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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Understanding relation between axis of least and maximum second moment
I was going through computer vision lecture video. You can find the pdf of this lecture here.
I was trying to understand how orientation of object corresponds to axis of least second moment aka ...
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Geometric interpretation of random effects design matrix
A fixed effects (FE) linear model can be understood in a precise way using concepts of geometry. Namely, the response vector in model space is projected onto the design subspace surface described by ...
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Describe a geometrical shape as a piecewise function
Consider a cube filled with random particles. Let's say the particles in cube are rotated around the z-axis through the center of the cube. Here, the rotation is proportional to the height of the cube,...
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Is it possible to uniformly draw points over a $D-1$ dimensional disk, given that one has an algorithm to draw over the $D$ dimensional sphere?
Suppose I have the following scenario:
And I am aware of an algorithm to draw uniformly from (in this case) the 2-sphere. Does this same algorithm readily extend to the situation where I randomly ...
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Geometric intuition for how ridge ($L_2$) regularization helps under multicollinearity
We have some nice posts (1, 2 and likely more) illustrating multicollinearity geometrically. Now, ridge regression ($L_2$ regularization) is known to be a remedy of multicollinearity. What is the ...
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Estimation of MLE - request for clarification
I am refering to an earlier post in Maximum Likelihood estimate of cell radius from observed cross section radii
I am not sure if I understood this question. What exactly is the meaning of ...
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Automatic selection of nuber or factors in PCA based on the Cattells scree plot in R
I would like to automatically select the number of factors after factor analysis (PCA). I mean the graphical selection method according to the Cattel criterion, which determines where the steepness of ...
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Penalize an attribute of points based on their distance from a line without use of any threshold
We have a set of points and a line. Each point in the set has a weight attribute that is an integer. How we could penalize this weight based on the distance of the point from the line without using ...
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Estimate weights given a weighted geometric median
Problem 1
Given a set of points and their weighted geometric median, find the weights associated with each point.
For instance, given the points ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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