Karhunen Loève expansion and distribution of non-Gaussian process maximum
In this note we show that when a second order random process is modeled through its truncated Karhunen Loève expansion and when the distribution of the random variables appearing in the expansion is approached by a Gaussian kernel, explicit relations for the mean number of up crossings, of the mean...
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| Veröffentlicht in: | Probabilistic engineering mechanics 2016-01, Vol.43, p.85-90 |
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| container_title | Probabilistic engineering mechanics |
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| creator | Poirion, Fabrice |
| description | In this note we show that when a second order random process is modeled through its truncated Karhunen Loève expansion and when the distribution of the random variables appearing in the expansion is approached by a Gaussian kernel, explicit relations for the mean number of up crossings, of the mean number of local maximums and more generally of Rice's moments can be derived in terms of Gaussian integrals. Several illustrations are given related to academic examples and natural hazards models.
•Numerical expressions are given to compute the distribution of a process and its derivative.•They are based on Karhunen Loève expansion and Gaussian mixture model.•Application to Rice's formula is given.•Application to the calculation of extreme is given.•Several examples illustrate the effectiveness of the approach. |
| doi_str_mv | 10.1016/j.probengmech.2015.12.005 |
| format | Article |
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•Numerical expressions are given to compute the distribution of a process and its derivative.•They are based on Karhunen Loève expansion and Gaussian mixture model.•Application to Rice's formula is given.•Application to the calculation of extreme is given.•Several examples illustrate the effectiveness of the approach.</description><identifier>ISSN: 0266-8920</identifier><identifier>EISSN: 1878-4275</identifier><identifier>DOI: 10.1016/j.probengmech.2015.12.005</identifier><language>eng</language><publisher>Elsevier Ltd</publisher><subject>Extreme value distribution ; Gaussian ; Illustrations ; Integrals ; Kernels ; Non-Gaussian ; Non-stationary ; Probabilistic methods ; Probability theory ; Random process ; Random processes ; Random variables ; Rice's series ; Simulation</subject><ispartof>Probabilistic engineering mechanics, 2016-01, Vol.43, p.85-90</ispartof><rights>2015 Elsevier Ltd</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c354t-975d8d5846e8df7092b762ed721438287416ab0b533f89551a3b933e86fd716a3</citedby><cites>FETCH-LOGICAL-c354t-975d8d5846e8df7092b762ed721438287416ab0b533f89551a3b933e86fd716a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0266892015300667$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65534</link.rule.ids></links><search><creatorcontrib>Poirion, Fabrice</creatorcontrib><title>Karhunen Loève expansion and distribution of non-Gaussian process maximum</title><title>Probabilistic engineering mechanics</title><description>In this note we show that when a second order random process is modeled through its truncated Karhunen Loève expansion and when the distribution of the random variables appearing in the expansion is approached by a Gaussian kernel, explicit relations for the mean number of up crossings, of the mean number of local maximums and more generally of Rice's moments can be derived in terms of Gaussian integrals. Several illustrations are given related to academic examples and natural hazards models.
•Numerical expressions are given to compute the distribution of a process and its derivative.•They are based on Karhunen Loève expansion and Gaussian mixture model.•Application to Rice's formula is given.•Application to the calculation of extreme is given.•Several examples illustrate the effectiveness of the approach.</description><subject>Extreme value distribution</subject><subject>Gaussian</subject><subject>Illustrations</subject><subject>Integrals</subject><subject>Kernels</subject><subject>Non-Gaussian</subject><subject>Non-stationary</subject><subject>Probabilistic methods</subject><subject>Probability theory</subject><subject>Random process</subject><subject>Random processes</subject><subject>Random variables</subject><subject>Rice's series</subject><subject>Simulation</subject><issn>0266-8920</issn><issn>1878-4275</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNqNkMlOwzAQhi0EEmV5h3DjkuAl3o6oYq_EBc6WY0_AVeMUO6nKG_EevBipyoEjp5Fm_kXzIXRBcEUwEVfLap36BuJbB-69opjwitAKY36AZkRJVdZU8kM0w1SIUmmKj9FJzkuMiSS1nqHHJ5vexwixWPTfXxsoYLu2MYc-Fjb6woc8pNCMw27Rt0XsY3lnx5yDjcXU7CDnorPb0I3dGTpq7SrD-e88Ra-3Ny_z-3LxfPcwv16UjvF6KLXkXnmuagHKtxJr2khBwUtKaqaokjURtsENZ6xVmnNiWaMZAyVaL6cTO0WX-9yp_2OEPJguZAerlY3Qj9kQRXkthCZskuq91KU-5wStWafQ2fRpCDY7fmZp_vAzO36GUDPxm7zzvRemXzYBkskuQHTgQwI3GN-Hf6T8AICDf9E</recordid><startdate>201601</startdate><enddate>201601</enddate><creator>Poirion, Fabrice</creator><general>Elsevier Ltd</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>201601</creationdate><title>Karhunen Loève expansion and distribution of non-Gaussian process maximum</title><author>Poirion, Fabrice</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c354t-975d8d5846e8df7092b762ed721438287416ab0b533f89551a3b933e86fd716a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Extreme value distribution</topic><topic>Gaussian</topic><topic>Illustrations</topic><topic>Integrals</topic><topic>Kernels</topic><topic>Non-Gaussian</topic><topic>Non-stationary</topic><topic>Probabilistic methods</topic><topic>Probability theory</topic><topic>Random process</topic><topic>Random processes</topic><topic>Random variables</topic><topic>Rice's series</topic><topic>Simulation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Poirion, Fabrice</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Probabilistic engineering mechanics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Poirion, Fabrice</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Karhunen Loève expansion and distribution of non-Gaussian process maximum</atitle><jtitle>Probabilistic engineering mechanics</jtitle><date>2016-01</date><risdate>2016</risdate><volume>43</volume><spage>85</spage><epage>90</epage><pages>85-90</pages><issn>0266-8920</issn><eissn>1878-4275</eissn><abstract>In this note we show that when a second order random process is modeled through its truncated Karhunen Loève expansion and when the distribution of the random variables appearing in the expansion is approached by a Gaussian kernel, explicit relations for the mean number of up crossings, of the mean number of local maximums and more generally of Rice's moments can be derived in terms of Gaussian integrals. Several illustrations are given related to academic examples and natural hazards models.
•Numerical expressions are given to compute the distribution of a process and its derivative.•They are based on Karhunen Loève expansion and Gaussian mixture model.•Application to Rice's formula is given.•Application to the calculation of extreme is given.•Several examples illustrate the effectiveness of the approach.</abstract><pub>Elsevier Ltd</pub><doi>10.1016/j.probengmech.2015.12.005</doi><tpages>6</tpages></addata></record> |
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| ispartof | Probabilistic engineering mechanics, 2016-01, Vol.43, p.85-90 |
| issn | 0266-8920 1878-4275 |
| language | eng |
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| source | Elsevier ScienceDirect Journals Complete |
| subjects | Extreme value distribution Gaussian Illustrations Integrals Kernels Non-Gaussian Non-stationary Probabilistic methods Probability theory Random process Random processes Random variables Rice's series Simulation |
| title | Karhunen Loève expansion and distribution of non-Gaussian process maximum |
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