This paper presents a detailed performance analysis for the kernel-based regularized pairwise learning model associated with a strongly convex loss. The robustness for the model is analyzed by applying an improved convex analysis method. The results show that the regularized pairwise learning model has better qualitatively robustness according to the probability measure. Some new comparison inequalities are provided, with which the convergence rates are derived. In particular an explicit learning rate is obtained in case that the loss is the least square loss.
{"title":"Error analysis of kernel regularized pairwise learning with a strongly convex loss","authors":"Shuhua Wang, B. Sheng","doi":"10.3934/mfc.2022030","DOIUrl":"https://doi.org/10.3934/mfc.2022030","url":null,"abstract":"This paper presents a detailed performance analysis for the kernel-based regularized pairwise learning model associated with a strongly convex loss. The robustness for the model is analyzed by applying an improved convex analysis method. The results show that the regularized pairwise learning model has better qualitatively robustness according to the probability measure. Some new comparison inequalities are provided, with which the convergence rates are derived. In particular an explicit learning rate is obtained in case that the loss is the least square loss.","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"18 1","pages":"625-650"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81807464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Almost sure convergence for the maxima and minima of strongly dependent nonstationary multivariate Gaussian sequences","authors":"Zhicheng Chen, Hongyun Zhang, Xinsheng Liu","doi":"10.3934/mfc.2022044","DOIUrl":"https://doi.org/10.3934/mfc.2022044","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"13 1","pages":"728-741"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85001335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The prediction interval is an important guide for financial organizations to make decisions for pricing loan rates. In this paper, we considered four models with bootstrap technique to calculate prediction intervals. Two datasets are used for the study and begin{document}$ 5 $end{document}-fold cross validation is used to estimate performance. The classical regression and Huber regression models have similar performance, both of them have narrow intervals. Although the RANSAC model has a slightly higher coverage rate, it has the widest interval. When the coverage rates are similar, the model with a narrower interval is recommended. Therefore, the classical and Huber regression models with bootstrap method are recommended to calculate the prediction interval.
The prediction interval is an important guide for financial organizations to make decisions for pricing loan rates. In this paper, we considered four models with bootstrap technique to calculate prediction intervals. Two datasets are used for the study and begin{document}$ 5 $end{document}-fold cross validation is used to estimate performance. The classical regression and Huber regression models have similar performance, both of them have narrow intervals. Although the RANSAC model has a slightly higher coverage rate, it has the widest interval. When the coverage rates are similar, the model with a narrower interval is recommended. Therefore, the classical and Huber regression models with bootstrap method are recommended to calculate the prediction interval.
{"title":"Prediction intervals of loan rate for mortgage data based on bootstrapping technique: A comparative study","authors":"Donglin Wang, Rencheng Sun, Lisa Green","doi":"10.3934/mfc.2022027","DOIUrl":"https://doi.org/10.3934/mfc.2022027","url":null,"abstract":"<p style='text-indent:20px;'>The prediction interval is an important guide for financial organizations to make decisions for pricing loan rates. In this paper, we considered four models with bootstrap technique to calculate prediction intervals. Two datasets are used for the study and <inline-formula><tex-math id=\"M1\">begin{document}$ 5 $end{document}</tex-math></inline-formula>-fold cross validation is used to estimate performance. The classical regression and Huber regression models have similar performance, both of them have narrow intervals. Although the RANSAC model has a slightly higher coverage rate, it has the widest interval. When the coverage rates are similar, the model with a narrower interval is recommended. Therefore, the classical and Huber regression models with bootstrap method are recommended to calculate the prediction interval.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"60 1","pages":"280-289"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85847242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fuzzy approximation based on $ tau- mathfrak{K} $ fuzzy open (closed) sets","authors":"Priti, A. Tripathi","doi":"10.3934/mfc.2023010","DOIUrl":"https://doi.org/10.3934/mfc.2023010","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"20 1","pages":"558-572"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82366259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory, so is it in learning theory. In this paper, we are going to construct a deep neural network with three hidden layers using sigmoid function to approximate and learn the piecewise smooth functions, respectively. In particular, we prove that the constructed deep sigmoid nets can reach the optimal approximation rate in approximating the piecewise smooth functions with controllable parameters but without saturation. Similar results can also be obtained in learning theory, that is, the constructed deep sigmoid nets can also realize the optimal learning rates in learning the piecewise smooth functions. The above two obtained results underlie the advantages of deep sigmoid nets and provide theoretical assessment for deep learning.
{"title":"Learning and approximating piecewise smooth functions by deep sigmoid neural networks","authors":"Xia Liu","doi":"10.3934/mfc.2023039","DOIUrl":"https://doi.org/10.3934/mfc.2023039","url":null,"abstract":"Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory, so is it in learning theory. In this paper, we are going to construct a deep neural network with three hidden layers using sigmoid function to approximate and learn the piecewise smooth functions, respectively. In particular, we prove that the constructed deep sigmoid nets can reach the optimal approximation rate in approximating the piecewise smooth functions with controllable parameters but without saturation. Similar results can also be obtained in learning theory, that is, the constructed deep sigmoid nets can also realize the optimal learning rates in learning the piecewise smooth functions. The above two obtained results underlie the advantages of deep sigmoid nets and provide theoretical assessment for deep learning.","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"298 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135699848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Autism spectrum disorder (ASD) is a type of mental health disorder, and its prevalence worldwide is estimated at about one in 100 children. Accurate diagnosis of ASD as early as possible is very important for the treatment of patients in clinical applications. ABIDE Ⅰ dataset as a repository of ASD is used much for developing classifiers for ASD from typical controls. In this paper, we mainly consider three types of correlations including Pearson correlation, partial correlation, and tangent correlation together based on different numbers of regions of interest (ROIs) from only one atlas, and then twelve deep neural network models are used to train 884 subjects with 5, 10, 15, 20-fold cross-validation on two types of split methods including stratified and non-stratified methods. We first consider six metrics to compare the model performance among the split methods. The six metrics are F1-Score, precision, recall, accuracy, and specificity, area under the precision-recall curve (PRAUC), and area under the Receiver Characteristic Operator curve (ROCAUC). The study achieved the highest accuracy rate of 71.94% for 5-fold cross-validation, 72.64% for 10-fold cross-validation, 72.96% for 15-fold cross-validation, and 73.43% for 20-fold cross-validation.
{"title":"Autism spectrum disorder (ASD) classification with three types of correlations based on ABIDE Ⅰ data","authors":"Donglin Wang, Xin Yang, Wandi Ding","doi":"10.3934/mfc.2023042","DOIUrl":"https://doi.org/10.3934/mfc.2023042","url":null,"abstract":"Autism spectrum disorder (ASD) is a type of mental health disorder, and its prevalence worldwide is estimated at about one in 100 children. Accurate diagnosis of ASD as early as possible is very important for the treatment of patients in clinical applications. ABIDE Ⅰ dataset as a repository of ASD is used much for developing classifiers for ASD from typical controls. In this paper, we mainly consider three types of correlations including Pearson correlation, partial correlation, and tangent correlation together based on different numbers of regions of interest (ROIs) from only one atlas, and then twelve deep neural network models are used to train 884 subjects with 5, 10, 15, 20-fold cross-validation on two types of split methods including stratified and non-stratified methods. We first consider six metrics to compare the model performance among the split methods. The six metrics are F1-Score, precision, recall, accuracy, and specificity, area under the precision-recall curve (PRAUC), and area under the Receiver Characteristic Operator curve (ROCAUC). The study achieved the highest accuracy rate of 71.94% for 5-fold cross-validation, 72.64% for 10-fold cross-validation, 72.96% for 15-fold cross-validation, and 73.43% for 20-fold cross-validation.","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, Karsli [15] estimated the convergence rate of the begin{document}$ q $end{document}-Bernstein-Durrmeyer operators for functions whose begin{document}$ q $end{document}-derivatives are of bounded variation on the interval begin{document}$ [0, 1] $end{document}. Inspired by this study, in the present paper we deal with the convergence rate of a begin{document}$ q $end{document}- analogue of the Stancu type modified Gamma operators, defined by Karsli et al. [17], for the functions begin{document}$ varphi $end{document} whose begin{document}$ q $end{document}-derivatives are of bounded variation on the interval begin{document}$ [0, infty ). $end{document} We present the approximation degree for the operator begin{document}$ left( { mathfrak{S}}_{n, ell, q}^{(alpha , beta )} { varphi}right)(mathfrak{z}) $end{document} at those points begin{document}$ mathfrak{z} $end{document} at which the one sided q-derivativesbegin{document}$ {D}_{q}^{+}{ varphi(mathfrak{z}); and; D} _{q}^{-}{ varphi(mathfrak{z})} $end{document} exist.
Recently, Karsli [15] estimated the convergence rate of the begin{document}$ q $end{document}-Bernstein-Durrmeyer operators for functions whose begin{document}$ q $end{document}-derivatives are of bounded variation on the interval begin{document}$ [0, 1] $end{document}. Inspired by this study, in the present paper we deal with the convergence rate of a begin{document}$ q $end{document}- analogue of the Stancu type modified Gamma operators, defined by Karsli et al. [17], for the functions begin{document}$ varphi $end{document} whose begin{document}$ q $end{document}-derivatives are of bounded variation on the interval begin{document}$ [0, infty ). $end{document} We present the approximation degree for the operator begin{document}$ left( { mathfrak{S}}_{n, ell, q}^{(alpha , beta )} { varphi}right)(mathfrak{z}) $end{document} at those points begin{document}$ mathfrak{z} $end{document} at which the one sided q-derivativesbegin{document}$ {D}_{q}^{+}{ varphi(mathfrak{z}); and; D} _{q}^{-}{ varphi(mathfrak{z})} $end{document} exist.
{"title":"Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation","authors":"H. Karsli, P. Agrawal","doi":"10.3934/mfc.2022002","DOIUrl":"https://doi.org/10.3934/mfc.2022002","url":null,"abstract":"<p style='text-indent:20px;'>Recently, Karsli [<xref ref-type=\"bibr\" rid=\"b15\">15</xref>] estimated the convergence rate of the <inline-formula><tex-math id=\"M2\">begin{document}$ q $end{document}</tex-math></inline-formula>-Bernstein-Durrmeyer operators for functions whose <inline-formula><tex-math id=\"M3\">begin{document}$ q $end{document}</tex-math></inline-formula>-derivatives are of bounded variation on the interval <inline-formula><tex-math id=\"M4\">begin{document}$ [0, 1] $end{document}</tex-math></inline-formula>. Inspired by this study, in the present paper we deal with the convergence rate of a <inline-formula><tex-math id=\"M5\">begin{document}$ q $end{document}</tex-math></inline-formula>- analogue of the Stancu type modified Gamma operators, defined by Karsli et al. [<xref ref-type=\"bibr\" rid=\"b17\">17</xref>], for the functions <inline-formula><tex-math id=\"M6\">begin{document}$ varphi $end{document}</tex-math></inline-formula> whose <inline-formula><tex-math id=\"M7\">begin{document}$ q $end{document}</tex-math></inline-formula>-derivatives are of bounded variation on the interval <inline-formula><tex-math id=\"M8\">begin{document}$ [0, infty ). $end{document}</tex-math></inline-formula> We present the approximation degree for the operator <inline-formula><tex-math id=\"M9\">begin{document}$ left( { mathfrak{S}}_{n, ell, q}^{(alpha , beta )} { varphi}right)(mathfrak{z}) $end{document}</tex-math></inline-formula> at those points <inline-formula><tex-math id=\"M10\">begin{document}$ mathfrak{z} $end{document}</tex-math></inline-formula> at which the one sided q-derivatives<inline-formula><tex-math id=\"M11\">begin{document}$ {D}_{q}^{+}{ varphi(mathfrak{z}); and; D} _{q}^{-}{ varphi(mathfrak{z})} $end{document}</tex-math></inline-formula> exist.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"21 1","pages":"601-615"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75357827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lyapunov type inequalities for nonlinear fractional Hamiltonian systems in the frame of conformable derivatives","authors":"Qi Zhang, J. Shao","doi":"10.3934/mfc.2023004","DOIUrl":"https://doi.org/10.3934/mfc.2023004","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given the increasing importance of discrete fractional calculus in mathematics, science engineering and so on, many different concepts of fractional difference and sum operators have been defined. In this paper, we mainly reviews some definitions of fractional differences and sum operators that emerged in the fields of discrete calculus. Moreover, some properties of those operators are also analyzed and compared with each other, including commutation rules, linearity, Leibniz rules, etc.
{"title":"A review of definitions of fractional differences and sums","authors":"Qiushuang Wang, R. Xu","doi":"10.3934/mfc.2022013","DOIUrl":"https://doi.org/10.3934/mfc.2022013","url":null,"abstract":"Given the increasing importance of discrete fractional calculus in mathematics, science engineering and so on, many different concepts of fractional difference and sum operators have been defined. In this paper, we mainly reviews some definitions of fractional differences and sum operators that emerged in the fields of discrete calculus. Moreover, some properties of those operators are also analyzed and compared with each other, including commutation rules, linearity, Leibniz rules, etc.","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"45 1","pages":"136-160"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89854411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article deals with shape preserving and local approximation properties of post-quantum Bernstein bases and operators over arbitrary interval begin{document}$ [a, b] $end{document} defined by Khan and Sharma (Iran J Sci Technol Trans Sci (2021)). The properties for begin{document}$ (mathfrak{p}, mathfrak{q}) $end{document}-Bernstein bases and Bézier curves over begin{document}$ [a, b] $end{document} have been given. A de Casteljau algorithm has been discussed. Further we obtain the rate of convergence for begin{document}$ (mathfrak{p}, mathfrak{q}) $end{document}-Bernstein operators over begin{document}$ [a, b] $end{document} in terms of Lipschitz type space having two parameters and Lipschitz maximal functions.
This article deals with shape preserving and local approximation properties of post-quantum Bernstein bases and operators over arbitrary interval begin{document}$ [a, b] $end{document} defined by Khan and Sharma (Iran J Sci Technol Trans Sci (2021)). The properties for begin{document}$ (mathfrak{p}, mathfrak{q}) $end{document}-Bernstein bases and Bézier curves over begin{document}$ [a, b] $end{document} have been given. A de Casteljau algorithm has been discussed. Further we obtain the rate of convergence for begin{document}$ (mathfrak{p}, mathfrak{q}) $end{document}-Bernstein operators over begin{document}$ [a, b] $end{document} in terms of Lipschitz type space having two parameters and Lipschitz maximal functions.
{"title":"Shape preserving properties of $ (mathfrak{p}, mathfrak{q}) $ Bernstein Bèzier curves and corresponding results over $ [a, b] $","authors":"V. Sharma, Asif Khan, M. Mursaleen","doi":"10.3934/mfc.2022041","DOIUrl":"https://doi.org/10.3934/mfc.2022041","url":null,"abstract":"<p style='text-indent:20px;'>This article deals with shape preserving and local approximation properties of post-quantum Bernstein bases and operators over arbitrary interval <inline-formula><tex-math id=\"M3\">begin{document}$ [a, b] $end{document}</tex-math></inline-formula> defined by Khan and Sharma (Iran J Sci Technol Trans Sci (2021)). The properties for <inline-formula><tex-math id=\"M4\">begin{document}$ (mathfrak{p}, mathfrak{q}) $end{document}</tex-math></inline-formula>-Bernstein bases and Bézier curves over <inline-formula><tex-math id=\"M5\">begin{document}$ [a, b] $end{document}</tex-math></inline-formula> have been given. A de Casteljau algorithm has been discussed. Further we obtain the rate of convergence for <inline-formula><tex-math id=\"M6\">begin{document}$ (mathfrak{p}, mathfrak{q}) $end{document}</tex-math></inline-formula>-Bernstein operators over <inline-formula><tex-math id=\"M7\">begin{document}$ [a, b] $end{document}</tex-math></inline-formula> in terms of Lipschitz type space having two parameters and Lipschitz maximal functions.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"97 1","pages":"691-703"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91053183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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