Plotting model manifolds

(Sethna, )
© 2024, James P. Sethna, all rights reserved.

Remember to not loop over the times: exp({1,2,3}) = [e^1,e^2,e^3]

y[θ_][t_]:= 1/Length[θ] Sum[Exp[-θ[[n]] t],{n,1,Length[θ]}]
ParametricPlot3D[{y[{θα,θβ}][...], y[...][...],...},{θα,0.,10.},{θβ,0.0,10}, PlotRange->{{0,1},{0,1},{0,1}}]

M=10000; (* or other large number *)
th2s = 10^(-2+4RandomReal[1,{M,2}]); (* Gives rho(x) \sim 1/x in [10^-2, 10^2] *)
points = Table[{y[th2s[[m]]][1/3],y[...][...],...}, {m,1,Length[th2s]}];
ListPointPlot3D[points, PlotRange->{{0,1},{0,1},{0,1}},BoxRatios->{1, 1, 1}]

th7s = 10^(...[...,{M,7}]);
points = Table[...];
ListPointPlot3D[...]

ts = Table[t,{t,0,10,0.5}];
(* Avoid error messages for underflow for late times, fast decays *)
Off[General::munfl ];
points = Table[...];
pca2 = PrincipalComponents[points];
pcaBiggest = pca2[[All,1;;3]];
ListPointPlot3D[pcaBiggest, PlotRange->{{-1,3},{-1,1},{-1,1}}, BoxRatios->{2, 1, 1},ViewPoint->Above]

points = Table[...];
pca7 = ...;
pcaBiggest = ...;
ListPointPlot3D[pcaBiggest, PlotRange->..., BoxRatios->...,ViewPoint->Above]
pcaNextThree = ...;
zoom = ...;
ListPointPlot3D[pcaNextThree, PlotRange->{{-zoom,zoom},{-zoom,zoom},{-zoom,zoom}}, BoxRatios->{1, 1, 1},ViewPoint->Above]