9
In addition to all that Bill Barth has already said above, let me mention that people often report the fastest of several runs. The rationale is that the actual run time is the ideal run time plus any number of slow downs resulting from other processes running, OS delays, network delays, etc. Since these are all noise we are not interested in, using the ...
8
$\mathbf{A}$ is an $(n+1) \times (n+1)$ matrix. It can be obtained as follows:
$\textbf{A} = \left[ \matrix{1 & 1 & 1 & \cdots & 1 \cr
x_0 & x_1 & x_2 & \cdots & x_{n} \cr
\vdots & \vdots & \vdots & \cdots & \vdots \cr
x_0^n & x_1^n & x_2^n & \cdots & ...
6
Histograms are not useful for high dimensional data. The curse of dimensionality affects one quite fast. As in your case if the grid is of size 7**6, you have on average one point in one bin. Kernel density estimator are better suited as long as you keep the kernel bandwidth large enough. In my experience the top hat kernel as k-nearest neighbor yields ...
6
This doesn't randomly sample points, but instead chooses representative points deterministically.
scipy.stats.norm.ppf(np.linspace(0, 1, 1000+2)[1:-1])
6
It's a question of how you choose $\lambda$ and $\Gamma$ (of course). Think for a moment about what happens if you choose $\Gamma=I$ and make $\lambda$ large: in that case you say that it is more important to you to minimize the regularization term $\|Ix\|^2=\|x\|^2$ than to minimize the misfit $\|Ax-b\|^2$. Obviously, making the term $\|x\|^2$ small means ...
5
You haven't specified the distribution of $x(t)$. I'll assume that you want to use a complex normal distribution, since that choice makes it reasonably easy to solve the problem and because this assumption is quite common in signal processing. I'll also discretize the problem so that you're generating a vector $X$ of $N$ entries with a specified complex ...
5
It's actually not all that complicated to calculate the Tracy-Widom CDF just from its definition: see On The Numerical Evaluation Of Fredholm Determinants by Folkmar Bornemann. The Wikipedia page gives the definition as
$$ F_2(s) = \det(I - A_s), $$
where $A_s$ is the integral operator on $[s,\infty)$ with the kernel
$$
\DeclareMathOperator{\Ai}{Ai}
K(x,y) =...
5
Would a decomposition of the form $A = XX^T$ suffice? This would be enough, e.g., if the end goal is sampling from the Gaussian distribution with this given covariance.
If so, you can use the following formula, which is quite similar to your approximation:
$$X = D^{1/2} + \frac{\sqrt{u^T D^{-1} u+1}-1}{u^T D^{-1} u} u u^T D^{-1/2}$$
This follows from ...
4
There are many reasons that a coarser simulation would give different results that a finer grained simulation. A few examples:
are boundary layers being resolved differently?
am I resolving new features (vortices / blockages to flow)
Thinking about a grid result as purely a convolution of a fine-grid result with a Gaussian will work very well in viscous ...
4
My experience in these types of problems is that almost any perturbation will do. I suggest you do not try to perturb $u$ or $p$, because generating a divergence-free $u$ with a perturbed component is more complicated than necessary. You can easily perturb $b$ with lots of different things including a small random field or a randomly-distributed set of ...
4
Geoffrey already answered the question, but I'd like to add another perspective to it. One of the things you will have to do one way or another is to debug code. It may not necessarily be that you have to debug anything that has to do with the random numbers themselves, but, say, a bug in the function evaluation that depends on the randomly selected sample.
...
4
There is no reason to use hardware random number generation for anything other than full cryptography. For everything else, including computational physics, pseudorandom generators are fine. I would suggest using the Random123 library of Salmon et al.: it's fast, trivially parallelizable, and strong (in particular stronger than Mersenne Twister).
It is ...
4
You can easily do this with for loops. Just use n to define your loop limits. Here's a fully functional solution for MATLAB (just define n and x first):
%pre-allocate A
A = zeros(n+1);
%first row:
for j=0:n
A(1,j+1)=sum(x.^j);
end
%rows 2 through n
for i=1:n
A(i+1,1:n)=A(i,2:n+1); %copy from previous row
A(i+1,n+1)=sum(x.^(n+i)); %compute last ...
4
Let $\mathbf{\theta}$ be a Gaussian random vector with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}_\mathbf{\theta}$. Let $\mathbf{p}_\theta$ denote the joint PDF.
Let $J_\mathbf{\theta}$ be the objective function, as its negative logarithm:
$$J_\mathbf{\theta}=-ln(\mathbf{p}_\theta)$$
By taking the partial derivatives w.r.t. $\theta_d$ and ...
4
Let $f(\omega)$ be your power spectrum. Then maybe something like
$$
\frac{\|f\|_{L^\infty}\|f\|_{L^0}}{\|f\|_{L^1}} = \frac{\mathrm{max}_{\omega\in\Omega} f(\omega)\cdot|\omega_{max}-\omega_{min}|}{\int_\Omega|f(\omega)|d\omega}.
$$
I know that $L^0$ isn't really good notation but I think it is useful for presenting this. This quantity is minimized by ...
3
Testing whether or not the mean is correct, or even if the histogram of your generated random variants "looks" like a certain distribution is not sufficient. Stick with much more rigorous test suites such as TestU01 or Diehard.
Also, you really only have TWO random numbers in each row, because of the constraint that they sum to 1.
This requires more ...
3
You don't need to sort your data first, if you have access to additional temporary storage, in which case, you should use a selection algorithm.
3
The quantity you are measuring currently is something akin to "prominence" which is a better formed topographical quantity than numerical one. The wiki page in that link describes all sorts of strange cases that arise, even without having any image/surface borders to worry about.
If you want to stick with this alignment metric I would suggest using a ...
3
From the Computer Science perspective I do not think that make sense to make an general statistical model for memory access time (latency) and memory bandwidth.
It does make sense to create an statistical model for a algorithm. That is because each algorithm has a specific memory access pattern, the memory access patterns are relevant to the cache hierarchy,...
3
It is imperative to compare many runs with refined meshes until you detect convergence. A single solution with no mesh refinement study shouldn't give you much confidence in your results.
Comparing runs with different fluid properties tells you something different. If you think that a set of runs with different viscosities is relevant to your ultimate ...
3
You may use Kahan summation algorithm [1]
The idea is to reschedule the sum operations in such a way precision loss is limited. The code is very simple (reproduced from [1] below).
If this does not suffice, you may use multiprecision representations, such as quad doubles [2]. They are supported by several languages / compilers (including GNU c). Finally, if ...
3
Mathematica has the TW distributions:
http://reference.wolfram.com/language/ref/TracyWidomDistribution.html
3
In general, your type of question would be called a "multivariate goodness of fit test". If $F(x_1,\ldots,x_n)$ is the $n$-dimensional CDF for the theoretical distribution, and the random variables $(X_1,\ldots,X_n)$ are a sample from your ODE, then
$$ Z_1 = F(X_1), \quad Z_2 = F(X_2\mid X_1), \quad\cdots\quad Z_n = F(X_n\mid X_1,\ldots,X_{n-1}) $$
are ...
3
(Converted to an answer from my comments and expanded.)
Basically you need to compute the fourth moments $E[x_i x_j y_k y_l]$ for all $i,j,k,l$, given the second moments. These fourth moments are not universally determined, but they depend on the distribution of your variables, even in the one-dimensional case (that's exactly the reason why kurtosis is a ...
2
This is just a linear fixed-point iteration
$$ X_t = M X_{t-1}$$
where
$$\DeclareMathOperator{\diag}{diag} M = AB + C + I = (\diag C)^{-1} B + C + I .$$
The convergence rate of this iteration is linear with worst-case convergence factor equal to the spectral radius of $M$ (therefore, if $\rho(M) > 1$, the method does not converge). Furthermore, if $\...
2
The problem you are discussing sounds like model order reduction (also called "model reduction"). Principal component analysis (also called Proper Orthogonal Decomposition, Karhunen-Loeve analysis, and other names) is one way of achieving model order reduction by imposing certain assumptions that reduce a problem to a singular value decomposition. There are ...
2
The AIC function need an 'lm' or 'glm' object (linear models).
See functions lm and glm
So just do : AIC(lm(logCPK~dataPOW))
2
See these lecture notes from the complex systems course taught by Prof. Peter Dodds. It has some great links to literature, etc..
In a nutshell, random multiplicative growth can lead to lognormal distributions and there are certain things one can enforce within the dataset (minimum number of occurrences among other things, see some of this wonderful ...
2
NumPy comes with a nifty random library with various distributions, including normal (Gaussian).
From the Numpy documentation:
mu, sigma = 0, 0.1 # mean and standard deviation
s = np.random.normal(mu, sigma, 1000)
which will give you 1000 normally distributed values with mean mu and standard deviation sigma.
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