Questions tagged [complex-numbers]
A complex number is of the form a + bi, where i is the square root of -1.
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Expected Value for Complex-Valued Random Variable
This question is part of Exercise 3.14 in the book The Analysis of Time Series: An Introduction with R, 7th Ed., by Chatfield and Xing.
Problem Statement: Suppose $\theta$ is uniformly distributed ...
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Interpreting complex roots of $R^2$
This is a common way to define $R^2$ in a regression problem.
$$
R^2=1-\left(\dfrac{
\overset{N}{\underset{i=1}{\sum}}\left(
y_i-\hat y_i
\right)^2
}{
\overset{N}{\underset{i=1}{\sum}}\left(
y_i-\bar ...
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Complex parameterizations of real-valued distributions
Suppose we have some random variable $X$ that takes values in $\mathbb{R}^n$, parameterized by $\theta \in \Theta$ where the parameter space $\Theta$ is finite-dimensional.
In almost all statistical ...
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A test do check the circularity of a complex variable [closed]
I need a way to measure the circularity of a complex random variable. A complex random variable is circular when its PDF depends only on its magnitude and does not depends on its angle.
For example, $...
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A Simple Toy ML problem that surprisingly fails to converge (or even "try"!)
This is a much simplified network from a real problem that, to me, has a surprising INability to learn a simple task via backprop, ie, it can't overfit or learn at all. This simple version has come at ...
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How to quantify the similarity between three sets of complex numbers? [closed]
I have multiple groups of measurements, each containing three sets of complex numbers (impedances of the same thing measured under three conditions). The Nyquist plots belows shows two of such groups.
...
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CDF for squared sum of Rayleigh random variables
In short, I am looking to estimate the distribution of
$ \eta = \sum_{i=1}^N (X_i - z_i)^2$, for each $X_i \sim \text{Rayleigh}(1)$ and constants $z_i$.
If $X_i$ were Gaussian, then this could be ...
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Understanding of bivariate Gaussian distributions in connection with complex random variable
Say that we want to model a complex-valued signal using the RV $S$, where $S$ can be expressed by it's real and imaginary part, i.e. $S = X + iY$, where $X$ and $Y$ are real-valued random variables.
...
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Putting a constraint on the output of the neural network
I am using a neural network to input some complex numbers and to obtain complex numbers. I converted the input complex numbers into real values by stacking the real part and imaginary parts as a ...
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Probability of Roots outside the unit disc
Consider a random polynomial $p(z)=\sum_0^n A_i z^i.$ where $A_i,i=0,12,3,..,n$ are iid uniform variables in the interval $(0,1).$ I want to show that that the probability of the root with min modulus ...
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Probability of a number being a bound for roots
.Consider the polynomial $p(z)=\sum_0^na_iz^k$ where $a_n=1$ and $a_k \sim N(0,1)$, $k=0,1,2,\dotsc,n-1.$
What is the probability that 2 will be a bound of the roots of the polynomial? How can we find ...
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Find correlation of complex data - xcorr and corrcoef vs direct multiplication
I'm coding a paper (on turbulent statistics) where it says find the time series correlation of complex data. It does that by taking the product of the vector and its complex conjugate (a velocity ...
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What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?
Note: I also posted this question on MATHEMATICS.
For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this ...
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Expectation involving i.i.d. complex Gaussian random vectors
$\boldsymbol{h_1}$ and $\boldsymbol{h_2}$ are i.i.d. circularly symmmetric complex Gaussian random vectors with zero mean and covariance matrix $\boldsymbol{K}$.
$ \boldsymbol{h_1} = \left [h_1(0),\...
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How to find the right scaling for exponential distribution
I have a complex Gaussian variable, $Z=X+jY$ with $X,Y \sim \mathcal{N}(0,\sigma^2)$, and I would like to find the parameter that scales the distribution of the squared magnitude $P=|Z|^2$. As ...
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Machine learning kernel with complex feature map
I have a question regarding my machine learning lecture where we had to decide whether $$K(x,y)=x_1y_1-x_2y_2$$ is a valid kernel (e.g. for a SVM). My intuition would say that it is a valid kernel ...
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Robust linear regression for complex valued data in R
Are there any existing R packages capable of performing a robust linear regression on complex valued data?
I have a set $Y$ of complex valued ($a + b i$) data, that are linearly dependent on another ...
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Error propagation of complex numbers
I have a problem with coordinate transformation of complex numbers and their covariances:
Let's say I want to do some statistic with a series of complex numbers and for this, I have to work in the ...
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Proof that variance is always greater than or equal to zero
It is common knowledge that: $$\begin{equation}\label{3} Var(X) \geq 0 \end{equation}$$ for every random variable $X$. Despite this, I do not remember seeing a formal proof of this.
Is there a proof ...
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Why use complex-valued random variables?
Edit: This question has been posted on Math.exchange here. To avoid duplication, please comment on the Math.exchange thread.
I am interested in random complex numbers and am trying to understand why ...
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Sum of squares for datasets valued in $\mathbb{R}^{m \times m}$ or $\mathbb{C}^{m\times m}$
Let us assume we have $k \times k$ matrix valued data and assume this is organized (possibly as time series):
$$ M_1, M_2, \ldots, M_n $$
Now, assume we are interested in writing down an error ...
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Can one uniformly generate complex numbers of absolute value less than a given constant $R \neq 1$? [duplicate]
Can one uniformly generate complex numbers of absolute value less than a given constant R?
This would appear to be equivalent to picking points $(x,y)$ uniformly in a disk of radius R, where $x$ is ...
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How to find variance of a complicated expression?
I have an equation given by
$$
\phi(k)=\sqrt{1-\rho^{2}}\sum_{j=1}^{k-1}\rho^{k-j-1}e(j)
$$
where $\rho$ has value between 0 to 1 and $e$ is modeled as $\mathcal{C}\mathcal{N}(0,\sigma^{2})$, i.e. ...
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Action of complex operator on (real) normal distribution
Let $\mathbf{x}$ be an N-dimensional (real) random variable following a multivariate normal distribution, $\mathbf{x}\sim \mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$. If $\mathbf{A}$ is a real ...
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Moment generating function of non-central Chi-squared distribution with complex mean?
I have random variables $(X_1, \dots, X_k)$ distributed independently according to normal distributions with complex means, i.e. $j\mu_i, i=1\dots k, j^2=-1$, with unit variances.
I want to study the ...
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Maximum likelihood estimation for the complex multivariate Gaussian
Background
Consider a multivariate Gaussian dataset $\mathbf{Y}$ with observations on $k$ individuals (rows) over $m$ variables (columns). The variables have covariance $\boldsymbol{\Sigma}$ (an $m\...
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Expectation over complex multiplication is multiplication of expectation
Suppose I have $\mathbb{E}\left[{\bf x} {\bf x}^H \right]$, where ${\bf x} \in \mathbb{C}^{N \times 1}$ is a random vector which has a uniform distribution, then can I say,
$\mathbb{E}\left[{\bf x} {\...
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Wald test for complex-valued data
Let $\mathbf{x} = [x_1,\ldots,x_N] \in \mathbb{C}^N$ be a set of complex-valued data and let $p(\mathbf{x};\alpha)$ their joint probability density function parametrized by the unknown complex scalar $...
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Using a surrogate model for the solution space of an optimization problem
I have an optimization problem: Given a complex $n\times n$ covariance matrix $C$ one must find a complex $n$-vector $v_C^\ast$ which (approximately) minimizes an objective $f_C(v)$ over all space. $...
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gam/gamm when response variable is complex
I would like to fit a generalized additive mixed model using mgcv or gamm4, but have a response variable consisting of complex numbers where y=a+1i*b. Is this possible, and if so are there any special ...
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Use of Complex Numbers in Statistics
I was asked recently if complex numbers were used in Statistics by a friend of mine who is an electrical engineer. Besides statistical applications in other fields (e.g. quantum mechanics) and ...
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Using complex number in non-negative matrix factorization (NMF)
In short, I wonder which kind of data can use complex number for NMF. And could an imaginary part possibly be a vector?
For detail, as I saw some papers used complex number in NMF (1), I think it ...
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Soft-thresholding for the LASSO with complex valued data
I'm currently implementing coordinate descent for the LASSO with complex-valued data. For this, one needs a complex version of the soft-thresholding operator, which seems hardly available on the net.
...
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How to find the expectation $\mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right]$?
Given the independent and complex Gaussian random variables $h$ and $w$, how does one can find the following expectation?
$$\mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right] = \int_{\mathbb{C}}\...
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Complex vs. Standard Neural Nets for Complex Data
I've seen some recent papers describing complex valued neural networks like this one. What I'm wondering is, rather than invent a new complex network architecture that takes a complex value as a ...
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AR(2) process- covariance stationary- complex roots
I am trying to check if this process is covariance stationary.
I have an AR(2) process given by: $Y_t(1-1.1L+0.8L^{2})=\epsilon_t$
I saw that to check if the process is stationary, instead of ...
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pca on polar coordinates?
I have a dataset composed of 123 rows (time bins) and 20 columns (variables)
The question I have is the following. each row,column pair has a radian value and a radius value. If I convert these pairs ...
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How to formally define a probability distributions over complex random variables?
Would that be just a probability over a bivariate real random variable, one representing the real part and another representing the imaginary part? How can I formally take moments of the complex ...
user188529
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Complex valued design matrix
In statistics design matrix is fundamental concept. It includes set of explanatory variables, for example in case of MRI data we use dc component, drift,physiological noise and so on. What will ...
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Multivariate distribution for products of random variables
Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
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Normalization of circularly-symmetric complex Gaussian distribution
I have a hard time describing my problem, but I'll try my best. It's all about the well-known zero-mean, circularly-symmetric, multivariate complex Gaussian distribution
$f(z)=\frac{1}{\pi^K\det(\...
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Independence of real and imaginary parts of the scalar product of multivariate complex normal vectors
Reading the paper "Distribution of Inner Product of Two Complex Gaussian Vectors and its Application to MPSK Performance" by Mallik R.K. I can not get one of the steps.
The author assumes two complex ...
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Analysis with complex data, anything different?
Say for example you are doing a linear model, but the data $y$ is complex.
$ y = x \beta + \epsilon $
My data set is complex, as in all the numbers in $y$ are of the form $(a + bi)$. Is there ...
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Estimation of X in Gaussian noise
Given
$\textbf{x}=[x_1 x_2 ... x_n]^T$ where $\textbf{x} \in \{ 0, a_1, a_2, a_3 \}^n, a_i \in \mathbb{C}$ and $\textbf{z} = \left\{z_1,z_2,\dots,z_n \right\}$ where $z_i \sim N(0,\sigma^2)$ is a ...
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