# 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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### Find all values of $a\in\mathbb{C}$ for which the equation $$x^4 +ax^2 +a^2x -1=0$$ has all roots of the same absolute value [migrated]

Let $\alpha, \beta, \gamma, \delta$ be the four roots of the equation, then $\alpha+\beta+\gamma+\delta=0$ and $\alpha\beta\gamma\delta=-1$. As all the roots have same absolute value, we can get that ...
1 vote
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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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125 views

### 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 ...
4k views

### Proof that variance is always greater than or equal to zero

It is common knowledge that: $$$$\label{3} Var(X) \geq 0$$$$ 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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1 vote
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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 ...
155 views

### 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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561 views

### 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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### 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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1 vote
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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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1 vote
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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 ...
447 views

### 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 ...
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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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1 vote