Questions tagged [symmetry]

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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 ...
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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 ...
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1 answer
44 views

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}...
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2 votes
1 answer
54 views

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 ...
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1 answer
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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 ...
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2 votes
1 answer
119 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 ...
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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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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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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
0 answers
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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 ...
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Dominant symmetry plane detection for point-based 3D models using weighted PCA

I'd like to apply the weighted PCA method for symmetry plane detection described in this article: https://www.hindawi.com/journals/am/2020/8861367/ Does anyone has an R script that can help me?
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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 ...
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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 &...
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2 votes
0 answers
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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}), \...
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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 ...
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1 answer
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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? ...
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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 ...
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1 vote
0 answers
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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 $$...
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2 votes
1 answer
162 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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4 votes
1 answer
730 views

Is the definition of symmetric distribution using cdf correct?

Based on wikipedia (https://en.wikipedia.org/wiki/Symmetric_probability_distribution), a distribution is symmetric about $x_0$ if and only if it is a distribution whose pdf(or pmf) $f(\cdot)$ ...
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1 vote
0 answers
29 views

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 ...
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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 ...
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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 ...
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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 ...
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1 answer
418 views

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 ...
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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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9 votes
1 answer
334 views

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 $...
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1 vote
0 answers
35 views

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
0 answers
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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 ...
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240 views

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 ...
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2 votes
1 answer
29 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 ...
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2 votes
1 answer
418 views

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 ...
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1 vote
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Asymmetric robust regression

What are the methods for robust regression with asymmetric distribution of outliers? I am specifically interested in equivalents of Huber and Tukey M-estimators. However, asymmetric heavy-tailed ...
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1 vote
1 answer
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Help - Expectations and Ratios

I would need your help for a problem I have and I don't know how to solve. I would like to know whether I could prove that : $$E[0.5X/(0.5X+0.25)] = E[0.5(1-X)/(0.5(1-X)+0.25)]$$ knowing that $E[X] =...
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1 answer
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Error distributions and consistent and unbiased OLS

If OLS estimator is unbiased and consistent, what does it imply about the distribution of error terms? In linear regression model: $ y_i = \boldsymbol{x_i' \beta} + \epsilon_i $ if the OLS estimator ...
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1 vote
1 answer
67 views

Is there a signed (ie anti-symmetric) version of SMAPE?

The symmetric mean absolute percent error (SMAPE) is a symmetrized version of percent error with the formula: $$\frac{200\%}{n}\sum_i\frac{|x_i - y_i|}{|x_i| + |y_i|}$$ SMAPE is symmetric: ...
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2 votes
2 answers
4k views

What is a symmetric distribution symmetric about if it has zero skewness? [duplicate]

We know that a distribution with zero Skewness are symmetric. A quick Google search or looking up in textbooks says that Symmetric distributions are distributions where the left side mirrors the ...
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What would be the output distribution of ReLu(X) activation (In case that the distribution of X is unknown)?

Suppose E[X]=0, var(X)=1 and we know X has a symmetric distribution, What would be the distribution of 𝑌=ReLU(𝑋)=max{0,𝑋}? I have seen this question What would be the output distribution of ReLu ...
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0 answers
57 views

Showing that if the PMF of $W$ is symmetric around zero then some parameters entering it are equivalent

Summary: In what follows, I specify the probability mass function (PMF) of a random variable $W$, depending on some parameters $(\lambda,\mu,\lambda',\mu')$. I would like your help to show that $$ \...
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0 votes
1 answer
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Definition of symmetric probability mass function [duplicate]

Consider the random variables $Y$ characterised by a probability mass function (PMF) as follows: $$ Y=\begin{cases} -2 & \text{ with probability $\frac{1}{2}$}\\ 2 & \text{ with probability $\...
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1 vote
1 answer
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Characterise the set of symmetric probability mass functions

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
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4 votes
1 answer
226 views

Symmetry group in posterior distribution/inference

Here's a scenario: Suppose I collect a dataset $\{x_i\}_{i=1}^k\subseteq\mathbb R$ of data points $x_i$, and I wish to explain it using a mixture of two Gaussians; assume the unknown parameters are ...
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5 votes
1 answer
173 views

What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID ...
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5 votes
2 answers
423 views

Identically distributed vs P(X > Y) = P(Y > X)

I've two related propositions which seem correct intuitively, but I struggle to prove them properly. Question 1 Prove or disprove: If $X$ and $Y$ are independent and have identical marginal ...
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3 votes
1 answer
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When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate distribution function $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that $$...
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