear regression x1,q= λq ,0+ λq ,1 x2, where x1,q is the dependent variable, x2 is the independent variable and ( λq ,0 , λq ,1 ) is the vector of parameters that is unknown [12–14].
Next, we form the average vector of parameters ( ωq ,0 , ωq ,1 ), where ωq ,0= βq ,0+2 λq ,0 , ωq ,1= βq ,1+2 λq ,1 . We build a linear regression x2,q=ωq ,0+ωq ,1 x1.
The resulting line is reduced to the form zq=ωq ,0+ωq ,1 x1− x2, where q is the number of local maxima.
Step 5. We build a classifier based on the constructed lines (Fig. 1f). Let ~Y={~yk , k =1 , K } be the set of classes, Z ={ zk ( X ) , k =1 , K } be the set of discriminant functions that separate classes ~yk. Then we define ∀ l :~yl= I ( X ) , l=1 , N , where Remarks. If the lines do not intersect at one point but form a triangle, the classification of the points in the middle of the triangle can be done as follows:
with existing methods, we will model a normal distribution with the parameters ⃗Θk={μ1,k , μ2,k , σ 1,k , σ 2,k , rk , ρk , N k }, k =1 , K , where ρk is the weighting factor of the k-th
K component of the mixture, ∑ ρk=1 , N k is the number of elements of the k-th component of the k=1 mixture [16–18].
We obtain Ω2, N ={ X l , yl ; l=1 , N }, where X ={x1 ; x2}, X ∈ R2 , y ={1 , … , K } is the class index.
K f ( X , ⃗Θ )=∑ ρk ηk ( X , ⃗Θk ) , k=1
2 2 2( 1−rk2 ) [ exp( −1 ( x1− μ1,k )
2 σ 1,k + ( x2− μ2,k )
2 σ 2,k − 2 rk ( x1− μ1,k )( x2− μ2,k ) ])
σ 1,k σ 2,k 2 π σ 1,k σ 2,k √1−rk2 It is necessary to find Ω2, N ={ X l ,~yl ; l=1 , N }, so that yl=~yl.
comparing the obtained arrangement of elements in clusters with the initial arrangement of elements (reference), it is possible to estimate the clustering error ε = N ε , where N is the total N
N number of elements, N ε=∑ Q ( X l ) is the number of elements that fell into a class different from l=1 the one they were in according to the modeling results, where 1 , yl ≠~yl Q ( X l )={0 , y =~y .
l l
method
method (SVM)
– – – + + – – + + – + .
+ + – – According to the results of the study, one of the main advantages of the proposed method is the ability to process an arbitrary amount of data due to the preliminary histogram approximation. In contrast to the support vector method, this method demonstrates robustness to the choice of initial parameters. Compared to the k-means method and the EM algorithm, the proposed method is robust to outliers, as it can identify anomalies as unlikely events and disregard them. In addition, the proposed method does not depend on a predefined number of clusters, as it is determined based on the number of local maxima of the density function. In addition, unlike the support vector method and the k-means method, this method does not require data normalization, since the polynomial spline approximation does not require strict assumptions about the shape of the data distribution [19].
Declaration on Generative AI While preparing this work, the authors used the AI programs Grammarly Pro to correct text grammar and Strike Plagiarism to search for possible plagiarism. After using this tool, the authors reviewed and edited the content as needed and took full responsibility for the publication’s content. [11] A. Ligun, A. Shumeiko, Asymptotic methods of curve reconstruction, NASU Institute of
[12] I. Ostroumov, et al., Modelling and simulation of DME navigation global service volume, Adv.
Space Res. 68(8) (2021) 3495–3507. [13] V. Tkachuk, et al., Using mobile ICT for online learning during COVID-19 lockdown,
(2021) 46–67. doi:10.1007/978-3-030-77592-6_3 [14] S. Gnatiuk, et al., Sparse generative embeddings of handwritten digits, in: International
[15] I. Ostroumov, et al., A probability estimation of aircraft departures and arrivals delays, in:
vol. 12950, 2021, 363–377. [16] Z. Hu, et al., Advanced method for compressing digital images as a part of video stream to preprocessing of UAV data before encryption, in: Advances in Computer Science for Engineering and Education VI. ICCSEEA 2023. Lecture Notes on Data Engineering and Communications Technologies, vol. 181, 2023, 371–381. doi:10.1007/978-3-031-36118-0_33 [17] I. Ostroumov, K. Marais, N. Kuzmenko, Aircraft positioning using multiple distance measurements and spline prediction, Aviation, 26(1) (2022) 1–10. [18] N. S. Kuzmenko, I. V. Ostroumov, K. Marais, An accuracy and availability estimation of aircraft positioning by navigational aids, in: 2018 IEEE 5th International Conference on
doi:10.1109/MSNMC.2018.8576276 [19] S. Gnatyuk, P. Prystavka, S. Dolgikh, Fairness audit and compositional analysis in trusted AI program, Advances in Computer Science for Engineering and Education VI, ICCSEEA 2023,
699. doi:10.1007/978-3-031-36118-0_62