Exponential Data Fitting and its Applications

Sum-of-exponentials models for time-resolved spectroscopy data

Author(s): Katharine M. Mullen and Ivo H. M. van Stokkum

Pp: 110-127 (18)

DOI: 10.2174/978160805048211001010110

* (Excluding Mailing and Handling)

Abstract

A model consisting of a sum of exponential functions is very useful for the description of time-resolved spectroscopy data. Each exponential term in the sum can often be associated with a given state of the physical system underlying the measurement, such as a protein excited by laser light. Then the exponential decay rate associated with each state describes the time profile of the contribution of the state to the observed data. The linear coefficients of the sum represent the relative amplitudes of the contributions of each state. When time-resolved spectroscopy data represent more than one wavelength, a linear coefficient is associated with each exponential decay term at each wavelength. In this chapter sum-of-exponentials models for time-resolved spectroscopy applications are reviewed. The parameter estimation problem of fitting the decay rates and linear coefficients of the sum under the least squares criteria is also reviewed, with attention to implementation of algorithms for model fitting. Case studies in fitting models to picosecond time-scale spectroscopic data illustrate the reviewed topics.


Keywords: Sum of exponentials; time-resolved spectroscopy

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      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 862 and `b`.`banner_from_date` <= 1740774798 and `b`.`banner_to_date` >= 1740774798 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      2.03msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 862
      • 3: 1740774798
      • 4: 1740774798
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='3ee27480e238f3c5d05feba6a7639703'
      1.21msalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:291
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Keywords
    1Keywords.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        x5XxQbMdt5VFtabJlubls8GGMbfBMKsEREKWJdI8
        uc_cart_id
        3ee27480e238f3c5d05feba6a7639703
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/968" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/968
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:16 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        []
        
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Fri, 28 Feb 2025 20:33:18 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6ImtyU1lUQWx5Qkg0Y1JiSEtCSWRtZEE9PSIsInZhbHVlIjoidWlwaGh5V1hoRk0xKytoVFdKSUhsQlhDMHBsOXFpS3h0L01nSW4xc0tGY1JPYVJHRERQS2pCRTkrZkxTdENKclBkTk9SSFp4UGVCdmlsNGtJeVpJZG4rZVppWGJlY1A2NEcvZ3QyOVNRa0taaDl1SkJqQXZ6UlZMc1BLcjBES2kiLCJtYWMiOiI0MzE3OWQzYzNmODhjYWQ1N2YyODA1NWU1YTdkMjA2OGMxZDkyOGFlYjJkOWZjNTI3ZDg0ODM2MDA5MTY2NWE4IiwidGFnIjoiIn0%3D; expires=Mon, 03 Mar 2025 08:33:18 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6ImtyU1lUQWx5Qkg0Y1JiSEtCSWRtZEE9PSIsInZhbHVlIjoidWlwaGh5V1hoRk0xKytoVFdKSUhsQlhDMHBsOXFpS3h0L01nSW4xc0tGY1JPYVJHRERQS2pCRTkrZkxTdENKclBkTk9SS" 1 => "alphaeurekaselectnet_session=eyJpdiI6ImVxUkQ1UUZmYUZIbzhJM1pTNHh1aGc9PSIsInZhbHVlIjoibVV3R2tJclFUSjgyelAxd0gyZ3FJY1JxeTdKdXptZVZRZG1QZ0l0VWdtZzM4NnlBMEg0M3JsNmNCWTdER1N3bWRUanNrZDRqdGliZXEwck43SzNMbXU2VUFCc0N3eG0vN2RoT3VRMGtwZVpwa082WDVhNE5kRzByUzM3YXBTLzUiLCJtYWMiOiI0ODNlOTFkYzllNWEyMzg0MDBjZjhkOGJkMDBlY2FlY2VkZmNlZDA0Y2QwNTgwNDFiMDRiNWE3ZjlhZTlmMDcxIiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6ImVxUkQ1UUZmYUZIbzhJM1pTNHh1aGc9PSIsInZhbHVlIjoibVV3R2tJclFUSjgyelAxd0gyZ3FJY1JxeTdKdXptZVZRZG1QZ0l0VWdtZzM4NnlBMEg0M3JsNmN" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6ImtyU1lUQWx5Qkg0Y1JiSEtCSWRtZEE9PSIsInZhbHVlIjoidWlwaGh5V1hoRk0xKytoVFdKSUhsQlhDMHBsOXFpS3h0L01nSW4xc0tGY1JPYVJHRERQS2pCRTkrZkxTdENKclBkTk9SSFp4UGVCdmlsNGtJeVpJZG4rZVppWGJlY1A2NEcvZ3QyOVNRa0taaDl1SkJqQXZ6UlZMc1BLcjBES2kiLCJtYWMiOiI0MzE3OWQzYzNmODhjYWQ1N2YyODA1NWU1YTdkMjA2OGMxZDkyOGFlYjJkOWZjNTI3ZDg0ODM2MDA5MTY2NWE4IiwidGFnIjoiIn0%3D; expires=Mon, 03-Mar-2025 08:33:18 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6ImtyU1lUQWx5Qkg0Y1JiSEtCSWRtZEE9PSIsInZhbHVlIjoidWlwaGh5V1hoRk0xKytoVFdKSUhsQlhDMHBsOXFpS3h0L01nSW4xc0tGY1JPYVJHRERQS2pCRTkrZkxTdENKclBkTk9SS" 1 => "alphaeurekaselectnet_session=eyJpdiI6ImVxUkQ1UUZmYUZIbzhJM1pTNHh1aGc9PSIsInZhbHVlIjoibVV3R2tJclFUSjgyelAxd0gyZ3FJY1JxeTdKdXptZVZRZG1QZ0l0VWdtZzM4NnlBMEg0M3JsNmNCWTdER1N3bWRUanNrZDRqdGliZXEwck43SzNMbXU2VUFCc0N3eG0vN2RoT3VRMGtwZVpwa082WDVhNE5kRzByUzM3YXBTLzUiLCJtYWMiOiI0ODNlOTFkYzllNWEyMzg0MDBjZjhkOGJkMDBlY2FlY2VkZmNlZDA0Y2QwNTgwNDFiMDRiNWE3ZjlhZTlmMDcxIiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6ImVxUkQ1UUZmYUZIbzhJM1pTNHh1aGc9PSIsInZhbHVlIjoibVV3R2tJclFUSjgyelAxd0gyZ3FJY1JxeTdKdXptZVZRZG1QZ0l0VWdtZzM4NnlBMEg0M3JsNmN" ] ]
        session_attributes
        0 of 0
        array:5 [ "_token" => "x5XxQbMdt5VFtabJlubls8GGMbfBMKsEREKWJdI8" "uc_cart_id" => "3ee27480e238f3c5d05feba6a7639703" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/968" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-28 20:33:18GET/chapter/968112183