I first begin by running the code below to tune a SVM:
tunecontrol <- tune.control(nrepeat=5, sampling = "fix",cross=5, performances=T)
tune_svm1 <- tune(svm,
Y ~ 1
+ X
, data = data,
ranges = list(epsilon = seq(epsilon_start
,epsilon_end
,(epsilon_end-epsilon_start)/10)
, cost = cost_start*(1:5)
, gamma = seq(gamma_start
,gamma_end
,(gamma_end - gamma_start)/5))
, tunecontrol=tunecontrol)
In tune_svm1$performances I have 330 observations containing all the values for epsilon, cost, and gamma that I stated in the ranges section of the above code as well as another column for the calculated error.
I'd like to generate a 3D surface plot for epsilon, cost, gamma, and error using three variables as X,Y,Z and the last for color. I've read on several resources for plot3d and persp but have had a lot of difficulty implementing.
If I try to follow the examples provided and use mesh to generate a mesh plot, I can only mesh together 3 of the 4 variables from tune_svm1$performances and saving the separate results for X,Y and Z as shown in the first link is difficult because the mesh is saved as an array, not a matrix. I've tried to hack a graph using the following code but the visual is nonsensical (probably because the order isn't being preserved by meshing each individually:
M1 <- mesh(tune_svm1$performances$epsilon[1:nrow(tune_svm1$performances)]
,tune_svm1$performances$cost[1:nrow(tune_svm1$performances)])
M2 <- mesh(tune_svm1$performances$epsilon[1:nrow(tune_svm1$performances)]
,tune_svm1$performances$gamma[1:nrow(tune_svm1$performances)])
M3 <- mesh(tune_svm1$performances$epsilon[1:nrow(tune_svm1$performances)]
,tune_svm1$performances$error[1:nrow(tune_svm1$performances)])
x <- M1$x ; y <- M1$y ; z <- M2$y ; c <- M3$y
surf3D(x,y,c, colvar = c)
What's the best way to approach this? Thank you.
Related
I am working with a biological model of the distribution of microbial biomass (b1) on a 2D grid. From the biomass a protein (p1) is produced. The biomass diffuses over the grid, while the protein does not. Only if a certain amount of protein is produced (p > p_lim), the biomass is supposed to diffuse.
I try to implement this by using a dummy cell variable z multiplied with the diffusion coefficient and setting it from 0 to 1 only in cells where p > p_lim.
The condition works fine and when the critical amount of p is reached in a cell, z is set to 1, and diffusion happens. However, the diffusion still does not work with the rate I would like, because to calculate diffusion, the face variable, not the value of the cell itself is used. The faces of z are always a mean of the cell with z=1 and its neighboring cells with z=0. I I, however, would like the diffusion to work at its original rate even if the neighbouring cell is still at p < p_lim.
So, my question is: Can i somehow access a faceVariable and change it? For example, set a face to 1 if any neigboring cell has reached p1 > p_lim? I guess this is not a proper mathematical thing to do, but I couldn't think of another way to simulate this problem.
I will show a very reduced form of my model below. In any case, I thank you very much for your time!
##### produce mesh
nx= 5.
ny= nx
dx = 1.
dy = dx
L = nx*dx
mesh = Grid2D(nx=nx,ny=ny,dx=dx,dy=dy)
#parameters
h1 = 0.5 # production rate of p
Db = 10. # diffusion coeff of b
p_lim=0.1
# cell variables
z = CellVariable(name="z",mesh=mesh,value=0.)
b1 = CellVariable(name="b1",mesh=mesh,hasOld=True,value=0.)
p1= CellVariable(name="p1",mesh=mesh,hasOld=True,value=0.)
# equations
eqb1 = (TransientTerm(var=b1)== DiffusionTerm(var=b1,coeff=Db*z.arithmeticFaceValue)-ImplicitSourceTerm(var=b1,coeff=h1))
eqp1 = (TransientTerm(var=p1)==ImplicitSourceTerm(var=b1,coeff=h1))
# set b1 to 10. in the center of the grid
b1.setValue(10.,where=((x>2.)&(x<3.)&(y>2.)&(y<3.)))
vi=Viewer(vars=(b1,p1),FIPY_VIEWER="matplotlib")
eq = eqb1 & eqp1
from builtins import range
for t in range(10):
b1.updateOld()
p1.updateOld()
z.setValue(z + 0.1,where=((p1>=p_lim) & (z < 1.)))
eq.solve(dt=0.1)
vi.plot()
In addition to .arithmeticFaceValue, FiPy provides other interpolators between cell and face values, such as .harmonicFaceValue and .minmodFaceValue.
These properties are implemented using subclasses of _CellToFaceVariable, specifically _ArithmeticCellToFaceVariable, _HarmonicCellToFaceVariable, and _MinmodCellToFaceVariable.
You can also make a custom interpolator by subclassing _CellToFaceVariable. Two such examples are _LevelSetDiffusionVariable and ScharfetterGummelFaceVariable (neither is well documented, I'm afraid).
You need to override the _calc_() method to provide your custom calculation. This method takes three arguments:
alpha: an array of the ratio (0-1) of the distance from the face to the cell on one side, relative to the distance from distance from the cell on the other side to the cell on the first side
id1: an array of indices of the cells on one side of the face
id2: an array of indices of the cells on the other side of the face
Note: You can ignore any clause if inline.doInline: and look at the _calc_() method defined under the else: clause.
I am trying to take numbers from two intervals in Julia. The problem is the following,
I am trying to create concentric spheres and I need to generate vectors of dimension equal to 15 filled with numbers taken from each circle. The code is:
rmax = 5
ra = fill(0.0,1,rmax)
for i=1:rmax-1
ra[:,i].=rad/i
ra[:,rmax].= 0
end
for i=1:3
ptset = Any[]
for j=1:200
yt= 0
yt= rand(Truncated(Normal(0, 1), -ra[i], ra[i] ))
if -ra[(i+1)] < yt <= -ra[i] || ra[(i+1)] <= yt < ra[i]
push!(ptset,yt)
if length(ptset) == 15
break
end
end
end
end
Here, I am trying to generate spheres with uniform random numbers inside of each one; In this case, yt is only part of the construction of the numbers inside the sphere.
I would like to generate points in a sphere with radius r0 (ra[:,4] for this case), then points distributed from the edge of the first sphere to the second one wit radius r1 (here ra[:,3]) and so on.
In order to do that, I try to take elements that fulfill one of the two conditions -ra[(i+1)] < yt <= -ra[i]
or ra[(i+1)] <= yt < ra[i], i.e. I would like to generate a vector with positive and negative numbers. I used the operator || but it seems to take only the positive part. I am new in Julia and I am not sure how to take the elements from both parts of the interval. Does anyone has a hit on how to do it?. Thanks in advance
I hope I understood you correctly. First, we need to be able to sample uniformly from an N-dimensional shell with radii r0 and r1:
using Random
using LinearAlgebra: normalize
struct Shell{N}
r0::Float64
r1::Float64
end
Base.eltype(::Type{<:Shell}) = Vector{Float64}
function Random.rand(rng::Random.AbstractRNG, d::Random.SamplerTrivial{Shell{N}}) where {N}
shell = d[]
Δ = shell.r1 - shell.r0
θ = normalize(randn(N)) # uniformly distributed N-dimensional direction of length 1
r = shell.r0 .* θ # scale to a point on the interior of the shell
return r .+ Δ .* θ .* .√rand(N) # add a uniformly random segment between r0 and r1
end
(See here for more info about hooking into Random. You could equally implement a new Distribution, but that's not really necessary.)
Most importantly, a truncated normal will not result in a uniform distribution, but neither will adding a uniform scaling into the right direction: see here for why the square root is necessary (and I hope I got it right; you should check the math once more).
Then we can just create a sequence of shell samples with nested radii:
rmax = 5
rad = 10.0
ra = range(0, rad, length=rmax)
ptset = [rand(Shell{2}(ra[i], ra[i+1]), 15) for i = 1:(rmax - 1)]
(This part I wasn't really sure about, but the point should be clear.)
I am trying to do up a 3D array plot in R.
I already have an array built up and defined with the corresponding z-values
e.g. CVHSP500 = array(0,c((nHSP500-N),N))
So now I am trying to do a 3D array plot with it. I decided to go with persp3d(CVHSP500,col = "lightblue",) and have obtained a rather decent plot.
3D Image
So there are obviously some issues with this plot.
1) The coordinates are not defined correctly.
Reading up online on the usage of persp3D, and other R programming functions/packages like slice3D, they all require x, y and z to be separate list.
I don't understand how to match the values of x and y to the respective z, and since persp3D works perfectly without me having to do that, I decided to use persp3D.
But I will need to insert coordinates for it, but I have no idea how to.
2) Any advice how do I color the plots for different ranges of z?
The ones online all seem to have to refer to individual x, y and z lists and some form of advanced modification which I can't really understand. This light blue color looks okay but it would be good for different ranges of z as well though.
Thanks for the help. Much appreciated.
To transform a 2D array representing z for each (x,y) into 3 vectors x, y and z, you can do this:
CVHSP500 = array(0,c((nHSP500-N),N))
x <- rep(1:(nHSP500-N),N)
y <- rep(1:N,(nHSP500-N))
z <- CVHSP500
dim(z) <- (nHSP500-N)*N
I have a plotting question regarding boxplots (using base graphics).
I have several arrays of data which I wish to turn into box plots and compare. The arrays reflect different experiments and what I would like to show is the base results and the percentage difference for the experiments (on one plot!). I.e. the base results on the 1st y axis and the % diff on the second y axis:
base <- array(runif(12*24*3), dim=c(12,24,3))
exp1 <- array(runif(12*24*3), dim=c(12,24,3))
exp2 <- array(runif(12*24*3), dim=c(12,24,3))
exp3 <- array(runif(12*24*3), dim=c(12,24,3))
exp4 <- array(runif(12*24*3), dim=c(12,24,3))
# calc p.diff
p.diff <- function(mod,base) {
100.0*((mod-base)/base) }
a <- p.diff(exp1,base)
b <- p.diff(exp2,base)
c <- p.diff(exp3,base)
# combine the % diff arrays
exps <- list(a,b,c)
# plot the results
boxplot(base, xlim=c(1,4), col="gray", xaxt="n", ylab="Base values", outline=FALSE)
axis(side=1, 1:4, labels=c("base","% exp1","% exp2","% exp3") )
par(new=TRUE)
boxplot(exps, col="red", ylim=c(-200,200), outline=FALSE, axes=FALSE)
axis(4)
grid()
This almost works but I don't get the positioning of the different box plots right (if you run my example you will see what I mean). So is there a better way to control the placement of the box plots? Or a better way to produce a similar type of figure?
Edited (1): You need to define the rigth sequences for the X axis. So that the plots don't overlap. Just try to play with it.
I think the labels of the X axes are not at the right place? I don't know a more elegant way of doing it but here is a solution:
# plot the results
boxplot(base, xlim=c(1,4), col="gray", xaxt="n", ylab="Base values", outline=FALSE)
axis(side=1,1,labels=('base'))
par(new=TRUE)
boxplot(exps, col="red", ylim=c(-200,200), outline=FALSE, axes=FALSE)
axis(4)
axis(side=1,1:3,labels=c("% exp1","% exp2","% exp3"))
grid()
So I added every label after creating the boxplot. First plot the base and label it, then plot exps and label it. Does it solve your problem?
Edit: Just to be more clear, You are adding a new plot with 3 values, that is why axis(side=1,1:3,labels=c("% exp1","% exp2","% exp3")) is from 1 to 3...
Edited (2):
Why don't you use multi rows in the plot and try to plot 2 graphs? Here is an example with your data:
#divide your plottin area into 2 columns with one row.
par(mfrow = c(1, 2))
# plot the results
boxplot(base, col="gray", xaxt="n", ylab="Base values", outline=FALSE,axes=FALSE)
axis(2)
axis(side=1,1,labels=('base'))
segments(0,0,1,0)
boxplot(exps,col="red", xaxt="n", ylim=c(-200,200), outline=FALSE, axes=FALSE)
axis(4)
axis(side=1,at=(1:3),labels=c("% exp1","% exp2","% exp3"))
you can have more information about it from here
Given the vector size N, I want to generate a vector <s1,s2, ..., sn> that s1+s2+...+sn = S.
Known 0<S<1 and si < S. Also such vectors generated should be uniformly distributed.
Any code in C that helps explain would be great!
The code here seems to do the trick, though it's rather complex.
I would probably settle for a simpler rejection-based algorithm, namely: pick an orthonormal basis in n-dimensional space starting with the hyperplane's normal vector. Transform each of the points (S,0,0,0..0), (0,S,0,0..0) into that basis and store the minimum and maximum along each of the basis vectors. Sample uniformly each component in the new basis, except for the first one (the normal vector), which is always S, then transform back to the original space and check if the constraints are satisfied. If they are not, sample again.
P.S. I think this is more of a maths question, actually, could be a good idea to ask at http://maths.stackexchange.com or http://stats.stackexchange.com
[I'll skip "hyper-" prefix for simplicity]
One of possible ideas: generate many uniformly distributed points in some enclosing volume and project them on the target part of plane.
To get uniform distribution the volume must be shaped like the part of plane but with added margins along plane normal.
To uniformly generate points in such volumewe can enclose it in a cube and reject everything outside of the volume.
select margin, let's take margin=S for simplicity (once margin is positive it affects only performance)
generate a point in cube [-M,S+M]x[-M,S+M]x[-M,S+M]
if distance to the plane is more than M, reject the point and go to #2
project the point on the plane
check that projection falls into [0,S]x[0,S]x[0,S], if not - reject and go to #2
add this point to the resulting set and go to #2 is you need more points
The problem can be mapped to that of sampling on linear polytopes for which the common approaches are Monte Carlo methods, Random Walks, and hit-and-run methods (see https://www.jmlr.org/papers/volume19/18-158/18-158.pdf for examples a short comparison). It is related to linear programming, and can be extended to manifolds.
There is also the analysis of polytopes in compositional data analysis, e.g. https://link.springer.com/content/pdf/10.1023/A:1023818214614.pdf, which provide an invertible transformation between the plane and the polytope that can be used for sampling.
If you are working on low dimensions, you can use also rejection sampling. This means you first sample on the plane containing the polytope (defined by your inequalities). This later method is easy to implement (and wasteful, of course), the GNU Octave (I let the author of the question re-implement in C) code below is an example.
The first requirement is to get vector orthogonal to the hyperplane. For a sum of N variables this is n = (1,...,1). The second requirement is a point on the plane. For your example that could be p = (S,...,S)/N.
Now any point on the plane satisfies n^T * (x - p) = 0
we assume also that x_i >= 0
With these given you compute an orthonormal basis on the plane (the nullity of the vector n) and then create random combination on that bases. Finally you map back to the original space and apply your constraints on the generated samples.
# Example in 3D
dim = 3;
S = 1;
n = ones(dim, 1); # perpendicular vector
p = S * ones(dim, 1) / dim;
# null-space of the perpendicular vector (transposed, i.e. row vector)
# this generates a basis in the plane
V = null (n.');
# These steps are just to reduce the amount of samples that are rejected
# we build a tight bounding box
bb = S * eye(dim); # each column is a corner of the constrained region
# project on the null-space
w_bb = V \ (bb - repmat(p, 1, dim));
wmin = min (w_bb(:));
wmax = max (w_bb(:));
# random combinations and map back
nsamples = 1e3;
w = wmin + (wmax - wmin) * rand(dim - 1, nsamples);
x = V * w + p;
# mask the points inside the polytope
msk = true(1, nsamples);
for i = 1:dim
msk &= (x(i,:) >= 0);
endfor
x_in = x(:, msk); # inside the polytope (your samples)
x_out = x(:, !msk); # outside the polytope
# plot the results
scatter3 (x(1,:), x(2,:), x(3,:), 8, double(msk), 'filled');
hold on
plot3(bb(1,:), bb(2,:), bb(3,:), 'xr')
axis image