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
13 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
34 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.
0
votes
0answers
11 views

Is there work for statistical properties analysis of road curvatures?

I looking for an analysis about the most suitable stochastic process to model distribution of road curvatures and its properties. I have been googling for several days and could not find it yet. For ...
3
votes
0answers
56 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 ...
0
votes
0answers
24 views

Does skipgram(word2vec) decreases the euclidean distance or increases cosine similarity between similar words?

The skip-gram model tends to predict the surrounding words or in other words, it tries to maximize the co-occurrence in the output of the Network. According to my knowledge, this makes a similar word ...
1
vote
0answers
18 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 ...
1
vote
0answers
41 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
53 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 ...
0
votes
0answers
19 views

Probability that some coefficients will be zero with lasso regression

Maybe it is duplicate, but I don't know how to ask a right question. Suppose I fixed some lambda, for instance lambda = 1. I draw the usual "square" |x| + |y| = 1. Now I want to find probability, ...
4
votes
0answers
18 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
281 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
23 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} ...
1
vote
1answer
23 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,...
0
votes
0answers
56 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
16 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
63 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
37 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 ...
0
votes
0answers
141 views

How are functional margin and geometric margin used in SVM?

I believe, geometric margin is euclidean distance between the point and hyperplane, whereas the functional margin just gives the confidence. At which stage is geometric margin and functional margin ...
1
vote
0answers
21 views

Mean distance from the centre to any point in a sphere and a cylinder [closed]

What is the mean distance from the centre to any point within a sphere of radius r? What is the mean distance from the centre to any point within a cylinder of radius r and length l?
1
vote
1answer
118 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
120 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
80 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
130 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
108 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
649 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
934 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
155 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 ...
0
votes
0answers
9 views

Fitting data (and getting normal vectors) to planes that can be vertical [duplicate]

I have many individual datasets of $(x,y,z)$ coordinates. I'd like to see if the coordinates form a plane (and if they do, I'd then like to compute a unit normal to the plane). My original method was ...
1
vote
1answer
52 views

SVM in a 2D plane

I was just watching this tutorial about Support Vector Machines, and I came to a halt because of the following problem. Given that $\vec{w}$ is a vector perpendicular to a hyperplane separating two ...
4
votes
1answer
120 views

What is the physical intuition behind the equality $\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar x) x_i$?

Suppose that $x_1,\ldots,x_n$ are real numbers, and let $\bar x$ denote the average $\frac{\sum_i x_i}n.$ I know how to prove on paper that the equality $$\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar ...
4
votes
2answers
390 views

Average absolute value of a coordinate of a random unit vector?

Let $\vec x$ be a random unit vector (that is, a random vector on the unit sphere). Let $x_i$ be the $i$'th coordinate (if it is easier, you can assume we are in 3-dimensional space). What is the ...
1
vote
1answer
65 views

Can CCA model any linear transformation?

I have recently been looking into canonical correlation analysis (CCA) as a way to map between different spaces. As I understand it, CCA maps data from both distinct spaces to a common (possibly ...
4
votes
0answers
57 views

Metrics to identify the presence of more than one circle in a set of (x,y) coordinates

I have a set of $(x,y)$ coordinates that represent any number of circles in a plane. What I am trying to do is determine whether there is 1 or more circles present in the data set. This would normally ...
4
votes
2answers
3k views

How does a quadratic kernel look like?

Based on visualized decision boundaries, we have to decide what kind of classifier has generated it. One example is shown in the image below - this is from a quadratic kernel Support Vector Machine (...
1
vote
0answers
37 views

Algorithm to find smallest ellipse containing p% of points in 2D plane?

This question is the same one I want answered: https://stackoverflow.com/questions/26810092/how-to-find-the-smallest-ellipse-covering-a-given-fraction-of-a-set-of-points-in Except I need to either ...
1
vote
0answers
24 views

Pairwise distances between 3 random points in a ball

Say I have a 3-D ball with radius $R$ and I randomly pick points from the volume of the ball. If I randomly pick two points, the probability density function of inter-point distances $P(r_{12})$ is ...
5
votes
1answer
886 views

Ellipse formula from points

Question: Best way to derive an ellipse formula when given bunch of points (say 60 points which when connected draws an ellipse)? Background: Using R Momocs library function conf_ell which returns ...
7
votes
1answer
319 views

Is the expectation of the sufficient statistics $S(X)$ transverse the whole space in an exponential family?

An exponential family is defined using two ingredients: - a base density $q_0(x)$ - a number of sufficient statistics $S_i(x)$ The family is all probability densities which can be written as: $$ q(x| ...
7
votes
2answers
405 views

How is the spherical elevation angle distributed when $(x,y,z)$ are uniformly and normally chosen?

As a follow up to How the polar coordinate, $\theta$, is distributed when $(x,y) \sim U(-1,1) \times U(-1,1)$ and if $(x,y) \sim N(0,1)\times N(0,1)$? Assume $(x,y,z) \sim U(-10,10) \times U(-10,10) \...
0
votes
0answers
31 views

mean and least error on the line

Suppose I have a line on which i have points in non-decreasing order. My intuition tells me that if I want to minimize the squared mean error on some subset of 3 (or other number) points, I would like ...
2
votes
4answers
358 views

How can I solve this joint probability problem?

Two points are dropped at random onto the unit interval, creating three sub intervals this way. Find the probability that the central sub interval will be two times shorter than the right ...
4
votes
3answers
816 views

Distribution of quadratic equation roots where coefficients are generated uniformly

We draw two points $p$ and $q$ at random from the interval $[−1, 1]$. Let $x_1$ and $x_2$ denote the roots of the equation $x^2 + px + q = 0$. Find the probability that, $x_1, x_2 \in \...
3
votes
1answer
189 views

Temporal Multi Dimensional Scaling

Let's say I apply a multidimensional scaling(MDS) to a dynamic dataset of $n$ points (eg, time series). At each step I will obtain a projection (in 2/3D) of the $n$ points. If nothing meaningful ...
0
votes
1answer
84 views

What is Geometrical Probability? [closed]

Is it a technique where Geometry is used to solve probabilistic problems? Is it a kind of probability which grows Geometrically when we conduct experiments? Is it a kind of distribution? I am ...
0
votes
0answers
79 views

Finding the non-linear curve that minimize the (sum of squared) distance to a set of point

I'm looking for the best approach to find the non-linear curve that minimize the mean square of perpendicular distance from a set of points. In essence, I would like to do non-linear major axis ...
3
votes
1answer
70 views

Geometry of Rejection regions for tests (NP lemma, Karl-Rubin, UMPU, LMP, etc)?

The Neyman-Pearson lemma, the Karlin-Rubin theorem, and the other for UMPU tests for the exponential family, etc. They all define a most powerful Rejection Region (RR), in a certain class of tests. I ...
3
votes
1answer
167 views

Centroid of nearest-neighbours on a hypersphere as a method for applying crossover in genetic algorithms

I am currently building a genetic algorithm to tune n parameters where n will probably be in the range of ...
1
vote
0answers
17 views

Learning areas of intersections

I will try my best to make this question clear. I have many circles intersected randomly ( wireless communication application WSN) , i want to let each circle knows the intersection area with its ...
1
vote
0answers
505 views

How to train neural net to output a 2D polygon?

Lets consider a net, that should map image to an arbitrary four-sided 2D polygon (all vertices are scaled to (0,0) - (1,1)). We need a loss function with a gradient to train it. If we try to use some ...
7
votes
1answer
4k views

Data space, variable space, observation space, model space (e.g. in linear regression)

Suppose we have the data matrix $\mathbf{X}$, which is $n$-by-$p$, and the label vector $Y$, which is $n$-by-one. Here, each row of the matrix is an observation, and each column corresponds to a ...