# Questions tagged [covariance]

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### What's the exact definition of the power spectral density function?

I learned from my signal processing course that PSD is the Fourier transform of the autocorrelation function. $$\mathscr{F}\Big\{\mathrm{E}\big[x(t)x(t+\tau)\big]\Big\}$$ Today I took my statistic ...
36 views

### Recursive estimation of signal variance of strongly colored noise

I have a quasi stationary but time varying stochastic signal of which I would like to recursively (adapting to its time dependence) estimate the variance. I believe the signal could be modeled like ...
18 views

### Confusion with Complex Gaussian process with Auto-covariance

I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ...
10 views

### How to use HAC for Time Series Dataframes with Covariance

I have a question concerning time series. I have a Dataframe with several Columns, each containing a time series. I need a Covariance Matrix of the different columns. However, my columns are ...
1 vote
65 views

### Why is the total noise variance less than the sum of individual noise variances?

I have three random variables: $Y$: my data $Y_n$: my data corrupted by additive white Gaussian noise (AWGN) $Y_{nc}$: my noisy data corrupted by a non-linear transformation $\mathcal{C}$. I have ...
44 views

### Finding correlation coefficient of two dependent random variables

I know when there are two independent random variables X and Y, the correlation coefficient is But what if the two random variables are dependent, for example, Y-X and X+Y, how can I represent them ...
1 vote
126 views

### Initial Process Covariance in 1-D Kalman Filter

Having a bit of confusion about what the initial process covariance (P) should be. Assume a 1-D tracking problem where I am measuring the distance/position of a static object. Would P not just be ...
1 vote
226 views

### Intuition for $\mathbf{P} = \mathbf{0}$ in steady-state when $\mathbf{Q} = \mathbf{0}$ (Kalman filter)

Consider the following discrete-time system: \begin{equation} \mathbf{x}(k+1) = \mathbf{A}_d \mathbf{x}(k) + \mathbf{B}_d \mathbf{u}(k) \end{equation} \begin{equation} y(k) = \mathbf{C}_d \mathbf{x}(k)...
64 views

### Related to wireless channel variance?

I am reading wireless research papers and found that mostly the wireless channel is considered to be Zero mean circularly symmetric Complex Gaussian (ZMCSCG) with variance 1 or 2 or 5 or 10 etc. My ...
1 vote
106 views

### Mann-Kendall Trend Test

I'm using the pyMannKendall package in Python to detect the presence of slope of any given waveform. More details of the package here: https://github.com/mmhs013/pyMannKendall I'm not exactly able to ...
52 views

1 vote
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### Variance of circular vector correlation

Say I have two zero-mean vectors, with a size of N and the translation between them is k. Say the image signal std is $\sigma_s$ and the noise std is given by $\sigma_n$. What is the variance of the ...
1 vote
143 views

### Whitening Filter for Noise Estimation

The signal I'm working on includes some noise which is I assume Gaussian white noise with zero mean and an arbitrary variance. So, it is like; ...
126 views

### Finding the error in the total integrated intensity of a fitted 2D Gaussian

I have been trying to fit signals to a 2D Gaussian function, and while I have bene able to use sciKit-image's curve_fit function to find the covariance matrix for ...
290 views

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### Estimating variance in arbitrary, periodic signal

I have a periodic signal $x[m], m \in [0;M-N+1]$ made of modulated templates $s[n],~ n \in [0;N-1],~ N \ll M = NK$ of finite energy and support (i.e. zero outside of its defined interval, which does ...
905 views

### Is there a difference what measurement units use in covariance matrix

The R matrix in the Kalman filter contains measurement noise. Diagonal elements of the matrix is the power of standard deviation. Is there a difference what measurement unit to use for standard error ...
414 views

### Why does it mean that the process/signal is not stationary when its variance varied with time? [closed]

Why does it mean that the process/signal is not stationary when its variance varied with time? that is, $VAR[X(t)]= \alpha \times t$,$t$ is time,and $\alpha$ is a constant,then $X(t)$ is not the WSS ...
4k views

### What Is the Difference Between PCA and Karhunen Loeve (KL) Transform?

I have been reading about Karhunen-Loeve (KL) transform. I see that when it is used to reduce dimension the procedure is identical to PCA, that is, for both methods the covariance matrix of the data ...
3k views

### [Python ]How can I improve my 1D Kalman Filter estimate?

I have written the following code to smooth an (almost) linear function: ...
100 views

### Interpretation of Eigen Values of covariance matrix

I am trying to obtain an intuitive understanding of Eigen Values of covariance matrix and have used a few layman terms because I fully do not understand the concept yet. The following is the code ...
1 vote
128 views

### Sensor fusion under unknown correlations: can covariance intersection account for delays?

Of late, there has been some interest in cooperative estimation algorithms in robotics, where the information sources are usually sensors such as cameras. When multiple robots observe surrounding ...
568 views

### Estimate standard deviation of random-walk using Kalman filter

I'm new to Kalman filters so this might be a stupid question. I created a Kalman filter that takes in time series observations and estimates the mean of that time series. This is simply modeling a ...
113 views

### 2-D parameter vector: Cramer Rao lower bound

Given a 2-D parameter vector, $\mathbf{X} = [x_1, x_2]$, let the corresponding $2\times2$ Fisher Information Matrix be $\mathbf{F}$. The Cramer-Rao Lower Bound (CRLB) is the inverse of the FIM. I ...
1 vote
1k views

### Is it possible to estimate variance of noise for a step answer signal?

I know there is not possible to find the true noise of a measured signal. The only way to "find" the noise is to estimate the noise. Noise has the mean 0, but the variance varies. So assume that we ...
1k views

### How to calculate the Diagonal loading factor evaluate calculate the inversion of a covariance matrix

I am programming a Generalised Likelihood Ratio Test (GLRT) detector. When it comes to inverting a covariance matrix $Ri$, I need to do a diagonal loading to fix the problem of sigularity of this ...
207 views

### Difference between $\mathbb{E}[\mathbf{x} \mathbf{x}^{\rm{H}}]$ and $\mathbb{E}[(\mathbf{x}-\boldsymbol{\mu}) (\mathbf{x}-\boldsymbol{\mu})^{\rm{H}}]$

Let us have a random vector $\mathbf{x} \sim \mathcal{CN} (\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} \neq \mathbf{0}$. What can we say about the relationship between the elements ...
So, I have multidimensional time-series $X \in R^{(d \times T)}$, and I want to determine the covariance matrix of that signal in a specific frequency band. I might filter the signal to that specific ...
Given a Wide Sense Stationary Multi Variate (Vector) Random Process $\boldsymbol{x} \left[ n \right]$ its Auto Covariance Matrix Function is given by:  {R}_{x, x} \left[ m \right] = \mathbb{E} \...