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Questions tagged [symmetry]

symmetry or asymmetry in data or in distributions

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Can a non-symmetrical distribution have the same areas under the PDF in the two sides around the mean?

I was thinking about symmetrical vs. non-symmetrical distributions and I found myself stuck in a thought that I had never thought before. We know that symmetrical distributions like Normal, Laplace, ...
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Enforcing symmetries "for bag-of-vector" data in XGBoost or random forest - geodata example for illustration

I'll give a concrete toy problem, then give some comments on what sorts of abstractions I care about. Toy problem: Each person $i$ in my dataset has a phone, and every once in a while the phone will ...
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Asymmetric errors on cluster size distributions

I have a question that might seem simple, but I'm unsure how to address it. For an analysis on pixel detectors, I am extracting a histogram similar to the ones attached, focusing on the cluster size ...
Massi88's user avatar
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Background reason for the terms ‘Isotropic’ and ‘Anisotropic’ in the context of GNN Message Passing

I’m reading a paper on Graph Neural Networks (GNNs) that uses the terms ‘isotropic’ and ‘anisotropic’ in the context of message passing. I understand that these terms originate from physics, chemistry,...
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12 votes
4 answers
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Should point estimates for a parameter always be exactly in the middle of their 95% CI or does it depend on the distribution? [duplicate]

I'm modelling some count data using negative binomial regression with glm.nb in R. I've noticed that my point estimates are quite consistently not at the midpoint of the 95% CIs and wondering if this ...
Anthony Fish Hodgson's user avatar
7 votes
1 answer
863 views

Why must a product of symmetric random variables be symmetric?

I was reading about weight initialization in neural networks (He et. al, 2015) when I came across this statement: "If we let $w_{l-1}$ have a symmetric distribution around zero and $b_{l} = 0$, ...
BlackKnight's user avatar
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Is multinomial logistic regression symmetric?

Simple linear regression is symmetric in the sense that, if I regress $Y$ on $X$ or $X$ on $Y$, I get the same $R^2$ and result from the overall $F$-test. ...
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How to design a Gaussian Process which respects input order symmetry

I am trying to design a Gaussian Process model for optimizing experimental parameters. In particular, an experiment requires n parameterizations of points on the 2D Cartesian plane. For a particular ...
cmk's user avatar
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1 answer
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Symmetry assumption in the Wilcoxon-Mann-Whitney test

The Wilcoxon-Mann-Whitney test requires that two distributions are symmetrical How can I check this assumption by using a hypothesis test and how apply it in R? If this assumption is not met what test ...
Statistical scientist's user avatar
12 votes
4 answers
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Does no correlation but dependence imply a symmetry in the joint variable space?

I was looking through the answers to this question, and all of them seem to have some form of symmetry between the variables. I'll walk through the examples in that question so you can see what I mean....
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Symmetry of standardized regression coefficient (=Pearson correlation) in linear regression

Suppose 2 continuous variables X and Y. Their Pearson correlation equals 0.8. This correlation is symmetric (it does not assume a dependent or independent variable). We proceed to a linear regression, ...
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Verification of proof that a point of symmetry is mode (Casella-Berger 2.27)

I was solving the following problem 2.27 from Statistical Inference by Casella-Berger. 2.27 Let $f(x)$ be a pdf, and let $a$ be a number such that if $a\geq x\geq y$, then $f(a)\geq f(x) \geq f(y)$, ...
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Standard deviation of symmetric data

Within my field a recent study suggested to use the symmetric properties of certain image datasets to improve signal to noise ratio (SNR). I will spare you the details, but in the end one can get a ...
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5 votes
1 answer
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Difference between Symmetrically normalized Laplacian matrix versus graph laplacian matrix

I am trying to understand the graph laplacian matrix in Graph Convolution networks. To get a basic understanding of graph laplacian matrix I am referring to this https://mbernste.github.io/posts/...
Jose_Peeterson's user avatar
1 vote
1 answer
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The Wilcoxon signed-rank test without symmetry caused by one outlier

I am comparing two algorithms on the same input data. Now I want to see whether the difference in output is significant. For this I need to use the Wilcoxon signed-rank test, since my data is paired ...
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In estimating $X + Y$, is it helpful if I know random variables $X$ and $Y$ are identically and independently distributed?

Suppose I have $$X \sim Dist_1$$ $$Y \sim Dist_2$$ and I want to estimate $X + Y$. I can sample from $Dist_1$ and $Dist_2$ and generate samples for $X + Y$. So far so good. Now suppose I discover that ...
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Is there a statistical test for matrix symmetry? [closed]

I have collected some data and done some processing until the question I'm faced with is - "is the data matrix $X$ a symmetric matrix?". Note that elements of $X$ represents event counts. ...
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Is a bivariate copula relevant in this physics setting manifesting uniform univariate marginals--and, if so, how can it be constructed?

To quickly place our probabilistic (copula) question in its subject matter setting, we note that a fundamental concept in quantum theory is that of entanglement QuantumEntanglement. The states of ...
Paul B. Slater's user avatar
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What are examples of symmetric copulas $f(x,y)=f(y,x)$ having relative minima for $f(x,x)$?

In a previous posting on this site RepulsiveBehavior I attempted to detail a quantum-information-theoretic separability/entanglement problem I am pursuing. Detailed issues of sampling sizes for a data ...
Paul B. Slater's user avatar
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1 answer
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Index for the shape of a distribution (necessary or sufficient condition?)

I have a confusion about the main statistical indicators for the shape of a distribution. Let me consider first the Fisher index defined as $$\rho=\frac{\frac{1}{n}\sum_{i=1}^n(x_i-\bar x)^3}{\sigma^3}...
user268193's user avatar
2 votes
1 answer
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Will MSE-based estimator generate symmetric residuals if the error has got symmetric support (not distribution)?

This question is more specific than :my old question Take follow regression model: $y=f(x)+e$ Where $e\sim D$ with a such symmetric support $A=(-a,a)$, not symmetric distribution. Now given a data set ...
Davi Américo's user avatar
2 votes
1 answer
63 views

Will quadratic-based estimation (not necessarily MSE) always generate a symmetric residuals after training it?

These are error's empirical distribution for XGB, RF and kNN, the last one have taken on another dataset. Neither of them is normally distribuited but they all are symmetric. None of used algorithms ...
Davi Américo's user avatar
2 votes
1 answer
346 views

Empirical Implications of Unbiased Estimators

I am familiar with the layperson explanation of an unbiased estimator as follows: if we repeat an experiment under identical conditions many times, the average value of the estimate will be close to ...
Mr Saltine's user avatar
12 votes
6 answers
2k views

Example distribution where 74% of probability is above the mean

Watching Why You Should Want Driverless Cars On Roads Now, at 8:14 Derek Muller claims: Surveys show 74 % of people believe they are above average drivers. This claim motivates my question, but some ...
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2 votes
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Output of ANN with zero initialized weights represents what?

In class we discussed that if the weights of an ANN (standard feed forward NN in binary classification setting [0,1]) are initialized all at zero, the ANN fails to break symmetrie and therefore, the ...
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1 answer
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Does conditional symmetry imply mean independence? [duplicate]

suppose I have two random variables $X$ and $Q$. $Q$ is conditionally symmetrically distributed about zero, i.e., its density satisfying satisfying $f(-q|X=x)=f(q|X=x)$ for every $q\in \Omega_{Q|X}$...
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1 vote
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A parameter to differentiate multimodal density plots

I am trying to find a parameter that would summarize the shape of a density plot where: an insight into the symmetry is given (not a priority); and how regular/irregular the multi modals are For ...
jack kelly's user avatar
0 votes
1 answer
268 views

3D symmetry plane using PCA [duplicate]

I'm trying to compute the symmetry plane of a 3D mesh representing an animal footprint in R. I've ran a PCA on the 5755 points that are making up the 3D mesh (see below): The output of the PCA is ...
Antoine Marchal's user avatar
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1 answer
120 views

When may the Kernel Trick Matrix be non symmetric?

Ridge Regression can be expressed as $$\hat{y} = (\mathbf{X'X} + a\mathbf{I}_d)^{-1}\mathbf{X}x$$ where $\hat{y}$ is the predicted label, $\mathbf{I}_d$ the $d \times d$ identify matrix, $\mathbf{x}$ ...
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Bisymmetric covariance matrix in Auto-Regressive Model

When I learning Time Series, about the Auto-Regressive model (AR) of order $p$: $$x_t=\alpha_1x_{t-1}+\dots+\alpha_px_{t-p}+w_t,$$ where $w_t$ is a time series of white noises. The textbook (Paul &...
kelvin hong 方's user avatar
2 votes
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How to find the symmetric kernel for the given U-statistic?

The U-statistic is given by \begin{equation} \widehat{\Delta}=\frac{1}{\binom{n_1}{2}\binom{n_2}{2}}\sum_{1\leq i_1<i_2\leq n_1}\sum_{1\leq j_1<j_2\leq n_2}f(X_{i_1},X_{i_2},Y_{j_1},Y_{j_2}), \...
reeba mary's user avatar
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Data augmentation for traditional machine learning algorithms

Data augmentation suffices multiple purposes, I would list a few here: Increasing dataset size: The data is just fragment/stand-in trying to represent reality, having more data should thus result in ...
Imago's user avatar
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3 votes
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What is "symmetry" in evaluation metrics

I'm seeing Mean absolute percentage error (MAPE) is not symmetric. Tried to understand what is symmetry here but didn't find a good answer online. Can I ask: What is symmetry in evaluation metrics? ...
Cherry Wu's user avatar
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Why doesn't $(e^{A})^{-1} = e^{-A}$ hold for a symmetric matrix in Python?

$e^A$ is just the $A$ matrix with all of its elements exponentiated, called a matrix exponential. It follows that the inverse $(e^{A})^{-1} = e^{-A}$ for square matrices, although I could find nothing ...
develarist's user avatar
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4 votes
1 answer
194 views

Can we use Kullback-Leibler in either direction as a loss function for probabilistic classifiers?

Suppose we are learning a probabilistic classifier $q(x)$ approximating a true distribution $p(x)$. One natural similarity measure between distributions $p$ and $q$ is the Kullback-Leibler distance $$...
user118967's user avatar
3 votes
1 answer
248 views

About unique determination of symmetric point (or center) of a distribution based on pdf or cdf

Suppose we have a distribution that is known to be continuous and symmetric, and is otherwise unknown. We want to decide whether it is actually centered at zero using an equation involving pdf or cdf. ...
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5 votes
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Is the definition of symmetric distribution using cdf correct?

Based on Wikipedia, a distribution is symmetric about $x_0$ if and only if it is a distribution whose pdf(or pmf) $f(\cdot)$ satisfies $f(x_0+x)=f(x_0-x)$ for any $x\in \mathbb{R}$. Can we also ...
ExcitedSnail's user avatar
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1 vote
0 answers
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Efficiently sampling a symmetric posterior with MCMC

I am using MCMC (via emcee) to sample a posterior distribution $p(\vec\theta|Y)$ where $\vec\theta = (\theta_1, \theta_2, \ldots)$ are parameters for a physical model of the process generating an ...
Alexis's user avatar
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0 answers
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Symmetry of distribution function is defined as $f(x-a)=f(-(x-a))$, then expectation is $'a'$. i.e $E(X)=a$ [duplicate]

I came across this statement in a book. While I know, how to prove $E(X) = a$ is using $f(x+a)=f(x-a)$. I cannot seem to prove it using $f(x-a)=f(-(x-a))$. I keep going on in a loop, no matter what I ...
Doubts's user avatar
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1 vote
1 answer
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Is 0 the unique center for the mixture density of $N(-a,\sigma^2)$ and $N(a,\sigma^2)$, each with weight 0.5? [duplicate]

Suppose $f_{-a}(x)$ is the pdf for $N(-a,\sigma^2)$ and $f_{a}(x)$ is the pdf for $N(a,\sigma^2)$. Let $f(x)=0.5f_{-a}(x)+0.5f_{a}(x)$ be the mixture density. Is $c=0$ the unique center for $f(x)$ in ...
ExcitedSnail's user avatar
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1 vote
1 answer
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How to analyse association between two (paired) sets of measurements, when arbitrary to which set each member of a pair belongs

I have pairs of measurements and need advice to select a measure of association between the measurements. The special aspect that has me confused is the symmetry: there is no reason to allocate a ...
hutch's user avatar
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0 votes
1 answer
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Correlation is a symmetric measure, but scatter plot matrix shows asymmetric dependence

The correlation matrix demonstrates that correlation is a symmetric measure: $\rho(X,Y) = \rho(Y,X)$ since the lower off-diagonals are mirror images of the upper off-diagonals. The scatterplot matrix ...
develarist's user avatar
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1 vote
0 answers
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If I want to model a bivariate distribution that is symmetric about (0,0) using copula, what copulas can I use?

If I want to model bivariate data $\{X_i,Y_i\}_{i=1}^{n}$ using copula. The true joint density of $(X,Y)$ denoted as $f_{XY}(,)$ is unknown, but I know it's symmetric about (0,0) in the sense that $f_{...
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10 votes
1 answer
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If I know the density I'm estimating is symmetric about 0, how to impose this restriction in my kernel density estimator?

Suppose I'm interested in estimating the unknown smooth density of $X$ denoted by $f(\cdot)$ using data $\{X_i\}_{i=1}^{n}$. Suppose I also know that $f(\cdot)$ is symmetric about 0 in the sense that $...
ExcitedSnail's user avatar
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1 vote
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Which second order kernel is symmetric, has bounded support and satisfy $\int x^2 k(x)dx=1$?

Suppose $k(\cdot)$ is a univariate kernel function of order 2 in the sense that $\int x k(x)dx=0$, and $\int x^2 k(x)dx\neq0$. $k(\cdot)$ equals 0 outside a bounded interval, and $k(-x)=k(x)$ for any $...
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2 votes
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If the joint density $f_{X_1,...,X_n}(x_1,...,x_n)$ is symmetric about the origin, does this imply that each marginal cdf $F_{X_i}(0)=1/2$?

If the joint density $f_{X_1,...,X_n}(x_1,...,x_n)$ is symmetric about the origin in the sense that for any $(x_1,...,x_n)$, it holds that $f_{X_1,...,X_n}(x_1,...,x_n)=f_{X_1,...,X_n}(-x_1,...,-x_n)$ ...
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Symmetrization in Proof of Hoeffding's Lemma

This alternative proof of a slightly weaker version of Hoeffding's Lemma features in Stanford's CS229 course notes. What's notable about this proof is its use of symmetrization. However, I find this ...
minch's user avatar
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What is the expected cost of using LDA?

Suppose that you observe $(X_1,Y_1),...,(X_{100}Y_{100})$, which you assume to be i.i.d. copies of a random pair $(X,Y)$ taking values in $\mathbb{R}^2 \times \{1,2\}$. I have that the cost of ...
user avatar
2 votes
1 answer
65 views

In this case, no problem for initializing weights in deep learning networks to 0

Deep learning textbooks say that initializing all weights of neural networks to 0 will be problematic as it breaks symmetry. I tried with a simple 1-layer neural network but found such is not the ...
zell's user avatar
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3 votes
1 answer
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Which test(s) are alternative to Mann-Whitney test for non-parametric continues data when symmetry assumption is violated

From here and here I see that we cannot use Mann-Whitney test if symmetry assumption is violated. Which test(s) can we use instead of Mann-Whitney test for non-parametric continues data if symmetry ...
vasili111's user avatar
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