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Test des Pull request #1

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2182bae
Add the matrical form of the ODE + test github
weber-s Jan 20, 2017
e54a882
Deleted remaining EOF...
weber-s Jan 20, 2017
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29 changes: 27 additions & 2 deletions KalmanFilter-LinearPredatorPreyModel.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -16,6 +16,31 @@
"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",
"$$\n",
"\\begin{pmatrix}\n",
"0.9 & 0.2 &0 \\\\\n",
"-0.2 & 1 & 1\\\\\n",
"0 & 0& 1\n",
"\\end{pmatrix}\n",
"\\cdot\n",
"\\begin{pmatrix}\n",
"x_0\\\\ x_1\\\\ x_2\n",
"\\end{pmatrix}_{k+1}\n",
"=\n",
"\\begin{pmatrix}\n",
"x_0\\\\ x_1\\\\ x_2\n",
"\\end{pmatrix}_k\n",
"$$\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."
Expand Down Expand Up @@ -361,7 +386,7 @@
"metadata": {
"kernelspec": {
"display_name": "Python 2",
"language": "python",
"language": "python2",
"name": "python2"
},
"language_info": {
Expand All @@ -374,7 +399,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.10"
"version": "2.7.13"
}
},
"nbformat": 4,
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