Questions tagged [convolution]

Convolution is a function-valued operation on two functions $f$ and $g$: $\int _{-\infty }^{\infty }f(\tau )g(t-\tau )d\tau$. Often used for obtaining the density of a sum of independent random variables. This tag should also be used for the inverse operation of deconvolution. DO NOT use this tag for convolutional neural networks.

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155 views

Autocorrelation of convolution integral

Work out the autocorrelation $r_Y(\tau) = E[Y(t)Y(t+\tau)]$ with $Y(t) = \int_{-\infty}^{\infty} h(t-u) x(u)$ and $X$ a WSS, ergodic process I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ ...
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110 views

Help in convolution operation

I am stuck in understanding how the output of convolution operation is obtained. Can somebody please show the steps? The question is an image is processed by applying 3*3 mean filter . What is the ...
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2answers
15k views

Sum of Bernoulli variables with different success probabilities [duplicate]

Let $x_i$ be independent Bernoulli random variables with success probabilities $p_i$. That is, $x_i=1$ with probability $p_i$ and $x_i=0$ with probability $1-p_i$. Is there a closed expression or an ...
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211 views

Deconvolution of sum results in negative numbers

Given $T=G+A$ where $A$ and $G$ are independent random variables, I'd like to estimate the distribution of $G$ given empirical (measured) distributions of $T$ and $A$. Of note: all three random ...
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1answer
359 views

Convolution of difference distribution doesn't conserve expectation - faulty assumption?

I have observations of a system which I hypothesize is defined as $T = A + G$ where $T,A, G$ are random variables. I've measured the empirical continuous distributions $T$ and $A$, and the question is,...
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2answers
1k views

Distribution of the convolution of squared normal and chi-squared variables?

the following problem came up recently while analyzing data. If the random variable X follows a normal distribution and Y follows a $\chi^2_n$ distribution (with n dof), how is $Z = X^2 + Y^2$ ...
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3answers
876 views

Back propagation in Convolutional neural networks

I am trying to understand how CNN works. I want to use them in object recognition task. I thouhgt that CNN is unsupervised networks. My main question is how can I implement the back propagation ...
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596 views

Two-sample bootstrap?

I have two independent samples of observations. From each sample I produce a statistic. Let's denote these as $\theta_1$ and $\theta_2$. I'd like to test the hypothesis that $H_0: \Theta_1=\Theta_2$, ...
6
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1answer
97 views

Express product as convolution? Or, given $f(s)$, find $g$ satisfying $f(s)=\mathbb{E}[g(X)]$ where $X\sim \mathcal{N}(0,s^2)$

Given a function $f(\mu)$ (satisfying certain properties), it is possible to find a function $g(x)$ such that $f(\mu)=\int_{-\infty}^{\infty} g(x)\phi(x-\mu) dx $, where $\phi$ is the standard normal ...
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1answer
2k views

Standard deviations and confidence intervals (weighted) running average

My question is related to this one. I am calculating averages, actually as many as I have samples because I calculate a running average, and for equal weighting I know how to calculate the $95\%$ CI, ...
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2answers
4k views

Difference of two random variable distributions

I have two sets of random variables. I have generated two CDFs for them. Two of the CDFs are plotted graphically. I need to find the difference in distribution of the two CDFs. I have learned about ...
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Training a convolutional neural network

Based on my research on convolution neural networks, every other layer in such a network has a subsampling operation, in which the resolution of the image is reduced so as to improve generalization of ...
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455 views

Sum of absolute values of T random variables

Where X is a r.v. following a symmetric T distribution with 0 mean and tail parameter $\alpha$. I am looking for the distribution of the n-summed variable $ \sum_{1 \leq i \leq n}|x_i|$. $Y=|X|$ ...
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1answer
296 views

PDF of sum of one continuous logistic and one Bernoulli random variables

I need some help people. I have a logistic r.v. $V\sim \Lambda (0, \frac {\pi^2}{3})$ and a Bernoulli $Z \sim B(p_z)$, independent. I have also $c\in R$, $\tilde Z = cZ$, and I have to find the ...
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369 views

Hypothesis testing with exponential family

I'm interested in running hypothesis tests for a variety of members of the exponential family with continuous support, for different values of the parameter/s, for a sample of n i.i.d random variables ...
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1answer
4k views

“Kernel density estimation” is a convolution of what?

I am trying to get a better understanding of kernel density estimation. Using the definition from Wikipedia: https://en.wikipedia.org/wiki/Kernel_density_estimation#Definition $ \hat{f_h}(x) = \...
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59 views

Finding the normal divisor of a random variable

I have an iid sample $x_1, ..., x_n$ from a random variable $X$ which itself is a convolution $X = Z + \mathcal{N}(0, \sigma^2)$. Neither distribution of $Z$ nor parameter $\sigma^2$ are known (we ...
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1answer
375 views

Deconvolution with fourier transform or characteristic function?

Let us consider the following model: $$Y_j = X_j + \epsilon_j \hspace{15pt} j=1, ..., n$$ Where $Y_j$ is a noisy signal, $\epsilon_j$ is the noise which is independend from the signal $X_j$. We have ...
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2answers
507 views

Negative value in density deconvolution

I have a set of samples from measurement. It can be expressed as random variable $Z = X + Y$. So that $Z$ is observable, but $X,Y$ is unobservable. According to prior knowledge and histogram of $Z$, I ...
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1answer
408 views

Is output of Deamer deconvolution not a density?

I have a Model Y= X+e and need the density of X. The deamer package deconvolves the density for X, but if I use the simpsons rule to integrate this density, I get values which are above 1. The ...
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233 views

How to explain certain patterns appearing after kernel averaging?

Having a 2D map filled uniformly by random values (Figure:top-left), the next maps are kernel averaged with a kernel of sizes ...
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156 views

About kernel based estimates

Kernel based operations are common in a variety of applications, such as image processing (e.g., blurring), generating smoothed estimation maps, and so on. A common approach is to select four ...