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3 premier chap de 'alea'

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.gitignore
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**/__pycache__
**/.ipynb_checkpoints/
**/.DS_Store
**/build
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.idea/inspectionProfiles/Project_Default.xml
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<inspection_tool class="PyCompatibilityInspection" enabled="true" level="WARNING" enabled_by_default="true">
<option name="ourVersions">
<value>
<list size="1">
<list size="3">
<item index="0" class="java.lang.String" itemvalue="3.5" />
<item index="1" class="java.lang.String" itemvalue="3.6" />
<item index="2" class="java.lang.String" itemvalue="3.7" />
</list>
</value>
</option>
......
.idea/misc.xml
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<option name="languageLevel" value="ES6" />
</component>
<component name="ProjectRootManager" version="2" project-jdk-name="Python 3.6" project-jdk-type="Python SDK" />
<component name="PythonCompatibilityInspectionAdvertiser">
<option name="version" value="3" />
</component>
</project>
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.idea/workspace.xml
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alea/00-python_de_base.ipynb
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alea/01-numpy.ipynb
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alea/08-Chaine_de_Markov.ipynb
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"np.set_printoptions(precision=3,suppress=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Préliminaires d'algèbre linéaire\n",
"\n",
"Attention, en numpy les vecteurs, les matrices lignes et les matrices colonnes sont des objets différents:\n",
"\n",
"* vecteur.shape = (?)\n",
"* matrice_ligne.shape = (1,?)\n",
"* matrice_colonne.shape = (?,1)\n",
"* matrice_quelconque.shape = (?,?)\n",
"\n",
"Observez bien les sorties consoles: les vecteurs s'écrivent avec 1 crochet, les matrices avec 2 crochets.\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Multiplication matricielle\n",
"\n",
"`np.matmul()` s'applique uniquement entre matrices."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"mat_col\n",
" [[1.]\n",
" [1.]\n",
" [1.]]\n",
"\n",
"mat_lin\n",
" [[1. 1. 1.]]\n",
"\n",
"mat_square\n",
" [[1. 1. 1.]\n",
" [1. 1. 1.]\n",
" [1. 1. 1.]]\n",
"\n",
"mat_square × mat_col\n",
" [[3.]\n",
" [3.]\n",
" [3.]]\n",
"\n",
"mat_lin × mat_square\n",
" [[3. 3. 3.]]\n",
"\n",
"mat_lin × mat_square × mat_col\n",
" [[9.]]\n"
]
}
],
"source": [
"size=3\n",
"mat_col=np.ones(shape=[size,1])\n",
"mat_lin=np.ones(shape=[1,size])\n",
"mat_square=np.ones(shape=[size,size])\n",
"\n",
"print(\"\\nmat_col\\n\",mat_col)\n",
"print(\"\\nmat_lin\\n\",mat_lin)\n",
"print(\"\\nmat_square\\n\",mat_square)\n",
"\n",
"print(\"\\nmat_square × mat_col\\n\",np.matmul(mat_square,mat_col) )\n",
"print(\"\\nmat_lin × mat_square\\n\",np.matmul(mat_lin,mat_square))\n",
"print(\"\\nmat_lin × mat_square × mat_col\\n\", np.matmul(np.matmul(mat_lin,mat_square),mat_col))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"sinon on peut utiliser `np.dot()` qui permet les multiplications matrice × vecteur, matrice × matrice, vecteur × vecteur (=produit scalaire)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"vec0 × vec1\n",
" 6.0\n",
"\n",
"vec0 × mat_square\n",
" [3. 3. 3.]\n",
"\n",
"mat_square × vec0\n",
" [3. 3. 3.]\n",
"\n",
"mat_square × mat_square\n",
" [[3. 3. 3.]\n",
" [3. 3. 3.]\n",
" [3. 3. 3.]]\n"
]
}
],
"source": [
"size=3\n",
"vec0=np.ones(shape=[size])\n",
"vec1=2*np.ones(shape=[size])\n",
"mat_square=np.ones(shape=[size,size])\n",
"\n",
"print(\"\\nvec0 × vec1\\n\",np.dot(vec0,vec1))\n",
"print(\"\\nvec0 × mat_square\\n\",np.dot(vec0,mat_square))\n",
"print(\"\\nmat_square × vec0\\n\",np.dot(mat_square,vec0))\n",
"print(\"\\nmat_square × mat_square\\n\",np.dot(mat_square,mat_square))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"et si vous avez python 3.5+, vous pouver utilisez l'opérateur @ qui rend les codes plus lisibles"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"vec0 × vec1\n",
" 6.0\n",
"\n",
"vec0 × mat_square\n",
" [3. 3. 3.]\n",
"\n",
"mat_square × vec0\n",
" [3. 3. 3.]\n",
"\n",
"mat_square × mat_square\n",
" [[3. 3. 3.]\n",
" [3. 3. 3.]\n",
" [3. 3. 3.]]\n"
]
}
],
"source": [
"print(\"\\nvec0 × vec1\\n\",vec0 @ vec1)\n",
"print(\"\\nvec0 × mat_square\\n\",vec0 @ mat_square)\n",
"print(\"\\nmat_square × vec0\\n\",mat_square @ vec0)\n",
"print(\"\\nmat_square × mat_square\\n\",mat_square @ mat_square)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"***Exo:*** Multipliez matriciellement des matrices de taille non-compatible.\n",
"Extrayez la partie intéressante du message d'erreur."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### inverse et transposée \n",
"\n",
"Notez que le pseudo-inverse permet d'inverser les matrices non-inversibles (testez). "
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"mat\n",
" [[1. 0. 2.]\n",
" [0. 1. 0.]\n",
" [0. 0. 1.]]\n",
"\n",
"mat^T\n",
" [[1. 0. 0.]\n",
" [0. 1. 0.]\n",
" [2. 0. 1.]]\n",
"\n",
"mat^(-1)\n",
" [[ 1. 0. -2.]\n",
" [ 0. 1. 0.]\n",
" [ 0. 0. 1.]]\n",
"\n",
"mat^(-1)-pseudo-inverse\n",
" [[ 1. 0. -2.]\n",
" [ 0. 1. 0.]\n",
" [ 0. 0. 1.]]\n"
]
}
],
"source": [
"size=3\n",
"mat=np.zeros(shape=[size,size])\n",
"for i in range(size):\n",
" mat[i,i]=1\n",
"mat[0,size-1]=2\n",
"\n",
"\n",
"print(\"\\nmat\\n\",mat )\n",
"print(\"\\nmat^T\\n\",mat.T)\n",
"print(\"\\nmat^(-1)\\n\",np.linalg.inv(mat))\n",
"print(\"\\nmat^(-1)-pseudo-inverse\\n\",np.linalg.pinv(mat))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### vecteur propre et valeur propre\n",
"\n",
"Souvenez-vous qu'en anglais valeur/vecteur 'propre' c'est 'eigen' value/vector"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"mat\n",
" [[0.085 0.287 0.206 0.422]\n",
" [0.191 0.262 0.253 0.295]\n",
" [0.244 0.07 0.06 0.626]\n",
" [0.383 0.001 0.161 0.454]]\n",
"\n",
"eigVal\n",
" [ 1. 0.122 -0.195 -0.066]\n",
"\n",
"eigVec\n",
" [[-0.5 -0.416 -0.418 0.39 ]\n",
" [-0.5 -0.763 0.307 0.432]\n",
" [-0.5 0.402 -0.739 -0.813]\n",
" [-0.5 0.287 0.43 -0.036]]\n"
]
}
],
"source": [
"size=4\n",
"\"\"\"une matrice aléatoire auquel on fait subir une opération de normalisation (laquelle?)\"\"\"\n",
"mat=np.random.uniform(low=0,high=10,size=[size,size])\n",
"rowSum=np.sum(mat,axis=1)\n",
"\"\"\" mat[i,j]=mat[i,j]/rowSum(i) \"\"\"\n",
"for i in range(size): mat[i,:]/=rowSum[i]\n",
"\n",
"eigVal,eigVec=np.linalg.eig(mat)\n",
"print(\"\\nmat\\n\", mat )\n",
"print(\"\\neigVal\\n\",np.real(eigVal))\n",
"print(\"\\neigVec\\n\", np.real(eigVec))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"***A vous:*** \n",
"\n",
"* Pourquoi 1 est-il valeur propre ? Quel est le vecteur propre associé? `np.linalg.eig` donne-t-il les vecteurs propres à droite ou à gauche? Sont-ils disposés en ligne ou bien en colonne?\n",
"\n",
"* Toutes les matrices sont-elles diagonalisables? A quoi servent les np.real()? \n",
"\n",
"* laquelle de ces deux formules est la bonne ?\n",
"\n",
" Diag = P^-1 @ mat @ P \n",
"\n",
"ou\n",
"\n",
" Diag = P @ mat @ P^-1\n",
" \n",
"Vérifiez avec python. Aide: Pour transformez un vecteur en matrice diagonale, utilisez `np.diag(vecteur)`"
]
},
{
"cell_type": "markdown",
"metadata": {},
......
%% Cell type:markdown id: tags:
# Chaine de Markov
%% Cell type:code id: tags:
``` python
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
np.set_printoptions(precision=3,suppress=True)
```
%% Cell type:markdown id: tags:
## Préliminaires d'algèbre linéaire
Attention, en numpy les vecteurs, les matrices lignes et les matrices colonnes sont des objets différents:
* vecteur.shape = (?)
* matrice_ligne.shape = (1,?)
* matrice_colonne.shape = (?,1)
* matrice_quelconque.shape = (?,?)
Observez bien les sorties consoles: les vecteurs s'écrivent avec 1 crochet, les matrices avec 2 crochets.
%% Cell type:markdown id: tags:
### Multiplication matricielle
`np.matmul()` s'applique uniquement entre matrices.
%% Cell type:code id: tags:
``` python
size=3
mat_col=np.ones(shape=[size,1])
mat_lin=np.ones(shape=[1,size])
mat_square=np.ones(shape=[size,size])
print("\nmat_col\n",mat_col)
print("\nmat_lin\n",mat_lin)
print("\nmat_square\n",mat_square)
print("\nmat_square × mat_col\n",np.matmul(mat_square,mat_col) )
print("\nmat_lin × mat_square\n",np.matmul(mat_lin,mat_square))
print("\nmat_lin × mat_square × mat_col\n", np.matmul(np.matmul(mat_lin,mat_square),mat_col))
```
%%%% Output: stream
mat_col
[[1.]
[1.]
[1.]]
mat_lin
[[1. 1. 1.]]
mat_square
[[1. 1. 1.]
[1. 1. 1.]
[1. 1. 1.]]
mat_square × mat_col
[[3.]
[3.]
[3.]]
mat_lin × mat_square
[[3. 3. 3.]]
mat_lin × mat_square × mat_col
[[9.]]
%% Cell type:markdown id: tags:
sinon on peut utiliser `np.dot()` qui permet les multiplications matrice × vecteur, matrice × matrice, vecteur × vecteur (=produit scalaire)
%% Cell type:code id: tags:
``` python
size=3
vec0=np.ones(shape=[size])
vec1=2*np.ones(shape=[size])
mat_square=np.ones(shape=[size,size])
print("\nvec0 × vec1\n",np.dot(vec0,vec1))
print("\nvec0 × mat_square\n",np.dot(vec0,mat_square))
print("\nmat_square × vec0\n",np.dot(mat_square,vec0))
print("\nmat_square × mat_square\n",np.dot(mat_square,mat_square))
```
%%%% Output: stream
vec0 × vec1
6.0
vec0 × mat_square
[3. 3. 3.]
mat_square × vec0
[3. 3. 3.]
mat_square × mat_square
[[3. 3. 3.]
[3. 3. 3.]
[3. 3. 3.]]
%% Cell type:markdown id: tags:
et si vous avez python 3.5+, vous pouver utilisez l'opérateur @ qui rend les codes plus lisibles
%% Cell type:code id: tags:
``` python
print("\nvec0 × vec1\n",vec0 @ vec1)
print("\nvec0 × mat_square\n",vec0 @ mat_square)
print("\nmat_square × vec0\n",mat_square @ vec0)
print("\nmat_square × mat_square\n",mat_square @ mat_square)
```
%%%% Output: stream
vec0 × vec1
6.0
vec0 × mat_square
[3. 3. 3.]
mat_square × vec0
[3. 3. 3.]
mat_square × mat_square
[[3. 3. 3.]
[3. 3. 3.]
[3. 3. 3.]]
%% Cell type:markdown id: tags:
***Exo:*** Multipliez matriciellement des matrices de taille non-compatible.
Extrayez la partie intéressante du message d'erreur.
%% Cell type:markdown id: tags:
### inverse et transposée
Notez que le pseudo-inverse permet d'inverser les matrices non-inversibles (testez).
%% Cell type:code id: tags:
``` python
size=3
mat=np.zeros(shape=[size,size])
for i in range(size):
mat[i,i]=1
mat[0,size-1]=2
print("\nmat\n",mat )
print("\nmat^T\n",mat.T)
print("\nmat^(-1)\n",np.linalg.inv(mat))
print("\nmat^(-1)-pseudo-inverse\n",np.linalg.pinv(mat))
```
%%%% Output: stream
mat
[[1. 0. 2.]
[0. 1. 0.]
[0. 0. 1.]]
mat^T
[[1. 0. 0.]
[0. 1. 0.]
[2. 0. 1.]]
mat^(-1)
[[ 1. 0. -2.]
[ 0. 1. 0.]
[ 0. 0. 1.]]
mat^(-1)-pseudo-inverse
[[ 1. 0. -2.]
[ 0. 1. 0.]
[ 0. 0. 1.]]
%% Cell type:markdown id: tags:
### vecteur propre et valeur propre
Souvenez-vous qu'en anglais valeur/vecteur 'propre' c'est 'eigen' value/vector
%% Cell type:code id: tags:
``` python
size=4
"""une matrice aléatoire auquel on fait subir une opération de normalisation (laquelle?)"""
mat=np.random.uniform(low=0,high=10,size=[size,size])
rowSum=np.sum(mat,axis=1)
""" mat[i,j]=mat[i,j]/rowSum(i) """
for i in range(size): mat[i,:]/=rowSum[i]
eigVal,eigVec=np.linalg.eig(mat)
print("\nmat\n", mat )
print("\neigVal\n",np.real(eigVal))
print("\neigVec\n", np.real(eigVec))
```
%%%% Output: stream
mat
[[0.085 0.287 0.206 0.422]
[0.191 0.262 0.253 0.295]
[0.244 0.07 0.06 0.626]
[0.383 0.001 0.161 0.454]]
eigVal
[ 1. 0.122 -0.195 -0.066]
eigVec
[[-0.5 -0.416 -0.418 0.39 ]
[-0.5 -0.763 0.307 0.432]
[-0.5 0.402 -0.739 -0.813]
[-0.5 0.287 0.43 -0.036]]
%% Cell type:markdown id: tags:
***A vous:***
* Pourquoi 1 est-il valeur propre ? Quel est le vecteur propre associé? `np.linalg.eig` donne-t-il les vecteurs propres à droite ou à gauche? Sont-ils disposés en ligne ou bien en colonne?
* Toutes les matrices sont-elles diagonalisables? A quoi servent les np.real()?
* laquelle de ces deux formules est la bonne ?
Diag = P^-1 @ mat @ P
ou
Diag = P @ mat @ P^-1
Vérifiez avec python. Aide: Pour transformez un vecteur en matrice diagonale, utilisez `np.diag(vecteur)`
%% Cell type:markdown id: tags:
Attention, lorsqu'on part d'une matrice symétrique (ou hermitienne), pour obtenir la diagonalisation dans une base othornormale il faut utiliser `np.linalg.eigh` (h-comme hermitienne). Vérifiez-le!
%% Cell type:markdown id: tags:
## Définition intuitive d'une chaine de Markov
%% Cell type:markdown id: tags:
***NB:*** Le mot "chaine de Markov" (tout court) est pour nous synonime de "chaine de Markov homogène en temps".
On considère un ensemble dénombrable $E$ : l'ensemble des "états". Par exemple $E=\{0,1,2,3,4,5\}$. On considère aussi que les éléments de $E$ sont des sommets d'un graphe, dont les flèches sont pondérées.
![graph_ponder](./img/graph_ponder.png)
Une chaine de Markov $t\to X_t$ est un processus aléatoire qui se ballade en suivant le graphe. Au temps $t=0$ elle se trouve en une sommet $X_0$ donné. Si au temps $t$ elle se trouve en $X_t$, elle tire une des flèches partant de $X_t$, avec une probabilité proportionnelle aux pondérations, et elle suit cette flèche pour arriver à un nouvel état $X_{t+1}$
Notations:
* Le poids mis sur la fléche allant de $x$ à $y$ est noté $P(x,y)$.
* Quand il n'y a pas de flèche entre $x$ et $y$, on note $P(x,y)=0$.
* La matrice $\big ( P(x,y) \big)_{x,y \in E}$ est appelée matrice de transition.
***Remarques:*** Ici, les étiquettes que l'on a mises sur les états n'ont aucune importance. On aurait pu tout aussi bien prendre $E=\{A,B,C,E,D,F\}$. Dans les exemples suivants, le fait que les indexes soient des entiers aura son importance.
%% Cell type:code id: tags:
``` python
def premier_chaine():
P=np.zeros([6,6])
P[0,3]=4
P[1,0]=2.1
P[2,1]=2.5
P[3,1]=3
P[3,4]=2
P[4,1]=2
P[4,2]=0.2
P[4,5]=6
P[5,2]=7.3
sumLine=np.sum(P,axis=1)
""" P[i,j]=P[i,j]/sumLine[i] """
P/=np.expand_dims(sumLine,axis=1)
return P
print(premier_chaine())
```
%%%% Output: stream
[[0. 0. 0. 1. 0. 0. ]
[1. 0. 0. 0. 0. 0. ]
[0. 1. 0. 0. 0. 0. ]
[0. 0.6 0. 0. 0.4 0. ]
[0. 0.244 0.024 0. 0. 0.732]
[0. 0. 1. 0. 0. 0. ]]
%% Cell type:code id: tags:
``` python
def markov_from_P(t_max,P,x0):
X=np.zeros(t_max,dtype=np.int)
X[0]=x0
for t in range(t_max-1):
X[t+1]=np.random.choice(a=range(len(P)),p=P[X[t],:])
return X
t_max=300
P=premier_chaine()
X=markov_from_P(t_max,P,3)
plt.figure(figsize=(20,3))
plt.plot(range(t_max),X);
plt.grid()
```
%%%% Output: display_data