Questions tagged [geometry]

For on-topic questions involving geometry. For purely mathematical question about geometry it is better to ask on the math SE site

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Why can the hyperdimensional plane be discribed as $\textbf{w} \cdot \textbf{x} - b$ for support vector machines

So given the picture and the related definitions from this answer: How does the equation $\textbf{w} \cdot \textbf{x}^{(i)} - b = -1$ hold for several vectors $\textbf{x}^{(i)}$ when $\textbf{w} \...
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Meaning of Canonical Correlation Analysis Orthogonal Constraint

Hello Folks - I'm trying to understand canonical correlation analysis (CCA). I found that it's useful for finding linear projections such that the projection results are maximally correlated. However, ...
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Using Machine Learning to Create Periodic Paths

Question: How can I use Machine Learning to predict the right initial conditions $(P_0,V_0)$, given the angles of the triangular table, that will result in periodic paths in the triangular billiard ...
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Need help interpreting an answer about the Cauchy distribution and Huygens principle

I'm trying to understand the answer here, which provides a physical interpretation for why the Cauchy distributions mean doesn't exist makes the following statement: If a unit light source is located ...
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Does the existence of gradient in any function necessarily imply the existence of a subgradient at that point?

First , I apologize if the question is not supposed to be here, or if it is off topic for the subjects dealt with in here. I was reading on subgradients, with respect to convex functions in the ...
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Grouping by similarity of variables as weights (2 variables into one index) and locations (using python): Weighted location clustering with an index

I'm currently in a problem. My task is to group n subzones into larger m zones (n>m, but for this case is not very important to tell the numbers, just looking for an adequate approach). Also, for ...
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1 vote
1 answer
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What optimization algorithms are best at traversing complicated geometries, and what trade-offs exist between different algorithms?

Hi all :] so short version is just the title, and specifically as pertains to those algorithms included in R's optimx::optimx() under ...
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how to determine the equation of a hyperplane (W and b) using only support vectors points without a Lagrange multiplier?

In n-dimensional, assuming I have support vectors points (number of sv < n), can I find the hyperplan equation (w and b) that separates 2 sets of data using only support vectors points without a ...
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How is the set of probability distributions on $m$ values an $m-1$-dimensional simplex? [duplicate]

In Zuk et al. 2012, they claim: The set of probability distributions on $m$ values is the $m-1$-dimensional simplex denoted $S_m$. Probability distribution functions are familiar to me. I understand ...
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Calculating covariance matrix from given vertices of triangles

I am given 3D vertices of triangles and I want to build the covariance matrix from it. These vertices are pure geometric vertices no noise. I can easily get it from libraries of python but I wanted to ...
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Why does my simulation of nearest neighbors as circle origins provide different non-contact probability compared to theoretical?

I am simulating spatially distributed points in $\mathbb{R}^2$ with intensity $\lambda$ (units 1/area), which act as circle origins with radii being a random variable $R_k$. Given the distance to the $...
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Impractical question: is it possible to find the regression line using a ruler and compass?

The ancient greeks famously sought to construct geometrical relationships using only a ruler and a compass. Given a set of points in a two dimensional plane, is it possible to find the OLS line using ...
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How to find the average distance between randomly distributed points in a rectangle?

Assume there are n points randomly distributed in a rectangle (x being the height y being the width) shown below in the figure. I would like to calculate the average distance between 2 random red ...
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Geometric properties of independence between random variables [closed]

Suppose $Y$ and $X$ are two independent random variables. Are there any geometrical properties (of PDF/CDF/PMF/etc.) that capture this independence? I know that the Support has to be rectangular and ...
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Does the distribution $f(x) \propto (1-x^2)^{n/2}$ have a name?

The distribution $f(x) \propto (1-x^2)^{n/2}$ for $-1 \leq x \leq 1$ It occurs in a problem like Law of the norm of the empirical mean of uniforms on the sphere? It relates to intersections of high ...
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Calculating the mean or median "whereabout" of a joystick

I have recently collected data using the NASA TLX tool, which includes a tracking task, where subjects have to keep the joystick deadcenter while having to react to multiple tasks. The raw data looks ...
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Finding a Projection Plane in Dimensionality Reduction (e.g., Multidimensional Scaling)

I have a set of data points in high-dimensional space that I wish to map onto a lower dimension (3D or 2D). Question : How do I obtain the Projection (Hyper)Plane (e.g., its normal vector or its set ...
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How to find the weighted midpoint between n-points?

I was hoping if someone could guide me to the right algorithm to find the weighted midpoint of n-points. The photo attached below perfectly describes my problem. Let's just say we have three points ...
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2 votes
1 answer
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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 ...
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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 ...
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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
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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 ...
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Expectation of differences between arcs on a circle

Consider a circle with a circumference of $n$. On this circle, I define two arcs of length $k<n$, $A_1$ and $A_2$. The centres of the two arcs are uniformly distributed on the circle. Let $\Omega_{...
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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 ...
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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 ...
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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 ...
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2 votes
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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 ...
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4 votes
2 answers
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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 ...
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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 ...
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191 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 ...
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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 ...
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6 votes
2 answers
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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&\...
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1 vote
1 answer
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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:...
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2 votes
1 answer
570 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 ...
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4 votes
2 answers
191 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 ...
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4 votes
1 answer
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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 ...
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1 vote
1 answer
312 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 ...
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2 votes
1 answer
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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 &...
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1 vote
1 answer
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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 ...
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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 $\Omega\subseteq B$ ...
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2 votes
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Boundary point errors in PCA projection using sklearn

I am preparing a small example of a projection using python, numpy and sklearn to perform <...
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4 votes
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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 ...
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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 ...
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4 votes
1 answer
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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. ...
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2 votes
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MLE for an ellipse

I'm not at all a statistician, but in the course of my work I've come across a non-trivial maximum likelihood estimation problem, and I'm looking for ideas and/or references on how to approach it. ...
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1 vote
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Maximize volume of convex hull [closed]

Suppose I have N points (labeled 1, 2, ..., k, ..., N) in D dimensions. I'd like to choose ...
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1 answer
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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.
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3 votes
0 answers
133 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 ...
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1 vote
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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 ...
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2 votes
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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 ...
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