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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What is a mathematic rigorous definition of "blue noise"?

Let $d\in\mathbb N$, $I$ be a finite nonempty set, $(x_i)_{i\in I}\subseteq[0,1)^d$, $(w_i)_{i\in I}\subseteq[0,\infty)$ with $\sum_{i\in I}w_i=1$ and $$\sigma:=\sum_{i\in I}w_i\delta_{x_i}.$$ I ...
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Gaussian white noise model for beginners

It is a classical result that the "Gaussian white noise model" is asymptotically equivalent to nonparametric regression and density estimation [1,2]. While this is clearly an important ...
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Remove Additive Zero Mean Gaussian Noise

I have six values, and each value is corrupted by Additive-Zero-Mean-Gaussian Noise with var = 0.05). Each value is range from 0 to 1. Is there anyway for me to remove these additive noise?
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residuals with non-zero mean

Once you learned your forecasting model, it is necessary to check if the residuals are a white noise: the mean is zero and no autorrelation. If the residuals have mean m, the rule is to add m to all ...
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Is a moving average model fitted to white noise?

I don't understand the following definition of a moving average model from Hyndman 2021, Forecasting: principles and practice A moving average model uses past forecast errors in a regression-like ...
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Is this a white noise? Can I use ARCH/GARCH models on this?

I am trying to find out if I can use ARCH/GARCH models. To my knowledge, to use ARCH/GARCH models you should have autocorrelation and this correlogram should not be a white noise. How can I know if ...
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why is it important for the random component in the time series model to be white noise?

I am curious to know why is it important to ensure that the random component in the time series model is white noise? what is the significance behind it? Also, if the random component is white noise, ...
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Variance of random walks in time series analysis [duplicate]

“For a random walk stochastic process, the variance is infinite.” Do you agree? Why?
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White noise in time series models

In virtually every time series model out there (e.g. AR(1)), there is always the existence of a white noise term $\varepsilon_{t}$ like $y_{t} = \delta + \phi y_{t-1} + \varepsilon_{t}$. I don't ...
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Evaluation of trend removal methods

Let's assume the following time series model where mt is a deterministic trend and Yt is a random noise component: Xt = mt + Yt According to my understanding a trend removal method is considered to ...
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RMS and PSD of Sinusoid + Noise

This is a basic question on computing the statistics of a combined signal: the sum a stochastic signal (eg noise), and a deterministic signal (eg sinusoid). How to use the RMS equation, to find ...
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Determining overfitting model by computing variance in prediction error

I have a data set for regression, with a set of input features and 1 response variable. To confirm if a trained model has overfitted, we can see if the train error << test error at untrained ...
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Show that two white noise definitions are equivalent

Given $(\varepsilon_t)\sim WN(0,\sigma^2)$ a white noise. By definition $$E(\varepsilon_t)=0,\,\, E(\varepsilon_t^2)=\sigma^2 \quad \forall t$$ and $$E(\varepsilon_t \varepsilon_s) = 0, \quad s\neq t$$...
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Do variance of discrete white noise components cancel each other out if they have opposite signs? [duplicate]

Suppose y-= u + en - e0 and var(ei)=a2 for i>=0. So would var(y-) be equal to 0 since u(mean is a constant number and has 0 variance) and var(en - e0)= a2-a2=0?
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Why is the difference between 2 time series drawn from the same process not White Noise?

I take the difference between 2 time series (each with 200,000 observations) drawn from the same ARMA(2,1) process and find that (at least the first 1000 observations of) this difference looks like ...
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White Random Process(White Noise) and Mean Value

Can someone give a definition of a white random process? I was trying to understand the penultimate paragraph of this answer Is it necessary for white noise to have zero mean which states that there ...
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Strange and very low results in Ljung-Box test

I have been reading some posts about the Ljung-Box test and I am applying it to some of my databases. However, I am not really understanding the outputs, I think I am doing something wrong. I have a ...
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Types of noise processes and the one assumed in arima() estimation in R

Here is a time series class defining white noise incorrectly as an independent sequence of random variables. source Aside from the widespread mix-up of White noise and iid noise, a further ...
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Prewhitening before cross-correlation

i have a question concerning cross correlation and prewhitening of two time series. I understand the common procedure to avoid spurious correlation is to model your input series x and filter the ...
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Prove that a process is memoryless (simple example)

Given the following stochastic process \begin{equation} x_t = \frac{u_t}{\sum_{s=1}^{t-1} u_s} \end{equation} where $u_t \overset{i.i.d.}{\sim} \mathcal{N}(0,\sigma^2)$, $\sigma^2<\infty$, and ...
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Yule-Walker equations (ARMA(1,1))

My professor assigned me this task: Let $\{X_T\}$ be causal $ARMA(1,1)$ process, i.e. $X_t - \varphi X_{t-1} = Z_t + \theta Z_{t-1}, \ \ Z_t \sim WN(0,\sigma^2)$. i) Using Yule-Walker equations, ...
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How is adding noise to training data equivalent to regularization?

I've noticed that some people argue that adding noise to training data equivalent to regularizing our predictor parameters. How is this the case? Some of the examples listed on SE discussing this ...
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If $\varepsilon_t$ is white noise, then $\eta_t = \varepsilon_t - \varepsilon_{t-1}$ is white noise?

If $\varepsilon_t$ is white noise, then $\eta_t = \varepsilon_t - \varepsilon_{t-1}$ is white noise ? By the fact that $\varepsilon_t$ is white noise, I don't know what I can say about $\varepsilon_{t-...
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Tranforming Breusch-Godfrey and Ljung-Box Tests: Canonical Rotation of Portmanteau Tests

Breusch-Godfrey and Ljung-Box appear to be optimizations of Chi-Square. Recently I noticed a library in R that transforms a Chi-Square to a Phi Four Point Coefficient (...
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How to detect a sporadic time series?

Could someone give me some suggestions as to how I can go about trying to detect if a time series is sporadic or not? Are there any tests for the same? Also, I think sporadic series are quite ...
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Standard deviation growth of discrete Brownian motion?

In my current project, I have a collection of $N$ i.i.d. samples of a multivariate standard Gaussian distribution in $D$-dimensional space. My ultimate goal is to gradually perturb the standard ...
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Correlated residuals - white noise analysis

I want to check and analyze the goodness of my compressed sensing reconstruction algorithm (fit) using the autocorrelation of the residuals. Given my problem, I guess there are two ways to do this: ...
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How do moving averages work to make predictions?

A moving average model is where $\epsilon_t$ is white noise that is $cov(\epsilon_t,\epsilon_{t-h})=0$, $var(e_t)=\sigma^2$, $E(e_t)=0$. How is it related to predict future values by averaging ...
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Can a white noise process be predicted?

I'm trying to better understand why white noise processes can't be predicted. I understand that these processes have mean zero and no correlaation between it's values at different times. I'm looking ...
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Time series forecasting - Residuals not white noise

This is my first message on CrossValidated to get some insights on an issue I am facing while trying to model properly a time series. I am relatively new to this science so please brace with me. My ...
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Covariance of a Stationary Process

Let $Y_t$ be a stationary process such that $Y_1 = a_1$ and $Y_2 = \theta a_1 + a_2$, where $\theta$ is a parameter and $a_t$ is the white noise process with mean 2 and variance $\sigma^2_a = 0.5$. ...
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Prediction error for ARMA process

Let $X(t)= \phi X_{t-12}+Z_t+\theta Z_{t-1}$ where $Z_t\sim WN(0,1)$. I need to find prediction error for projecting $X_t$ onto $H_{t-3}(X)$ (Hilbert space). So, I know that $X_t \perp P_{H_{t-3}}X_t$ ...
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Time series, linear process, white noise

Let $X_n = \sum\limits_{j=0}^{\infty} a_jZ_{t-j}$, where $Z_t$ is a (weak) white noise $(0,\sigma^2)$ and $a_j \in L^2$. Prove that ACF $\gamma_X(h)\longrightarrow 0$ as $h\longrightarrow +\infty$. So:...
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Finding the variance of a process generated by white noise

Given that $a_t \sim WN(2, 0.5)$, I have generated the process defined by $$Y_1 = a_1$$ $$Y_t = \theta Y_{t - 1} + a_t$$ to be: $$Y_t = \theta^{t - 1}a_1 + \theta^{t - 2}a_2 + \cdots + \theta a_{t - 1}...
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Higher power than 2 for white noise time series?

Let $\{Yt\}$ given by $Y_{t} = Z_{t}$ With $Z_{t} \sim{N}(0,\sigma^{2})$ What are $E[Y_t^{3}]$ and $ E[Y_t^{4}]$?
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White noise that is not strictly stationary

For an exercise I was asked to provide an example of a white noise sequence that is not strictly stationary. I found in multiple sources that an example of this is $$X_t = \sin(2\pi t U) $$ where $t$ ...
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ARIMA(0,0,0) model but residuals not white noise

I have a dataset where I am trying to fit an ARIMA model to a stock return - the data set is stationary. I have used the Auto.Arima function to select appropriate AR and MA terms, and BIC selects ...
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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 term for uncorrelated is i.i.d (identically and independently ...
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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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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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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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In a VAR model, why are white noises correlated with each other in the reduced form model

With my understanding, we have 2 different models when studying VAR processes (Vector Autoregression). We have $$ \begin{bmatrix} y_{t} \\ x_{t} \end{bmatrix} = \begin{bmatrix} c_{1,0} \\ c_{2,0} ...