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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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The State of White Noise Processes

Assuming we have a random process indexed by time, and the values assumed by the random process are its state. So a random process has a discrete or continuous state and a discrete or continuous ...
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Are stationary processes non-predictable, and non-stationary ones predictable?

I am reading A canonical analysis of multiple time series by Box and Tiao (1977). In the abstract of the paper, the authors mention: The least predictable components are often nearly white noise ...
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Why does differencing White Noise induce autocorrelation of $-0.5$?

I am curious about the following problem. Let's have a variable given by white noise, $$y_t \sim \operatorname{NID}(0,1).$$ Let's say we difference it, $$\Delta y_t = y_t - y_{t-1}.$$ And now, if we ...
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Asymptotic distribution of white noise ACF

I (encountered this in my lecture) wonder why do we want the autocorrelation of our residuals to be mostly within 2 s.d. as a sign that residuals are consistent white noise? More specifically why do ...
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Does white noise guarantee that $X_{t-1}$ is uncorrelated with $u_t$?

I have a question about the properties of white noise in a time series context. Specifically, I want to know: If we assume that the error term $u_t$ in a time series model is white noise, does this ...
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How to interpret Allan Variance curves that do not follow the canonical shape

I am currently working on characterizing the noise sources of a Global Navigation Satellite System (GNSS) sensor using an Allan Variance plot, which is commonly employed to analyze frequency stability ...
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Should I consider it white noise?

I have an hourly time series that I want to forecast. Prior to modelling it, I tested it for random walk. The ACF and PACF plots for the time series are as follows: Since the PACF has a high value at ...
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An ARMA model with white noise errors, that are ARCH? (How is that possible?)

First my assumption was that ARMA models take only the autocorrelation of the time series into consideration but not of the error terms (wrong!). But this assumption is wrong! As the within ARMA ...
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Deriving parameters of gaussian noise that will lead to bounded random walk

Given a bound on random walk, I am trying to derive the parameters of a normal distribution of noise, which when added to a signal and integrated will lead to random walk within specified bounds. My ...
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Investigating the Impact of Additive Gaussian Noise on EEG Signal Classification: Analyzing the Relationship between Augmented and Original Data

Definition: I have conducted research on EEG signal classification, specifically focusing on distinguishing between two different classes using raw EEG signals. Data availability poses a significant ...
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Prove that white noise + normality = independence

If the time series process is linear, then the ARIMA model is specified. The residuals from this model are $(1.)$ no autocorrelation $(2.)$ mean equals zero $(3.)$ constant variance. We say that this ...
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What color financial time series are there? [closed]

There is a folklore white noise hypothesis related to (and equivalent to some forms of) the efficient market hypothesis in finance -see references below. But are there some asset pairs whose return ...
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ACF and PACF vs Ljung Box test

I have a time series with realized sales prices on monthly basis in a large European city which comes as an index and I would like to do 1 period ahead forecasting. I have run ADF and KPSS for unit ...
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ARIMA(0,0,0) residuals are the same as the timeseries data

I am very new to analyzing and forecasting timeseries data so apologies if this question has an answer too obvious. I am trying to find the residuals between a stationarized price data and white noise ...
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CLT for sums of Fourier transform of white noises r.v

Define $I_n(\lambda_j) = \frac{1}{2\pi n} |\sum_{t = 1}^n Z_t e^{it\lambda_j}|^2 = \frac{1}{2 \pi} \sum_{h = - \infty}^{\infty} \hat{\gamma}_n(h) e^{ih\lambda_j}$ where $Z_t$ is a $WN \sim (0, \sigma^...
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Why isn't the pseudo-inverse the best choice in my linear estimation problem?

Context: I have a problem of the following form. Let $\boldsymbol{\theta}\in\mathbb{R}^n$ be a fixed vector I want to estimate. Let $\mathbf{M}\in\mathbb{R}^{m\times n}$ be a matrix with $m>n$ and $...
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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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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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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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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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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 ...
ColorStatistics's user avatar
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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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1 vote
1 answer
325 views

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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3 answers
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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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Why are the standardised residuals of a GARCH process a white noise process?

Suppose I have a GARCH process: $$X_t = \mu_t+\varepsilon_t$$ $$\varepsilon_t=\sigma_tz_t$$ where $z_t$ is some iid zero mean, unit variance random variable, and: $$\sigma_t^2=α_1 \varepsilon_{t−1}^2 +...
Douglas Kennedy's user avatar
3 votes
1 answer
66 views

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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1 answer
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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 ...
Mathias Schinnerup Hejberg's user avatar
1 vote
1 answer
347 views

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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