Differences

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--- misc [2022/02/22 10:16] – [another statement] potthast
+++ misc [2023/03/28 09:14] (current) – external edit 127.0.0.1
@@ Line 9: / Line 9: @@
   where you can quickly type some ideas in LaTeX and the other
   person with internet connection can read it ...
+===== Da Redesign 1 =====
+$a$: all information on bonds, spots or collections or records of observations
+$dim_a$: dimension of bonds
+$y$: Observation vector, List of observation values
+$dim_y$: dimenension of obs
+$is$: Ispace, Interpolation space, list of interpolation points and variables
+$dim_i$: dimension of interpolation space
+$y-i-list$: values needed for y from $is$, this is a vector of some dimension $dim_{yi}$, referenced by vectors yii and yil of integers, where yii(jy) is the starting index and yil(jy) is the number of points needed for the observation with index jy.
+===== Da Redesign 2 =====
+==== Maths Notation ====
+Linearisierung
+$$
+f(x) = f(x_0) + df*(x-x_0), \hspace{1cm}
+q_i(t,q) = q_i(t_0,q_0) + \frac{dq_i}{dt} (t-t_0) + \frac{dq_i}{dq} (q-q_0)
+$$
+$$
+A = \left( \begin{array}{cc}
+& 0 \\
+& 1 \\
+v_1 & v_2
+\end{array} \right), \hspace{1cm}
+A^{T} = \left( \begin{array}{cc}
+& 0 & v_1\\
+& 1 & v_2
+\end{array} \right)
+$$
+$$
+x = \left( \begin{array}{c} t \\ q \end{array}\right), \hspace{1cm}
+A * x = \left( \begin{array}{c} t \\ q \\ q_i \end{array} \right)
+$$
+$z$ be the reflectance using $H_{s}$ MFASIS, then:
+$$
+H_{s} = (\alpha_1, \alpha_2, \alpha_3),
+\hspace{1cm} \left( \begin{array}{c} t \\ q \\ q_i \end{array} \right) =
+\left( \begin{array}{c} \alpha_1 z \\ \alpha_2 z \\ \alpha_3 z \end{array} \right) = H_s^T z
+$$
+$$
+A^T H^T z = A^T \left( \begin{array}{c} \alpha_1 z \\ \alpha_2 z \\ \alpha_3 z \end{array} \right)
+= \left( \begin{array}{c} (\alpha_1  + v_1 \alpha_3 ) z \\
+(\alpha_2  + v_2 \alpha_3 ) z
+\end{array} \right) \hspace{1cm}
+H_s * A = \left( \begin{array}{cc} (\alpha_1  + v_1 \alpha_3 ) & (\alpha_2  + v_2 \alpha_3 )
+\end{array} \right)
+$$
 ==== Maths Notation ====
@@ Line 309: / Line 369: @@
 $$
-\tilde{y} := \frac{y_{2}-y_{1}}{h}
+D{y} := \frac{y_{2}-y_{1}}{h}
 $$
@@ Line 316: / Line 376: @@
 $$
 {\bf y} := \left( \begin{array}{c}
-y_{1} \\ \tilde{y}
+y_{1} \\ D{y}
 \end{array} \right)
 $$
 and
 $$
-\| {\bf y} - {\bf y^{(o)}} \|^2 := \| y_{1} - y_{1}^{(o)} \|^2 + \| \tilde{y} - \tilde{y}^{(o)} \|^2
+\| {\bf y} - {\bf y^{(o)}} \|^2 := \| y_{1} - y_{1}^{(o)} \|^2 + \| D{y} - D{y}^{(o)} \|^2
 $$
 or the original one
@@ Line 333: / Line 393: @@
 A = \left( \begin{array}{cc}
 & 0 \\
--1/h ^ 1/h
+-\frac{1}{h} & \frac{1}{h}
 \end{array}\right)
 $$