Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm
Background: Low-rank approximation is used to interpret the features of a correlation matrix using visualization tools; however, a low-rank approximation may result in an estimation that is far from zero, even if the corresponding original value is zero. In such a case, the results lead to a misinte...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/5/579 |
| _version_ | 1827669134065795072 |
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| author | Kensuke Tanioka Yuki Furotani Satoru Hiwa |
| author_facet | Kensuke Tanioka Yuki Furotani Satoru Hiwa |
| author_sort | Kensuke Tanioka |
| collection | DOAJ |
| description | Background: Low-rank approximation is used to interpret the features of a correlation matrix using visualization tools; however, a low-rank approximation may result in an estimation that is far from zero, even if the corresponding original value is zero. In such a case, the results lead to a misinterpretation. Methods: To overcome this, we propose a novel approach to estimate a sparse low-rank correlation matrix based on threshold values. We introduce a new cross-validation function to tune the corresponding threshold values. To calculate the value of a function, the MM algorithm is used to estimate the sparse low-rank correlation matrix, and a grid search was performed to select the threshold values. Results: Through numerical simulation, we found that the false positive rate (FPR), interpretability, and average relative error of the proposed method were superior to those of the tandem approach. For the application of microarray gene expression, the FPRs of the proposed approach with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mspace width="0.277778em"></mspace><mn>3</mn></mrow></semantics></math></inline-formula> and 5 were <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.128</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.139</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.197</mn></mrow></semantics></math></inline-formula>, respectively, while the FPR of the tandem approach was <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.285</mn></mrow></semantics></math></inline-formula>. Conclusions: We propose a novel approach to estimate sparse low-rank correlation matrices. The advantage of the proposed method is that it provides results that are interpretable using a heatmap, thereby avoiding result misinterpretations. We demonstrated the superiority of the proposed method through both numerical simulations and real examples. |
| first_indexed | 2024-03-10T03:56:12Z |
| format | Article |
| id | doaj.art-83d211c98c6e4de787e05f7097d0b7b4 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T03:56:12Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-83d211c98c6e4de787e05f7097d0b7b42023-11-23T10:54:09ZengMDPI AGEntropy1099-43002022-04-0124557910.3390/e24050579Thresholding Approach for Low-Rank Correlation Matrix Based on MM AlgorithmKensuke Tanioka0Yuki Furotani1Satoru Hiwa2Department of Biomedical Sciences and Informatics, Doshisha University, Kyoto 610-0394, JapanGraduate School of Life and Medical Sciences, Doshisha University, Kyoto 610-0394, JapanDepartment of Biomedical Sciences and Informatics, Doshisha University, Kyoto 610-0394, JapanBackground: Low-rank approximation is used to interpret the features of a correlation matrix using visualization tools; however, a low-rank approximation may result in an estimation that is far from zero, even if the corresponding original value is zero. In such a case, the results lead to a misinterpretation. Methods: To overcome this, we propose a novel approach to estimate a sparse low-rank correlation matrix based on threshold values. We introduce a new cross-validation function to tune the corresponding threshold values. To calculate the value of a function, the MM algorithm is used to estimate the sparse low-rank correlation matrix, and a grid search was performed to select the threshold values. Results: Through numerical simulation, we found that the false positive rate (FPR), interpretability, and average relative error of the proposed method were superior to those of the tandem approach. For the application of microarray gene expression, the FPRs of the proposed approach with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mspace width="0.277778em"></mspace><mn>3</mn></mrow></semantics></math></inline-formula> and 5 were <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.128</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.139</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.197</mn></mrow></semantics></math></inline-formula>, respectively, while the FPR of the tandem approach was <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.285</mn></mrow></semantics></math></inline-formula>. Conclusions: We propose a novel approach to estimate sparse low-rank correlation matrices. The advantage of the proposed method is that it provides results that are interpretable using a heatmap, thereby avoiding result misinterpretations. We demonstrated the superiority of the proposed method through both numerical simulations and real examples.https://www.mdpi.com/1099-4300/24/5/579cross-validationproportional thresholdsparse estimation |
| spellingShingle | Kensuke Tanioka Yuki Furotani Satoru Hiwa Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm Entropy cross-validation proportional threshold sparse estimation |
| title | Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm |
| title_full | Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm |
| title_fullStr | Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm |
| title_full_unstemmed | Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm |
| title_short | Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm |
| title_sort | thresholding approach for low rank correlation matrix based on mm algorithm |
| topic | cross-validation proportional threshold sparse estimation |
| url | https://www.mdpi.com/1099-4300/24/5/579 |
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