Relation between the new model (the one in Sara’s paper) and the old one (the one in PLoS). It is analytical!

Name of this file: Cuaderno20120402T111150.m

Model definition

I define the new model with k=1, so that we have deltan

P=@(ax,ay,s,deltan) (1+(1+ax*s.^-deltan)./(1+ay*s.^deltan)).^-1;
Pold=@(a,s,deltan) (1+a*s.^-deltan).^-1;

Comparison for a=1

They are identical

figure
deltan=-10:10;
a=1;
s=2.5;
ax=1;
ay=1;
plot(deltan,Pold(a,s,deltan))
hold on
plot(deltan,P(ax,ay,s,deltan),’r–‘)

 

Comparison for a~=1

I check for a=10.

figure
deltan=-10:10;
a=10;
s=2.5;
ax=1;
ay=.1;
plot(deltan,Pold(a,s,deltan))
hold on
plot(deltan,P(ax,ay,s,deltan),’r–‘)

aes=10.^(-5:.1:5);
n_a=length(aes);
errores1=NaN(n_a);
for cx_a=1:n_a
for cy_a=1:n_a
errores1(cy_a,cx_a)=sum(abs(Pold(a,s,deltan)-P(aes(cx_a),aes(cy_a),s,deltan)));
end % cy
end % cx
[m,ind]=min(errores1(:));
[i,j]=ind2sub(size(errores1),ind);
ax=aes(j)
ay=aes(i)
figure
imagesc(log10(aes),log10(aes),errores1)
colorbar
figure
imagesc(log10(aes),log10(aes),log10(errores1))
colorbar
xlabel(‘log_{10}(a_x)’)
ylabel(‘log_{10}(a_y)’)
figure
plot(deltan,Pold(a,s,deltan))
hold on
plot(deltan,P(ax,ay,s,deltan),’r–‘)

ax =
10
ay =
0.1000

 

 

 

 

So we get a very clear minimum, with excellent agreement.

General search for a~=1

I repeat the whole procedure for different values of a.

a=10.^(-5:.5:5);
m=NaN(length(a),1);
ax=m;
ay=m;
for c_a=1:length(a)
aes=10.^(-5:.1:5);
n_a=length(aes);
errores2=NaN(n_a);
for cx_a=1:n_a
for cy_a=1:n_a
errores2(cy_a,cx_a)=sum(abs(Pold(a(c_a),s,deltan)-P(aes(cx_a),aes(cy_a),s,deltan)));
end % cy
end % cx
[m(c_a),ind]=min(errores2(:));
[i,j]=ind2sub(size(errores2),ind);
ax(c_a)=aes(j);
ay(c_a)=aes(i);
end % c_a
subplot(1,2,1)
plot(log10(a),log10(m))
xlabel(‘log_{10}(a_{old})’)
ylabel(‘log_{10}(error)’)
subplot(1,2,2)
hold off
plot(log10(a),log10(ax))
hold on
plot(log10(a),log10(ay),’r’)
xlabel(‘log_{10}(a_{old})’)
ylabel(‘log_{10}(a_{new})’)

 

figure
loglog(a,ax./ay,’.-‘)
xlabel(‘a_{old}’)
ylabel(‘a_x/a_y’)
figure
loglog(a,ax)
xlabel(‘a_{old}’)
ylabel(‘a_x’)
figure
loglog(a,ay)
xlabel(‘a_{old}’)
ylabel(‘a_y’)
figure
loglog(ax,ay)
xlabel(‘a_x’)
ylabel(‘a_y’)

 

 

 

 

So incredibly simple relation! There must be something analytical here…

Efectiviguonder

It turns out that

.

So it is analytical! We just have ax=a, ay=1/a.