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Project: FFLY
Views: 34

Ici je cherche à estimer a0a0,γ\gamma et σ\sigma. Auparavant on n'estimait pas sigma et du coup on était linéaire.

# Variables symbolique x,y,sigma = var("x,y,sigma") xn,yn=var("x_n,y_n") Yn=var("Y_n") G,a0,gamma,R=var("G_n a_0 gamma R") yb=function("yb",nargs=2,latex_name="\overline{Y_n}") # Fonction likelihood def L(yb): return -log(sqrt(2*pi)*sigma)-(1/2)*((Yn-yb(x,y))/sigma)**2 # Fonction Jacobienne def j(expr,sym): return [expr.diff(i) for i in sym] J= j(L(yb),[x,y,sigma])

La jacobienne du log likelihood sans connaitre la fonction Yn(x,y)\overline{Y_n}(x,y) et en dérivant par rapport à xx,yy et σ\sigma:

show(J)
[(YnYn(x,y))xYn(x,y)σ2\displaystyle \frac{{\left(Y_{n} - \overline{Y_n}\left(x, y\right)\right)} \frac{\partial}{\partial x}\overline{Y_n}\left(x, y\right)}{\sigma^{2}}, (YnYn(x,y))yYn(x,y)σ2\displaystyle \frac{{\left(Y_{n} - \overline{Y_n}\left(x, y\right)\right)} \frac{\partial}{\partial y}\overline{Y_n}\left(x, y\right)}{\sigma^{2}}, (YnYn(x,y))2σ31σ\displaystyle \frac{{\left(Y_{n} - \overline{Y_n}\left(x, y\right)\right)}^{2}}{\sigma^{3}} - \frac{1}{\sigma}]

La même en remplacant Yn(x,y)\overline{Y_n}(x,y) par Gna010γlog10(dn)RG_n-a_0-10\gamma\log_{10}(d_n)-R

def yb(x,y): d = sqrt((x-xn)**2+(y-yn)**2) return G-a0-10*gamma*log(d,10)-R J= j(L(yb),[x,y,sigma]) show(J)
[10(GnRYna010γlog((xxn)2+(yyn)2)log(10))γ(xxn)((xxn)2+(yyn)2)σ2log(10)\displaystyle \frac{10 \, {\left(G_{n} - R - Y_{n} - a_{0} - \frac{10 \, \gamma \log\left(\sqrt{{\left(x - x_{n}\right)}^{2} + {\left(y - y_{n}\right)}^{2}}\right)}{\log\left(10\right)}\right)} \gamma {\left(x - x_{n}\right)}}{{\left({\left(x - x_{n}\right)}^{2} + {\left(y - y_{n}\right)}^{2}\right)} \sigma^{2} \log\left(10\right)}, 10(GnRYna010γlog((xxn)2+(yyn)2)log(10))γ(yyn)((xxn)2+(yyn)2)σ2log(10)\displaystyle \frac{10 \, {\left(G_{n} - R - Y_{n} - a_{0} - \frac{10 \, \gamma \log\left(\sqrt{{\left(x - x_{n}\right)}^{2} + {\left(y - y_{n}\right)}^{2}}\right)}{\log\left(10\right)}\right)} \gamma {\left(y - y_{n}\right)}}{{\left({\left(x - x_{n}\right)}^{2} + {\left(y - y_{n}\right)}^{2}\right)} \sigma^{2} \log\left(10\right)}, (GnRYna010γlog((xxn)2+(yyn)2)log(10))2σ31σ\displaystyle \frac{{\left(G_{n} - R - Y_{n} - a_{0} - \frac{10 \, \gamma \log\left(\sqrt{{\left(x - x_{n}\right)}^{2} + {\left(y - y_{n}\right)}^{2}}\right)}{\log\left(10\right)}\right)}^{2}}{\sigma^{3}} - \frac{1}{\sigma}]