From 2182bae1879905a6f23e631ccbccd10aecf913df Mon Sep 17 00:00:00 2001
From: Samuel <samuel.weber@normalesup.org>
Date: Fri, 20 Jan 2017 15:33:02 +0100
Subject: [PATCH 1/2] Add the matrical form of the ODE + test github

---
 KalmanFilter-LinearPredatorPreyModel.ipynb | 34 ++++++++++++++++++++--
 1 file changed, 32 insertions(+), 2 deletions(-)

diff --git a/KalmanFilter-LinearPredatorPreyModel.ipynb b/KalmanFilter-LinearPredatorPreyModel.ipynb
index a40340c..99de199 100644
--- a/KalmanFilter-LinearPredatorPreyModel.ipynb
+++ b/KalmanFilter-LinearPredatorPreyModel.ipynb
@@ -16,6 +16,36 @@
     "If we do not know the source evolution, it can simply be simulated with persistence:\n",
     "$$ \\frac{dx_2}{dt} = 0 $$\n",
     "\n",
+    "If we use a forward Euler scheme, we have\n",
+    "$$\n",
+    "x_0(k+1) = 0.9 x_0(k) + 0.2 x_1(k)\\\\\n",
+    "x_1(k+1) = -0.2 x_0(k) + x_1(k) + x_2(k)\\\\\n",
+    "x_2(k+1) = x_2(k)\n",
+    "$$\n",
+    "\n",
+    "or in a matrical notation\n",
+    "\n",
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    "0.9 & 0.2 &0 \\\\\n",
+    "-0.2 & 1 & 1\\\\\n",
+    "0 & 0& 1\n",
+    "\\end{pmatrix}\n",
+    "\\cdot\n",
+    "\\begin{pmatrix}\n",
+    "x_0\\\\\n",
+    "x_1\\\\\n",
+    "x_2\n",
+    "\\end{pmatrix}_{k+1}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "x_0\\\\\n",
+    "x_1\\\\\n",
+    "x_2\n",
+    "\\end{pmatrix}_k\n",
+    "$$\n",
+    "\n",
+    "\n",
     "The model is implemented as a class, below. The `forecast` function of this class propagates in time not only the state ${\\bf x}=(x_0, x_1, x_2)$ of the model, but also the covariance matrix. This is made using a square-root decomposition of the covariance matrix and by propagating each column of this square-root. The class also includes basic plotting functions.\n",
     "\n",
     "Here, the problem tackled by data assimilation is to retrieve the source term ($x_2$) using observations of the other variables (predators, typically), and not knowing the seasonal dynamics of the source: We assume persistence."
@@ -361,7 +391,7 @@
  "metadata": {
   "kernelspec": {
    "display_name": "Python 2",
-   "language": "python",
+   "language": "python2",
    "name": "python2"
   },
   "language_info": {
@@ -374,7 +404,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython2",
-   "version": "2.7.10"
+   "version": "2.7.13"
   }
  },
  "nbformat": 4,

From e54a882c1564b6bce046fb8e5fd462a7364bd901 Mon Sep 17 00:00:00 2001
From: Samuel <samuel.weber@normalesup.org>
Date: Fri, 20 Jan 2017 15:35:54 +0100
Subject: [PATCH 2/2] Deleted remaining EOF...

---
 KalmanFilter-LinearPredatorPreyModel.ipynb | 9 ++-------
 1 file changed, 2 insertions(+), 7 deletions(-)

diff --git a/KalmanFilter-LinearPredatorPreyModel.ipynb b/KalmanFilter-LinearPredatorPreyModel.ipynb
index 99de199..1dd7fc4 100644
--- a/KalmanFilter-LinearPredatorPreyModel.ipynb
+++ b/KalmanFilter-LinearPredatorPreyModel.ipynb
@@ -33,19 +33,14 @@
     "\\end{pmatrix}\n",
     "\\cdot\n",
     "\\begin{pmatrix}\n",
-    "x_0\\\\\n",
-    "x_1\\\\\n",
-    "x_2\n",
+    "x_0\\\\ x_1\\\\ x_2\n",
     "\\end{pmatrix}_{k+1}\n",
     "=\n",
     "\\begin{pmatrix}\n",
-    "x_0\\\\\n",
-    "x_1\\\\\n",
-    "x_2\n",
+    "x_0\\\\ x_1\\\\ x_2\n",
     "\\end{pmatrix}_k\n",
     "$$\n",
     "\n",
-    "\n",
     "The model is implemented as a class, below. The `forecast` function of this class propagates in time not only the state ${\\bf x}=(x_0, x_1, x_2)$ of the model, but also the covariance matrix. This is made using a square-root decomposition of the covariance matrix and by propagating each column of this square-root. The class also includes basic plotting functions.\n",
     "\n",
     "Here, the problem tackled by data assimilation is to retrieve the source term ($x_2$) using observations of the other variables (predators, typically), and not knowing the seasonal dynamics of the source: We assume persistence."