%=============================================================================== % ifacconf.tex 2022-02-11 jpuente % Template for IFAC meeting papers % Copyright (c) 2022 International Federation of Automatic Control %=============================================================================== \documentclass{ifacconf} % In your preamble (before \begin{document}) \usepackage{amsmath} \usepackage{graphicx} % include this line if your document contains figures \usepackage{natbib} % required for bibliography \usepackage{float} \usepackage{booktabs} %% 三线表 \usepackage{diagbox} %% 斜线表头 \usepackage{multirow} %=============================================================================== \begin{document} \begin{frontmatter} \title{A Precise and Adaptive Graph Regularized Low Rank Representation Model for Recognizing Oil-bearing} % Title, preferably not more than 10 words. \thanks[footnoteinfo]{This work is supported by the National Natural Science Foundation of China (nos.70573101, 12071112, 11471102, 12101195), in part by basic research projects for key scientific research projects in Henan Province (no. 20ZX001) and in part by Natural Science Foundation of Henan Province for Youth (no. 202300410146) \\Corresponding author:Youlin Shang and Kehong Yuan, E-mail: mathshang@sina.com, yuankh2006@126.com. } \author[1,2]{Kehong Yuan} \author[3]{Youlin Shang} \author[4]{Haixiang Guo} \author[1]{Shaofei Zang} \author[1]{Zhonghua Liu} \address[1]{College of Information Engineering, Henan University of Science and Technology, Kaiyuan Avenue, Luoyang, 471023, Henan, China(e-mail: yuankh2006@126.com, mathshang@sina.com, faterdumk0732@sina.com, zangshaofei@163.com, lzhua\_217@163.com)} \address[2]{Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang, 471023, Henan, China} \address[3]{School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, Henan, China} \address[4]{College of Economics and Management, China University of Geoscience, Wuhan, 430074, Hubei, China} \begin{abstract} % Abstract of not more than 250 words. The recognition of oil-bearing formation is an important part in oil exploration, and recognition technology influences the predictive accuracy and efficiency. Low rank representation (LRR) has aroused much attention in the field of data mining. As a modified version, the low rank representation with adaptive graph regularization (LRR-AGR) exploits the global and local information of data for graph learning, and it simultaneously integrates distance regularization term, non-negative constraint and a rank constraint into the framework of LRR. However, how to balance these regularization terms according to the data greatly limits its clustering performance. To adaptively balance these regularization terms according to data and further improve the clustering performance, we propose a novel model named low-rank representation with adaptive parameters and graph regularization (LRR-APGR) in this paper. Firstly, a novel parameter optimization model is formulated and designed based on the framework of LRR-AGR and the feedback mechanism. Secondly, two global intelligent optimization algorithms, which can effectively solve the parameter optimization problem are presented based on particle swarm optimization (PSO) in multi-dimensional continuous space. Experimental results on the data oilsk81, oilsk83 and oilsk85 wells of Jianghan oil fields in China show that the proposed method can significantly improve the clustering performance and the predictive accuracy. \end{abstract} \begin{keyword} Low rank representation, graph regularization, rank constraint, particle swarm optimization . \end{keyword} \end{frontmatter} %=============================================================================== \section{Introduction} The recognition of oil-bearing formation is the process of recognizing the characters of each layer in the well. These characters include dry layer, water layer, inferior oil layer, oil layer\cite{WB:1}. In the fields of pattern recognition, according to whether use label information, data analysis technologies can be divided into three groups, i.e., supervised learning, unsupervised learning and semi-supervised learning\cite{WB:2}. In order to obtain a good predictive accuracy, which is the main task in the recognition of oil-bearing formation, it is necessary to appropriately use the correlative information of the data and data analysis technologies. The multidimensionality and heterogeneity of well log data is a very challenging problem in recognizing oil-bearing formation. It is found that reservoirs expand over hundreds to thousands of miles spatially in thin layers resulting in huge multidimensional data collection at well locations\cite{WB:3}. In addition, unconventional reservoirs are found to be complex and highly heterogeneous that are commonly characterized by their ultra-tight permeability. Therefore, different well log features and recognizing models are selected to deal with well log data. Self-organizing feature map neural network (SOM) model and six feature parameters were selected to identify the unconventional reservoir\cite{WB:4}, i.e., resistivity (RT), porosity (POR), permeability (PERM), shale content (Vsh), water saturation (Sw), and acoustic time (AC). Back propagation (BP) neural network model based on fuzzy approach degree and four parameters i.e., POR, Sw, AC and induction well log value (COND),were used to identify the reservoir oil and gas properties\cite{WB:5}. Genetic algorithms (GA) and fuzzy c-means (FCM) were used to reduce well log feature, and construct a model through BP neural network optimized by GA to predict the value of optimal feature subset\cite{WB:32}. Obviously, the acquisition of relevant features and the establishment of appropriate models are two critical steps in recognizing oil-bearing formation. It can be known that it is meaningful and beneficial for label information to construct predictive model and optimize feature selection. However, label information is difficult to be obtained in real applications. It is a very challenging problem to automatically manage these well log samples into different natural groups. Data clustering is the most favored technique to deal with this challenge for discovering the natural groups without any label information \cite{WB:2}. Because of pleasing efficacy in exploring low-dimensional subspace structures embedded in data, low-rank representation (LRR), as a subspace clustering technology, has attracted great interest in pattern analysis and signal processing communities\cite{WB:8}. LRR can better capture global structure of data and preserve the membership of the samples that belong to the same subspace, simultaneously partition data into different clusters with each cluster corresponding to a subspace\cite{WB:31}. It has become one of the new tools dealing with high-dimensional data, and it is widely used in image classification and clustering\cite{WB:33}, data recovery\cite{WB:14}, dimensionality reduction\cite{WB:3}, anomaly detection \cite{WB:34} and other fields. Non-negative and sparse constraints added to the representation coefficient matrix in LRR, a non-negative low-rank and sparse (NNLRS) model was proposed\cite{WB:9}, which can enhance physical interpretation for real data and capture the local dim linear relationships of data. By introducing the hypergraph Laplacian regularization term into the NNLRS model, the non-negative sparse hyper-Laplacian regularized low-rank representation (NSHLRR) model was proposed\cite{WB:8}, in which the intrinsic non-linear geometry information in data is further captured. According to spectral theory and prior cluster information, i.e., the number of clusters embedded in data, rank constraint is imposed on the Laplacian matrix in LRR model\cite{WB:35}, in which the learned graph is forced to have exactly connected components corresponding to clusters embedded in data. A non-negative weighted sparse distance constraint and a rank constraint were integrated into the framework of LRR model\cite{WB:10}, low-rank representation with adaptive graph regularization (LRR-AGR) model was proposed. LRR-AGR not only simultaneously captured the local structure and global structure of data, but also learned an optimal graph with clearly clustering structure. In fact, it is essential to exploit the intrinsic structure of data for data clustering task. In real world applications, the clustering performance of LRR-AGR model is depended on these different regularization terms, i.e., distance constraint term, low-rank constraint term, and rank constraint term. The influence of these regularization terms for the model are tuned by three parameters. Actually, these parameters balance the above terms in LRR-AGR model. How to adaptively balance these regularization terms in LRR-AGR model is an open problem\cite{WB:10}. However, adaptively balance these regularization terms, i.e., adaptively select optimal parameters for a dataset, can reduce the sensitivity and improve the clustering performance. As a result, aiming at well log data, a new approach which can adaptively select optimal parameters for LRR-AGR model to improve the clustering performance, is proposed in this paper. Inspired by the reference\cite{WB:47}, i.e., heuristic algorithms optimize the parameters in Lotka-Volterra system, we optimize the parameters of LRR-AGR model based on particle swarm optimization (PSO) algorithm. We propose a model named low-rank representation with adaptive parameters and graph regularization (LRR-APGR) in this paper. Firstly, parameter estimation for LRR-AGR is modeled as a multidimension optimization problem based on the framework of LRR-AGR model and the feedback mechanism between parameter values and clustering performance evaluation. Secondly, an effective searching strategy based on MPSO algorithm is presented to solve the optimization problem in multidimensional continuous space. The optimal parameters searched by PSO algorithm, matching with LRR-AGR model, are selected adaptively for each well log data. In summary, our main contributions lie in the following three aspects: (1) A modified LRR-AGR model with optimal parameters named LRR-APGR is proposed and applied successfully in oil-bearing recognition. (2) Instead of parameters of LRR-AGR model selected in a candidate range set, we model the parameter estimation as a multidimension continuous optimization problem based on the framework of LRR-AGR model and the feedback mechanism between parameter values and clustering performance evaluation. Moreover, an effective algorithm based on PSO algorithm for searching optimal solution in multidimension continuous space is designed to solve the optimization problem. (3) The proposed model has good interpretability and adaptability, in which the recognition ability in oil-bearing formation is mainly achieved by capturing various structures information through the combination of regularization terms and their interactions tuned by these parameters among various regularization terms. The remainder of this paper is organized as follows. In section 2, we give a brief description to several related works. Section 3 presents the proposed model about the parameter estimation of LRR-AGR and the mechanism in solving the optimization problem about this model. Section 4 analyzes the result of experiment based on data oilsk81, oilsk83, oilsk85, wells of Jianghan oil fields in China. Section 5 offers the conclusion of the paper and the next research direction. \section{ Related works}\label{sec2} \subsection{LRR model}\label{subsec2} According to the assumption that data in a high-dimensional space actually lie on the union of several linear subspaces, LRR model is proposed, aiming at finding the lowest-rank representation of all data for data clustering\cite{WB:31}. It has been shown that LRR model is an effective tool for subspace clustering because it is better at capturing the global structure of data. For the given data $X$, the LRR model is defined as rank minimization problem and solved by a following convex optimization problem through a good surrogate for rank function.\\ \begin{equation} \begin{array}{l} \mathop {\min }\limits_{Z,E} {\left\| Z \right\|_ * } + \lambda {\left\| E \right\|_1} \\ s.t.X = XZ + E \end{array} \end{equation} where ${\left\| Z \right\|_ * }$ is the nuclear norm of representation matrix $Z$, calculated as ${\left\| Z \right\|_ * } = \sum\limits_i^n {{\delta _i}}$. ${\delta _i}$ is the i-th singular value of matrix $Z$, $n$ is the number of samples in data, and $E$ is the error term used to model different noises, ${\left\| E \right\|_1}$ is the ${l_1}$ norm, defined as the sum of absolutes of all entries, and $\lambda $ is a penalty parameter for balancing the rank function and the ${l_1}$ norm.\\ Several methods have been proposed for solving the problem (1), such as augmented Lagrange multiplier method (ALM)\cite{WB:11}, and linearized alternating direction method with adaptive penalty (LADMAP)\cite{WB:12}. After obtaining an optimal solution ${Z^ * }$, each column of ${Z^ * }$ is normalizes as ${z_i} = {z_i}/{\left\| {{z_i}}. \right\|_\infty }$. Then the similarity graph ${W}={\lvert Z \rvert + {\lvert Z^T \lvert}} $ is calculated. Each element ${w_{ij}}$ denotes the similarity degree between samples ${x_i}$ and ${x_j}$, and then applies the spectral clustering algorithm to cluster data into different subspaces.\\ \subsection{LRR-AGR model}\label{subsec2} LRR model is better at capturing the global data structures (such as multiple clusters and subspaces). However, it may fail to discover the intrinsic geometric and discriminating structures of data\cite{WB:8}, which is essential to actual application. A model which well preserves the locality and similarity of data, has the potential to convey more discriminative information. In addition, if the representation coefficient value is negative, it will lead to lack physical interpretation in the real-world applications. By introducing the distance constraint, low-rank constraint, non-negative constraint and a rank constraint, LRR-AGR is proposed to learn an ideal graph\cite{WB:10}. The LRR-AGR model is formulated as follows\\ \begin{equation} \begin{array}{l} \mathop {\min }\limits_{Z,E} \sum\limits_{i,j}^n {\left\| {{x_i} - {x_j}} \right\|_2^2{z_{ij}} + {\lambda _1}{{\left\| Z \right\|}_ * } + {\lambda _2}{{\left\| E \right\|}_1}} \\ s.t.X = XZ + E,diag(Z) = 0,Z \ge 0,\\rank({L_z}) = n - c \end{array} \end{equation} where $\sum\limits_{i,j}^n {\left\| {{x_i} - {x_j}} \right\|_2^2{z_{ij}}} $ is the distance constraint, if $Z$ is non-negative, adaptively selecting few nearest neighbor samples for representation, which enables the model to guarantee the locality and sparsity, and the non-negative constraint on $Z$ aims to guarantee that each data point is in the middle of its neighbors, which can enhance physical interpretation for real data and embody the dependency among data points than otherwise, $c$ is the number of connected components in optimal graph corresponding to the clusters embedded in data, rank constraint $rank({L_z}) = n - c$, as a prior information, is imposed on the Laplacian matrix ${L_z}$ which is defined as ${{L_z} = D - (Z + {Z^T})/2}$, where $D$ is a diagonal matrix and its i-th diagonal element ${D_{ii}} = \sum\nolimits_j {({z_{ij}} + {z_{ji}})} /2$, ${\lambda _1},{\lambda _2}$ are positive penalty parameters for balancing the low-rank term, the weighted distance regularization term and the error term.\\ As the Laplacian matrix ${L_z}$ is positive semi-definite\cite{WB:13} and the facilitation of Theorem \cite{WB:15}, the minimization optimization problem (2) is converted into the following equivalent optimization problem: \begin{equation} \begin{array}{l} \mathop {\min }\limits_{Z,E} \sum\limits_{i,j}^n {\left\| {{x_i} - {x_j}} \right\|_2^2{z_{ij}} + {\lambda _1}{{\left\| Z \right\|}_ * } + {\lambda _2}{{\left\| E \right\|}_1}} + \\2{\lambda _3}\sum\limits_{i = 1}^c {Tr({F^T}LzF)}\\ s.t.X = XZ + E,diag(Z) = 0,Z \ge 0,\sum\nolimits_j {{z_{ij}} = 1,} \\{F^T}F = I \end{array} \end{equation} where $F = {[f_1^T,f_2^T, \cdots ,f_n^T]^T} \in {R^{n \times c}}$ is the set of $c$ eigenvectors corresponding to the first the $c$ smallest eigenvalues of ${L_z}$, ${Tr( \cdot )}$ is the trace operator, and ${\lambda _3}$ is also a positive penalty parameter used to balance the regularization term in the model. And the optimization problem (3) has been solved by alternating direction method (ADM) through constructing the augmented Lagrangian function \cite{WB:16}. \subsection{PSO algorithm and MPSO algorithm}\label{subsec3} PSO is an evolutionary computation technique\cite{WB:36}. Due to its simplicity of implementation and its ability to quickly converge to a reasonably acceptable solution, PSO has been successfully applied to solve nonconvex or combinatorial optimization problems that arise in many science and engineering domains\cite{WB:41}. As a swarm intelligent search algorithm, PSO aims to find the optimal solution by interacting and sharing information with neighbor particles. The information shared in PSO is obtained by the best positions visited earlier by itself and by any particle in the population so far, named as $pbes{t_i}$ and $gbest$, respectively. Each particle $i$ of the swarm has a current position in search space as ${x_i}$ and a velocity ${v_i}$. So, in the standard PSO algorithm, for each iteration $t$, the velocity and position update equation are described as follows.\\ \begin{equation} \begin{array}{lll} {v_{ij}}(t + 1) = {w_{ij}}{v_i}(t) + {c_1}{r_1}(pbes{t_{ij}}(t) - {x_{ij}}(t)) +\\ {c_2}{r_2}(gbes{t_{ij}}(t) - {x_{ij}}(t))\\ {x_{ij}}(t + 1) = {x_{ij}}(t) + {v_{ij}}(t + 1) \end{array} \end{equation} where ${w_{ij}}$ is the inertial weight,${c_1}$ and ${c_2}$ are acceleration positive constants, ${r_1}$, ${r_2}$ are random numbers in the range of [0,1] drawn from a uniform distribution, $j$ is the j-th dimension in solution space. To guarantee the search from global to local, inertia weight starts with a value close to 1 and linearly decreases to 0.4.\\ To solve the local optimum or stagnation problem in the standard PSO algorithm, a modified version of PSO (MPSO), which adds another behavioral term to the swarm is implemented to solve the parameters optimization problem of Lotka-Volterra system \cite{WB:44}. The formula of MPSO algorithm is updated as follows. \begin{equation} \begin{array}{l} {v_{ij}}(t + 1) = {w_{ij}}(t){v_i}(t) + {c_1}{r_1}(pbes{t_{ij}}(t) - {x_{ij}}(t)) +\\ {c_2}{r_2}(gbes{t_{ij}}(t) - {x_{ij}}(t)) + {c_3}{r_3}(pran{d_{ij}}(t) - {x_{ij}}(t))\\ {x_{ij}}(t + 1) = {x_{ij}}(t) + {v_{ij}}(t + 1) \end{array} \end{equation} where ${c_3}$ is also an acceleration positive constant, ${r_3}$ is the elements with random sequences in range [0,1]. $prand$ is the position of a selected randomly in swarm. To diversity and improve the search capability in the swarm, a new random term provided by $prand$ is added. The random particle's information can weaken the attraction of the $gbest$ position, and move the particles to a better location. \section{Low rank representation with adaptive graph regularization and adaptive parameters}\label{sec4} LRR-AGR learned an ideal graph with intrinsic structure of data because of integrating distance constraint, low-rank constraint and rank constraint into the framework of LRR. These constraints in LRR-AGR were balanced by three parameters, i.e., ${\lambda _1},{\lambda _2}$ and ${\lambda _3}$. Depending on the application and the characteristics of the problem, tuning all these parameters properly may lead to better results. It is necessary to balance these regularization terms adaptively in the recognition of oil-bearing formation. Thus, a parameter optimization model is formulated based on the framework of LRR-AGR and the feedback mechanism aiming at different well log data. Meanwhile, an optimal algorithm based on PSO and MPSO is implemented to solve the optimization problem. \subsection{The model of parameter optimization for LRR-AGR}\label{subsec5} As mentioned in formula (3), assuming the optimal matrix of ${Z^ * }$ and $E$ are given, there are three parameters, i.e., ${\lambda _1},{\lambda _2}$ and ${\lambda _3}$ affecting the performance of LRR-AGR. To obtain the best clustering performance, we need to optimize these parameters. In this paper, we introduce the feedback theory of a system to precisely control these parameters and regularization terms for aimed well log data. The LRR-AGR model is regarded as a recognition system. Clustering performance is considered as a state under the input of these parameters. The process of obtaining and evaluating the optimal representation matrix $Z$ is regarded as the executing agency. Fig.1 is the schematic diagram of obtaining optimal parameters to precisely control different regularization terms for LRR-AGR model. In the following, we describe quantitatively the model of parameter optimization in mathematical form according to above analysis, so as to achieve optimal parameters. Denote ${c}$ as the clusters of data, ${X_0}$ is initial input calculated by data $X$, $P$ is a parameter combination of ${\lambda _1},{\lambda _2},{\lambda _3}$. Actually, ${c}, P$ and ${X_0}$ are pre-programmed or calculated in LRR-AGR model, for a given data $X$, then the optimal representation matrix ${Z^*}$ is obtained through solving the problem (3) by ADM in LRR-AGR model. We denote this process as ${Z^ * } = F(X,{X_0},{c},P)$. To evaluate the optimal graph ${Z^ * }$ the two metrices function, i.e., clustering accuracy (ACC) and normalized mutual information (NMI)\cite{WB:46}, defined as following formula (8) and (9), are selected to evaluate the clustering result achieved by spectral clustering. We denote the evaluation and quantization process as $Y = G({Z^ * })$ where $Y$ is the evaluation value. In a word, under a given initial input, after obtaining an optimal graph ${Z^ * }$, for a data $X$, the clustering evaluation value $Y$ can be obtained and summarized as \\ \begin{equation} Y = G(F(X,{X_0},c,P))\\ \end{equation} Simultaneously, the parameter optimization problem is the inverse problem of (6), i.e., the following formula \begin{equation} {P^*} = \mathop {\arg \max }\limits_P (G(F(X,{X_0},c,P)))\\ \end{equation} \subsection{Solution of the model of parameter estimation}\label{subsec5} Parameter estimation is a process to obtain the parameter values of a mathematical model using sample data of a given system\cite{WB:47}. As an optimization method, the three parameters, i.e., ${\lambda _1},{\lambda _2}$ and ${\lambda _3}$ are chosen in a candidate parameter range set of $\left\{10^{-5}, 10^{-4}, 10^{-3}, 10^{-2}, 10^{-1}, 1,10^1, 10^2\right\}$\cite{WB:10}. According to the sensitivity of parameters to the clustering ACC, parameter ${\lambda _3}$ is firstly fixed, and then perform the LRR-AGR model to find the optimal ${\lambda _1}$ and ${\lambda _2}$ in a candidate domain. Then by similar strategy, parameters ${\lambda _1}$ and ${\lambda _2}$ are fixed, and then perform LRR-AGR method to find the optimal parameter ${\lambda _3}$ in a candidate domain. Lastly, the optimal combination of these parameters can be obtained in the 3D candidate space which is composed by the three candidate domains of parameters. In fact, the above method is a discretization method and the optimal parameters may not exist in the 3D candidate space. It is targeted at specific dataset corresponding research optimal method. In the recognition of oil-bearing formation, we propose an effective method to optimal parameters in a multi-dimensional continuous space. At first, we consider $P$ as the variable and detection signal, and change its value in 3-dimensional continuous space. Then, the clustering evaluation value $Y$ is regarded as state feedback describing the state information of recognition system under the detection signal of $P$. So, a feedback mechanism between $P$ and $Y$ is established, as shown in Fig.1, which is an effective measure to utilize state information to tune the detection signal $P$. As we know, under the above assumptions, there exists a $Y$ value corresponding with $P$, whichever the evaluation function is ACC or NMI. PSO is a swarm intelligent search algorithm based on particle's historical information (i.e., $pbest$ and $gbest$). The information of its own individual extremum and global extremum is shared among all particles and used to adjust new particle's position and velocity. Therefore, we explore the standard PSO algorithm and the MPSO algorithm to search optimal parameters. Fig.1 is a schematic diagram of obtaining an optimal parameter for LRR-AGR model based on the feedback mechanism and PSO algorithm. Fig.2 shows the flowchart for MPSO algorithm in solving the optimization problem. Here, the fitness function $f( \cdot )$ is a composition function of $F$ and $G$, formulated as (7), $c$ is the clusters of data $X$. \begin{figure}[h]% \centering \includegraphics[width=0.48\textwidth]{Fig-1.eps} \caption{A schematic diagram of obtaining an optimal parameter for LRR-AGR model based on the feedback mechanism and PSO search algorithm }\label{fig2} \end{figure} \begin{figure}[h]% \centering \includegraphics[width=0.48\textwidth]{Fig2.eps} \caption{ Flowchart of MPSO algorithm in solving the parameter optimization problem }\label{fig2} \end{figure} \section{The application of the proposed model in oil-bearing of reservoir recognition}\label{sec5} In this section, we conduct several experiments on the real data oilsk81, oilsk83, oilsk85 wells of Jianghan oil fields in China, to evaluate the model of LRR-AGR with optimal parameters. All experiments are performed on the software Matlab R2016b and Windows 10 system, hardware Intel Core i5-1135G7, 16GB ram. \subsection{Description of data and parameters setting in searching algorithm}\label{subsec5} Experiments are performed on three well log data from Jianghan oil fields in China. Table 1 shows the description of the real used data. There are six well log features and four class information when recognizing oil-bearing formation, i.e. acoustic travel time (AC), compensated neutron logging (CNL), resistivity (RT), porosity (POR), oil saturation (SO), permeability (PERM). Class information is dry layer, water layer, oil layer, and inferior oil layer. The quality of data is good without any missing values. In PSO and MPSO algorithm, we denote the population size, initial weight, acceleration constants and maximum iteration as $m$, $w$, $({c_1},{c_2},{c_3})$, ${t_{\max }}$, respectively. Table 2 shows the parameters setting in PSO and MPSO algorithm. Fig.2 shows the flowchart for MPSO algorithm in solving parameter optimization problem.\\ \begin{table}[h] \centering \caption{Statistic description on the well-logging datasets} \begin{tabular}{llll} \toprule Dataset & Variables & Classes & Total number of samples\\ \midrule oilsk81 & 6 & 4 & 31 \\ oilsk83 & 6 & 4 & 50 \\ oilsk85 & 6 & 4 & 34 \\ \bottomrule \end{tabular} \end{table} \begin{table}[h] \centering \caption{Parameter setting in PSO and MPSO algorithms} \begin{tabular}{lllll} \toprule Algorithm & $m$ & $w$ & $({c_1},{c_2},{c_3})$& ${t_{\max }}$\\ \midrule PSO & 150 & 0.7 & $(0.5,0.5,0)$ & 40 \\ MPSO & 150 & $0.7 - 0.4{\rm{ }}t/{t_{\max }}$ & $(0.5,0.5,0.6)$ & 40 \\ \bottomrule \end{tabular} \end{table} \begin{table}[h] \begin{center} \centering \caption{Optimal parameters searched by PSO and MPSO for three oilsk data (Part)} %\begin{minipage}{0.6\textwidth} %\caption{Optimal parameters searched by PSO and MPSO for three oilsk data (Part)}\label{tab2} \begin{tabular*}{0.56\textwidth}{@{\extracolsep{\fill}}lllllll@{\extracolsep{\fill}}} \toprule% & \multicolumn{2}{@{}c@{}}{oilsk81}& \multicolumn{2}{@{}c@{}}{oilsk83}& \multicolumn{2}{@{}c@{}}{oilsk85}\\\cmidrule{2-3}\cmidrule{4-5}\cmidrule{6-7} %& \multicolumn{3}{@{}c@{}}{Element 1} & \multicolumn{3}{@{}c@{}}{Element 2} \\\cmidrule{2-4}\cmidrule{5-7}% & PSO & MPSO & PSO &MPSO & PSO &MPSO \\ \midrule ${\lambda _1}$ & 0.3641415 & 0.0964888 & 0.2572347 & 0.5910087 & 0.4452192 & 0.9916791\\ ${\lambda _2}$ & 0.1279067 & 0.0111202 & 0.1462713 & 0.3038111 & 0.3982446 & 0.5919352\\ ${\lambda _3}$ & 0.0624397 & 0.0149865 & 0.1440339 & 0.1907449 & 0.2774069 & 0.3200301\\ \bottomrule \end{tabular*} %\end{minipage} \end{center} \end{table} \subsection{Evaluation metrics}\label{subsec5} In this paper, we use two metrics, i.e., clustering accuracy (ACC) and normalized mutual information (NMI) to evaluate the clustering performance of different algorithms. Moreover, ACC is also selected as fitness function in PSO and MPSO, which can evaluate the match between the three parameters and the LRR-AGR model. For a dataset $X$ with $n$ samples, ACC is calculated as follows: \begin{equation} ACC = \frac{{\sum\limits_i^n {\delta ({y_i},map({r_i}))} }}{n}\\ \end{equation} where ${r_i}$ and ${y_i}$ denote the cluster label obtained by clustering algorithm and the true label of sample ${x_i}$, respectively. Under the condition of ${x=y}$, then ${\delta (x,y) = 1}$, otherwise, ${\delta (x,y) = 0}$. $map( \cdot )$ is a permutation mapping function, used to map each prediction cluster label ${r_i}$ to the equivalent label according to the distribution of the true label \cite{WB:46}.\\ When the predicted cluster label $K'$ is obtained and the true label $K$ is given, NMI is defined as follows\cite{WB:46}. \begin{equation} NMI(K,K') = \frac{{MI(K,K')}}{{\max (H(K),H(K'))}} \end{equation} where $H(K),H(K')$ denote the entropy of labels $K$ and $K'$, respectively. Mutual information $MI( \cdot )$is calculated as follows \begin{equation} MI(K,K') = \sum\limits_{s \in K} {\sum\limits_{t \in K'} {p(s,t)} } {\log _2}\left( {\frac{{p(s,t)}}{{p(s)p(t)}}} \right) \end{equation} where $p(s,t)$ denotes the joint probability distribution of $s$ and $t$, $p(s)$ and $p(t)$ are the marginal probability of $s$ and $t$, respectively. Both ACC and NMI are in the range of [0,1], and the larger value of ACC or NMI is, the better the clustering performance is. \subsection{ Optimal parameters and clustering result}\label{subsec5} Aiming at three well log data oilsk81, oilsk83 and oilsk85 wells of Jianghan oil fields in China (see Tables 7-9), optimal parameters in LRR-AGR model is searched by PSO and MPSO, respectively. Since the recognizing accuracy is strongly expected in recognizing oil-bearing formation in PSO or MPSO algorithm, we select the ACC metric as fitness function to search optimal parameters. To facilitate statistical results, we reorder the well log data which is listed in Table 7-9, respectively. The samples which belong to the same cluster are ordered in sequence. Meanwhile, the clusters are displayed as follows: dry layer, oil layer, inferior oil layer, and water layer. Table 3 shows the optimal parameters corresponding to each dataset. Table 4 shows the statistic result of clustering performance by ACC metric with the model of LRR-APGR. Experiments are performed 20 times. We observe the best, the average and the worst result to compare their matching performance. According to the matching results, we determine the optimal parameters between the two sets of optimal parameters searched by PSO and MPSO. \begin{table}[h] \begin{center} \centering \caption{Comparison of clustering performance (ACC) with different optimal parameters} \begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}ccccccc@{\extracolsep{\fill}}} \toprule% %& \multicolumn{2}{@{}c@{}}{oilsk81}& \multicolumn{2}{@{}c@{}}{oilsk83}& \multicolumn{2}{@{}c@{}}{oilsk85}\\\cmidrule{2-3}\cmidrule{4-5}\cmidrule{6-7} & \multicolumn{3}{@{}c@{}}{MPSO} & \multicolumn{3}{@{}c@{}}{PSO} \\\cmidrule{2-4}\cmidrule{5-7}% Dataset & Best & Average & Worst &Best & Average &Worst \\ \midrule oilsk81 & 1 & 1 & 1 & 1 & 1 & 1\\ oilsk83 & 0.96 & 0.954 & 0.94 & 0.94 & 0.916 & 0.90\\ oilsk85 & 1 & 0.989 & 0.9705 & 1 & 0.9823 & 0.9705\\ \bottomrule \end{tabular*} %\end{minipage} \end{center} \end{table} Firstly, from Table 3, we can find that different well log data has different optimal parameters and each data can have various optimal parameters. Simultaneously, it is difficult to obtain optimal parameters for a dataset in a given candidate parameter range set. It is meaningful and necessary to adaptively select optimal parameters for a given dataset. Secondly, from Table 4, we can find the clustering performance is different for different optimal parameters. The clustering performance with the optimal parameters searched by MPSO is more significant. MPSO algorithm has advantages over PSO algorithm in searching global optimal solution due to adding random particle information to the population. It is the reason that it can move the particles to a better location and weaken the attraction of the ${P_g}$ position to local minima in the search space. However, all results show that MPSO or PSO algorithm can be a useful and powerful technique for parameter optimization of LRR-AGR model with an accurate performance, fast convergence process, and very low deviations. Above all, adaptively balancing these regularizations, i.e., global structure constraint, local structure constraint, and prior structure constraint can significantly improve clustering performance in recognition of oil-bearing formation.\\ To extensively demonstrate the effectiveness of the LRR-APGR model in recognizing the oil-bearing formation, we select some related clustering methods for comparison, i.e. K-means \cite{WB:48}, Normalized cut clustering method (Ncut)\cite{WB:49}, LRR\cite{WB:26}, NNLRS \cite{WB:9}, latent low-rank representation for subspace segmentation and feature extraction (LatLRR)\cite {WB:50}, Laplacian regularized LRR (LapLRR) \cite{WB:51}, NSHLRR \cite{WB:8}, LRR-AGR\cite{WB:10}, LRR with adaptive dictionary learning (ALRR)\cite{WB:52},and a hierarchical weighted low-rank representation (HWLRR)\cite{WB:53}. The two metrics of clustering performance are ACC and NMI. The result of recognition in oil-bearing for oilsk81, oilsk83, oilsk85 well-logging data are listed in Table 5 and 6, respectively. \begin{table}[h]\tiny \begin{center} \centering \caption{Clustering result comparison on oilsk81, oilsk83 with mean ACC and NMI and their standard deviations. Note: bold numbers denote the best results.} \begin{tabular*}{0.54\textwidth}{@{\extracolsep{\fill}}lcccccc@{\extracolsep{\fill}}lcccccc@{\extracolsep{\fill}}} \toprule% & \multicolumn{2}{@{}c@{}}{oilsk81}& \multicolumn{2}{@{}c@{}}{oilsk83} \\\cmidrule{2-3}\cmidrule{4-5}% Metrics & ACC & NMI & ACC & NMI \\ \midrule K-means & $72.10\pm6.30$ & $59.05\pm4.09$ & $54.00\pm1.72$ & $53.90\pm10.65$ \\ Ncut & $59.35\pm3.97$ & $42.64\pm1.44$ & $52.00\pm0.00$ & $28.67\pm0.00$ \\ LRR & $74.19\pm0.00$ & $53.89\pm0.00$ & $66.20\pm4.63$ & $60.84\pm1.72$ \\ LatLRR & $75.32\pm5.26$ & $59.66\pm2.61$ & $68.10\pm4.65$ & $61.11\pm2.37$ \\ NNLRS & $76.24\pm4.25$ & $58.77\pm2.65$ & $75.12\pm4.32$ & $62.22\pm1.23$ \\ LapLRR & $80.64\pm0.00$ & $52.74\pm0.00$ & $76.00\pm0.00$ & $60.14\pm0.00$ \\ NSHLRR & $79.03\pm2.87$ & $59.31\pm3.78$ & $80.00\pm0.00$ & $64.85\pm0.00$ \\ ALRR & $70.97\pm0.00$ & $66.80\pm0.00$ & $56.00\pm0.00$ & $59.95\pm0.00$ \\ HWLRR & $64.52\pm0.00$ & $57.20\pm0.00$ & $58.00\pm1.24$ & $56.31\pm0.00$ \\ LRR-AGR & $92.58\pm4.33$ & $81.39\pm2.94$ & $88.00\pm0.00$ & $69.68\pm0.00$ \\ LRR-APGR & $\bf {100.00}\pm0.00$ & $\bf{100.00}\pm0.00$ & $\bf{96.00}\pm0.00$ & $\bf{91.00}\pm0.00$ \\ \bottomrule \end{tabular*} %\end{minipage} \end{center} \end{table} \begin{table}[h]\tiny \begin{center} \centering \caption{Clustering result comparison on oilsk85 with mean ACC and NMI and their standard deviations. Note: bold numbers denote the best results.} \begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}lcccccc@{\extracolsep{\fill}}} \toprule% & oilsk85 \\\cmidrule{2-3}% \\multicolumn{2}{@{}c@{}}{oilsk85}%@{}{}%multicolumn{2}{c@}{oilsk85} Metrics & ACC & NMI \\ \midrule K-means & $51.62\pm10.69$ & $37.48\pm9.13$\\ Ncut & $52.94\pm0.00$ & $36.51\pm0.00$\\ LRR & $71.18\pm2.05$ & $60.51\pm0.80$\\ LatLRR & $73.55\pm0.00$ & $61.62\pm0.00$\\ NNLRS & $80.79\pm2.78$ & $62.45\pm0.49$\\ LapLRR & $63.97\pm2.31$ & $49.54\pm0.43$\\ NSHLRR & $81.91\pm1.08$ & $57.94\pm0.15$\\ ALRR & $50.00\pm0.00$ & $51.09\pm0.00$\\ HWLRR & $47.06\pm0.00$ & $42.38\pm0.00$\\ LRR-AGR & $85.29\pm0.00$ & $75.14\pm0.00$\\ LRR-APGR & $\bf{100.00}\pm0.00$ & $\bf{100.00}\pm0.00$\\ \bottomrule \end{tabular*} \end{center} \end{table} From Table 5 and 6, we can firstly find that LRR and its improved version have a better clustering performance than K-means, Ncut in most cases. It is inappropriate to cluster original data directly because it contains many redundant features even noises. It is beneficial to capture the intrinsic structure relationships of data for improving the clustering performance. Then, LapLRR, NSHLRR can obtain better clustering performance than LRR, LatLRR. By integrating Laplacian term into LRR model, the local structure of data is captured, which can guide model to learn a better graph. It is the reason that local structure contains sufficient discriminative information for data clustering. As another local structure constraint, distance constraint is added into the LRR-AGR model. Experiment results show that it is more effective than Laplacian term. By integrating distance constraint, non-negative constraint, rank constraint into the LRR model, the LRR-AGR model has potential to capture the global structure, the local structure, and prior structure information of data. Meanwhile, the results show that the intrinsic geometric structure of data and the prior connected structure are more suitable and necessary for the clustering task. Lastly, as the bold numbers shown in Table 5 and 6, the proposed method LRR-APGR is much better than all the other methods. This clearly demonstrates that optimal parameters adaptively balance these constraints according to data which are exploited in LRR-AGR model are helpful for recognizing the oil-bearing formation. In particular, the pleasing result is beneficial from the block-diagonal structure of the optimal representation matrix. As shown in Fig.3, we can see the optimal matrix obtained by LRR-APGR model has a relatively distinct block-diagonal structure, which is helpful for data clustering. As we can see, (a) and (c), the optimal matrices in Fig.3 are permuted according to the order of four true clusters as shown in Table 7 and Table 9. Meanwhile, only two samples which belong to inferior oil are wrongly clustered into the water layer, as shown in (b), while the remaining data are exactly consistent with true clusters as shown in Table 8. \begin{figure}[h] \begin{minipage}{0.32\linewidth} \vspace{3pt} \centerline{\includegraphics[width=\textwidth]{1.eps}} \centerline{(a)} \end{minipage} \begin{minipage}{0.32\linewidth} \vspace{3pt} \centerline{\includegraphics[width=\textwidth]{2.eps}} \centerline{(b)} \end{minipage} \begin{minipage}{0.32\linewidth} \vspace{3pt} \centerline{\includegraphics[width=\textwidth]{3.eps}} \centerline{(c)} \end{minipage} \caption{Illustration of optimal matrix produced by LRR-APGR for different well log data. } \label{fig4} \end{figure}\\ As above analysis, it is critical and useful to use the prior structure and capture the intrinsic structure of data, i.e., both local and global structure in recognizing the oil-bearing formation task. Meanwhile, we find that it is necessary to adaptively adjust parameters which are used to balance these regularizations describing corresponding structure information in real world applications. Therefore, the mechanism of parametric modeling and the optimization method are applicable and effective, and the proposed method LRR-APGR is more suitable for recognizing the oil-bearing formation. \section{Conclusions}\label{sec5} The recognition of oil-bearing in reservoir is a process that restores well log data to geological information about the reservoir category. It is a stage to reflect the achievements of logging interpretation, and it can improve recovery efficiency significantly and minimize uncertainties. A novel model LRR-APGR is proposed based on MPSO for high-dimensional and low-rank logging data, which is an improved version of LRR-AGR with optimal parameters. The parameters in the model of LRR-AGR is optimized by MPSO in continuous space for the well log data of oilsk81, oilsk83, and oilsk85. Experimental results on well log data show that the model of LRR-APGR has a higher accuracy in recognition and a stronger robustness than other models.\\ Although the results presented here are extremely encouraging, there is an issue that deserves in-depth study in the future. The optimal parameters used here only are found by MPSO according to different well log data. Therefore, the rules of the optimal parameter distribution are worth of studying. A mechanism that obtains the optimal parameters automatically should be investigated. \begin{thebibliography}{xx} % you can also add the bibliography by hand \bibitem[(Guo et al., 2011)]{WB:1} Guo, H., Liao, X., Zhu, K., Ding, C.,\& Gao, Y. (2011). 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\setlength{\tabcolsep}{1mm}{ \begin{tabular}{cccccccc} \toprule%[1.5pt] Layer & AC & CNL & RT & POR & SO& PERM & Conclusion \\ \midrule%[1pt] 1 & 195 & 7.5 & 13.0 & 6.0 & 0& 0& Dry \\ 2 & 225 & 10.0 & 7.3 &11.0 & 0 & 0 & Water\\ 3 & 230 &14.0 &5.5 &12.0 & 0 & 0 & Water\\ 4 &220 &9.0 &25.0 &9.0 & 56 & 1.3 & Oil\\ 5 & 225 & 8.0 &30.0 & 9.0 & 58 & 2.3 & Oil\\ 6 & 210 & 7.0 &26.0 & 6.0 & 0 & 0 & Dry\\ 7 & 220 & 8.0 &26.0 & 10.0 & 60 & 2.4 & Oil\\ 8 & 225 & 9.0 &30.0 & 10.0 & 62 & 2.5 & Oil\\ 9 & 195 & 4.0 &36.0 & 5.5 & 0 & 0 & Dry\\ 10 & 220 & 9.0 &30.0 & 9.0 & 61 & 1.7 & Oil\\ 11 & 217 & 7.5 &50.0 & 8.0 & 55 & 1.1 & Oil\\ 12 & 210 & 6.0 &130.0 & 7.0 & 48 & 0.7 & Inferior oil\\ 13 & 195 & 4.0 &100.0 & 5.0 & 0 & 0 & Dry\\ 14 & 195 & 4.0 &70.0 & 5.0 & 0 & 0 & Dry\\ 15 & 200 & 6.0 &90.0 & 6.0 & 0 & 0 & Dry\\ 16 & 200 & 4.0 &130.0 & 6.0 & 0 & 0 & Dry\\ 17 & 200 & 4.0 &90.0 & 5.0 & 0 & 0 & Dry\\ 18 &215 & 9.0 &25.0 & 9.0 &54 & 1.6 & Oil\\ 19 &195 & 4.0 &70.0 & 4.0 & 0 & 0 &Dry\\ 20 &200 & 6.0 &55.0 & 6.0 & 0 & 0 & Dry\\ 21 & 200 & 4.0 &100.0 & 5.0 & 0 & 0 & Dry\\ 22 & 240 & 13.5 &12.0 &12.0 &40 & 2.4 & Oil\\ 23 & 212 & 8.0 &36.0 &8.0 &60 &1.5 & Oil\\ 24 &197 & 6.0 &50.0 &6.0 & 0 & 0 & Dry\\ 25 & 202 & 6.0 &55.0 &7.0 & 52 &0.8 &Inferior oil\\ 26 &195 & 4.5 &50.0 &6.0 & 0 & 0 & Dry\\ 27 &203 &5.0 &45.0 &7.0 &46 &0.6 & Inferior oil\\ 28 &195 &6.0 &50.0 &6.0 &0 &0 & Dry\\ 29 &210 &7.5 &20.0 &8.0 &57 &1.2 & Oil\\ 30 &201 &6.0 &16.0 &7.0 &40 & 0.4 &Inferior oil\\ 31 &213 &9.5 &12.0 &9.0 &61 & 2 &Oil\\ \bottomrule \end{tabular}} \end{minipage} \end{center} \end{table} \begin{table}[h] \begin{center}\small \begin{minipage}{210pt} \caption{Log explanation of oilsk83 well}\label{tab2}% \setlength{\tabcolsep}{1mm}{ \begin{tabular}{cccccccc} \toprule%[1.5pt] Layer & AC & CNL & RT & POR & SO& PERM & Conclusion \\ \midrule%[1pt] 1 &225 &10 &4 &10 &0 &0 &Water\\ 2 &226 &10 &5 &10.5 &0 &0 &Water\\ 3 &220 &8.5 &6.6 &9.5 &0 &0 &Water\\ 4 &235 &12 &8.8 &10 &32 &0.4 &Inferior oil\\ 5 &226 &13 &8 &9 &35 &0.2 &Inferior oil\\ 6 &202 &10 &11 &7 &0 &0 &Dry\\ 7 &209 &12 &30 &3 &0 &0 &Dry\\ 8 &198 &8 &46 &4 &0 &0 &Dry\\ 9 &178 &0.8 &600 &1.5 &0 &0 &Dry\\ 10 &220 &9 &35 &10 &52 &1.8 &Oil\\ 11 &205 &6 &58 &8 &36 &0.5 &Inferior oil\\ 12 &216 &8.3 &40 &10 &55 &2.6 &Oil\\ 13 &197 &3.5 &120 &4 &0 &0 &Dry\\ 14 &236 &11 &17 &9 &51 &1.2 &Oil\\ 15 &213 &6 &40 &5 &0 &0 &Dry\\ 16 &235 &10 &30 &9.5 &52 &2.5 &Oil\\ 17 &202 &6 &60 &5 &0 &0 &Dry\\ 18 &206 &7 &40 &8 &50 &1.6 &Oil\\ 19 &192 &4 &130 &3 &0 &0 &Dry\\ 20 &210 &8 &40 &7.6 &53 &2.2 &Oil\\ 21 &205 &7.5 &50 &7 &36 &0.7 &Inferior oil\\ 22 &208 &5 &18 &7 &35 &0.8 &Inferior oil\\ 23 &225 &7 &15 &9 &50 &1.2 &Oil\\ 24 &190 &2 &53 &3 &0 &0 &Dry\\ 25 &212 &5 &30 &7 &30 &0.5 &Inferior oil\\ 26 &200 &4 &40 &2 &0 &0 &Dry\\ 27 &201 &4 &46 &2.9 &0 &0 &Dry\\ 28 &195 &3.5 &100 &3 &0 &0 &Dry\\ 29 &199 &11 &40 &1 &0 &0 &Dry\\ 30 &188 &3.8 &400 &2 &0 &0 &Dry\\ 31 &197 &6 &280 &3 &0 &0 &Dry\\ 32 &200 &6 &105 &5 &0 &0 &Dry\\ 33 &196 &6 &190 &3 &0 &0 &Dry\\ 34 &210 &11 &60 &8.5 &62 &2.6 &Oil\\ 35 &209 &9 &48 &8 &52 &1.6 &Oil\\ 36 &185 &1.6 &70 &1 &0 &0 &Dry\\ 37 &188 &4 &70 &2 &0 &0 &Dry\\ 38 &203 &8 &27 &7 &40 &0.8 &Inferior oil\\ 39 &192 &5.5 &98 &3 &0 &0 &Dry\\ 40 &190 &4 &100 &2 &0 &0 &Dry\\ 41 &191 &4.3 &105 &3 &0 &0 &Dry\\ 42 &188 &5 &70 &2 &0 &0 &Dry\\ 43 &210 &8.3 &30 &8 &60 &4 &Oil\\ 44 &185 &3.9 &85 &1 &0 &0 &Dry\\ 45 &190 &5 &23 &4 &0 &0 &Dry\\ 46 &211 &9.5 &10 &7.5 &61 &4.3 &Oil\\ 47 &199 &5.2 &14 &2 &0 &0 &Dry\\ 48 &205 &8 &12 &4 &0 &0 &Dry\\ 49 &200 &5 &18 &3 &0 &0 &Dry\\ 50 &211 &8.5 &9 &7.5 &50 &5 &Oil\\ % \bottomrule \end{tabular}} \end{minipage} \end{center} \end{table} \begin{table}[h] \begin{center}\small \begin{minipage}{210pt} \caption{Log explanation of oilsk85 well}\label{tab2}% \setlength{\tabcolsep}{1mm}{ \begin{tabular}{cccccccc} \toprule%[1.5pt] Layer & AC & CNL& RT & POR & SO& PERM & Conclusion \\ \midrule%[1pt] 1 &225 &15.1 &10.5 &10.7 &0 &3.2 &Water\\ 2 &224 &13.4 &16 &10.5 &0 &2.9 &Water\\ 3 &200 &11.9 &23 &4.8 &0 &0 &Dry\\ 4 &230 &13 &8.5 &11.3 &0 &3.5 &Water\\ 5 &245 &15.7 &12 &14.8 &48 &8.1 &Inferior oil\\ 6 &230 &17.5 &0 &11.3 &0 &3.8 &Water\\ 7 &203 &7.2 &18 &5.2 &0 &0 &Dry\\ 8 &201 &8.1 &20 &4.8 &0 &0 &Dry\\ 9 &208 &6.6 &16 &6.8 &35 &1 &Inferior oil\\ 10 &205 &9 &36 &6.1 &39 &0.9 &Inferior oil\\ 11 &200 &8.1 &33 &5 &0 &0 &Dry\\ 12 &195 &9.8 &34 &3.8 &0 &0 &Dry\\ 13 &175 &12.4 &360 &0.1 &0 &0 &Dry\\ 14 &190 &11.1 &100 &0.3 &0 &0 &Dry\\ 15 &200 &14 &50 &5 &0 &0 &Dry\\ 16 &195 &12.9 &90 &3.8 &0 &0 &Dry\\ 17 &199 &11.5 &100 &4.7 &0 &0 &Dry\\ 18 &190 &16.6 &100 &2.7 &0 &0 &Dry\\ 19 &180 &8.9 &300 &0.5 &0 &0 &Dry\\ 20 &230 &13 &40 &11.8 &59 &3.5 &Oil\\ 21 &200 &14.6 &160 &4.9 &0 &0 &Dry\\ 22 &215 &12.1 &80 &8.4 &60 &2.2 &Oil\\ 23 &188 &8.6 &90 &2.3 &0 &0 &Dry\\ 24 &188 &11.3 &150 &2.3 &0 &0 &Dry\\ 25 &200 &11.5 &165 &5 &0 &0 &Dry\\ 26 &190 &9.5 &180 &2.7 &0 &0 &Dry\\ 27 &198 &10.8 &60 &4.5 &0 &0 &Dry\\ 28 &195 &9.8 &90 &3.8 &0 &0 &Dry\\ 29 &193 &9.4 &35 &3.4 &0 &0 &Dry\\ 30 &195 &10 &32 &3.8 &0 &0 &Dry\\ 31 &195 &11.6 &390 &3.8 &0 &0 &Dry\\ 32 &197 &8.8 &100 &4.3 &0 &0 &Dry\\ 33 &207 &8.6 &60 &6.6 &46 &1.8 &Inferior oil\\ 34 &185 &10.3 &100 &1.6 &0 &0 &Dry\\ \bottomrule \end{tabular}} \end{minipage} \end{center} \end{table} % in the appendices. \end{document}