# Questions tagged [white-noise]

White noise is a random process whose "components each have a probability distribution with zero mean and finite variance, and are statistically independent".

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### White noise assumption in the autocorrelation proof

I followed the proof presented in Quantitative Risk Management: Concepts, Techniques and Tools by D. Duffie, S. Schaefer (proposition 4.9, pages 128-129). To arrive at the numerator for the ...
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### Are colored noises correlated / uncorrelated?

Let, x be a random variable (r.v) that is white Gaussian, has a flat power spectrum. y can be any colored noise. I think another ...
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### Probability density function for white Gaussian noise

in many signal processing text books and lectures we find that if we assume that the noise is white Gaussian then the probability density function itself takes the Gaussian form (see here for example) ...
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### Applying fitted model of input on output time-series in pre-whitening

I have two different time series (input time series and output time series)for doing cross correlation. When I fit input time series for pre-whitening, it has good fit for ARIMA(0,1,1). So for pre-...
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### Ljung-Box test to determine white noise of time-series (not residual)

I know we use Ljung-Box test to determine if the residuals are white noise or not. Can we use same test to determine if the time-series in itself is white - noise? This is to by-pass acf-pacf plotting,...
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### Show that $x_{t},y_{t}$ are jointly stationary, and interpretation of CAcovF, $\gamma_{XY}(h)$ not being symmetric for lags $h$

Consider two white noise processes $(w_{t})_{t}$~$WN(0,\sigma_{w}^{2})$ and $(u_{t})_{t}$~$WN(0,\sigma_{u}^{2})$ that are also independent of each other such that $y_{t}=w_{t}-\theta w_{t-1}+u_{t}$ ...
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### Second-order moments of a White Noise process

One of my textbooks on time-series analysis claims that Dependency in the second moments of the residuals contradicts the assumption of a constant, time-invariant variance. Thus [the residual] is not ...
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### Probability of periodic signal in gaussian white noise time series

Suppose I have time series data that is accurately modeled as gaussian white noise with mean and variance $\mu, \sigma^2$. I would like to write down a probability density characterizing the ...
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### Math behind Differencing: Is White Noise Stationary?

I'm just starting to learn the math behind stationarity and differencing, so I apologize if this is a silly question. Lets say I have a non-stationary time series process (pure random walk) defined by:...
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### Vector AR/ARMA Model [closed]

For a vector AR/ARMA model in practice, if there are k different time series in the vector, there are k corresponding Gaussian white noises as well. Is it realistic to assume that those k white noises ...
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### Is an independent process always a white noise process?

In econometrics, an independent process means that all values are independent of each other, but does this also mean that all independent processes are white noise processes? and is the reverse true?
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### Variance of a white noise process

How do I find the variance of this? $$y_t=M + \sum_{i=0}^n 0.5^i\varepsilon_{t-2i}$$ M is the mean.
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### ljung box test for space time model

i did white noise test using ljung-box for space time model. i used mq function but the output always change everytime i running. is it really like that? thank you library(MTS) wndb=matrix(rnorm(468),...
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### Is error term in MA model in univariate time series the same as white noise

I what to know if the error term referred to in moving average model of time series the same as white noise? which is usually define in r as ...
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### What is the relation between the serial correlation and white noise series?

I am a bit confused about why would I use the Ljung-Box test in order to determine whether a series is a white noise ? I know that the Ljung-Box test studies whether we have serial correlation. I ...
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### Outliers Kalman Filtering

This might not be the right place to ask this questions, but I figured it's more of a machine learning question. I am also asking on the pyro forum for brevity. I'm working with the simple extended ...
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### What happens when a time series is multiplied by an iid error

Let's say say I have a standard AR(1) process, apart from the fact that is multiplied by the $ε_i$~$iid (0,1)$. Would this affect the independence of the the AR(1) series? My intuition is no ...
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### How to compute error terms in moving average time series model? [duplicate]

Currently I am studying time series Moving Average model MA(q) $$X_t -\mu= \epsilon_t + \theta_1\epsilon_{t-1} + \theta_2 \epsilon_{t-2} + ... + \theta_q \epsilon_q$$ where $\theta_1,...,\theta_q$ are ...
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### Including regressors to improve forecasts on white noise

I am conducting some time series forecasts using quite limited data, 13 years annually. Basically, I am trying to forecast companies emission totals using historical values. The historical data ...
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I've generated a dataset of 100 elements from a 3-variate Gaussian distribution with parameters $\mu = 0$ and $\Sigma = \begin{pmatrix}1 & \rho_1 & \rho_2 \\ \rho_1 & 1 & \rho_1 \\ \... 1answer 204 views ### Wold Decomposition Theorem and Moving Average model - Error Terms I'm stuck with Wold's decomposition theorem in time series analysis. The theorem says that every stationary time series can be written as a sum of two components, one being entirely deterministic (... 0answers 15 views ### Do coefficients in moving average process add to 1? I'm studying weakly stationary stochastic processes, and I'm confused by the title of the "moving average" representation of such a process. Suppose that$y_t$is a weakly stationary stochastic ... 1answer 33 views ### Parameters of ARMA model In my professor's notes, it is written that if the variable$y$is explained with an ARMA($p$,$q$) model, then$y_t$(i.e.$y$at time t) depends on the most recent$p$lags of its own value and the ... 1answer 292 views ### Understanding the infinite sum of random variables I am doing a course on time series analysis, and am struggling with this definition: We call a weakly stationary process$\{X_t\}$invertible with respect to a white noise$\{\epsilon_t\}$if ... 1answer 134 views ### If the ACF of a time series is within the 95% bounds, is it white noise? I have a detrended series where the ACF and PACF has lags all within the 95% confidence bounds. This would suggest the series is a White Noise. However, fitting it to an ARMA model (in R) gives the ... 1answer 240 views ### How to determine mse of estimate from correlation matrix of estimate error? I have a model of an information transmission system Y = XH + N, where X is a diagonal matrix with the transmitted "symbols" (known), H is a column vector which distorts the transmitted symbols and N ... 0answers 105 views ### Can the sum of several time-series be a white noise process, when the individual time series are not? Intuitively, I think that it is possible for a sum of time series to be white noise, when the individual time series are not. Reason I am asking, is because I want to know if it's useful to ... 0answers 34 views ### Continuous white noise I have been reading about topics that include statistics, and slowly start to get a grasp. However sometimes it seems to me that I dont get the simplest concepts. White noise seems to have several ... 0answers 49 views ### How to determine the line which a time-series fluctuating around Let$X_t=1+3t+0.5X_{t-1}+ \epsilon_t$be a trend-stationary model, where$\epsilon$is a white noise, which has zero expected value and standard deviation. Which line is the time series fluctuating ... 3answers 681 views ### Where does the white noise come from in MA(q) model? I'm having trouble understanding the intuition of the moving average model. How is summing up a bunch of white noises related to predicting your particular time series data? Suppose I have a MA(q) ... 0answers 32 views ### Using a neural network for a regression problem, where the model to be learned suffers from awgn I have currently a neural network to learn a (relatively non complex) system model (vector regression). Its problem is that the outputs of the system suffer from arbitry additional white gaussian ... 1answer 217 views ### Mean and Correlation of a First-Order ARCH(1) Process For a first-order ARCH(1) process $$Y_t = \epsilon_t(\alpha_0 + \alpha_1Y_{t-1}^2)^{1/2}$$ $$t \in \mathbb{Z}$$ $$\alpha_0, \alpha_1 > 0$$$ \{\epsilon_t\}_{t \in \mathbb{Z}} $and$Y_t$is ... 0answers 2k views ### “Add White Gaussian Noise with SNR” vs. “Add 5% Gaussian Noise” I have a noise-free dataset which is a vector of numbers,$\mathbf{d}$, with length$N$. I want to "add noise" to this data. My understanding is that there are two ways to do this. 1) Add some ... 1answer 559 views ### How to Interpret these ACF/PACF plots Would it be safe to regard this time series data as a white noise? Here's the dataset I used for computing the ACF/PACF Date 2008-05-23 0.323555 2008-10-15 0.650817 2009-03-11 -0.193327 ... 0answers 120 views ### Correcting for noise in gene expression data I have a training set of RT-qPCR gene expression data (not run in triplicate) for a batch of samples with two phenotypes$A$and$B$on which I've trained a logistic regression classifier. I also ... 2answers 526 views ### ARMA models and residual series Assuming that model is correct, why does the residual series of an ARMA model resemble a white noise process? 0answers 67 views ### Covariance Functions of Stationary Gaussian Random Processes I am trying to solve this question: Suppose that$n(t)$,$−∞ < t < ∞$is a stationary Gaussian random process with covariance function$E\{n(t)n(t-\tau)\} = \delta(\tau) + {5 \over 4}e^{-\left|\...
I am creating an artificial dataset corresponding to different noise levels. This is to simulate results of a recognition software (e.g. face recognition). For example, for $noise_{level} = 0.1$, the ...