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Questions tagged [mahalanobis-distance]

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Calculating Mahalanobis parameters from x data

Roughly speaking, the Mahalanobis distance $d(x) = \sqrt{(x - \mu)^T * \Sigma^{-1} * (x - \mu)}$ is the distance between a multivariate Gaussian distribution ($\mu$, $\Sigma$) and a point. If the ...
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Encoding the position information in real distance metric

The Euclidean distance doesn't preserve the exact position information. For example, the distance of the points (3,1) and (1,3) would be the same from the origin. Is there any distance metric which ...
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Proof That the Mahalanobis Distance is $\ge 0$

I was just introduced to the Mahalanobis distance between two vectors $\mathrm{\mathbf{X}}$ and $\mathrm{\mathbf{Y}}$ of random variables: $$|| \mathrm{\mathbf{X}} - \mathrm{\mathbf{Y}}||_{\Sigma} = (...
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Calculating Mahalanobis-Distance on Basis of condensed clusters

So I'm not shure if there exists a method to solve my problem, but here it is: I have around 100 classes and from each class I have 20'000 reference-feature-vectors with length 10. So until now I've ...
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Convert a vector of distances to a normalized vector of similarities

I'm struggling to find a way to solve this problem. I have derived a $m \times n$ matrix containing in each row the Mahalanobis distance from a certain centroid. So at the end I have $m$ rows each ...
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Why is this right?

How do they get 3? And what is coplanar distance
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How to find a covariance matrix of minimal volume including all data points by Mahalanobis distance?

Let $v_i$ be real vectors from $R^D$, for $i=0..N-1$. How to find a real symmetric positive-definite matrix $\Sigma$ so that $v_i^T \Sigma^{-1}v_i \leq 1$ for any $v_i$ and $det\ \Sigma$ is minimal? ...
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What is the Mahalanobis Distance pf y relative to the distribution of x?

I'm learning about the Mahalanobis Distance, and I had a question regarding an interesting problem: Let $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ be a Gaussian random vector with mean $\mu = \...
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Minimum Mahalanobis Distance on a Linear Regression Problem.

I am reading a paper, "A Morphable Model For The Synthesis Of 3D Faces" paper I have a question about eq(3). Solving a linear regression problem with data $D=(S_i, T_i)$ and target values $\mu_i$, ...
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About the convexity of distance metrc problems.

Suppose that we consider a pseudo-distance in $\mathbb{R}^n$ (so we admit that different points can have zero distance between them) that comes from a dot product. It is known that these distances ...
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Positive-definite matrix for positiveness in Mahalanobis distance

I am trying to prove that the Mahalanobis distance $d(x,y)$ is always positive, that is: $\forall{x,y \in E}\,{[0 \le d(x,y)]}$. To do so, I need to demonstrate that: $0\le\sqrt{(x-y)^TS^{-1}(x-y)}$, ...
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Mahalanobis whitening in a multivariate Gaussian kernel

I'm a biologist trying to understand math. It would help me out no end if somebody could clarify my misunderstandings. Question 1 is causing me the most grief. The image below is taken from the paper ...
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Average standardized distance between units?

Let $$ \pmb{X} = \begin{pmatrix} x_{11}& \cdots& x_{1j}& \cdots& x_{1p} \\ \vdots& & \vdots & & \vdots \\ x_{i1} & \cdots & x_{ij} & \cdots & x_{ip} \\ \...
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Proving that Mahalanobis norm is a norm indeed

While reading this thread I wanted to prove that Mahalanobis norm $\lVert{x-y}\rVert_m$ is a norm indeed. The norm is defined like so: $\lVert{x-y}\rVert_m=d(x-y,0)=d(x,y)=\sqrt{(x-y)^T S (x-y)}=\sqrt{...
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Optimization problem with Mahalanobis distance

This question is a follow-up of one of my previous questions: Optimizing a vector equation Let $x$ and $b$ be two vectors of real numbers in k dimensional space. Let $W$ be a k-by-k matrix of real ...
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PDF of Mahalanobis distance of a multivariate random normal variable

Suppose $x$ is a multivariate normal random variable with mean $\mu$ and covariance matrix $C$. Let $h(x)$ be a the Mahalanobis distance of $x$ given by $h(x)=\sqrt{(x-\mu)^TC^{-1}(x-\mu)}$. Then how ...
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Composite cost function using angular distance between vectors and distance between points

I am trying to come up with a way to find the transformation parameters between two sets of planes and would like to get a cost function C which incorporates two ...
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“we note that the matrix Σ can be taken to be symmetric, without loss of generality”

I'm reading the book Pattern Recognition and Machine Learning by Christopher Bishop, and on page 80, with regard to the multivariate gaussian distribution: $$ \mathcal{N}(\mathbf{x} | \boldsymbol{\mu}...
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distance from a sphere

My question is how far would I have to be away from a sphere to see it in its totally. Using a view of vision of 120 degrees. radius of 3.959 miles I am trying to work out how close an astronaut ...
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Space Time distance metrics

Am working to cluster in Space-Time, and thinking of Space Time metrics as follow: Distance(X,Y) = DistanceDiff**2 + coeff * TimeDiff**2 Am not sure if this is ...
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Billiard power and direction algorithm

Board clear Image Board with 2 sample shot Important: White ball and black ball are exactly one(1) pixle not bigger Hello. I want to write algorithm that computer play billiard and hit specific ...
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Hessian matrix of the mahalanobis distance wrt the Cholesky decomposition of a covariance matrix

I'm stuck with the following problem: I have to compute the second derivative (hessian matrix) of the mahalanobis distance $$ [x-\mu]^{T} \Sigma^{-1} [x-\mu] $$ wrt to the Cholesky decomposition of ...
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Finding similarity of strings using distance function : Bounding the distance function?

I want to know if 2 binary strings $s$ and $t$ each of $d$ length (dimension) and N = 2 (the alphebet) in this case 0 and 1 are similar to each other or not using the following distance function where ...
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distance travelled

A troop 5 metres long starts marching. A soldier at the end of the file steps out and starts marching at a higher speed. On reaching the head of the column , he immediately turns around and marches ...
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Calculate probability of distance for d-dimensional normal

Is there any simple way to calculate the probability of distance in the following form for d-dimensional normal distribution? $P(||\mathbf{x}-\mathbf{\mu}||^2>||\mathbf{x}-\mathbf{a}||^2)$, where $...
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Bhattacharya Distance on Distributions (Matrices) with Different Number of Variables (Dimensions)

We have two matrices, $A$ and $B$, representing two different probability distributions, with dimensions, $m*n$ and $k*n$, respectively. How can we calculate the Bhattacharya distance or another ...
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Bhattacharya Distance (or A Measure of Similarity) — On Matrices with Different Dimensions

We have a series of observations of different properties (such as heart rate or blood sugar level and others as well) across different days from different people from different geographical regions. ...
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Mahalanobis distance

Suppose there is a function $f$, for which we know the inequality $$f(r)\leq r$$ is true, where $r=||x-y||_2=\sqrt{(x-y)^T (x-y)}$ is the Euclidean distance. If now we use the Mahalanobis distance $...
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Optimize to Find the Mahalanobis Distance to Minimize the Term

I have an optimization problem defined as following: Assuming we have a data set $ { \left\{ \left( {x}_{i}, {y}_{i} \right) \right\}}_{i = 1}^{N} $ where $ {x}_{i} \in {\mathbb{R}}^{d} $ and $ {y}_{...
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Malalanobis distance between two multivariate Gaussian distributions

Let $\mathbf{x}\in\Bbb{R}^n$ be an $n$-dimensional real vector distributed normally with mean vector $\mu\in\Bbb{R}^n$ and covariance matrix $\Sigma$; i.e. $\mathbf{x}\sim\mathcal{N}(\mu,\Sigma)$. ...