Optimizing multi-classifier fusion for seabed sediment classification using machine learning

ABSTRACT

Seabed sediment mapping with acoustical data and ground-truth samples is a growing field in marine science. In recent years, multi-classifier ensemble models have gained prominence for classification problems by combining several base classifiers. However, traditional ensemble methods do not consider the confidence scores of base classifiers, leading to suboptimal fusion when there are conflicting predictions. The current study introduces a novel optimization strategy that enhances the ensemble’s accuracy by constructing an ideal ensemble predicted probability matrix based on the fusion of predicted probabilities of the base classifiers, to improve seabed sediment mapping. The proposed approach not only addresses the limitations of traditional ensemble methods but also significantly increases the ensemble’s performance. The proposed approach demonstrates significant accuracy improvements. On the under-sampled dataset, it achieves 73.5% improvement compared to individual classifiers (random forest, decision tree, support vector machine), surpassing their respective accuracies. On the standard dataset, the ensemble model attains an accuracy of 79.1%, surpassing individual classifiers. Employing over-sampling techniques further elevates accuracy to 94.9%, exceeding the individual classifier performances. The proposed method is evaluated on acoustical data obtained from the Irish Sea. The proposed method outperforms base classifiers in terms of accuracy, F1 score, and the Kappa coefficient.

KEYWORDS:

• Bayesian theory
• differential evolution optimization
• feature extraction
• machine learning
• seabed sediment classification

1. Introduction

The ocean floor, covering 71% of Earth's surface, harbours useful resources, delivers essential ecosystem services, and offers diverse habitats for numerous species (Hosack et al. 2006; McGonigle et al. 2011). Seabed maps are crucial for ocean monitoring and management, primarily providing information on seafloor topography and habitats. Accurate habitat mapping aids in determining and measuring the human impact on the seafloor by offering data on the physical features, spatial distribution, and ecological roles of biological communities and habitats (Kostylev et al. 2001). Seabed sediments principally consist of the seabed's surface components, which include rocks and surface sediments (Damveld et al. 2019; Gary Greene, Cacchione, and Hampton 2017; Zajac 2008). Comprehensive and high-quality investigations of seabed sediments can enhance marine scientific research, resource development, engineering projects, environmental conservation, and military security across various fields (Barbagelata et al. 1991; Diesing et al. 2020; A. Z. Leon et al. 2020). For successful benthic environment evaluation, obtaining accurate information on seabed sediment distribution through accurate seabed maps is crucial. However, precise and appropriate maps and spatial models that accurately represent the seabed substrate types, habitats, and properties, which are useful for research, resource management, conservation, and spatial planning, are limited in the literature (Ierodiaconou et al. 2011). As a result, seabed sediment classification has progressively garnered increased interest from experts and researchers (McGonigle and Collier 2014).

Machine learning (ML) has been used for seabed sediment classification using multi-beam echosounder data has yielded remarkable results. ML is often adopted to perform both supervised and unsupervised classification of the seabed (Frederick, Villar, and Michalopoulou 2020; Viala, Lamouret, and Abadie 2021; Wan et al. 2022; Zhang et al. 2022). Recent studies that have used ML to study the seabed include Liu et al. (2019), in which the sediment types of the Caminada dredge pit in the eastern portion of the submarine sandy Ship Shoal of the Louisiana inner shelf of the United States (USA) were identified using multiple machine learning classifiers and the performance of multiple supervised classification methods was analysed. Zhao et al. (2020) suggested and verified a hybrid machine learning framework for predicting water levels in five representative high-sediment load reaches of China's Lower Yellow River (LYR). Wang et al. (2021) used the XBoost algorithm to study offshore seabed sediment classification based on particle size parameters. In their work, they concluded that the developed model offered an efficient method for offshore sediment classification using Folk's scheme and grain-size parameters and could serve as an additional approach for studying sedimentary environments. To accomplish precise prediction and mapping of geographic seabed sediment information, Cui et al. (2021) created a deep learning model based on feature optimization. By integrating ten major stream sediment geochemical datasets, principal component analysis (PCA) and self-organizing maps (SOMs) were used to delineate major lithologies in the Jining area, Inner Mongolia, China (G. Wu et al. 2021). ML algorithms that have been utilized in the field of acoustic sediment classification include random forest (RF), support vector machine (SVM), k-nearest neighbour (KNN) and artificial neural networks (ANNs) (Bressan et al. 2020; He et al. 2022; Lüdtke et al. 2012; McLaren, McIntyre, and Prospere 2019; Mudiyanselage et al. 2022; Prasad et al. 2020). Despite their strong performance and successes in classification tasks, ML algorithms require the support of other approaches to effectively and fully execute their tasks (Morgan and Jacobs 2020). An example is the normalization of data before classification (Singh and Singh 2020) when using ML algorithms such as SVM and KNN, since SVM and KNN algorithms are sensitive to large variations in data points. Other approaches that help solve the ML algorithm problem also occur after classification has been performed (Diesing and Stephens 2015; Gudiyangada Nachappa et al. 2020). In particular, post classification approaches are designed to help improve overall classification performance and usually eliminate errors that occur within individual classifiers by aggregating their decisions.

Multi-classifier ensemble learning is one such approach, in which the decision scores of many classifiers are combined to predict the final class label of an input sample. A multi-classifier model is designed to capture the key properties of all of its constituent models, outperforming individual base classifiers. Such models are resilient because ensembling reduces the dispersion or spread of the base models’ predictions. The multi-classifier model reduces the variation in the prediction errors of the basis classifiers by adding some bias to the competing base learners (Divyabarathi et al. 2022; Galvez et al. 2022; Kim et al. 2022).

Recent works have used rule-based majority voting aggregators to fuse multiple ML algorithms to classify seabed sediments. Yang et al. (2022) performed marine oil spill detection using deep learning models. Wan et al. (2022) also classified seabed sediment using a decision fusion method. The rule-based voting method has shown great results in improving classification. However, in some cases there are conflicting predictions among several classifiers, making it difficult to provide reliable results (F. Leon, Floria, and Badica 2017). Such cases occur when two or more classifiers predict the wrong class, and the aggregator simply uses that class, regardless of whether it is the wrong prediction. Furthermore, the classifiers utilize the predictions directly without the confidence score of the base classifiers. Finally, the quality of the fusion results is affected by the number of target classes (Tulyakov et al. 2008).

To address the aforementioned challenges in previous studies, the current study proposes a multi-classifier approach with ML algorithms to effectively classify seabed sediments. The proposed approach utilizes the confidence score of the base classifiers to make predictions. According to (Carpenter et al. 1999; Foody, Boyd, and Sanchez-Hernandez 2007), confidence scores are important in multi-classifiers due to the following: (1) agreement between classifiers and (2) correlations between the predicted accuracies. To implement this approach, the fusion process is formulated as an optimization problem where the accuracy of the fused classifier is maximized. The objective function is based on Bayesian theory (Bernardo and Smith 2008), where weights are assigned to each base model to yield an optimal posterior matrix with higher accuracy than the base classifiers. The question lies in which optimal weight can yield the best ensemble probability matrix with the highest accuracy for the fused model. Based on the formulated objective function, the differential evolution (DE) algorithm is used to optimize the whole process (Storn 1996). The algorithm explores the search space efficiently through mutation, crossover and selection operators to guide the population towards the global optimum. Notably, DE boasts several advantages, including its suitability for optimizing non-linear, non-convex, and non-differentiable objective functions, making it well-suited for the accuracy-based optimization (J. Parihar and Malik 2022; Priyavada and Kumar 2023; Zhao et al. 2022). Furthermore, DE’s population-based approach proves beneficial when dealing with problems involving multiple variables, such as optimizing the weights for three classifiers. Additionally, DE has relatively few control parameters, is straightforward to implement, and has stronger global optimization ability than other evolutionary algorithms. These characteristics make DE a good choice for maximizing the objective function representing multi-classifier accuracy by finding the optimal weight for fusing the base classifier confidence scores. The proposed approach is trained on acoustical data acquired from the Irish Sea. Additionally, multiple tests are performed to show the robustness and efficacy of the proposed approach.

The contribution of this paper lies in the development of an efficient and optimal multi-classifier ensemble method that fuses the base classifiers based on their confidence scores, as opposed to conventional methods that directly fuse base classifiers based on the predicted score values.

2. Materials and methods

2.1. Data

The Irish Sea, located in the middle of northwest Europe’s continental shelf, borders the North Atlantic to the north and south (see Figure 1(a)). The depth of the water in this region varies, with coastal regions measuring less than 20 m deep and interior sections reaching more than 100 m deep. The western portion of the Irish Sea is deeper (>100 m), with weaker tidal energy and greater saline levels. The Irish Sea’s benthic habitat is influenced by a wide range of factors, resulting in a variety of complex, diverse, and dynamic physical conditions and biological communities, ranging from stony reefs to deep mud basins. The seafloor has a variety of landforms and underwater deposits. Acoustic sediment classification benefits greatly from studying materials such as silty clay, clayey silt, argillaceous sand, sandy mud, silt, gravel, mudstone, bedrock, and coarse sand, providing valuable insights into acoustic sediment categorization. The flow of water in the Irish Sea is mostly northwards, with a flow rate of approximately $8\mathrm{k}{\mathrm{m}}^3\mathrm{per}\mathrm{day}$. However, there is some unpredictability in the northwards exchange of seawater. The Irish Sea's water is thoroughly mixed, resulting in a vertically homogeneous water column throughout the year.

Figure 1. The study area and the acquired acoustical data. (a) Location of the study area (Irish Sea). (b) Bathymetric data obtained from the acoustic data. (c) Backscatter data, representing the seabed texture, and the locations of grab samples colour-coded by Folk class (see Table 1).

To better understand the geological and geographical characteristics of the Irish Sea, acoustical data were obtained between 9 February 2012 and 21 February 2012, and archived with the Marine Environmental Data and Information Network (MENIM), which is a part of the British Geological Survey (BGS) data archive centre. The acoustical data are presented in Figure 1(a), with Figure 1(b) showing the bathymetric data and Figure 1(c) representing the backscatter data.

To support the classification of acoustical sediment, samples were collected using a grab method from various locations, as displayed in Figure 1(c), and subsequently analysed based on Folk's classification scheme. Table 1 provides a summary of the number and types of ground-truth samples obtained. Notably, a total of 34 grab samples were secured, with the most commonly occurring classes being muddy sand (nine samples), slightly gravelly muddy sand (five samples), and gravelly mud (five samples). The spatial distribution of these samples, as depicted in Figure 1(c), ensures comprehensive coverage of the range of acoustic conditions and seabed sediment types prevalent in the study area. These ground-truth sediment samples are of utmost significance because they provide crucial training data for machine learning classifiers to establish associations between acoustics and sediment properties.

Table 1. Grabbed samples of the study area based on Folk’s classification.

Sediment classes (labels)	Size of samples
Muddy Sand, mS	9
Slightly gravelly sand, (g)S	3
Slightly gravelly mud, (g)M	4
Slightly gravelly muddy sand, (g)mS	5
Gravelly muddy sand, gmS	2
Gravelly mud, gM	5
Mud, M	3
Sand, S	1
Gravelly sand, gS	2

2.2. Proposed approach

This paper presents a novel approach for tackling the challenges of classifying seafloor sediments discussed in the introduction. The proposed methodology employs a multi-classifier strategy that utilizes a combination of machine learning algorithms to accurately classify various seabed sediment types. The approach effectively blends the confidence scores of the base classifiers to enhance the accuracy of the multi-classifier model. Figure 2 represents the framework for the proposed approach. The proposed approach is divided into three stages: feature extraction and object-based segmentation of acoustical data, training of base classifiers, and classifier fusion optimization. Initially, significant acoustical features are retrieved from the multibeam data, and training examples are generated using object-based image analysis. The aforementioned examples are then used to train the three base classifiers: RF, SVM, and DT. Finally, by maximizing an objective function reflecting the multi-classifier’s accuracy, a differential evolution optimization technique is used to discover the ideal fusion of the base classifiers.

Figure 2. Schematic of the overall multi-classifier framework for this study. The three main parts include data preparation (feature extraction and object-based segmentation), classification, and optimization.

2.2.1. Feature extraction

To accurately identify the type of sediment present, feature extraction is crucial in establishing a stable mapping relationship with the raw acoustic data. The multibeam backscatter intensity feature is the most commonly used for feature extraction in studies on habitat mapping, as it provides the foundation for correct sediment identification. Furthermore, bathymetry and bathymetry-derived variables offer an intuitive representation of seafloor topography. The bathymetric features extracted for training in this paper encompass properties such as aspect, general curvature, geomorphons, landforms, morphometric features, the multiscale topographic position index, the multiresolution index of valley bottom flatness, plan curvature, profile curvature, slope, the terrain ruggedness index, the topographic position index, variance, and the vector ruggedness measure. Given that the topography of the seafloor is highly correlated with the distribution of different seabed sediment types, it has been demonstrated that the combination of backscatter intensity features and topographic features is more effective in the classification of seabed sediment (Marsh and Brown 2009). The study combines both bathymetry and backscattering to extract features for sediment classification, with extracted backscatter features including the angular second moment, contrast, energy, entropy, GLCM contrast, GLCM entropy, and inverse difference moment for sediment classification. The resulting extracted acoustic features for this study, after eliminating features with excessive noise and low separability between different classes, are summarized in Table 2.

Table 2. The extracted features from the acoustical data consist of 21 features, including both textures and topographic features.

Feature	Definition	Mathematical formula	Citation reference
Angular second moment	A measure of textural homogeneity in an image, based on the gray level co-occurrence matrix (GLCM).	$\sum \sum P(i,j{)}^2$	(Haralick, Dinstein, and Shanmugam 1973)
ASPECT	The orientation of the steepest slope on a surface, usually represented in degrees from north.	$\arctan \left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}y},{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}xy}}}\right)$	(Goulden et al. 2016)
CONTRAST	A measure of the local variation in an image, which highlights the differences between neighbouring pixel values.	$\sum \sum (i-j{)}^2\ast P(i,j)$	(Haralick, Dinstein, and Shanmugam 1973)
ENERGY	A measure of the uniformity or orderliness in an image, based on the GLCM.	$\sum \sum P(i,j{)}^2$	(Haralick, Dinstein, and Shanmugam 1973)
ENTROPY	A measure of the randomness or disorder in an image, based on the GLCM.	$\sum \sum P(i,j)\ast \log (P(i,j))$	(Haralick, Dinstein, and Shanmugam 1973)
General Curvature	A measure of the overall curvature of a surface, which can help identify landforms such as ridges and valleys.	$2\ast \left({\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{x}^2}\ast {\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{y}^2}-\left({\displaystyle \frac{\mathrm{d}{z}^2}{\mathrm{d}x}\ast \mathrm{d}{y}^2}\right)}}\right)$	(Zevenbergen and Thorne 1987)
Geomorphons	A pattern-based method for identifying and classifying landforms, based on local terrain relief.	Analyzes spatial patterns and elevation data to categorize landforms.	(Jasiewicz and Stepinski 2013)
GLCM_contrast	A measure of the contrast in an image, based on the GLCM.	$\sum \sum (i-j{)}^2\ast P(i,j)$	(Haralick, Dinstein, and Shanmugam 1973)
GLCM_entropy	A measure of the entropy in an image, based on the GLCM.	$\sum \sum P(i,j)\ast \log (P(i,j))$	(Haralick, Dinstein, and Shanmugam 1973)
Inverse difference moment	A measure of the local homogeneity in an image, based on the GLCM.	$\sum \sum {\displaystyle \frac{P(i,j)}{(1+{(i-j)}^2)}}$	(Haralick, Dinstein, and Shanmugam 1973)
LANDFORMS	Physical features on the Earth's surface, such as mountains, valleys, and plains.	-Descriptive features of the Earth's surface.	-(A Weiss 2001)
Morphometric features	Quantitative properties of landforms, such as slope, aspect, and curvature, used to analyze and classify terrain.	- Derived from elevation data and spatial relationships.	- (Jasiewicz and Stepinski 2013)
Multiscale topographic position index (mTPI)	A measure of the relative position of a location within the surrounding terrain, calculated at multiple scales.	${\displaystyle \frac{\mathrm{elevation}-\mathrm{mean}\mathrm{\_}\mathrm{elevation}}{\mathrm{standard}\mathrm{\_}\mathrm{deviation}\mathrm{\_}\mathrm{elevation}}}$	(A Weiss 2001)
Multiresolution index of valley bottom flatness (MRVBF)	A measure of the flatness of valley bottoms, based on the local elevation and curvature.	${\displaystyle \frac{{\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{x}^2}+{\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{y}^2}}}}{1+{\left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}\right)}^2+{\left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}y}}\right)}^2}}$	(Gallant, research, and 2003)
Plan curvature	A measure of the curvature of a surface in the horizontal plane, which can help identify convergent and divergent flow patterns.	$2\ast \left({\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{x}^2}+{\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{y}^2}-{\left({\displaystyle \frac{\mathrm{d}{z}^2}{\mathrm{d}x}\ast \mathrm{d}y}\right)}^2}}\right)$	(Zevenbergen and Thorne 1987)
Profile curvature	A measure of the curvature of a surface in the vertical plane, which can help identify convex and concave slopes.	${\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{x}^2}+{\displaystyle \frac{{\mathrm{d}}^{2}z}{\mathrm{d}{y}^2}}}$	(Zevenbergen and Thorne 1987)
SLOPE	The steepness of a surface, usually represented as the angle between the horizontal plane and the tangent to the surface at a given point.	$\arctan (\sqrt{{\left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}\right)}^2+{\left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}y}}\right)}^2}$	(Goulden et al. 2016)
Terrain ruggedness index (TRI)	A measure of the variability in elevation within a specified area, which can help identify rough or smooth terrain.	$\sqrt{\left({\displaystyle \frac{\mathrm{\Delta }{x}^2+\mathrm{\Delta }{y}^2+\mathrm{\Delta }{z}^2}{n-1}}\right)}$	(Riley, DeGloria, and Elliot 1997)
Topographic position index (TPI)	A measure of the relative position of a location within the surrounding terrain, based on elevation differences.	${\displaystyle \frac{\mathrm{elevation}-\mathrm{mean}\mathrm{\_}\mathrm{elevation}}{\mathrm{standard}\mathrm{\_}\mathrm{deviation}\mathrm{\_}\mathrm{elevation}}}$	(Guisan, Weiss, and Weiss 1999)
VARIANCE	A measure of the dispersion of pixel values in an image, which can help identify areas of high or low variability.	$\sum \sum {\displaystyle \frac{{(l(i,j)-\mu )}^2}{N}}$	–
Vector ruggedness measure (VRM)	A measure of terrain ruggedness based on the variability in the orientation of slope and aspect vectors, which can help identify complex or simple terrain.	$\sqrt{\left(1-{\left({\displaystyle \frac{\sum X}{N}}\right)}^2-{\left({\displaystyle \frac{\sum Y}{N}}\right)}^2-{\left({\displaystyle \frac{\sum Z}{N}}\right)}^2\right)}$	(Sappington, Longshore, and Thompson 2007)

Note: In the mathematical expressions provided within Table 2, ‘i’ and ‘j’ denote the respective row and column positions in the co-occurrence matrix, ‘z’ signifies the elevation value, ‘P(i,j)’ represents the probability of adjacent pixels having gray levels ‘i’and ‘j’, ‘μ’ stands for the mean pixel value, and ‘N’ indicates the total pixel count.

2.2.2. Object-based image analysis.

The first stage in object-based image analysis (OBIA) is segmentation, a semi-automated procedure that separates image data into individual segments. Segmentation that successfully delimits the characteristics of interest is crucial to the success of object-based image analysis. OBIA operates on the basis of the idea that images should not be analysed based on individual pixels but rather on image segments, where an image segment is a collection of pixels. Furthermore, the image’s segments are hierarchically structured. The segments of an image are identical to the pixels at the pixel level. A finer level of visual objects is produced using segmentation, beginning at the pixel level. Image segments can be segmented further into intermediate and coarse levels to provide even more granular analysis. There is a plethora of data that can be derived from these image segments to form their characteristics. Afterwards, during the classification process, derived data are used. Statistics regarding the geometry, texture, or average colour of any of an object's layers constitute image segment characteristics.

The present study employs a hierarchical process for segmenting the study area into homogenous sections. The OBIA process is utilized to assist in the selection of training samples for the ML classifiers based on the location of the grabbed sample data. An assumption is made that generated training points located within the identical segment as a grabbed sample data point represent sediments of the same type. The segmentation process utilizes the multi-resolution segmentation (MRS) algorithm to group pixels into segments based on a predefined homogeneity value, known as the scale parameter. In addition, the shape parameter controls the influence of segment shape on the segmentation, while the compactness factor regulates the smoothness of the image object outline produced by the segmentation. Accurate segmentation provides an ideal outline of the targets of interest. However, often, many features of heterogeneous classes occur within the same segment, or one feature is demarcated by multiple segments. The spectral difference algorithm is employed to merge neighbouring segments where the difference between mean layer intensities is below a user-defined value. The pixel-based object resizing algorithm can also contract or expand image object boundaries until a specific geometry is achieved or a particular image layer value is met.

The following parameters are utilized for multi-resolution segmentation: The segmentation contains all 21 characteristics from Table 2, with a scale of 250, shape set to 0.5, and compactness set to 0.5. The resulting segmentation is shown in Figure 3.

Figure 3. OBIA segmentation with sediment sample location. Based on this segmentation, training and validation samples are generated for the ML model for learning.

2.2.3. Base classifiers

The proposed seabed sediment classification using an adaptive differential evolution ensemble of machine learning classifiers is presented in this section. First, segments shown in Figure 3, generated using the OBIA approach described above, are divided into training and testing samples, which are fed to the machine learning base classifiers. Three different machine learning algorithms with diverse backgrounds are used so that proper evaluations can be made. Confidence scores are extracted from these classifiers, and their fusion is performed by optimization to yield the best overall accuracy with respect to the individual accuracies of the base classifiers. The base classifiers used are as follows:

1. Random Forest

Random Forest (Breiman 2001) is an ensemble learning technique that builds many decision trees during training and produces the mean prediction (regression) or mode of the classes (classification) of the individual trees. The underlying principle of RF is to average the outputs from many decision trees, each trained with a random subset of the data, to enhance the performance and decrease overfitting of a single decision tree. RF can be expressed mathematically as $D=\{(x_1,y_1),(x_2,y_2),\dots ,(x_N,y_N),i=1,2,\dots ,N\}$, where $N$ indicates the number of instances, D represents the dataset and $(x_i,y_i)$ represent inputs and labels, respectively. A bootstrap sample of the dataset is used to train each of the $T$ decision trees that make up the random forest classifier. The predictions from all decision trees that received the most votes are the output of the random forest classifier: (1) $C_{\mathrm{RF}}(\mathbf{x})=\arg \underset{k}{\max }\sum _{t=1}^T⊮(C_t(\mathbf{x})=k)$(1) where $k$ represents the class label and $C_t(\mathbf{x})$ reflects the prediction made by the t-th decision tree for the input $\mathbf{x}$.

(2)	Decision Tree

Each internal node in a decision tree (Webb et al. 2011) corresponds to a feature, each branch to a decision rule, and each leaf node to an outcome or class label. Decision trees resemble flowcharts. Decision trees use recursive binary splitting to create the tree structure, aiming to create the purest partitions at each level. Gini impurity and entropy are the two most typical splitting criteria. The decision tree classifier $C_{\mathrm{DT}}$ converts an input feature vector $x$ to a class label by following the decision rules from the root node to a leaf node for a given dataset $D=\{(x_1,y_1),(x_2,y_2),\dots ,(x_N,y_N)\},i=1,2,\dots ,N$, where N indicates the number of instances, D represents the dataset and $(x_i,y_i)$ represent inputs and labels, respectively. The majority class of the instances in that partition is represented by the label given to the leaf node.

(3)	Support Vector Machine

The goal of SVM (Cristianini and Ricci 2008; Vapnik 1995), a supervised learning algorithm, is to identify the best hyperplane that maximizes the distance between two classes in the feature space. The best hyperplane, or support vector, is the one with the greatest distance to the closest data points from both classes in a linearly separable dataset. Let $D=\{(x_1,y_1),(x_2,y_2),\dots ,(x_N,y_N)\},i=1,2,\dots ,N$, where N indicates the number of instances, D represents the dataset and $(x_i,y_i)$ represent inputs and labels, respectively, be a dataset with binary class labels $y_i\in -1,1$. The SVM classifier can be formulated as the following optimization problem: (2) $\underset{\mathbf{w},b}{\min }\left({\displaystyle \frac{1}{2}|\mathbf{w}{|}^2}\right)\mathrm{subject}\mathrm{to}{y}_i({\mathbf{w}}^{\mathrm{\top }}{\mathbf{x}}_i+b)\ge 1,\mathrm{for}i=1,\dots ,N$(2) where $b$ stands for the bias term and $\mathbf{w}$ stands for the weight vector normal to the hyperplane. Using kernel functions, the SVM may be expanded to translate input data to a higher-dimensional space where a linear separator can be discovered in the case of non-linearly separable data. The SVM classifier's output is represented as follows: (3) $C_{\mathrm{SVM}}(\mathbf{x})=\mathrm{sign}({\mathbf{w}}^{\mathrm{\top }}\varphi (\mathbf{x})+b)$(3) where $(\mathbf{x})$ refers to the input data point, $\mathbf{w}$ represents the weight vector, $b$ is the bias term and $\varphi (\mathbf{x})$ represents the feature mapping or transformation applied to the input data point.

2.2.4. Differential evolution optimization

Differential evolution (DE) (Storn 1996) is an optimization algorithm that uses a population to look for and find optimal solutions by utilizing mutation, crossover, and selection operators. DE excels at maximizing functions that are not linear, convex, or differentiable. The proposed method utilizes DE to fine-tune the multi-classifier's weights (p) for maximum accuracy on a validation set. The p for each classifier is used as an optimization variable and weight when fusing the classifiers. The main steps of DE can be summarized as follows:

First, create an initial population $P^{(0)}=x_1^{(0)},x_2^{(0)},\dots ,x_N^{(0)}$ of $N$ size, where each element $x_i^{(0)}$ is a possible solution within the search space. Next, create a new population with a mutated version of each target vector $x_i^{(t)}$ by applying the mutation operator to obtain $v_i^{(t)}$. The most popular mutation strategy, DE/rand/1, where ‘DE’ stands for differential evolution, is the overarching optimization algorithm. ‘rand’ indicates that this strategy involves random selection of individuals from the population, and ‘1’ signifies that it uses a single difference vector in the mutation process, which is defined as follows: (4) $v_i^{(t)}=x_{r1}^{(t)}+F(x_{r2}^{(t)}-x_{r3}^{(t)})$(4) where $r1$, $r2$, and $r3$ are distinct random indices from $1,2,\dots ,N$ and $F$ is the scaling factor. Crossover is the next step after mutation. Here, based on the target vector $x_i^{(t)}$and its corresponding mutant $v_i^{(t)}$, a new trial vector $u_i^{(t)}$ is created using the crossover operator. The most widely used crossover scheme is binomial crossover, which is expressed as (5) $u_{ij}^{(t)}=\{\begin{array}{cc}{v}_{ij}^{(t)},& \mathrm{if}\mathrm{ran}{\mathrm{d}}_j\le CR\mathrm{or}j=j_{\mathrm{rand}}\\ x_{ij}^{(t)},& \mathrm{otherwise}\end{array}$(5) where $\mathrm{ran}{\mathrm{d}}_j\sim u(0,1)$, CR represents the crossover probability, and $j_{\mathrm{rand}}$ is a randomly selected index of range $1,2,\dots ,N$. Finally, a search criterion using greedy selection is performed. The search is performed for the objective function values corresponding to the target vector ${\mathbf{x}}_{\mathit{i}}^{(0)}$and its trial vector ${\mathbf{u}}_i^{(t)}$: (6) ${\mathbf{x}}_i^{(t+1)}=\{\begin{array}{cc}{\mathbf{u}}_i^{(t)},& \mathrm{if}f({\mathbf{u}}_i^{(t)})\le f({\mathbf{x}}_i^{(t)})\\ {\mathbf{x}}_i^{(t)},& \mathrm{otherwise}\end{array}$(6) In addition to finding the optimal classifier weights (p) that maximize validation accuracy, the weights can be inverted to give higher weight to poorer performing classifiers. The inversion provides a robustness check on the optimization by testing whether weighting the least accurate classifiers can still improve the multi-classifier performance.

To achieve a high objective value representing the accuracy of the multi-classifier model, the study employs an adaptive search space bound ($\mathrm{bound}$) approach. By dynamically modifying the search space based on the best solutions found, the adaptive search approach enhances optimization algorithms and improves the balance between exploration and exploitation, accelerates convergence, and enhances the algorithm's robustness. The search space for classifier weights, $p$, ranges from $0$ to $1$. To facilitate convergence, these bounds adapt to the best solutions. The adaptive bounds approach concentrates the search around promising solutions while still allowing exploration, thus improving optimization performance. The adaptive bounds can be expressed mathematically as follows: (7) $\{\begin{array}{c}n\mathrm{\_}\mathrm{lowe}{\mathrm{r}}_i=min(x_{ki}-\delta ,0)\\ n\mathrm{\_}\mathrm{uppe}{\mathrm{r}}_i=max(x_{ki}+\delta ,1)\end{array}$(7) for $i=1,2,\dots ,m$, where $m$ is the number of base classifiers, $\delta$ determines the maximum allowable change and $x_{ki}$ is the current value in the i-th dimension. From (7), the new bounds for the search space are given as: (8) ${\mathrm{n}}_{\mathrm{bounds}}=[(\mathrm{new}\mathrm{\_}\mathrm{lowe}{\mathrm{r}}_1,\mathrm{new}\mathrm{\_}\mathrm{uppe}{\mathrm{r}}_1),\dots ,(\mathrm{new}\mathrm{\_}\mathrm{lowe}{\mathrm{r}}_{\mathrm{m}},\mathrm{new}\mathrm{\_}\mathrm{uppe}{\mathrm{r}}_{\mathrm{m}})]$(8)

2.2.5. Bayesian-based objective function

The objective function of the optimization approach described in the previous section applies Bayesian theory to combine the classifiers (RF, DT, and SVM) in an optimal manner. The predicted probability matrixes of the base classifiers are fused together and then weighted by the weights (p) in the multi-classifier. The objective function is then formulated as follows: Given a set of classifier confidence scores, $C=[C_1,C_2,C_3,\dots ,C_m]$ and their corresponding probabilities $P=[P_1,P_2,\dots ,P_m{]}^{\mathrm{\top }}$, for class $k$, the multi-classifier's posterior probability is calculated as: (9) $P(y=k\mid \mathbf{x},\mathbf{p})=\sum _{j=1}^m{p}_{j}P(y=k\mid \mathbf{x},C_j)$(9) In Equation (9), $P(y=k|\mathbf{x},\mathbf{p})$ represents the probability that a given data point $\mathbf{x}$ belongs to class $k$ based on the weighted combination of the base classifiers. $m$ represents the number of base classifiers in the multi-classifier fusion, represents the $j\mathrm{th}$ base classifier, and represents the weights assigned to the $j\mathrm{th}$ base classifier.

The class with the greatest predicted probability is the class selected for an instance. To evaluate the robustness of the optimized $p$, an inversion test is performed where the lowest performing classifiers receive the highest weight. The inverted weights are obtained by ranking the classifiers by validation accuracy and assigning priorities proportional to the inverse of their rank. The objective function and DE optimization remain the same, using the inverted (${\mathrm{p}}^{-1}$) instead of $\mathbf{p}$. (10) $C_{\mathrm{ensemble}}(\mathbf{x},\mathbf{p})=\arg \underset{k}{\max }\mathrm{P}(y=k\mid \mathbf{x},\mathbf{p})$(10) The approach aims to determine an optimized $\mathbf{p}$, i.e. ${\mathbf{p}}^{\ast }$, that leads to the highest accuracy of the multi-classifier model on a validation set. Therefore, the optimization problem is posed as follows: (11) $p\ast =\arg \underset{k}{\max }\mathrm{Accuracy}(C_{\mathrm{ensemble}}(X_{\mathrm{val}},\mathbf{p}))$(11) where $p$ is the vector of weights for each classifier, $X_{\mathrm{val}}$ is the dataset, and $C_{\mathrm{ensemble}}$ is the predicted output of the multi-classifier model based on the weighted combination of classifier probabilities according to $p$. Additionally, an optimization algorithm has been developed, and it is presented in Table 3. The optimal weighted vector ($\mathbf{p}$) is sought using the differential evolution approach, considering class probability matrices from various classifiers, including RF, DT, and SVM.

Table 3. Algorithm for differential evolution optimization.

Input: Class probability matrices of random forest (RF), decision tree (DT), and support vector machine (SVM): ${\mathrm{P}}_{\mathrm{RF}},{\mathrm{P}}_{\mathrm{DT}},{\mathrm{P}}_{\mathrm{SVM}}$.
Output: Optimal weighted vector: $p$.
1	Initialize the weight vector $p=[p_1,p_2,p_3]$ with the constraint $\sum _{\mathrm{i}=1}^3{p}_1=1$.
2	Compute the combined probability matrix ${\mathrm{P}}_{\mathrm{E}}=p_1{\mathrm{P}}_{\mathrm{RF}}+p_2{\mathrm{P}}_{\mathrm{DT}}+p_3{\mathrm{P}}_{\mathrm{SVM}}$.
3	Obtain the ensemble’s prediction $y_i$ by taking the class with the highest probability in ${\mathrm{P}}_{\mathrm{E}}$.
4	Define the ensemble’s negative accuracy $\mathrm{L}(p)$as the negative of the correct predicted proportion.
5	Maximize $\mathrm{L}(\mathrm{w})$ using the Differential Evolution algorithm to find the optimal weight vector $p\ast$.
6	For i = 1 to the number of instances (N) do
7	For j = 1 to the number of classes(M) do
8	Compute the ensemble probability ${\mathrm{P}}_{\mathrm{E}}$ as the weighted sum of base classifiers’ probabilities: ${\mathrm{P}}_{{\mathrm{E}}_{\mathrm{ij}}}=p_1^{\ast }{\mathrm{P}}_{\mathrm{R}{\mathrm{F}}_{\mathrm{ij}}}+p_2^{\ast }{\mathrm{P}}_{\mathrm{D}{\mathrm{T}}_{\mathrm{ij}}}+p_3^{\ast }{\mathrm{P}}_{\mathrm{SV}{\mathrm{M}}_{\mathrm{ij}}}$.
9	end
10	end
11	Obtain the final ensemble predictions $y_i^{\ast }$ by taking the class with the highest probability in the optimized ${\mathrm{P}}_{\mathrm{E}}$.
12	Evaluate the performance of the ensemble classifier using the final predictions $y_i^{\ast }$ and the true class label.

2.3. Implementation

In performing sediment classification using the ensemble method defined, each of the grabbed sample datasets in Table 1 is identified and located in a separate, unique segment after the OBIA segmentation (see Figure 2). No two sample points are identified in the same OBIA segment. Each segment therefore defines the homogenous region for that sample type. Based on the location of each grabbed sample point, 500 data points are generated per segment, amounting to 17,000 over the study area. Features from Table 2 are then extracted for each point. The dataset is split into portions of 70:30 for the training and validation sets, respectively. The three classifiers – RF, DT, and SVM – are trained to obtain the predicted probabilities. A few cases are evaluated to test the robustness of the proposed method: (1) Using the standard dataset, the classifiers are trained on the training set and validated on the validation set. (2) To account for data imbalance, split training using under-sampling and over-sampling of the dataset is performed using the imbalanced-learn library in Python. (3) Finally, training using 10-fold cross-validation is applied. The predicted probabilities per classifier are computed. The objective function computes the negative ensemble accuracy using the weights of the predicted class probabilities from the base classifiers. The weights are determined, which are the optimization variables for the DE algorithm.

The parameters used by the DE algorithm for optimization are as follows: the strategy parameter is ‘DE/rand/1’, the population size for optimization is $1000$ data points for split training and over-sampled training and 500 for under-sampled training, the tolerance level is set to ${10}^{-8}$, mutation equals (0.5, 1), and the optimization is run for a configured number of 1000 iterations. Each weight value is limited to [0, 1]. The best weights are changed at each iteration by either adjusting the margin or keeping the boundaries inside the [0, 1] interval. The DE algorithm can be applied by coupling the objective function with the adaptive bounds (7) – (8). The best possible weights are acquired following the completion of optimization. The fused model is obtained by computing the predictions of the validation dataset using the acquired optimal weights. To perform the optimization, the ‘differential_evolution’ package of SciPy is employed (package: from scipy.optimize import differential_evolution). The package provides tools and functions necessary to effectively implement the DE algorithm for optimization.

The approach using DE proves to be successful. The adaptive bounds mechanism reduces the number of iterations while achieving convergence at the global maximum. The same DE approach and parameters are used for split training, imbalance training, and cross-validation training. The final prediction is evaluated on the basis of accuracy, $F1$ score, and the kappa coefficient. Additionally, two correlation tests are used to determine the performance of the multi-classifier model: Pearson’s correlation test and Spearman’s correlation test. The aim is to obtain a positive correlation between the confidence score and the predicted accuracy.

2.4. Evaluation metrics

The metrics of evaluation employed to ascertain the efficacy of the suggested technique are now defined.

Accuracy refers to the proportion of samples that have been classified correctly in relation to the size of the dataset. The formula utilized to compute accuracy is presented in (12), and it takes into account four fundamental variables: true positives (TPs), true negatives (TNs), false positives (FPs), and false negatives (FNs). True positives represent instances correctly classified as belonging to the target class, while true negatives are instances correctly classified as not belonging to the target class. False positives correspond to instances incorrectly classified as belonging to the target class, and false negatives are instances incorrectly classified as not belonging to the target class. (12) $\mathrm{Accuracy}={\displaystyle \frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{TN}+\mathrm{FP}+\mathrm{FN}}}$(12) The F1 score (F1) is computed as the harmonic average between the precision and recall scores. Its calculation is performed using the following formula: (13) $F1={\displaystyle \frac{2\times \mathrm{Precision}\times \mathrm{Recall}}{\mathrm{Precision}\times \mathrm{Recall}}}$(13) where F1 represents the F1 score, Precision denotes precision, and Recall stands for recall. Precision measures the proportion of true-positive predictions among all positive predictions made by the model. Recall, on the other hand, calculates the proportion of true-positive predictions among all actual positive instances in the dataset. By taking the harmonic average of precision and recall, the F1 score provides a single value that combines both measures, making it particularly useful when there is a need to balance precision and recall in a classification task. It serves as an important indicator of a model's ability to make accurate positive predictions while minimizing false positives and false negatives.

The kappa coefficient ($\kappa$) is a numerical indicator utilized to measure the level of agreement between the ground-truth data and the predicted outcomes. This measure can be computed using the following formula: (14) $\kappa ={\displaystyle \frac{P_o-P_e}{1-P_o}}$(14) where $\kappa$ represents the kappa coefficient, $P\mathrm{\_}o$ is the relative observed agreement between the ground-truth and predicted outcomes and $P\mathrm{\_}e$ is the probability of agreement occurring by chance.

3. Results

3.1. Optimization results

The optimization approach adopted in this study aims to maximize the accuracy of the resulting multi-classifier model with respect to the weights. In other words, finding an optimal posterior probability matrix is based on the fusion of the base classifiers’ prediction probabilities. The results of the optimization based on the method described in Section 2 are presented in this section.

The optimization results are presented for three cases – under-sampled, standard, and over-sampled – to demonstrate the robustness of the proposed approach. Table 4 presents the optimal weight values, which are assigned to each classifier as described in Section 2.2.5 and Table 3, along with their corresponding accuracies on the training dataset obtained after the three different cases: over-sampled, training on the standard, and under-sampled. To check for the effectiveness of the optimization, the weights are also inversed (see Section 2.2.4 and 2.2.5) to give more attention to the poorest-performing classifiers. The results for the inverted weights are also given in Table 4. Table 5 presents changes in bounds observed during optimization compared to the initial bounds, where the bound adaptation is defined in Section 2.2.4. Figure 4 shows the objective value, convergence, and the steps taken during optimization.

Figure 4. Summary of the optimization results. The left column shows plots of the objective function over optimization iteration. The centre column shows the optimization steps for achieving the optimal weights, and the right column provides a 3D visualization of the convergence. The rows correspond to the different datasets: (a) under-sampled, (b) standard, and (c) over-sampled.

Table 4. Optimization results on all training datasets (under-sampled, standard, and over- sampled).

	Under-sampled		Standard		Over-sampled
	Weight ($p$)	Inverse Weight ($p^{-1}$)	Weight ($p$)	Inverse Weight ($p^{-1}$)	Weight ($p$)	Inverse Weight ($p^{-1}$)
RF	0.8438	0.0457	0.9167	0.0222	0.9225	0.015
DT	0.0906	0.4198	0.0496	0.4114	0.066	0.1473
SVM	0.0655	0.5344	0.0336	0.556	0.0113	0.842
Objective Value (Accuracy values)	73.656	73.651	79.139	79.4	95.01	95.2

Note: Both the weight ($p$) and inverse weight ($p^{-1}$) values, in addition to the objective value after convergence, are presented.

Table 5. Search space bounds on the weights acquired for each optimization are presented in the table below.

Under-sampled				Standard				Over-sampled
Initial bounds		Final bounds		Initial bounds		Final bounds		Initial bounds		Final bounds
0.01	1	0.30182	1	0.01	1	0.459	1	0.01	1	0.3896	1
0.01	1	0	0.587	0.01	1	0	0.5519	0.01	1	0	0.5636
0.01	1	0	0.5635	0.01	1	0	0.5351	0.01	1	0	0.5109

Note: The initial bound is used at the start of the optimization, and the final bounds are used after convergence.

3.2. Classification and fusion results

To evaluate the performance of the proposed multi-classifer approach, the accuracy, F1 score, and $\kappa$ are compared to those of the individual base classifiers on the validation dataset. The validation dataset used for evaluation is described in Section 2.3. The results are presented across three cases – under-sampling, standard, and over-sampling – to demonstrate robustness. Tables 6–8 present a comparison of the accuracy, F1 score, and $\kappa$ and AUC concerning the proposed approach and each of the base classifiers. All results in this section are evaluated based on the validation dataset described in Section 2.3. More specifically, Table 6 gives the under-sampled training results. Table 7 compares the proposed and base classifiers using the standard dataset. Finally, Table 8 presents the over-sampled result. The resulting confusion matrix is presented in Figure 5, and the AUC is also shown for the cases.

Figure 5. Confusion matrix. (a) Under-sampled training (b) Standard (c) Over-sampled training.

Table 6. Classification results for under-sampled training.

	Accuracy (%)	F1(%)	$\kappa$	AUC
Proposed	73.481	73.395	0.7016	95.93
RF	71.814	71.744	0.6941	95.2
DT	60.814	61.047	0.5591	78.45
SVM	63.318	62.9631	0.5858	90.83

Table 7. Classification results for the standard dataset.

	Accuracy (%)	F1(%)	$\kappa$	AUC
Proposed	79.1372	78.8379	0.75	95
RF	76.7647	76.4406	0.74	94
DT	70.0098	70.0143	0.64	78
SVM	70.9607	70.7934	0.65	90

Table 8. Classification results for the over-sampled dataset.

	Accuracy (%)	F1 (%)	$\kappa$	AUC
Proposed	94.9711	94.887	0.95	1.0
RF	92.8559	92.7707	0.94	1.0
DT	92.288	92.0907	0.91	96
SVM	78.5728	77.9301	0.75	95

Additional assessment was conducted utilizing 10-fold cross-validation on the standard data, whereby the accuracy, F1 score, and $\kappa$ for each fold are documented in Table 9. The uniformity in performance across these folds demonstrates the resilience of the proposed methodology. Moreover, the favourable correlation values obtained from Pearson’s and Spearman’s tests, as presented in Table 10, evince a strong association between the anticipated accuracy and classifier confidence. The multi-classifier confidence in Table 10 refers to the confidence scores (C) output by the multi-classifier for each validation sample, as defined in Section 2.2.5. The correlation is computed between these confidence scores (C) and the prediction accuracy for each validation sample. In other words, the confidence score vector is compared to a vector of 0s and 1s representing whether each prediction was correct over the validation set. This aligns with the motivation for the confidence-based fusion strategy. Finally, Figure 6 visually portrays the resulting seabed sediment classification maps generated by the proposed multi-classifier approach in contrast to the individual classifiers.

Figure 6. The generated seabed sediment maps after classification. (a) represents the sediment map for the proposed approach. (b) Sediment map for RF. (c) represents the SVM sediment map. (d) Sediment map for DT classification.

Table 9. Classification results for the 10-fold cross-validation training on the standard dataset.

Fold	Accuracy (%)				F1 (%)				$\kappa$
	Proposed	RF	DT	SVM	Proposed	RF	DT	SVM	Proposed	RF	DT	SVM
1	80.82	79.12	70.35	71.06	80.7	78.94	70.48	70.96	0.76	0.75	0.62	0.66
2	82.75	80.58	72.41	72.7	81.46	80.25	72.3	72.66	0.78	0.77	0.67	0.68
3	79.76	79.24	71.12	71.94	79.49	78.96	71.08	71.66	0.76	0.75	0.66	0.67
4	82.06	81.47	71.53	73.65	82.72	81.16	71.31	73.18	0.79	0.78	0.66	0.69
5	81.65	80.41	73.18	73.24	81.44	80.12	73.13	73.06	0.78	0.77	0.68	0.68
6	81.94	81.53	71.12	72.29	81.81	81.38	71.09	72.11	0.78	0.78	0.66	0.67
7	79.29	79.06	70.59	70.06	79.04	78.75	70.65	69.84	0.76	0.75	0.66	0.65
8	80.65	78.94	71	71.47	80.64	78.97	71.11	71.64	0.77	0.75	0.66	0.66
9	81.65	80.41	71.35	70.71	81.38	80.19	71.23	70.44	0.78	0.77	0.67	0.66
10	79.47	78.88	71.24	72.41	81.38	80.19	71.23	70.44	0.76	0.75	0.66	0.67
Mean	81	79.96	71.39	71.95	81.01	79.89	71.36	71.6	0.77	0.76	0.66	0.67
Std	1.13	0.99	0.79	1.08	1.03	0.9	0.74	1.11	0.01	0.01	0.02	0.01

Table 10. Pearson’s and Spearman’s correlation tests between the confidence (C) and predicted accuracy of the proposed multi-classifiers defined in Section 2.

	Under-sampled	Standard-	Over-sampled
Pearson's Correlation	0.675	0.688	0.725
Spearman's Correlation	0.696	0.688	0.55

4. Discussion

The optimization strategy developed and used in this paper aims to produce a multi-classifier model with the highest accuracy. The approach combines the confidence scores of the base classifiers into a single posterior probability matrix to accurately classify seabed sediments. This section discusses the results in Section 3 using the method outlined in Section 2.

As shown in Figure 4, all optimization processes successfully converged at the global maximum. The optimization results in Figure 4 (left column) illustrate that training converged faster than the configured number of 1000 iterations defined in Section 2.3, while achieving a higher objective value (accuracy) than the base classifiers. The successful convergence observed in Figure 4 is a result of implementing the adaptive bounds approach during optimization, as detailed in Section 2.2.4 (7)–(8) (Rauf, Bangyal, and Lali 2021; K. Farda and Thammano 2022; Wu et al. 2022). The purpose of including this information in Table 5 is to illustrate how the bounds on the weights were defined and how they evolved during the optimization process. These bounds are crucial in shaping the optimization problem and controlling the search space for the weights assigned to the classifiers, which, in turn, influence the optimization algorithm’s behaviour. Tighter bounds can restrict the search space, while wider bounds allow for more flexibility. The adaptive nature of these bounds is underscored by their relevance:

• Initially, the algorithm explores a broader weight range. As it iterates and converges, it refines these bounds. This dynamic adjustment is akin to a self-regulating mechanism, where the algorithm adapts its search space based on observed improvements or deteriorations in accuracy.

• The different datasets (under-sampled, standard, and over-sampled) present varying challenges due to their distinct data characteristics. As a result, the optimization algorithm may adapt its weight bounds differently for each dataset. This adaptability showcases the algorithm’s ability to tailor its strategy to the specific nuances of the dataset at hand.

• Understanding how weight bounds evolve offers insight into the degree of autonomy granted to the optimization algorithm and can be perceived as a manifestation of the algorithm’s decision-making procedure, where it dynamically determines the constraints required to achieve optimal results (see Figure 4).

• The evolution of bounds directly impacts the optimization’s efficiency and effectiveness. Tighter final bounds indicate more precise convergence, while broader bounds may imply a more exploratory optimization process.

The results of the objective values presented in Table 4 indicate that the optimization effectively improves the classification performance of the multi-classifier fusion compared to individual classifiers. Table 4 displays the optimized weight for each classifier on each training dataset. The higher objective accuracy achieved demonstrates that the proposed approach successfully learns an improved weighted combination of the classifier probabilities compared to equal weights. As shown in Table 4, the proposed optimization achieves higher training accuracy than the individual classifiers for each dataset. Tables 6–8 demonstrate the improvement in the multi-classifier generalization through higher validation metrics than those of the base classifiers across the different cases. Specifically, the results reveal the following: (1) For under-sampled training, the objective value of 73.656% is higher than the 71.81%, 60.81%, and 63.31% values for RF, DT, and SVM, respectively. (2) With standard dataset training, the objective value of 79.139% exceeds the values of 76.76%, 70.0%, and 70.96% for RF, DT, and SVM, respectively. (3) In the context of over-sampled training, the 95.01% objective value is greater than the respective values for RF (92.85%), DT (92.28%), and SVM (78.57%). As the quantity of data increases, evidenced by Batista, Prati, and Monard (2004), the enhancement in performance is evident not only in accuracy but also in F1 score, κ, and AUC – critical performance indicators for classification tasks.

From an optimization perspective, the weights can be used to describe regions of high importance in the search space based on the confidence scores. In other words, based on the confidence score, the optimization is able to adjust the weights and select the predictions with the highest confidence score, thereby increasing the likelihood of finding the optimal global maximum. The adjustment is relevant when there are conflicting predictions among several base classifiers. The use of confidence scores in the optimization approach helps overcome the limitations of the traditional deterministic multi-classifier approach (Wan et al. 2022; Yang et al. 2022).

The positive values of the correlation test given in Table 10 show that all three strategies demonstrate a positive association between predicted accuracy and the multi-classifier’s confidence, supporting the notion that more confidence in prediction is linked to better performance. According to (Carpenter et al. 1999; Foody, Boyd, and Sanchez-Hernandez 2007), confidence scores are important in multi-classifiers due to (1) agreement between classifiers and (2) correlations between the predicted accuracies. The argument is also supported by Kuncheva (2004), who discussed the methods for achieving high multi-classifier performance by taking advantage of the positive correlation between the confidence score and the predicted accuracy.

The consistent accuracy of both weight and inverse weight findings across all trainings shown in Table 4 provides strong evidence for the usefulness of inversing the weight in this setting. The inverse weights are used as a control to prioritize the least efficient classifiers, which boosts the efficiency of the algorithm even further. The approach excels when the multi-classifier model is made up of diverse base classifiers with overlapping capabilities. Even if they perform poorly, the weaker classifiers in such a multi-classifier model may hold information that improves the overall predictive accuracy (Bortolotti 2017; Dietterich 2000). The multi-classifier approach using inverse weights achieves high performance across a wide range of training situations by dynamically adjusting the weights given to individual base classifiers based on their individual performance.

The improved performance of the multi-classifier over the individual classifiers confirms the claim presented by Diesing and Stephens (2015) that the mapping of maritime habitats is presently transitioning from expert analysis by interpretation to automatic seabed categorization techniques. While several methods and classifiers have been tried, there is no agreement on ‘what works best’, and there is unlikely to be a single solution that meets all requirements. Even though individual classifiers did perform well based on Tables 6–8, their combination neutralizes their individual errors and enhances total performance because each model has the capacity to learn unique patterns from the underlying dataset (Chen et al. 2021).

The 10-fold cross-validation results further support the robustness of the proposed method. Based on the values in Table 9, the proposed approach outperforms the base classifiers in each fold. It should be noted that while cross-validation is performed on the standard, it achieves higher metric scores than the standard training approach in Table 7. The high average metric scores found in the cross-validation underline the significance of using a suitable quantity of training data to obtain improved results. This observation also explains the high performance in the over-sampled results in Table 8 compared to that of the under-sampled and standard training results. The consistent success across various folds suggests that the technique generalizes well to new data and is not overly sensitive to the choice of training data.

The resulting sediment map for the four classifiers is shown in Figure 6. It appears that the proposed approach shows more distinct sediment classes. Upon visual inspection, the proposed multi-classifier approach shown in Figure 7 produces sediment class distributions that appear more spatially consolidated and distinct than those in the individual classifier maps. For example, the mS class shows more continuity along the central region without fragmentation in the proposed map. Additionally, the (g)mS and (g)M areas are more confined to the eastern section without intermixing with other classes. Overall, the sediment class boundaries are better delineated in the proposed multi-classifier map, with less fragmentation and intermixing of classes due to misclassifications.

Figure 7. The figure illustrates discrepancies in sediment maps between the proposed method and the base classifiers. The figure is divided into four sub-figures, each representing a distinct area of interest. Sub-figure (a) corresponds to the area of interest in the sediment map generated by the proposed method, while sub-figures (b), (c), and (d) correspond to the areas of interest in the sediment maps generated by RT, SVM, and DT classifiers, respectively.

Additionally, the significant correlation coefficients derived from Pearson’s and Spearman’s tests in Table 10 provide compelling evidence for a robust connection between the predicted accuracy and classifier confidence scores. The strong Pearson's correlations, with coefficients equal to or exceeding 0.675, clearly indicate a powerful positive linear association. In parallel, the elevated Spearman’s values also substantiate a pronounced monotonic relationship between accuracy and confidence. The correlation test aligns perfectly with the underlying rationale of employing confidence scores to enhance fusion, since heightened confidence values exhibit a strong correlation with more precise predictions.

Multiclassifier models enable the integration of the key attributes from all participating classifiers, resulting in superior performance compared to individual models. Over time, numerous well-known multi-classifier techniques have been introduced in prior research. Table 11 provides a comprehensive comparative analysis of our proposed methodology in relation to previously established approaches found in the literature. The comparison in Table 11 encompasses various scenarios, including under-sampled, standard, over-sampled, and cross-validation (cv) cases.

Table 11. Presents a comparative analysis of the proposed methodology with previously established approaches in the literature across various scenarios, encompassing under-sampled, standard, over-sampled, and cross-validation (cv) cases.

Method	Accuracy
	Under-sampled	standard	Over-sampled	cv
Wan et al. 2022	70.24	75.22	93.26	79.82
Yang et al. 2022	69.5	75.86	93.4	79
Proposed	73.39	79.13	94.97	81

5. Conclusion

The prediction abilities of various basic classifiers (random forest, decision tree, and support vector machine) were combined in this study to produce a multi-classifer approach for precisely classifying seafloor sediments. The goal of the optimization strategy was to increase the multi-classifier's performance. The proposed multi-classifier approach outperformed each individual basic classifier in terms of accuracy, F1 score, the $\kappa$, and the AUC. The 10-fold cross-validation results demonstrated the generalizability of the method by showing that the classification is not overly dependent on training data selection and may generate consistent results across different data divisions. In the future, different optimization methods can be tested to further improve the proposed approach.

Acknowledgments

The authors want to thank the British Geological Survey (BGS) provides the multi-beam dataset and ground-truth samples.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are openly available in ‘public geographic science database belongs to the British Geological Survey (BGS)’ at http://mapapps2.bgs.ac.uk/geoindex_offshore/home.html.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China under grant [number 52201400], and supported by Shandong Provincial Natural Science Foundation under grant [number ZR202111260306], and supported by Special Projects for Promoting High Quality Economic Development (Marine Economic Development) in Guangdong Province under grant (GDNRC[2023]42), and Funding of Key Laboratory of Submarine Geoscience, Ministry of Natural Resources, under grant number KLSG2203.