session 1: iterated map applets

lectures_beamer/S1E1_Modelling.tex

@@ -52,13 +52,49 @@
         }
     \end{frame}
 
+    \begin{frame}{Filtering and smoothing}
+
+    \end{frame}
+
+    \begin{frame}{Bayesian inference}
+        Overview of the bayesian setup: prior, model, posterior
+    \end{frame}
+
     \begin{frame}{Representing our understading of the process}
-        Explain what types of models we can use we the tools we are going to learn
+        Explain what types of models we can use with the tools we are going to learn
     \end{frame}
 
     \interludeframe{\textbf{\large Coffee break: 20 minutes}}
 
     \section{Iterated maps}
+    \begin{frame}{1D Example}
+        \only<1>{
+            \[
+                x_k = x_{k-1}
+            \]
+        }
+        \only<2>{
+            \[
+                x_k = x_{k-1} + \epsilon \quad \epsilon \gaussian(0, \sigma^2)
+            \]
+        }
+    \end{frame}
+
+    \begin{frame}{2D Example}
+        \only<1>{
+            \begin{align*}
+                \bm{x}_k = A \bm{x}_{k-1}\\
+                A = \alpha \begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix} \quad 0 < \alpha < 1
+            \end{align*}
+        }
+        \only<2>{
+            \begin{align*}
+                \bm{x}_k = A \bm{x}_{k-1} + \begin{bmatrix}0 \\ \epsilon \end{bmatrix} \quad \epsilon \gaussian(0, \sigma^2)\\
+                A = \alpha \begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix} \quad 0 < \alpha < 1
+            \end{align*}
+        }
+
+    \end{frame}
 
     \section{Ordinary differential equations}
 
@@ -70,7 +106,7 @@
 
     \section{Gaussian Distribution}
     % Gaussian distribution in 1D
-    % Gaussian distribution in 2D
+    % Gaussian distribution in 2D: applet
     % Gaussian distribution in more dimensions
     % Linear transformation in 1D: analytic and visual
     % Nonlinear transformation in 1D: visual

mfiles/s_iterated_map1D.m

+## Copyright (C) 2021 - Juan Pablo Carbajal
+##
+## This program is free software; you can redistribute it and/or modify
+## it under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or
+## (at your option) any later version.
+##
+## This program is distributed in the hope that it will be useful,
+## but WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+## GNU General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with this program. If not, see <http://www.gnu.org/licenses/>.
+
+## Author: Juan Pablo Carbajal <ajuanpi+dev@gmail.com>
+
+pkg load statistics
+
+noise_std = 0.0;
+steps     = 200;
+x0        = 0;
+replicas  = 10;
+
+function y = map(x, sd=0)
+  y = x + normrnd(0, sd);
+endfunction
+
+figure()
+
+for rep=1:replicas
+x    = zeros(steps+1, 1);
+x(1) = x0;
+for i=2:steps+1
+  x(i) = map(x(i-1), noise_std);
+endfor
+
+
+hold on;
+
+plot(x, 'o', 'markersize', 2)
+endfor
+
+grid
+
+xlabel('iteration')
+ylabel('x')
+
+axis tight
+

mfiles/s_iterated_map2D.m

+## Copyright (C) 2021 - Juan Pablo Carbajal
+##
+## This program is free software; you can redistribute it and/or modify
+## it under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or
+## (at your option) any later version.
+##
+## This program is distributed in the hope that it will be useful,
+## but WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+## GNU General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with this program. If not, see <http://www.gnu.org/licenses/>.
+
+## Author: Juan Pablo Carbajal <ajuanpi+dev@gmail.com>
+
+pkg load statistics
+
+noise_std = 0.01;
+steps     = 200;
+x0        = [0 0.1];
+replicas  = 3;
+
+function y = map(x, sd=0)
+  A = [0 -1; 1 0] * 0.05;
+  y = x + x * A;
+  y(2) +=  normrnd(0, sd);
+endfunction
+
+figure ()
+ax1 = subplot (2, 2, [2, 4]);
+axis(ax1, "equal")
+hold(ax1, 'on')
+ax2 = subplot (2, 2, 1);
+hold(ax2, 'on')
+ax3 = subplot (2, 2, 3);
+hold(ax3, 'on')
+
+for rep = 1:replicas
+  x       = zeros(steps+1, 2);
+  x(1, :) = x0;
+  for i=2:steps+1
+    x(i, :) = map (x(i-1, :), noise_std);
+  endfor
+
+  plot(ax1, x(:,1), x(:,2), 'o', 'markersize', 2)
+  plot(ax2, x(:,1), 'o', 'markersize', 2)
+  plot(ax3, x(:,2), 'o', 'markersize', 2)
+endfor
+
+grid(ax1)
+grid(ax2)
+grid(ax3)
+
+xlabel(ax1, 'x_1')
+ylabel(ax1, 'x_2')
+ylabel(ax2, 'x_1')
+xlabel(ax3, 'iteration')
+ylabel(ax3, 'x_2')
+
+
+axis(ax1, "tight")
+axis(ax2, "tight")
+axis(ax3, "tight")
+
+