I am trying to visualize a higher dimensional vector field. Is there a way to do this. I asked this question here, I was told to post it here.
As an example, one can use
$$\begin{eqnarray} \dot{x}&=& \left(J-R\right)x \end{eqnarray}$$
where $x\in \mathbb{R}^5$ $J $ is a skew-symmetric matrix and $R$ is a positive semi-definite matrix.
PS: The only idea I came so far is: somehow I have to use t-sne, but I really don't know how.
Somehow I have to use t-sne, but I really don't know how.
Since you have a PhD my answer will be brief, it's quite a lengthy subject.
The aim of dimensionality reduction is to preserve as much of the significant structure of the high-dimensional data as possible in the low-dimensional map. For high-dimensional data that lies on or near a low-dimensional, non-linear manifold it is usually more important to keep the low-dimensional representations of very similar datapoints close together, which is typically not possible with a linear mapping.
T-distributed Stochastic Neighbor Embedding (t-SNE) is a way of converting a high-dimensional data set into a matrix of pair-wise similarities. t-SNE is capable of capturing much of the local structure of the high-dimensional data very well, while also revealing global structure such as the presence of clusters at several scales.
Stochastic Neighbor Embedding (SNE) starts by converting the high-dimensional Euclidean distances between datapoints into conditional probabilities that represent similarities. The similarity of datapoint $x_j$ to datapoint $x_i$ is the conditional probability, $p_{j|i}$, that $x_i$ would pick $x_j$ as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at $x_i$. For nearby datapoints, $p_{j|i}$ is relatively high, whereas for widely separated datapoints, $p_{j|i}$ will be almost infinitesimal (for reasonable values of the variance of the Gaussian, $σ_i$).
The cost function used by t-SNE differs from the one used by SNE in two ways: (1) it uses a symmetrized version of the SNE cost function with simpler gradients that was briefly introduced by Cook et al. (2007) and (2) it uses a Student-t distribution rather than a Gaussian to compute the similarity between two points in the low-dimensional space. t-SNE employs a heavy-tailed distribution in the low-dimensional space to alleviate both the crowding problem and the optimization problems of SNE.
A few t-SNE tutorials:
Laurens van der Maaten's (author of the original paper) webpage has implications in: Matlab, CUDA, Python, Torch, Julia, Java, R, JavaScript and more ...
Distill's interactive t-SNE dimensionality reduction webpage: "How to Use t-SNE Effectively".
GoogleTechTalks Video - "Visualizing Data Using t-SNE".
CrossValidated.SE is probably a better site to ask about statistics visualization.
