Questions tagged [distance]

Measure of distance between distributions or variables, such as Euclidean distance between points in n-space.

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15 views

Is Euclidean distance the same as distance-from-correlation as $d(x, y) = \sqrt{2m[1 - r(x, y)]}$

I found in a couple of documents (e.g. this) that the Euclidean distance $d(x, y) = \sqrt{\sum_{i = 1}^{n}{(x_i - y_i)^2}}$ can be obtained from correlation coeffcient if $x$ and $y$ are standardised ...
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Similarity Metric Validation

I want to score a number of similarity metrics, i.e. given a function s(x,y) which returns a number that is higher the more similar x and y are. I'm want to objectively score a number of different ...
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Gaussian Mixture Models and distance matrix

I have a (euclidean) distance matrix and I want to perform GMM clustering. I read in another post (gaussian mixture model - approximate a matrix) that I could apply MDS or PCA to this matrix and use ...
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MDS as input for K-means

I have a distance matrix, which I know is in the Euclidean space. Then I performed PCA on the distance matrix to obtain MDS or PCoA result matrix. I read in multiple posts that I could use this as ...
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Dissimilarity matrix, Euclidean distance and Kmeans

I was wondering whether the Sklearn Kmeans and AgglomerativeClustering with Ward linkage takes as input regular Euclidean ...
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Wasserstein distance for categorical data? Relationship to TVD?

Is the Wasserstein distance applicable for categorical data? e.g. if we have the distribution of different coloured balls in two bags, ...
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23 views

Is the set of distribution $\{ X | \max_t |f_X(t) - f_Y(t)| \leq \epsilon \}$ convex, where f is the cdf or inverse cdf?

I'm trying to figure out if the set is convex, where the maximum difference between cdf(or inverse cdf) of X and a reference distribution Y is smaller than $\epsilon$. 1. Let $f_X(t)$ denote the cdf ...
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i.i.d data for computing distance correlation

When computing an estimate of the distance correlation of random vectors $X$ and $Y$ using paired samples of the vectors, should the samples be i.i.d? Can I have data in which $X_i$ and $X_j$ are ...
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Probability of points less than a fixed distance apart in a vector space

I have a distribution $D$ of points in a normed vector space (it's $\mathbb{R}^n$ using the $L_\infty$ norm, but I don't think that matters). In this particular space, points that are less than a ...
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Computing the most representative sample of a random variable

Let $X$ be a real-valued random variable and $n > 0$. Using numerical methods, how can we find the vector $\vec v$ of $n$ real numbers that is most characteristic of $X$, in the sense that the ...
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Measure of distance between two survey responses

I've found some survey data where respondents answer 63 question by giving a response for each question between 0-10 (0 for strong disagreement, 10 for strong agreement). So I can view every ...
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Similarity/dissimilarity matrix over classes

I am trying to get something like a confusion matrix for different classes, but without training a model. The idea is to use some kind of distance between classes. The data set is like this, just for ...
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J-S Divergence as a Percentage

Is there a way to interpret the Jensen-Shannon divergence which is normalized to be between 0 and 1 between two probability distributions as a "percent difference", i.e., there is a x% ...
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Clustering - Distance Metric for Comparing Short Lists of Terms (non-repeating, no frequency)

Clustering involves using some distance or similarity metric. What is the best way to score the similarity of these small sets of words? Criteria: These are technical terms which are extracted from ...
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Using $\chi^2$-Distance for Histogram-Comparison with 0-Valued Elements leads to NaNs

I want to compare two histograms by using the $\chi^2$-distance. There is a definition in the OpenCV-library. Also this question gives a lot of insight about this distance-metric. Because i didn't ...
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K-Means transform function does not match pairwise_distances_argmin_min centar calculation

I need to be able to select first N most representative points from each cluster calculated by K-means. To do so, I am aiming to calulate the distance of each point to its own cluster center and take ...
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Specific proof related to MDS distance matrix

Given a symmetric, positive semidefinite matrix A, and matrix D, where $D_{ij}=A_{ii}-2A_{ij}+A_{jj}$, prove that there exist n vectors {$\vec{v_1},...,\vec{v_n}$} such that $D_{ij}=||\vec{v_i}-\vec{...
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Product of two probability density function

Suppose $f$ and $g$ are two probability density functions. I have seen economists use $\int f(x)g(x) dx$ as some kind of similarity measure. For example, Jaffe (1986) uses sum of product of two ...
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Distance of a vector to a sample of a distribution?

Is there a better measure than Euclidean distance for my problem? I have two functions e() and f(x). x is a vector of continuous values. The process e() outputs a variable z. Subsequently, a variable ...
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Is there a metric to identify whether samples are uniformly distributed when number of samples is small?

Suppose I have a small number of samples drawn from an unknown distribution $\{X_1,X_2,...,X_n\}$, where $0\le X_i \le L$, and $3\le n \le10$. I want to identify a metric to understand how far these ...
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Finding a sub-population from dataset matching another target dataset

Let's say one has a finite collection of i.i.d. samples from an unknown source distribution $S=\{x_{i} | i \in [1,n_{S}], x_{i} \sim p_{X_{S}}(x)\}$. Where each $x$ is multidimensional and has ...
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Distance matrix time series analysis? (Ecology/diversity)

I am trying to analyze a time series of ecological data. Each time point in the series is a matrix of animals-by-foods (that they were observed to eat). For each of these time points, I compute the ...
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two times square in distance calculation on one example?

I read a book on Kernels, See the following example. Why the authors take square two times here? what is the logic?
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Best approximation of the Mahalanobis distance by standardized Euclidean distance

I am looking for the best way to approximate the Mahalanobis distance by the standardized Euclidean distance, which would reduce the number of the required multiplications. The easiest way is the ...
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Does Wasserstein distance require the source and target distributions to have the same mass?

If we minimize the Wasserstein loss, $$W_1 (P_S, P_T) = \underset{\gamma \in \Pi}{\text{min }} \sum_{x,y} |x-y|\gamma(x,y)$$ which means we are looking for the coupling $\gamma(x,y)$ that minimizes ...
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How to compute Gower distance manually?

Consider the following tuples: a = {1, 0, 13, apple} b = {1, 1, NA, pear} c = {0, 1, 12, apple} The first two elements for each observation are binary, the third ...
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Can any distance be expressed by the product scalar or inner scalar? [duplicate]

I know that a distance could be expressed by inner scalars or scalar products. Is this true for all metrics (i.e., respecting the three axioms: the identity of indiscernible, symmetry, triangle ...
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If entropy is the underlying measure for KL-divergence, what is the underlying measure for the Wasserstein distance?

If entropy is the basis measure underlying KL-divergence (aka relative entropy), what is the basis measure underlying the Wasserstein distance?
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Best way to compare time series data similiarity

I have two time series'. One represents ground truth heart rate in a a subject (top in blue), and the other a prediction of heart rate using a novel method (bottom in orange). It looks like this: ...
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44 views

Properties of Sinkhorn distance

I am reading the paper by Cuturi http://www.marcocuturi.net/Papers/cuturi13sinkhorn.pdf and I am curious about the properties of Sinkhorn distance and wondering what properties of the Sinkhorn ...
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What is the intuitive difference between Wasserstein-1 distance and Wasserstein-2 distance?

What is the intuitive difference between Wasserstein-1 distance and Wasserstein-2 distance, and how to know which one to use?
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174 views

What are the advantages of Wasserstein distance compared to Jensen-Shannon divergence?

What is the practical difference between Wasserstein metric and Jensen-Shannon divergence? Wasserstein metric is also referred to as Earth mover's distance. From Wikipedia: Wasserstein metric is a ...
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Brier score: $L_1$ instead of $L_2$ [duplicate]

Assume that we have some count data $x_{1}, \dots, x_{n}$, which take values $\{1, \dots, m\}$ and we have some estimator of the probability mass function, $\hat{\mathbf{p}} = (\hat{p}_{1}, \dots, \...
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Intended interpretation of one-mode, three-way (dis)similarities?

I have what I think is a very simple question, the answer has just eluded me so far. A two-way similarity, $s_{ij}$ (for objects $i$ and $j$) can be interpreted fairly straightforwardly as the degree ...
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Which is the 'best' method to detect outliers from a PCoA analysis?

I'm trying to find a suitable method for detection of outliers from a PCoA output. This analysis is used to visualize the results from a distance matrix between a set of sample applying the Chemical ...
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48 views

How to compare two set of PMFs?

I'm facing some challenge and I don't know the correct approach for this. I'm having two sets of PMFs $S_1, S_2$ and I need to compare (distance like Jensen–Shannon) $S_1$ with $S_2$. What's the best ...
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Euclidian distance vs cosine similarity

Currently I'm working on facial recognition. If I use encoding/feature vectors of 2 images which method will prove more accuracy, L2 norm or cosine similarity and why? I read "ICA performs ...
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Why clustering in a linear scale using correlation based distance gives better results than clustering in a log2 scale?(PAM clustering)

I have questions regarding cluster analysis. I am trying to cluster data made up of proteins. (23 columns and 1800 rows) I have the data in a log2 scale, some variables range between 2-10 and others ...
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1answer
57 views

Statistical distance between two matrices

The statistical distance between two probability distributions can be measured with $f$-divergences such as the KL-divergence. The statistical distance between two clusters can be measured with ...
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Are there any linkage functions that can handle signed dissimilarity matrix?

I know that a "distance" matrix is a symmetric positive matrix where the diagonal is zero. A "dissimilarity" matrix, to my understanding, is a generalization of a distance matrix. ...
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What is the appropriate metric for determining distance / dissimilarity of sparse, high dimensional data in PCA space?

I'm working with scRNA-seq data (~96% sparse, high dimensional), and am trying to determine distances between the cells in PCA space - NOT for the specific purpose of clustering. The principal ...
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Outlier detection in high-dimensional longitudinal data

I'm having a longitudinal dataset with a large number of variables where I would like to use a ML algorithm to inspect possible outliers. What are the techniques you would use for this? I've seen a ...
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1answer
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Coordinates from noisy distance matrix?

I have a black box in which I know there is a 1D line and points along this line, and as output from this box I can get out a distance matrix for the points, but I know there is noise in the estimate ...
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Is there any divergence like Jensen-Shannon for two vectors which are not distribution? [duplicate]

I know that the Jensen-Shannon is defined as a divergence between two or more distributions ($P_1,P_2,...P_k$). But, instead of distributions, I have some multiplications of two distributions (so they ...
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35 views

Difference between two multi-variable datasets

I have multiple activities that a person performs, and the corresponding multidimensional values [$x$, $y$, $z$ - coordinates] for two devices and their readings. I need to find the difference between ...
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48 views

What is the correct way to implement Jensen-Shannon Distance?

I'm trying to use this code to compute the Jensen-Shannon distance: ...
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How important is triangle inequality for statistical estimators?

(Pearson's) correlation is a measure of co-dependence that does not fulfill certain axioms such non-negativity and triangle inequality. In layman's terms, how would you describe what triangle ...
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The first principal component line minimizes the sum of the squared perpendicular distances between each point and the line [duplicate]

I am currently studying An Introduction to Statistical Learning, corrected 7th printing, by Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani. Chapter 6.3.1 Principal Components ...
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direction of outlier detected by the Mahalanobis distance

Mahalanobis distance provides a value that might be used for the detection of outliers. My question: how to calculate the direction of the outlier (as a vector)? A simple answer would be to use the ...
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1answer
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Sensitivity of KL Divergence

I am very new to the concept of KL divergence. Although I have grasped the fundamental formulations, I have a confusion comparing the KL divergence across the different distributions. Suppose I have 3 ...

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