{"id":628,"date":"2019-07-11T16:48:25","date_gmt":"2019-07-11T15:48:25","guid":{"rendered":"http:\/\/35.193.178.118\/?page_id=628"},"modified":"2019-07-30T11:36:23","modified_gmt":"2019-07-30T10:36:23","slug":"measurement-function-pt1","status":"publish","type":"page","link":"https:\/\/research.reading.ac.uk\/fiduceo\/archive\/tutorials\/measurement-function-pt1\/","title":{"rendered":"Measurement function"},"content":{"rendered":"\r\n<h1 class=\"wp-block-heading\">Introducing the measurement function<\/h1>\r\n\r\n\r\n\r\n<figure class=\"wp-block-video\"><video src=\"http:\/\/35.193.178.118\/wp-content\/uploads\/2019\/07\/Recipe%201%20Emma.mp4\" controls=\"controls\" width=\"300\" height=\"150\"><\/video>\r\n<figcaption>What is a measurement function? Why is it important? What do we use it for? <a href=\"https:\/\/www.bensound.com\/\">Music by Bensound<\/a><\/figcaption>\r\n<\/figure>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n\r\n\r\n\r\n<p>As we saw in the video above, the ultimate aim of uncertainty analysis is to obtain an estimate of the uncertainty associated with the measured value of a measurand. In most cases, the measurand <span class=\"katex-eq\" data-katex-display=\"false\">(Y) <\/span> is not measured directly but is instead obtained from a number of input quantities <span class=\"katex-eq\" data-katex-display=\"false\"> X_i <\/span> via a mathematical relationship that we will call the \u2018measurement function\u2019, as shown below.<\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-634\" src=\"https:\/\/research.reading.ac.uk\/fiduceo\/wp-content\/uploads\/sites\/129\/2019\/07\/tutorial-1-pt1-fig1-1-1024x408.png\" alt=\"\" width=\"731\" height=\"293\" \/><\/figure>\r\n\r\n\r\n\r\n<p>Often, we are able to explicitly write the measurement function in terms of an analytic expression of the form:<\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\">y=f(x_{1}, x_{2}, x_{3} &#8230; x_{n})<\/span><\/p>\r\n\r\n\r\n\r\n<p>where <em>y,<\/em> which is an estimate of the measurand, <span class=\"katex-eq\" data-katex-display=\"false\"> Y <\/span> ,\u00a0is obtained from estimates, <span class=\"katex-eq\" data-katex-display=\"false\"> x_{i} <\/span> ,\u00a0of the input quantities, <span class=\"katex-eq\" data-katex-display=\"false\"> X_{i} <\/span> , via the functional relationship <span class=\"katex-eq\" data-katex-display=\"false\"> f <\/span> . \u00a0There are, however, cases in which it is necessary to define the measurement function in a different way, for example as an iterative solution of a model implemented through code.<\/p>\r\n\r\n\r\n\r\n<p>Each input quantity may be influenced by one or more error effects (each of which has an associated probability distribution) leading to uncertainty in its estimate, <span class=\"katex-eq\" data-katex-display=\"false\"> u(x_{i}) <\/span> .\u00a0The aim of uncertainty analysis is to use this information to establish the combined standard uncertainty associated with our estimate of the measurand, <span class=\"katex-eq\" data-katex-display=\"false\"> u_{c}(y) <\/span> . A summary of the approach towards this aim is shown schematically in the figure below. A summary of the approach towards this aim is shown schematically in the figure below.<\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-639\" src=\"https:\/\/research.reading.ac.uk\/fiduceo\/wp-content\/uploads\/sites\/129\/2019\/07\/tutorial-1-pt1-fig2-1-2-1024x333.png\" alt=\"\" width=\"747\" height=\"243\" \/><\/figure>\r\n\r\n\r\n\r\n<p>In practice, we typically combine individual standard uncertainties mathematically, using the Law of Propagation of Uncertainty, which is given by the \u2018Guide to the Expression of Uncertainty in Measurement\u2019 (the GUM), as:<\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\">u^{2}_{c}(y) = \\sum_{i=1}^n c^{2}_{i}u^{2}(x_{i}) + 2 \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n c_i c_j u(x_i, x_j)<\/span><\/p>\r\n\r\n\r\n\r\n<p>where:<\/p>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li><span class=\"katex-eq\" data-katex-display=\"false\"> u_c(y) <\/span> is the combined standard uncertainty associated with our estimate of the measurand, <span class=\"katex-eq\" data-katex-display=\"false\"> Y <\/span><\/li>\r\n<li><span class=\"katex-eq\" data-katex-display=\"false\"> u(x_i) <\/span> is the standard uncertainty in our estimate of the input quantity <span class=\"katex-eq\" data-katex-display=\"false\"> X_i <\/span><\/li>\r\n<li><span class=\"katex-eq\" data-katex-display=\"false\"> c_i <\/span> and <span class=\"katex-eq\" data-katex-display=\"false\"> c_j <\/span> are sensitivity coefficients<\/li>\r\n<li><span class=\"katex-eq\" data-katex-display=\"false\"> u(x_i, x_j) <\/span> is the estimated covariance associated with <span class=\"katex-eq\" data-katex-display=\"false\"> x_i <\/span> and <span class=\"katex-eq\" data-katex-display=\"false\"> x_j <\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>In cases in which there is no error correlation between the input quantities, the second term in the Law of Propagation of Uncertainty is not required, and the equation can be simplified to:<\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\">u^{2}_{c}(y) = \\sum_{i=1}^n c^{2}_{i}u^{2}(x_{i})<\/span><\/p>\r\n\r\n\r\n\r\n<p>We\u2019ll look at the Law of Propagation of Uncertainty and its constituent terms in more detail as we progress through the following lessons, but first, let\u2019s look at the measurement function in more detail and introduce the concept of a \u2018plus zero\u2019 term.<\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Introducing the measurement function &nbsp; &nbsp; As we saw in the video above, the ultimate aim of uncertainty analysis is to obtain an estimate of the uncertainty associated with the&#8230;<a class=\"read-more\" href=\"&#104;&#116;&#116;&#112;&#115;&#58;&#47;&#47;&#114;&#101;&#115;&#101;&#97;&#114;&#99;&#104;&#46;&#114;&#101;&#97;&#100;&#105;&#110;&#103;&#46;&#97;&#99;&#46;&#117;&#107;&#47;&#102;&#105;&#100;&#117;&#99;&#101;&#111;&#47;&#97;&#114;&#99;&#104;&#105;&#118;&#101;&#47;&#116;&#117;&#116;&#111;&#114;&#105;&#97;&#108;&#115;&#47;&#109;&#101;&#97;&#115;&#117;&#114;&#101;&#109;&#101;&#110;&#116;&#45;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#45;&#112;&#116;&#49;&#47;\">Read More ><\/a><\/p>\n","protected":false},"author":219,"featured_media":0,"parent":305,"menu_order":18,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"__cvm_playback_settings":[],"__cvm_video_id":"","footnotes":""},"coauthors":[6],"class_list":["post-628","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.8.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Measurement function - Fiduceo<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/research.reading.ac.uk\/fiduceo\/archive\/tutorials\/measurement-function-pt1\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Measurement function - Fiduceo\" \/>\n<meta property=\"og:description\" content=\"Introducing the measurement function &nbsp; 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