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--- parametrization [2024/06/20 14:58] – angelegt pschlue1
+++ parametrization [2024/06/20 14:59] (current) – pschlue1
@@ Line 1: / Line 1: @@
- <font 11pt/inherit;;#c00000;;inherit>**Parameterization:**</font>
+====== Parameterization ======
- <font 11pt/inherit;;#000000;;inherit>In a model, quantity</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i A CKGE_TMP_i</font>
+In a model, quantity A is modelled. A is influenced by unknown B. Parameterization is to express B using A and/or other known quantities, such that A can be estimated. Parameterization schemes usually consist of parameters to be determined empirically.\\
- <font 11pt/inherit;;#000000;;inherit>is modelled.</font>
+Example, we model population density ρ using model (conservation equation) dρ/dt = s, where s is birth-death rate, but unknown. Because of this, the model is not “closed”. Since s is too difficult to estimate, we express s using ρ, e.g., s = r ρ, where r is a parameter. The model is now\\
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i A CKGE_TMP_i</font>
+dρ/dt = r ρ\\
- <font 11pt/inherit;;#000000;;inherit>is influenced by</font>
+which is the Mathus (1798) population growth theory. We use parameterization to describe our understanding of the processes. It is a vital technique for  representing cross-scale and cross-compartment interactions in complex systems. (YS, 12.06.2024).
- <font 11pt/inherit;;#000000;;inherit>unknown</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i B CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>.</font>
- <font 11pt/inherit;;#000000;;inherit>Parameterization is to express</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i B CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>using</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i A CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>and/or other known quantities</font>
- <font 11pt/inherit;;#000000;;inherit>, such that</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i A CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>can be estimated</font>
- <font 11pt/inherit;;#000000;;inherit>.</font>
- <font 11pt/inherit;;#000000;;inherit>Parameterization scheme</font>
- <font 11pt/inherit;;#000000;;inherit>s usually</font>
- <font 11pt/inherit;;#000000;;inherit>consist of parameters</font>
- <font 11pt/inherit;;#000000;;inherit>to be</font>
- <font 11pt/inherit;;#000000;;inherit>determined empirically.</font>
- <font 11pt/inherit;;#000000;;inherit>Example, we model population density</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i ρ CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>using model (</font>
- <font 11pt/inherit;;#000000;;inherit>conservation equation</font>
- <font 11pt/inherit;;#000000;;inherit>)</font>
- <font 11pt/inherit;;#000000;;inherit>d</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i ρ CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>/d</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i t CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>=</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i s CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>, where</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i s CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>is birth-death rate</font>
- <font 11pt/inherit;;#000000;;inherit>, but unknown. Because</font>
- <font 11pt/inherit;;#000000;;inherit>of this,</font>
- <font 11pt/inherit;;#000000;;inherit>the</font>
- <font 11pt/inherit;;#000000;;inherit>model is</font>
- <font 11pt/inherit;;#000000;;inherit>not “closed”. Since</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i s CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>is too difficult to estimate</font>
- <font 11pt/inherit;;#000000;;inherit>,</font>
- <font 11pt/inherit;;#000000;;inherit>we express</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i s CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>using</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i ρ CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>, e.g.,</font>
- <font 11pt/inherit;;#000000;;inherit>s CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>=</font>
- <font 11pt/inherit;;#000000;;inherit>r  CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>ρ CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>,</font>
- <font 11pt/inherit;;#000000;;inherit>where</font>
- <font 11pt/inherit;;#000000;;inherit>r CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>is</font>
- <font 11pt/inherit;;#000000;;inherit>a parameter.</font>
- <font 11pt/inherit;;#000000;;inherit>The model is</font>
- <font 11pt/inherit;;#000000;;inherit>now</font>
- <font 11pt/inherit;;#000000;;inherit>d</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i ρ CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>/d</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i t CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>=</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i r CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>CKGE_TMP_i ρ CKGE_TMP_i</font>
- <font 11pt/inherit;;#000000;;inherit>which</font>
- <font 11pt/inherit;;#000000;;inherit>is the Mathus (1798) population growth theory.</font>
- <font 11pt/inherit;;#000000;;inherit>We use p</font>
- <font 11pt/inherit;;#000000;;inherit>arameterization</font>
- <font 11pt/inherit;;#000000;;inherit>to describe our understanding of the processes. It</font>
- <font 11pt/inherit;;#000000;;inherit>is</font>
- <font 11pt/inherit;;#000000;;inherit>a vital technique for</font>
- <font 11pt/inherit;;#000000;;inherit>representing cross-scale and cross-compartment interactions</font>
- <font 11pt/inherit;;#000000;;inherit>in complex systems</font>
- <font 11pt/inherit;;#000000;;inherit>.</font>
- <font 11pt/inherit;;#000000;;inherit>(</font>
- <font 11pt/inherit;;#00b0f0;;inherit>YS, 12.06.2024</font>
- <font 11pt/inherit;;#000000;;inherit>).</font>