Questions tagged [geometry]

For on-topic questions involving geometry. For purely mathematical questiona about geometry it is better to ask on math SE https://math.stackexchange.com/

Filter by
Sorted by
Tagged with
1
vote
0answers
8 views

How many points needed to compute the Homography? [migrated]

I'm working on a project where i'm using planar homography. As seen in the above image, every point gives two equations and since the homography matrix has 8 degree of freedom, 4 points are enough to ...
2
votes
1answer
38 views

Non-Euclidean analogue to MSE loss

The most basic machine learning model called OLS uses the RSS (squared loss) or its average, mean squared error (MSE), for its loss function, which is aligned with Euclidean geometry. What is the ...
2
votes
0answers
29 views

Why do we need to triangulate a convex polygon in order to sample uniformly from it?

Suppose I want to uniformly sample points inside a convex polygon. One of the most common approaches described here and on the internet in general consists in triangulation of the polygon and generate ...
1
vote
0answers
17 views

What are the principal components of a set of points lying on a circle? [closed]

So if there are a set of points lying on a circle (2 dimensional), what will be the principal components given that the variance of points vary equally along any 2 perpendicular directions
0
votes
0answers
5 views

Instability in Calculating Mahalanobis Distance

I am trying to calculate Mahalanobis distance from a point to a cluster of points. The code below does that. ...
4
votes
0answers
28 views

How to do statistics on different geometries?

The original motivation: Let's say, we have two particles, doing random one motion on an infinite 1-D lattice $\mathbb{Z}$, and we are interested in the autocorrelation of the trajectory of each ...
0
votes
0answers
18 views

Distribution of the dot product between random unit vectors [duplicate]

Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases: $X,X'$ independent with uniform distribution on the sphere $...
0
votes
0answers
19 views

$\mathbb{G}=(1,B,T,B \times T)$ is an orthogonal design

I have an orthogonal system with 2 factors, B and T. I know that B and T are orthogonal geometric. But I also need to show that that the system $\mathbb{G}=(1,B,T,B \times T)$ is an orthogonal ...
1
vote
0answers
24 views

PCA on polar coordinates (0°-360°)

I have a data set where one of the variables is in polar coordinates ($\varphi$ = 0° to 360° degrees) and PCA was applied in order to reduce dimensions. As far as I know, using PCA on polar ...
7
votes
1answer
187 views

What is the geometric relationship between the covariance matrix and the inverse of the covariance matrix?

The covariance matrix represents the dispersion of data points while the inverse of the covariance matrix represents the tightness of data points. How is the dispersion and tightness related ...
0
votes
1answer
29 views

graph convolution network

I am trying to understand papers and lectures on graph convolution networks but whenever I open some paper, I get lost on the very first page. I started with some videos like this and this and papers ...
2
votes
0answers
17 views

Approximate the mean area of 2D Voronoi cell

Consider a random uniform distribution of $N$ points in $2D$ space bounded by $[0, 1]$ in both dimensions. Example: If I want to estimate the mean area of their Voronoi cells, I have to obtain the ...
3
votes
2answers
53 views

Understanding sufficient statistics geometrically

Consider the distribution $\mathcal{P} = \mathcal{N}(\mu, 1)$, where the variance is known but the mean is unknown. Let $X_1,X_2\sim P$ i.i.d. In this case $T = X_1+X_2$ is a sufficient statistic. I ...
4
votes
0answers
102 views

Geometric interpretation of the difference between the means (ANOVA)

First of all, a disclosure: I'm a medical doctor trying to understand statistics for research. Coming from a non-mathematical background I can do many mistakes. I've read some traditional books of ...
0
votes
0answers
74 views

Bivariate normal probability of being inside ellipse

Assume that $\mathbf{X}$ is a bivariate normal random variable $$\mathbf{\mu} = E\mathbf{X} = \begin{bmatrix} 0 \\ 2 \end{bmatrix} \ \text{and} \ \Sigma = Cov \ \mathbf{X} = \begin{bmatrix} 3 & 1 ...
2
votes
0answers
38 views

Inverse sampling a direction over the hemisphere of a surface?

Let $S^2:=\{x\in\mathbb R^3:|x|=1\}$ denote the unit 2-sphere, $$\omega_{x\to y}:=\frac{y-x}{|y-x|}\;\;\;\text{for }x,y\in\mathbb R^3\text{ with }x\ne y,$$ $M\subseteq\mathbb{R}^3$ be the disjoint ...
6
votes
2answers
150 views

Expected triangle area from normal distribution

3 points are randomly selected from a multinormal distribution $\mathcal{N}(\vec{0},\Sigma)$ in $\mathbb{R^3}$ with $\Sigma=\begin{pmatrix}\sigma^2&0 &0 \\0&\sigma^2&0\\0&0&\...
0
votes
1answer
7 views

Intervals from an underdetermined nonnegative linear system

I'm working on a problem in genomics that yields the following puzzle. Let $b\in \mathbb R^I$, $t$ and $p\in \mathbb R^J$, and $s \in \mathbb R^{I\times J}$. Suppose $t,b,p$ are known. Further suppose:...
2
votes
1answer
272 views

How do I interpret the angles of two concentration ellipses?

Consider a map with two concentration ellipses like this below. The Vomit_y group is (almost?) perfectly vertical, while the Vomit_n group seems to be oriented at about 45 degrees. I understand that ...
4
votes
2answers
77 views

Poisson process on an $n$-sphere

I have an algorithm that embeds data points into Euclidean space. If I norm these points then they will lie on the unit $n$-sphere, where $n+1$ is the dimensionality of the embedding space (generally ...
2
votes
1answer
725 views

what means to be outside unit circle?

I am trying to study time series without a great math background and I came across the next problem: When checking for stationarity I check the roots, and if they are not on the unit circle, then it ...
1
vote
1answer
47 views

What is the probability that you can form a triangle with these three line segments?

Two numbers are randomly selected between $(0,1)$, uniformly and independently distributed. What is the probability that the three resulting line segments, which are obtained by cutting the interval ...
2
votes
1answer
42 views

How to generate data such that an equation needs to hold?

Can I create or generate $\{y_i\}_{i=1}^{4}$ data set such that this equation holds $$ \sum_{i=1}^{4}\sum_{j=1}^{4}m_{ij}y_{i}y_{j}=6 $$ where $$ m=\left[ \begin{array}{cccc} 13 & 12 & 3 &...
1
vote
1answer
50 views

D-optimal DOE suggest repeated samples

I tried to generate a D optimal design but the design output sounds very weird to me. I have a (real) process and I`d like to explore 3 factors, but the process have a lot of constraints so I provide ...
0
votes
0answers
82 views

Sampling from/near boundary of a region in R^n

Suppose $\Omega$ is a region in $\mathbb R^n$, and suppose we are given a function $\chi(x)$ with $\chi(x)=1$ if $x\in \Omega$ and $\chi(x)=0$ otherwise. If it helps we can assume $\Omega\subseteq B$ ...
3
votes
0answers
119 views

Geometric interpretation of Cholesky Decomposition

I understand that a square matrix, say $A$, can be thought of as a linear transformation within the same space. I could be as simple as basis change or some other transformation. In this way of ...
1
vote
0answers
37 views

Is there a geometric interpretation for the surprising result of $E[x(y-\mu_y)] = COV(x,y)$?

I was playing around with non-central second moments, and noticed that $E[x(y-\mu_y)] = E[(x-\mu_x + \mu_x)(y-\mu_y)] = COV[x,y] + E[\mu_x(y-\mu_y) = COV[x,y] + 0$. I find this very surprising. It ...
4
votes
1answer
51 views

Bounding data by two parallel lines with minimum distance between them

I have a set of data samples that approximately follow a straight line in 2D. I need to find two parallel lines that are spaced as close as possible such that all of the samples lie between the lines. ...
1
vote
0answers
26 views

Maximize volume of convex hull [closed]

Suppose I have N points (labeled 1, 2, ..., k, ..., N) in D dimensions. I'd like to choose ...
1
vote
1answer
245 views

How to calculate the midpoint between 2 latitude and longitude coordinates in R? [closed]

I have two sets of lat and long coordinates. I would like to find the midpoint between them. Not sure how to calculate this in R.
3
votes
0answers
78 views

What is the geometric meaning of correlation matrix

I recently read this article explaining the geometric meaning of covariance matrix. http://www.visiondummy.com/2014/04/geometric-interpretation-covariance-matrix/ My question is : is there an ...
1
vote
0answers
21 views

Expected number of quadrilaterals

There are N points on the plane and the probability that the probability that two points are connected is p. What is the expected number of Quadrilaterals you can find? Assume that there is no three ...
2
votes
0answers
46 views

Is the Franke-Wolfe algorithm the same as Manifold optimization?

The Frank-Wolfe optimization algorithm describes optimization over a constrained domain. In the Manifold Optimization literature (e.g. [1]) a Gradient-Step is done using an exponential map. This maps ...
2
votes
0answers
78 views

Limitations to using the principle of least squares to fit an ellipse to experimental data?

I have used the method described below many times without issue to determine the major and minor axes of the "data ellipse". However, for this set of data (Fig. A) the method described below is ...
4
votes
0answers
104 views

How would one carry out principal component analysis in the Beltrami-Klein model of hyperbolic space?

This is a follow up to a comment by @whuber in response to this question: Is there an extension of PCA for data embedded in hyperbolic spaces? Sorry, I would have posted it there as a comment but it ...
1
vote
1answer
790 views

Drawing 95% ellipse over scatter plot

The context is regression analysis using Eviews, but first I wanted to create a few scatter plots and overlay error ellipses on them. Eviews doesn't support that kind of graph ornamentation so I am ...
4
votes
1answer
73 views

Sampling from the surface of a sphere in n dimensions with specific centre

I have a point $p$ on the surface of a unit sphere. I want to sample points from the surface of the sphere, such that the probability density of a point $q$ on the surface is given by \begin{equation} ...
2
votes
1answer
34 views

When is the result of multidimensional-scaling unique up to isometry?

What conditions on the ambient space and/or the given matrix of dissimilarities guarantee that all point configurations that minimise the error function of multidimensional scaling (MDS) are congruent,...
3
votes
1answer
429 views

Expected value as an orthogonal projection

I'm reading a paper in which the expected value of a random variable, $\mathbb{E}[X]$, is characterized as an orthogonal projection. This is on page 10. I've seen the geometric interpretation of ...
0
votes
1answer
22 views

Location of “x standard deviation(s)” for multivariate normals

In 1D with scalar parameters $(\mu,\sigma^2)$, it is common to represent normally distributed data with error bars spanning $[\mu-\sigma, \mu+\sigma]$. In n-D with parameters $(\mu, \Sigma)$, where $\...
0
votes
1answer
69 views

Is an geometric approach of support vector machine part of supervised learning?

The support vector machine can be approximated geometrically by enclosing the data of the classes by convex hulls. Then you can e.g. using the Rotating Calipers, you can search the widest band/border ...
0
votes
2answers
49 views

Can I create a test dataset with known errors to validate accuracy assessment? [closed]

I developed a procedure to measure the geometric accuracy of 3D building models based on the similarity to a 3D point cloud. Therefore I created mainly two quality criteria. The result of my automatic ...
4
votes
1answer
280 views

Geometric interpretation of mathematical expectation of a random variable

Is there any nice geometric interpretation of the mathematical expectation of a random variable (preferably based on density or cumulative density plot)? (For example, median has a nice geometric ...
2
votes
0answers
134 views

Vapnik-Chervonenkis Dimension

My question has to do with the VC dimension of the class of convex polygons with $m$ vertices. A solution to this problem is given in the following: Advanced Algorithms. Call the class of all ...
2
votes
0answers
159 views

Lower bounds on covering numbers for sparse vectors

Consider the set $S_k$, which is defined as the subset of $k$-sparse vectors in the unit sphere in $d$ dimensions: $$ S_k \triangleq \left\{ x \in \mathbb{R}^d : \| x \|_2 = 1, \, \left|\operatorname{...
1
vote
0answers
331 views

ML techniques to classify simple polygons

I'm curious about suitability of Machine Learning techniques to classify grayscale images of polygons into categories defined by number of their sides, which will be kept small. Images will be "...
5
votes
1answer
124 views

Covering the unit sphere with sparse vectors

I'm looking for references for covering the $d$-dimensional unit sphere $$ \mathbb{S}^{d - 1} = \left\{ x \in \mathbb{R}^d : \| x \| = 1 \right\} $$ I'm trying to cover $\mathbb{S}^{d-1}$ with ...
5
votes
0answers
848 views

Why word embeddings learned from word2vec are linearly correlated

I was playing with CBOW from the word2vec program downloaded from https://code.google.com/archive/p/word2vec/ with some sequence data (peptides in this case). I was trying to get embeddings for amino ...
1
vote
2answers
6k views

What type of data are dates?

According to Yale: Categorical variables represent types of data which may be divided into groups (Lacey M, 1997) To me, dates do not fit this definition. They are ordinal, as one date is ...
6
votes
0answers
199 views

Is there any geometric intuition on least absolute deviation regression?

There are a lot of geometric intuitions for regression with least square, e.g., projection, orthogonal, etc. (This and this answers are good examples.) Is there similar geometric intuition for least ...