adding linear interpolation to a fitness equation

So I'm creating a dynamic state-variable model and I want to add in a forgetting rate for an informational state, however that'll make the values non-integers and thus I need to do linear interpolation. How do I add that in to my fitness equation?

Sorry for the overwhelming amount of code

%constants
m1=0.9;
m2=0.3;
n1=0.3;
n2=0.9;
b=0.9;
crit=3;
cap=15;
term=20;

%arrays and zeros

opt=zeros(cap,cap, cap,term);
Fd=zeros(cap, cap, cap,term);
Fdd=zeros(cap,cap,cap,term);
Fdb=zeros(cap,cap, cap,term);
Fdbb=zeros(cap,cap,cap,term);
Ft=zeros(cap,cap,cap,term);

x=linspace(1,cap,cap)';
z=linspace(1,cap,cap)';
y=linspace(1,cap,cap)';
f=@(x,y) x-y;
M=f(z.',y);

p=zeros(size(M));
p2=zeros(size(M));
p3=zeros(size(M));
p4=zeros(size(M));
p5=zeros(size(M));
%z and y
z1=z+1; %dem went to patch 1 and found food
z1(z1>cap)=cap;
z2=z-1; %dem went to patch 1 and didn't find food
z2(z2<1)=1;

zp=(z1+1); %dem went to patch 1 and found food; obs went to patch 1 and found food
zp(zp>cap)=cap;
zpp=(z1-1); %dem went to patch1 and found food; obs went to patch 1 and didn't find food
zpp(zpp<1)=1;
zd=(z2+1); %dem went to patch1 and didn't find food; obs went to patch 1 and found food
zd(zd>cap)=cap;
zdd=(z2-1); %dem went to patch 1 and didn't find food; obs went to patch 1 and didn't find food
zdd(zdd<1)=1;
zg=(z+1); %obs went to patch 1 and found food
zg(zg>cap)=cap;
zgg=(z-1); %obs went to patch 1 and didn't find food
zgg(zgg<1)=1;

y1=y+1; %dem went to patch 2 and found food
y1(y1>cap)=cap;
y2=y-1; %dem went to patch 2 and didn't find food
y2(y2<1)=1;

yp=(y1+1); %dem went to patch 2 and found food; obs went to patch 2 and found food
yp(yp>cap)=cap;
ypp=(y1-1); %dem went to patch 2 and found food; obs went to patch 2 and didn't food
ypp(ypp<1)=1;
yd=(y2+1); %dem went to patch 2 and didn't find food; obs went to patch 2 and found food
yd(yd>cap)=cap;
ydd=(y2-1); %dem went to patch 2 and didn't find food; obs went to patch 2 and didn't find food
ydd(ydd<1)=1;
yg=(y+1); %obs went to patch 2 and found food
yg(yg>cap)=cap;
ygg=(y-1); %obs went to patch 2 and didn't find food
ygg(ygg<1)=1;

%physical states
xp=x-1-1+3; %obs watched dem and found food
xp(xp>cap)=cap;
xpp=x-1-1; %obs watched dem and didn't find food
xpp(xpp<1)=1;

xd=x-1+3; %obs didn't watch and found food
xd(xd>cap)=cap;
xdd=x-1; %obs didn't watch and didn't find food
xdd(xdd<1)=1;

%specify fitness
Fd(x<=crit,:, :, :)=0;
Fd(x>crit,:,:,:)=1;
Fdb(x<=crit,:,:,:)=0;
Fdb(x>crit,:,:,:)=1;

%specify prob matrix of going to patch 1 for obs
    for j=1:15
    for k=1:15
            
                if M(j,z1(k))>0
                    p(j,z1(k))=0.9;
                elseif M(j,z1(k))<0
                    p(j,z1(k))=0.1;
                else
                    p(j,z1(k))=0.5;
                end
                if M(j,z2(k))>0
                    p2(j,z2(k))=0.9;
                elseif M(j,z2(k))<0
                    p2(j,z2(k))=0.1;
                else
                    p2(j,z2(k))=0.5;
                end
                if M(y1(j),k)>0
                    p3(y1(j),k)=0.9;
                elseif M(y1(j),k)<0
                    p3(y1(j),k)=0.1;
                else
                    p3(y1(j),k)=0.5;
                end
                if M(y2(j),k)>0
                    p4(y2(j),k)=0.9;
                elseif M(y2(j),k)<0
                    p4(y2(j),k)=0.1;
                else
                    p4(y2(j),k)=0.5;
                end
                if M(y(j),z(k))>0
                    p5(y(j),z(k))=0.9;
                elseif M(y(j),z(k))<0
                    p5(y(j),z(k))=0.1;
                else
                    p5(y(j),z(k))=0.5;
                end
    end
    end

%fitness equations for watching a dem (Fdd) and not watching a dem (Fdbb)
for tt=19:-1:1
    for i=1:15
    for j=1:15
        for k=1:15
            
        Fdd(i,j,k,tt)= b*(... 
            m1*(...
                p(j,z1(k))*(n1*Fd(xp(i),zp(j),y(k),tt+1) + (1-n1)*Fd(xpp(i),zpp(j),y(k),tt+1))+...
                (1-p(j,z1(k)))*(n2*Fd(xp(i),z1(j),yg(k),tt+1) + (1-n2)*Fd(xpp(i),z1(j),ygg(k),tt+1)))+...
                (1-m1)*(... 
                     p2(j,z2(k))*(n1*Fd(xp(i),zd(j),y(k),tt+1) + (1-n1)*Fd(xpp(i),zdd(j),y(k),tt+1))+...
                (1-p2(j,z2(k)))*(n2*Fd(xp(i),z2(j),yg(k),tt+1) + (1-n2)*Fd(xpp(i),z2(j),ygg(k),tt+1))...
                    ))+...
            (1-b)*( ...
                m2*(...
                p3(y1(j),k)*(n1*Fd(xp(i),zg(j),y1(k),tt+1) + (1-n1)*Fd(xpp(i) ,zgg(j),y1(k),tt+1))+ ...
                (1-p3(y1(j),k))*(n2*Fd(xp(i),z(j),yp(k),tt+1) + (1-n2)*Fd(xpp(i) ,z(j),ypp(k),tt+1)))+ ...
                (1-m2)*(p4(y2(j),k)*(n1*Fd(xp(i),zg(j),y2(k),tt+1) + (1-n1)*Fd(xpp(i),zgg(j),y2(k),tt+1))+...
            (1-p4(y2(j),k))*(n2*Fd(xp(i),z(j),yd(k),tt+1) + (1-n2)*Fd(xpp(i),z(j),ydd(k),tt+1))));
        Fdbb(i,j,k,tt)= p5(j,k)*( ...
                    n1*Fdb(xd(i),zg(j),y(k),tt+1) + (1-n1)*Fdb(xdd(i),zgg(j),y(k),tt+1) ...
                    )+...
                (1-p5(j,k))*( ...
                    n2*Fdb(xd(i),z(j),yg(k),tt+1) + (1-n2)*Fdb(xdd(i),z(j),ygg(k),tt+1)) ...
                   ;

   %optimal decision and fitness
            if Fdd(i,j,k,tt)>Fdbb(i,j,k,tt)
        opt(i,j,k,tt)=1;
        Ft(i,j,k,tt)=Fdd(i,j,k,tt);

            elseif Fdd(i,j,k,tt)<Fdbb(i,j,k,tt)
        opt(i,j,k,tt)=2;
        Ft(i,j,k,tt)=Fdbb(i,j,k,tt);
            elseif Fdd(i,j,k,tt)==Fdbb(i,j,k,tt)
                opt(i,j,k,tt)=3;
                Ft(i,j,k,tt)=Fdd(i,j,k,tt);
            
end
end 
end 
end
end 

I would like to add in a forgetting rate so that it looks more like this:

l=0.95;

zp=l*(z1+1); %dem went to patch 1 and found food; obs went to patch 1 and found food
zp(zp>cap)=cap;
zpp=l*(z1-1); %dem went to patch1 and found food; obs went to patch 1 and didn't find food
zpp(zpp<1)=1;
zd=l*(z2+1); %dem went to patch1 and didn't find food; obs went to patch 1 and found food
zd(zd>cap)=cap;
zdd=l*(z2-1); %dem went to patch 1 and didn't find food; obs went to patch 1 and didn't find food
zdd(zdd<1)=1;
zg=l*(z+1); %obs went to patch 1 and found food
zg(zg>cap)=cap;
zgg=l*(z-1); %obs went to patch 1 and didn't find food
zgg(zgg<1)=1;

%this would be added to the y's as well

How do I edit the fitness equation to add in linear interpolation?

note that Fdd is a 4-D matrix, because it has a physical state (x), two infomational states (z and y), and then time (t)

most of the interpolation I see modifies 1-D matrices

[Sam Chak]

@Kitt: So I'm creating a dynamic state-variable model and I want to add in a forgetting rate for an informational state, however that'll make the values non-integers and thus I need to do linear interpolation. How do I add that in to my fitness equation?

Hi, to be honest, I'm a bit confused. Based on the keyword 'state-variable', I assume you want to create a state-space model, which is typically described by ordinary differential equations. Is that correct?

If so, how are the 'forgetting rate' and 'fitness equation' related to the state-space model? I'm trying to understand the connection between them.

[Kitt]

Basically, there are two patches an individual can choose to forage at and I'm trying to model an individual's fitness if they decide to watch a demonstrator forage at a patch before the individual forages at a patch.

So my model is about an individual's fitness based on a physical state, and two informational states throughout 20 timesteps.

The entire fitness equation is:

for tt=20:-1:1
    for i=1:15
    for j=1:15
        for k=1:15
            
        Fdd(i,j,k,tt)= b*(... 
            m1*(...
                p(j,z1(k))*(n1*Fd(xp(i),zp(j),y(k),tt+1) + (1-n1)*Fd(xpp(i),zpp(j),y(k),tt+1))+...
                (1-p(j,z1(k)))*(n2*Fd(xp(i),z1(j),yg(k),tt+1) + (1-n2)*Fd(xpp(i),z1(j),ygg(k),tt+1)))+...
                (1-m1)*(... 
                     p2(j,z2(k))*(n1*Fd(xp(i),zd(j),y(k),tt+1) + (1-n1)*Fd(xpp(i),zdd(j),y(k),tt+1))+...
                (1-p2(j,z2(k)))*(n2*Fd(xp(i),z2(j),yg(k),tt+1) + (1-n2)*Fd(xpp(i),z2(j),ygg(k),tt+1))...
                    ))+...
            (1-b)*( ...
                m2*(...
                p3(y1(j),k)*(n1*Fd(xp(i),zg(j),y1(k),tt+1) + (1-n1)*Fd(xpp(i) ,zgg(j),y1(k),tt+1))+ ...
                (1-p3(y1(j),k))*(n2*Fd(xp(i),z(j),yp(k),tt+1) + (1-n2)*Fd(xpp(i) ,z(j),ypp(k),tt+1)))+ ...
                (1-m2)*(p4(y2(j),k)*(n1*Fd(xp(i),zg(j),y2(k),tt+1) + (1-n1)*Fd(xpp(i),zgg(j),y2(k),tt+1))+...
            (1-p4(y2(j),k))*(n2*Fd(xp(i),z(j),yd(k),tt+1) + (1-n2)*Fd(xpp(i),z(j),ydd(k),tt+1))));

With Fdd being the fitness equation if an individual watches a demonstrator forage and Fd being a matrix of fitness that's based on the physical state, which is the first dimension. When x<= 3 Fd is 0 and when x>3 Fd is 1.

cap=15;
term=20;

Fd=zeros(cap, cap, cap,term);

Fd(x<=crit,:, :, :)=0;
Fd(x>crit,:,:,:)=1;

That matrix is then put into the fitness equation Fdd that accounts for all possible combinations of a demonstrator foraging between two patches and the observer foraging between two patches.

The patch an individual forages at is based on their information on patch 1, state z, - their information on patch 2, state y. They then either find food or they don't find food which changes their physical state, state x.

I'm trying to add in a forgetting rate, which will change the information states z and y to non-integers (I'll be multiplying them by 0.95, which I'm modeling after something I've seen in a similar paper).

I'm really sorry if this is more confusing than helping. I'm new to matlab and I'm trying to make this model after learning about stochastic dynamic modeling in a class. I'm kind of frankenstiening this model off of a model that was used in my class. It's totally possible I'm using terms incorrectly.

Didn't realize at the beginning of this that adding in a seemingly simple choice of watching a demonstrator would lead to a very complicated model.

[Sam Chak]

Hi @Kitt

No worries, and there's no need to apologize. It was actually my mistake to confuse the dynamic state-variable model with a state-space representation. Thank you for clarifying that for me.

However, if you could revise/update your Question/Problem to include some mathematical equations for the fitness, demonstrator forage, and informational states, it would allow others to verify your code against the provided mathematical equations.

[Kitt]

Alright I revised it to inlude the whole code so that it runs. I feel bad including it all, since that's kind of one of the rules not to do, but there's just a lot of moving parts! hopefully this makes it more clear what it is I'm trying to do.

Answer 1

[Nipun]

Hi Kitt,

I understand that you want to add linear interpolation to a fitness equation in MATLAB. Here's how you can do it using interp1 for linear interpolation within your fitness function.

Assume you have a set of data points and you want to interpolate these points within your fitness function. Here’s a concise example:

Example Data Points

x = [1, 2, 4, 5];
y = [1, 4, 2, 3];

Fitness Function with Linear Interpolation

function fitness = myFitnessFunction(xq)
    % Given data points for interpolation
    x = [1, 2, 4, 5];
    y = [1, 4, 2, 3];
    
    % Linear interpolation
    yq = interp1(x, y, xq, 'linear');
    
    % Define your fitness equation using interpolated values
    fitness = sum(yq.^2); % Example fitness equation: sum of squares of interpolated values
end

Usage

% Query points
xq = [1.5, 3, 4.5];

% Calculate fitness
fitness_value = myFitnessFunction(xq);
disp(fitness_value);

Replace the fitness equation (sum(yq.^2)) with your actual fitness equation as needed. This example demonstrates how to use interp1 for linear interpolation within a custom fitness function.

Hope this helps.

Regards,

Nipun