Questions tagged [covariance]
The covariance tag has no usage guidance.
91
questions
3
votes
1
answer
219
views
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 ...
- 33
2
votes
0
answers
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 ...
- 21
0
votes
0
answers
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 ...
- 107
0
votes
0
answers
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
1
answer
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 ...
- 57
0
votes
0
answers
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
1
answer
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 ...
- 61
1
vote
1
answer
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)...
- 75
0
votes
0
answers
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 ...
- 37
1
vote
0
answers
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 ...
- 11
0
votes
0
answers
52
views
Optimal weighting in case of unknown noise variances
Let's consider $n$ multiple measurements $x_i[j]$ of the same desired signal $s[j]$ but with different uncorrelated zero-mean additive noise $n_i[j]$ with different variances $\sigma_i^2$:
$$
x_1[j] = ...
3
votes
2
answers
105
views
Generalized correlation coefficients
Assume there are given two Gaussian random vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ of equal length $N$ with corresponding means $\boldsymbol{\mu}_x$, $\boldsymbol{\mu}_y$ and covariance matrices ...
- 175
6
votes
1
answer
708
views
Kalman Filter State Covariance Matrix for Non Constant Process Noise Matrix in PyKalman
I'm experimenting with the pykalman Python library to learn about Kalman Filters. In the code below, I'm generating a random walk where each step is the last step ...
- 243
3
votes
1
answer
419
views
How to determine covariance matrices $\mathbf P$, $\mathbf Q$, and $\mathbf R$ in Extended Kalman Filter
I am implementing an Extended Kalman-Filter and an Unscented Kalman-Filter for state and parameter estimation of a conveyer belt system. The problem is that I don't really know how to determine the ...
- 31
0
votes
1
answer
607
views
Ideally Lowpass-filtered White Gaussian Noise: question about a derivation of variance and covariance
Consider white gaussian noise $\omega(t)$ that has passed through and ideal lowpass filter with bandwidth $B$. This filter has the impulse response
$$h(t) = 2B\text{sinc}(2Bt)$$
Sampling the filter ...
- 147
4
votes
1
answer
2k
views
Kalman Filter: Why do we decrease the state uncertainty regardless of the current measurement?
I'm struggling with fully understanding the concept behind a Kalman filter.
For the sake of simplicity, let's ingore the input variable $u$ and assume constant process $Q$ and measurement noise $R$.
...
- 41
0
votes
1
answer
39
views
Random function covariance
I was studying signal processing and I was frequently asked to verify if a certain covariance is possible for a given random function. I tried to check by verifying the property to prove it:
${γ}_{xx}(...
- 3
0
votes
1
answer
67
views
Variance Due to white noise input
I have the problem below. It sounds simple but for some reason I have been stuck on it for a long time and don't know what am doing wrong.
Am trying to solve this using correlations.
So we all know ...
- 303
1
vote
0
answers
64
views
Colored noise and adjusting covariance matrix with change of frequency
Assume a signal sampled at 2Hz.
Assume it is a colored noise with known PSD.
The measurement equation to be used in a Kalman filter requires a covariance matrix.
If I want to sample at 9Hz (for ...
- 11
4
votes
2
answers
181
views
Matched filter for "amplitude SNR" vs power SNR
I am working on an application for which a matched filter seems like the right concept. In the derivation for the matched filter which I went through (here), they define the SNR as the ratio of ...
- 43
6
votes
2
answers
650
views
Quadratic Programming with Linear Equality Constraints
I need to solve an equality constrained minimization problem as give below
$$\min_{\textbf{w}} \mathbf{w}^TR\mathbf{w} $$
such that
$$X\mathbf{w} = \mathbf{1}$$
where $R\in \mathbb{R}^{n\times n}$ is ...
- 331
1
vote
2
answers
2k
views
Covariance Matrix Polar to Cartesian
Let $R$ and $\Theta$ be zero mean independent Gaussian random variables with variances $\sigma_R^2$ and $\sigma_\Theta^2$ respectively. Hence their covariance matrix would be diagonal give by
$$C_{R,\...
- 331
1
vote
1
answer
39
views
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 ...
- 1,084
1
vote
0
answers
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;
...
- 147
2
votes
0
answers
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 ...
0
votes
1
answer
290
views
Problem implementing KL expansion with square-exponential kernel: output just looks like Gaussian
I am trying to implement the Karhunen Loeve expansion for a 1-D Gaussian random field with a square-exponential kernel.
Specifically, I know that a Gaussian process has a KL expansion $\hat{U}=\sum_{...
- 101
3
votes
1
answer
4k
views
Generate signals with a particular variance and SNR
Consider a system model of the form: $y_n = ax_n + v_n$ where $x_n$ is the input that is corrupted by $v_n$ which is an Additive White Gaussian Noise of zero-mean and variance 1 for $n = 1,2,...,N$ ...
- 291
0
votes
0
answers
105
views
Proof of weak stationary random process autocovariance always goes to zero?
Professor told me that if a random process is weak stationary, and it does not feature any periodic component, then its autocovariance always goes to zero.
I can intuitively understand it, however, ...
- 21
2
votes
1
answer
679
views
Variance of function of random variable
Is their an easier way to find variance of function of random variable?
Till now what I am doing is first find probability density function of (function of random variable) then integrate over range.
- 175
3
votes
0
answers
331
views
Role of Riccati Equations in Kalman Filter Design
I am working on a Kalman Filter (KF) design problem and I am struggling to understand the role of the Riccati equations in the design process of a KF.
Some sources discuss the importance of Riccati ...
- 131
3
votes
1
answer
722
views
Kalman Filter State Covariance Matrix
If I have a discrete time process model of the form:
$$x_{k+1} = x_{k} + v_{k}\cos(\theta_{k})dt$$
$$y_{k+1} = y_{k} + v_{k}\sin(\theta_{k})dt$$
$$v_{k+1} = v_{k} $$
$$\theta_{k+1} = \theta_{k}$$
...
- 131
3
votes
1
answer
142
views
Covariance block of MUSIC Algorithm
I am implementing MUSIC algorithm in Verilog, and I need to implement the blocks in following order
$$\text{Input signal}{\longrightarrow}\text{Co-variance/Correlation matrix}{\longrightarrow}\text{...
- 31
0
votes
1
answer
289
views
Why discretize a continuous transition matrix in Kalman Filter?
In the Kalman filter toolbox at http://becs.aalto.fi/en/research/bayes/ekfukf/cwpa_demo.html the example code shows that a function lti_disc is called, which is essentially a matrix exponential ...
- 487
-1
votes
1
answer
112
views
About the variance of the coefficient of narrowband noise when using signal-space representation
Assume that n(t) is a white Gaussian noise process with zero-mean and power spectrum density $N_0/2$. By using the signal-space representation, it can be expressed as:
$$n(t) =\sum_{j=1}^N n_j \phi_j(...
- 59
1
vote
2
answers
349
views
Should I pass Kalman Filter absolute or offset-from-mean sensor values?
I'm using Kalman filters to segment the loudness of an acoustic signal from surrounding noise. The problem I've encountered is that muffled or faulty microphones measuring 'silence' (-70dB, -69dB, -...
- 131
0
votes
1
answer
875
views
Covariance matrix associated with random DC level in Gaussian noise
Given a signal
$x[n] = A + w[n]$
where $A$ is a Gaussian random variable and $w[n]$ is Gaussian white noise, then the covariance matrix of the signal is given by
$[C(\sigma^2_A)]_{ij}=E[x[i-1]x[j-...
- 105
0
votes
1
answer
255
views
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 ...
- 13
2
votes
2
answers
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 ...
- 388
0
votes
1
answer
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 ...
- 159
10
votes
2
answers
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 ...
0
votes
2
answers
3k
views
[Python ]How can I improve my 1D Kalman Filter estimate?
I have written the following code to smooth an (almost) linear function:
...
0
votes
0
answers
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 ...
- 269
1
vote
0
answers
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 ...
- 143
2
votes
0
answers
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 ...
- 121
0
votes
0
answers
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 ...
- 97
1
vote
2
answers
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 ...
- 403
0
votes
1
answer
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 ...
7
votes
2
answers
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 ...
- 171
2
votes
1
answer
1k
views
How to estimate covariance matrix using Fourier representation?
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 ...
0
votes
1
answer
2k
views
Show That the Power Spectrum Density Matrix Is Positive Semi Definite (PSD) Matrix
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} \...
- 19.3k