There is a mapping function $\Phi :{ℝ}^d\to H$, where $H$ represents the Hilbert space. The kernel function $K\left(x,X_i\right)=\langle \Phi \left(x\right),\Phi \left(X_i\right)\rangle$ can be defined as the inner product of the Hilbert space resulting from the mapping. By examining the KDE from this viewpoint, the function can be formulated as the following equation:

$\tilde{\varphi }\left(x\right)=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\langle \Phi \left(x\right),\Phi \left(X_i\right)\rangle =\langle \Phi \left(x\right),\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\Phi \left(X_i\right)\rangle$(3)

According to the Equation (3), the KDE can be understood as the inner product between the mean of $\Phi \left(X_i\right)$ and the mapping of $\Phi \left(x\right)$. Due to the property of the RKHS, the estimated KDE value is equal to the mean of the sample mapping in the RKHS. The average value is highly affected by the extreme values among the samples’ mapping and needs to be calculated by using a robust estimator [18] . Replace the sample mean with a robust M-estimation.

$\hat{\eta }=\underset{\eta \in H}{\text{argmin}}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\psi \left(\Vert \Phi \left(X_i\right)-\eta \Vert \right)$(4)

By using the $\hat{\eta }$ and weighted form, the RKDE formula is expressed as follows.

$\begin{array}{c}\hat{\varphi }\left(x\right)=\langle \Phi \left(x\right),\hat{\eta }\rangle =\langle \Phi \left(x\right),\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{a}_i\Phi \left(x_i\right)\rangle \\ =\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{a}_i\langle \Phi \left(x\right),\Phi \left(X_i\right)\rangle =\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{a}_{i}K\left(x,X_i\right)\end{array}$(5)

The resulting kernel density estimator is provided in a weighted form.

$\hat{\varphi }\left(x\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{a}_i{K}_h\left(X,X_i\right)$(6)

In contrast to the conventional KDE, this expression incorporates weighted samples, eliminating the use of equal weighting. The weights adhere to the conditions $a_i\ge 0$, ${\displaystyle {\sum }_{i=1}^n{a}_i}=1$, with a tendency to provide lower weights to outlier data points. And $\psi$ in the Equation (4) is the robust loss function, and the value of $\Vert \Phi \left(X_i\right)-{\eta }^{\left(k\right)}\Vert$ can be calculated by this way.

$\begin{array}{c}{\Vert \Phi \left(X_i\right)-{\eta }^{\left(k\right)}\Vert }^2=\langle \Phi \left(X_i\right)-{\eta }^{\left(k\right)},\Phi \left(X_i\right)-{\eta }^{\left(k\right)}\rangle \\ =\langle \Phi \left(X_i\right),\Phi \left(X_i\right)\rangle -2\langle \Phi \left(X_i\right),{\eta }^{\left(k\right)}\rangle +\langle {\eta }^{\left(k\right)},{\eta }^{\left(k\right)}\rangle \end{array}$(7)

There is also ${\eta }^{\left(k\right)}={\displaystyle {\sum }_{j=1}^n{a}_j^{\left(k-1\right)}\Phi \left(X_j\right)}$, at the same time.

$\langle \Phi \left(X_i\right),\Phi \left(X_i\right)\rangle =K\left(X_i,X_i\right)$

$\langle \Phi \left(X_i\right),{\eta }^{\left(k\right)}\rangle ={\displaystyle \underset{j=1}{\overset{n}{\sum }}{a}_j^{\left(k-1\right)}K\left(X_i,X_j\right)}$(8)

$\langle {\eta }^{\left(k\right)},{\eta }^{\left(k\right)}\rangle ={\displaystyle \underset{j=1}{\overset{n}{\sum }}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_j^{\left(k-1\right)}{a}_i^{\left(k-1\right)}K\left(X_j,X_i\right)}}$

M-estimator is a robust estimator that extends the maximum likelihood estimator to handle contamination in the distribution [19] . The M-estimator effectively resolved the Equation (1), which resembles the maximum likelihood estimator but employs a distinct loss function. The maximum likelihood estimator for multivariate functions can be used to substitute the likelihood function with $\rho$ in the following manner.

$\hat{\theta }=\underset{\theta }{\arg \max }\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\ln f\left(x_i;\theta \right)=\underset{\theta }{\arg \min }\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\rho \left(\Vert x_i-\theta \Vert \right)$(9)

The function $\rho$ is strictly convex or a continuous symmetric function in the positive semi-axis. If $\rho$ is derivable, the M-estimator can also be expressed as the following equation.

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\psi \left(x_i-\hat{\theta }\right)=0$(10)

This evaluation function is derivative-related to the function, and the solution of Equation (10) is equivalent to Equation (9) if $\rho$ is a convex function and its derivative exists everywhere. By selecting appropriate evaluation functions $\psi$, M estimation can achieve efficient estimation while ensuring essential robustness performance. Standard robust estimation functions include the Huber and the Hampel loss functions. The Huber function is defined in the following equation:

$\rho \left(u\right)=\{\begin{array}{ll}\frac{u^2}{2},\hfill & \text{if}\left|u\right|\le \gamma \hfill \\ \gamma \left|u\right|-\frac{{\gamma }^2}{2},\hfill & \text{if}\gamma <\left|u\right|\hfill \end{array}$(11)

Then, we can make the $\psi$ function.

$\psi \left(u\right)=\{\begin{array}{ll}-\gamma ,\hfill & \text{if}u<-\gamma \hfill \\ u,\hfill & \text{if}-\gamma \le u\le \gamma \hfill \\ \gamma ,\hfill & \text{if}u>\gamma \hfill \end{array}$(12)

To control the parameters $\gamma$, for outliers, at that time, it is a constant, but when $\Vert x-\eta \Vert <\gamma$, and the $\psi \left(u\right)=u$, it increases monotonically. Using the Huber activation function, the objective function in Equation (6) can be calculated using an iterative algorithm to optimize weights.

$\psi \left(u\right)=\{\begin{array}{ll}u,\hfill & 0\le u<{\gamma }_1\hfill \\ {\gamma }_1,\hfill & {\gamma }_1\le u<{\gamma }_2\hfill \\ \frac{{\gamma }_3-u}{{\gamma }_3-{\gamma }_2},\hfill & {\gamma }_2\le u<{\gamma }_3\hfill \\ 0,\hfill & {\gamma }_3\le u\hfill \end{array}$(13)

Algorithm 4.2 outlines the computational procedure for the kernelized iterative reweighting algorithm. During the update, the weights are calculated based on Equation (8), Equations. (5), and (9). After each iteration, the weights of the samples are normalized and reintroduced into the formula for further calculations. The computation terminates and outputs the optimal weights if the value is below a specified threshold.

$\mathbb{E}{\hat{\varphi }}_{-i}\left(x_i\right)={\displaystyle \int \hat{\varphi }\left(x\right)f\left(x\right)\text{d}x}$(17)

the equation Equation (18) implies an approximation.

$\text{UCV}\left(h\right)=R\left(\hat{\varphi }\right)-\frac{2}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\hat{\varphi }}_{-i}\left(x_i\right)$(18)

Hence, the expected value of UCV is equal to the expected value of ISE subtracted by the constant. Therefore, an alternative term for this approach is unbiased cross-validation. The generalization of least-squares cross-validation to arbitrary dimensions can be achieved by using the multivariate product kernel. It is constructed as follows: let X be an $n\times d$ data matrix of random vectors $x_1,x_2,\cdots ,x_n$, where $X_i$ are independent observations sampled from a multivariate density $f\left(x\right)$ of dimension d. Let $x_{ij}$ denote the jth entry of $x_i$. Given is the multivariate product kernel estimator of $f\left(x\right)$

$\hat{\varphi }\left(x\right)=\frac{1}{2n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j=1}{\overset{d}{\prod }}\frac{1}{h_j}K\left(\frac{x_{ij}}{h_j}\right)}}$(19)

where $h_j$ is the smoothing parameter for the jth dimension. The smoothness of $\hat{\varphi }\left(x\right)$ when using the standard Normal kernel for K is demonstrated

$R\left(\hat{\varphi }\right)=\frac{1}{{\left(2\sqrt{\pi }\right)}^{d}n{h}_1\cdots h_d}+\frac{1}{{\left(2\sqrt{\pi }\right)}^d{n}^2{h}_1\cdots h_d}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j\ne i}{\overset{n}{\sum }}\text{exp}\left(-\frac{1}{4}{\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right)}}$(20)

where ${\delta }_{ijk}=\left(x_{ik}-x_{jk}\right)/h_k$. The second part of the UCV estimator in Eq. (18) is given explicitly by the coming equation.

(21)

For the sake of simplicity, the factor $n-1$ has been substituted with a large n. The multivariate least-squares cross-validation function, $\text{UCV}\left(h_1,\cdots ,h_d\right)$, is obtained.

$\begin{array}{c}\text{UCV}\left(h_1,\cdots ,h_d\right)=\frac{1}{{\left(2\sqrt{\pi }\right)}^{d}n{h}_1\cdots h_d}+\frac{1}{{\left(2\sqrt{\pi }\right)}^d{n}^2{h}_1\cdots h_d}\\ \times {\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j\ne i}{\overset{n}{\sum }}\left[\text{exp}\left(-\frac{1}{4}{\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right)-\left(2\times 2^{d/2}\right)\text{exp}\left(-\frac{1}{2}{\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right)\right]}}\end{array}$(22)

The BCV method integrates bias correction into the UCV method [22] for addressing potential biases that may occur from the iterative exclusion of individual data points. The following equation expresses the mean integrated squared error (MISE).

$\text{MISE}\left(h\right)={\displaystyle {\int }_{{ℝ}^d}E{\left(\hat{\varphi }\left(x\right)-f\left(x\right)\right)}^2\text{d}x}$(23)

The general form of BCV using the asymptotic form is the following equation [23] .

$\begin{array}{c}\text{BCV}\left(h_1,\cdots ,h_d\right)=\frac{1}{{\left(2\sqrt{\pi }\right)}^{d}n{h}_1\cdots h_d}+\frac{1}{4n\left(n-1\right)h_1\cdots h_d}\\ \times {\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j\ne i}{\overset{n}{\sum }}\left[{\left({\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right)}^2-\left(2d+4\right)\left({\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right)+\left(d^2+2d\right)\right]{\displaystyle \underset{k=1}{\overset{d}{\prod }}\varphi \left({\delta }_{ijk}\right)}}}\end{array}$(24)

Extensive research has been conducted on applying bootstrap in univariate kernel density estimation. A simplistic method for utilizing the bootstrap involves resampling $x_1^{\text{*}},\cdots ,x_n^{\text{*}}$ from the original dataset and subsequently creating a bootstrap density estimate in the following manner.

${\hat{\varphi }}^{\text{*}}\left(x\right)=\frac{1}{nh}{\displaystyle \underset{i=1}{\overset{n}{\sum }}K\left(\frac{x-x_1^{\text{*}}}{h}\right)}=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{K}_h\left(x-x_1^{\text{*}}\right)}$(25)

From Equation (23), the global MISE criterion can be computed from the point-wise MSE criterion.

$\text{MSE}\left(X\right)=E\left[{\left(\hat{\varphi }\left(x\right)-f\left(x\right)\right)}^2\right]$(26)

This formula’s naive bootstrap estimate of $\text{MSE}\left(x\right)$expresses.

${\text{MSE}}^{\text{*}}\left(X\right)=E^{\text{*}}\left[{\left({\hat{\varphi }}^{\text{*}}\left(x\right)-\hat{\varphi }\left(x\right)\right)}^2\right]$(27)

However, with the naive bootstrap, we can obtain the following,

$E^{\text{*}}{\hat{\varphi }}^{\text{*}}\left(x\right)=E{K}_h\left(x-x_1^{\text{*}}\right)=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{K}_h\left(x-x_i\right)}=\hat{\varphi }\left(x\right)$(28)

here, $x_1^{\text{*}}$ has a discrete uniform distribution over the original data. Thus, the simple bootstrap estimate of the MSE in Equation (27) fails because the bias, $E^{\text{*}}{\hat{\varphi }}^{\text{*}}\left(x\right)-\hat{\varphi }\left(x\right)$, vanishes and only the variance is estimated.

Reducing bias has been approached in many ways, limiting the resampling size to be less than the original sample size and building the estimator [24] [25] , and it is recommended using smoothed bootstrap to calculate MSE with Equation (27) [26] . To clarify, $x_1^{\text{*}}$ is resampled from the kernel estimate $\hat{\varphi }\left(x\right)=n^{-1}{\displaystyle {\sum }_{i=1}^n{K}_h\left(x-x_i\right)}$ instead of the original data. The smoothed bootstrap technique using a Normal kernel does not require resampling because the calculations can be written in a closed form. The new MISE of the estimator is the following form,

$\text{M}\hat{\text{I}}SE\left(h\right)={\displaystyle {\int }_{R^d}{E}^{\text{*}}{\left\{{\hat{\varphi }}^{\text{*}}\left(x\right)-\hat{\varphi }\left(x\right)\right\}}^2\text{d}x}$(29)

where $\hat{\varphi }\left(x\right)$ is the multivariate product kernel estimator given in Equation (19). ${\hat{\varphi }}^{\text{*}}\left(x\right)$ is a multivariate produce kernel estimator calculated with data sampled from $\hat{\varphi }\left(x\right)$, and the expectation, $E^{\text{*}}$, is taken for the density $\hat{\varphi }\left(x\right)$. The aim is to choose h to minimize the value of Equation (29).

By employing the standard kernel for K in Equation (19) and evaluating Equation (29), it derives the multivariate bootstrap criterion function minimize across $h_1,\cdots ,h_d$. We eliminated the $i=j$ terms in the sums to reduce the inflated variance of this resampling strategy [27] [28] .

$\begin{array}{c}B\left(h_1,\cdots ,h_d\right)=\frac{1}{{\left(2\sqrt{\pi }\right)}^{d}n{h}_1\cdots h_d}+\frac{1}{{\left(2\sqrt{\pi }\right)}^d{n}^2{h}_1\cdots h_d}\\ \times {\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j\ne i}{\overset{n}{\sum }}[\frac{n-1}{2^{d/2}n}\text{exp}\left\{-\frac{1}{8}{\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right\}+\text{exp}\left\{-\frac{1}{4}{\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right\}}}\\ -\frac{2\times 2^{d/2}}{3^{d/2}}\text{exp}\left\{-\frac{1}{6}{\displaystyle \underset{k=1}{\overset{d}{\sum }}{\delta }_{ijk}^2}\right\}]\end{array}$(30)

CCV leverages components’ advantages while minimizing their downsides. It analyses complex, noisy intelligence data well. This combination completes the RKDE bandwidth selection analysis. From Equation (22), Equations. (24) and (30), CCV can be expressed as the following equation.

$\text{CCV}=\text{UCV}\left(h_1,\cdots ,h_d\right)+\text{BCV}\left(h_1,\cdots ,h_d\right)+B\left(h_1,\cdots ,h_d\right)$(31)

The HHO algorithm is then applied to optimize the CCV objective function. HHO mimics the hunting strategies of Harris hawks and is particularly effective in exploring the parameter space for optimal solutions [29] . The implementation involves initializing a population of hawks, each representing a potential solution in the bandwidth search space. The fitness of each hawk is evaluated based on the CCV error, and their positions are iteratively updated to converge toward the optimal bandwidth. This optimization process is crucial as it ensures that the bandwidth selection is not only practical but also efficient, leading to improved classification performance. This hunting method comprises two phases, which are exploration and exploitation. Each phase mimics hawk predation behaviors [30] . Time decreases the prey’s escape energy, represented by $EG$ in the following formula.

$EG=2E{G}_0\left(1-\frac{t}{T}\right)$(32)

Here, $E{G}_0$ is equal to $2r_1-1$ that represents the initial energy state, $r_1$ is a random value $\left(0,1\right)$, and $E{G}_0$ fluctuates between $\left[-1,1\right]$ during each iteration. The maximum and current iteration numbers are T and t, respectively. Harris Hawks capture tactics depend on $EG$. They are in the exploration phase when $\left|EG\right|$ is greater than 1. Otherwise, they are in the exploitation phase.

In the exploration phase, hawks follow target animals and other flock members’ erratic prey-seeking movements to establish their location [31] . The following equation describes the mathematical model.

$p\left(t+1\right)=\{\begin{array}{ll}{p}_{rand}\left(t\right)-r_1\left|p_{rand}\left(t\right)-2r_{2}p\left(t\right)\right|,\hfill & q\ge 0.5\hfill \\ \left(p_{prey}\left(t\right)-p_m\left(t\right)\right)-r_3\left[LB+r_4\left(UB-LB\right)\right],\hfill & q<0.5\hfill \end{array}$(33)

In each iteration, $r_1,r_2,r_3,r_4$, and q, which are random values in $\left[0,1\right]$. Searching upper and lower boundaries are $UB$ and $LB$. In the next iteration, $p\left(t+1\right)$ represents the hawks’ position, $p_{prey}\left(t\right)$ represents the optimal position of prey, $p\left(t\right)$ is the current hawk position, $p_{rand}$ means a randomly selected hawk, and $p_m\left(t\right)$ is the average position of the current population. The following formula calculates the average position of hawks,

$p_m\left(t\right)=\frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{p}_i\left(t\right)$(34)

where $p_i\left(t\right)$ represents each hawk’s location in the iteration t.

In the exploitation phase, hawks chase and catch the victim during this period. It can be modeled by four predatory behaviors: soft encirclement, aggressive encirclement, gradual swift fall, and gradual rapid descent. HHO chooses a different strategy based on the $EG$ and the probability r.

When $\left|EG\right|\ge 0.5$, $r\ge 0.5$, the HHO carries out the soft besiege. In this stage, the prey still has the energy to escape, and the hawks use a weak encircle strategy to deplete it and launch a surprise attack. The following equation models the behavior

$p\left(t+1\right)=\text{Δ}p\left(t\right)-EG\left|J\times p_{prey}\left(t\right)-p\left(t\right)\right|$(35)

and

$\begin{array}{l}\text{Δ}p\left(t\right)=p_{prey}\left(t\right)-p\left(t\right)\\ J=2\left(1-r_5\right)\end{array}$(36)

where $\text{Δ}p\left(t\right)$ is the difference between the prey position and the Harris Hawks’ current position in iteration t, $r_5$ is a random value between (0, 1), and J is the rabbit’s random jump strength when escaping. To replicate rabbit motions, J changes randomly in each iteration.

When $\left|EG\right|<0.5$, $r\ge 0.5$, the HHO executes the hard besiege. In this stage, the prey has no energy to escape, and the Harris hawks use a hard encircling to hunt the prey for a final assault. The current position is updated according to the following equation.

$p\left(t+1\right)=p_{prey}\left(t\right)-EG\left|\Delta p\left(t\right)\right|$(37)

When $\left|EG\right|\ge 0.5$, $r<0.5$, the HHO fulfills the soft besiege with progressive rapid dive. In this stage, the prey has enough energy to escape from the eagles, and the Harris hawks will gradually dive and softly surround the target. The levy function is used in the position update process to simulate the prey’s escape mode and jumping action. This process is modeled as follows equation.

$\begin{array}{l}Y=p_{prey}\left(t\right)-EG\left|J\times p_{prey}\left(t\right)-p\left(t\right)\right|\\ Z=Y+S\times LF\left(D\right)\end{array}$(38)

D is the problem dimension and S is the D-dimensional random row vector. LF is a levy flight function expressed as Eq. (39).

$LF\left(\beta \right)=0.01\times \frac{u\ast \sigma }{{\left|v\right|}^{\frac{1}{\beta }}},\sigma ={\left(\frac{r\left(1+\beta \ast \text{sin}\frac{\pi \beta }{2}\right)}{r\left(\frac{1+\beta }{2}\right)\ast \beta \ast 2^{\frac{\beta -1}{2}}}\right)}^{\frac{1}{\beta }}$(39)

r is the escape probability, v and u are random values in the range [0, 1], and the specificity is 1.5. The final hawk’s position is determined as in Equation (40).

$p\left(t+1\right)=\{\begin{array}{ll}Y,\hfill & \text{if}F\left(p\left(t\right)\right)>F\left(Y\right)\hfill \\ Z,\hfill & \text{if}F\left(p\left(t\right)\right)>F\left(Z\right)\hfill \end{array}$(40)

where F is the fitness function.

When $\left|EG\right|<0.5$, $r<0.5$, the HHO carries out the hard besiege with a progressive rapid dive. In this stage, the prey is exhausted and has low escape energy; the Harris Hawks surround the prey through the hard besiege with a progressive rapid dive. They attempt to reduce the distance between their average position and the target prey. The position update formula is expressed in this equation.

$p\left(t+1\right)=\{\begin{array}{ll}Y,\hfill & \text{if}F\left(p\left(t\right)\right)>F\left(Y\right)\hfill \\ Z,\hfill & \text{if}F\left(p\left(t\right)\right)>F\left(Z\right)\hfill \end{array}$(41)

where Y and Z are updated by Equation (42).

$\begin{array}{l}Y=p_{prey}\left(t\right)-EG\left|J{p}_{prey}\left(t\right)-p_m\left(t\right)\right|\\ Z=Y+S\times LF\left(D\right)\end{array}$(42)

The algorithm 5.2 relates the basic flow of Harris Hawks Optimization.

The HHO-based optimization bandwidth was used as the bandwidth value for all estimators in the experimental comparison of function fitting effects. The MISE metric compares the differences between the estimated results and the proper distribution function. The experiences are done for both one-dimensional and two-dimensional data. For the one-dimensional comparison experiments, 300 sample points are generated using a standard normal distribution with 0 as the mean value, 0.5 as the variance value, and 40 outlier sample points using a uniform distribution in the interval from 6 to 17. In this case, $\gamma$ in the Huber function is taken from the 75% quantile of the mapping after each calculation, and the parameters ${\gamma }_1$, ${\gamma }_2$ and ${\gamma }_3$ of the Hampel function are taken as the 25%, 50% and already 75% quantiles, respectively. Figure 3 shows that the traditional KDE has the highest density values in the outlier section, even when the bandwidth value is optimized. This result means that the main density components are shown less accurately. Compared to the Hampel function, the red curve of the RKDE with the Huber function is the closest to the original reference function curve among all the fitted curves. Although there is a non-zero density value of the outliers corresponding to the red curve for the fitting part of its outliers, the fitting of the principal components is optimal compared to the other methods. The RKDE using the Hampel loss function completely suppresses the density values of outlier points for one-dimensional data. Still, there is a significant deviation in the fitting of the principal function.

From the one-dimensional experimental results, the RKDE using the Huber function is the most effective in fitting the one-dimensional data. We made 1000 sample points with the distribution F₁, which has a Gaussian function mean of [0,0] and variance of [1, 0; 0, 1]. We also made 1000 samples with the distribution F₂, which has a Gaussian function mean of [5.5, 5.5], and variance of [1, 0; 0, 1], and 1000 samples with the distribution with a Gaussian function mean of [−2, 6.5] and variance of [2, 0; 0, 2]. The Gaussian distribution F₃ generates 100 outlier sample points. ORKDE’s bandwidth values of both KDE and RKDE are from HHO-optimized bandwidth [0.34, 0.32]. Compared with Figure 3(b) of the reference function, the estimation of traditional KDE in Figure 3(c) produces estimations value in the region of the outlier points, which leads to some error in the central part, and the estimation results using ORKDE are more accurate in Figure 3(d) . Table 2 gives the parameters, outlier points, and number of samples for generating 1-D and 2-D data distributions. The experimental results in Table 3 show that the MISE value of ORKDE is the smallest among all types of fitting methods.

This part of the experiment mainly compares the classifications using ORKDE-NB, RKDE-NB, KDE-NB, and common classifiers. The data in this part is simulated data, mainly used to obey several different distributions of data in different data sets, a simulation of binary classification, and a triple classification problem, which takes into account the different dimensions of the data, the dimensions in this experiment range from two-dimensional, three-dimensional, to four-dimensional. The parameter control of outliers in this classification process, $L_o=\frac{L_1+L_2+\cdots +L_{cn}}{cn}$, $S_o=\frac{S_1+S_2+\cdots +S_{cn}}{cn}$ specific parameters are shown in Table 5 . Meanwhile, the ten-fold method is used for different datasets to verify the classifier’s effectiveness. The comparison of the effectiveness of each classifier for two-dimensional T-distribution is given separately in Table 4 .

Tables 6-9 give the results of testing the mean accuracy of the algorithms based on the simulated generation of relevant distribution experimental data in Table 5 . In addition to comparing the relevant Bayesian classifiers, basic classifiers such as SVM, Decision Tree (DT), Random Forest (RF), XGboost, etc, are used. Among them, SVM adopts RBF kernel, and its parameter adopts lattice search method, $c\in \left[-8,8\right]$, $g\in \left[-8,8\right]$, with a step size of 0.1. Decision Tree adopts the C4.5 model combined with dichotomization to deal with continuous attributes. The parameters of the random forest were set in the experiment as criterion = “gini”, maxdepth = None, minsamplessplit = 2, minsamplesleaf = 1, bootstrap = True. The parameters in the XGboost model were set as learningrate = 0.3, gamma = 2.5, maxdepth = 6, minchildweight = 3, maxdeltastep = 0.5; with the classification results in Tables 6-9 , among the classification accuracies of various dimensions in the binary classification, and tertiary classification, the classification accuracies of ORKDE- NB has the best classification accuracy in all cases. As the data dimensions increase, the accuracy of the same classifier decreases; this experimental result also fully demonstrates the effectiveness of the method proposed in this paper.

The ORKDE method demonstrates significant advantages over traditional kernel density estimation methods. Its robustness to outliers is achieved through the combination of M-estimation and a weighted kernel density estimator, resulting in more accurate density estimates. Additionally, the adaptive bandwidth selection facilitated by CCV and HHO enhances the estimator’s ability to adjust to the underlying data distribution, making it more versatile in real-world applications. In terms of applicability, the ORKDE method is designed to accommodate various data distributions, ensuring that the kernel density estimates accurately reflect the