Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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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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 anything procedurally different when working with such…
bill_e
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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 besides some characteristic functions, I couldn't…
StatsStudent
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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 of the above inequality? What if we include the…
OliverVD
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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}}\int_{\mathbb{C}}{\frac{|h|^4}{|h+w|^2}}…
Felipe Augusto de Figueiredo
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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 single channel, why not just separate the real and…
Austin
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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 since we can display it with: $$\Phi(x)=(x_1,…
Slim Shady
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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. circularly symmetric complex Gaussian. How do I…
charu
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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 like to find the $n$ dimensional PDF for a new set…
mrkprc1
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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 set $X$. I need to find the (complexed valued)…
QuantumJesus
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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 random variable $$ Z = \sum_{i=1}^k X_i^2,…
user2843539
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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 procedures I need to implement in order to…
sea83
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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 to a complex number I have effectively take a…
user448573
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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 random variable (just take moments separately and…
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 happen if the design matrix is complex valued? Does…
Vendetta
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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),\cdots,h_1(N-1) \right ]^T$ and $ \boldsymbol{h_2} =…
user36
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