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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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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19 views

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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21 views

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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18 views

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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33 views

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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26 views

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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139 views

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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24 views

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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62 views

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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30 views

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} ...
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Interpreting Ljung-Box white-noise test p-value

Good evening all, I am having some trouble understanding the Ljung-Box white-noise test p-value from SAS Forecast. So there are lags where the p-value exceeds 0.05, meaning that we fail to reject the ...
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Is an ARIMA(0,0,0) with non-zero mean equivalent to white noise?

Is an ARIMA with a non-zero mean equivalent to white noise? If not, how should the mean be interpreted?
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48 views

Forecasting with ARIMA models

I want to do a rolling window forecast on a time series but it seems the series is white noise ARIMA (0,0,0) with non-zero mean. But when I difference the dataset and model it with an ARIMA(0,1,1) I ...
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What is the difference between white noise and “strict” white noise?

The normwhn.test package in R has a function to test if data is white noise. The documentation of the function states the following : Performs an Univariate Test ...
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38 views

Statsmodel Ljung-Box test for non-zero mean

I have few time series, which I would like to model using ARIMA. But plotting ACF and PACF indicates these series are probably white noise. I am using Ljung-Box test to confirm it. I am implementing ...
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36 views

In ARMA model, the white noise is uncorrelated, but why there are autocorrelation existing between them?

I learnt the ARMA model allows the autocorrelation between noise. but I check the definition of white noise is a sequence of uncorrelated, fixed mean random variables. how can the uncorrelated random ...
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Does the procedure of progressively adding noise have a name?

I want to make a qualitative estimate of the amount of noise in my dataset $X$. For that purpose, I use a certain information estimator $F(X)$ (for example, mutual information between two different ...
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59 views

Time Series assumptions for iid $\epsilon$

thanks for reading my post. I know its fundamental and rather easy qns but I'm seriously struggling. Please help me, thank you very much! Let $\boldsymbol{X}$ have a distribution with mean $\mu$ and ...
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179 views

Proof of white noise process

I have the process $y_t=e_t+ae_{t-1}e_{t-2}$ where $e_t$ is iid with mean of $0$ and variance $\sigma^2$ How do I go about mathematically proving that this is a white noise process?
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412 views

Autocovariance of White Noise

I'm studying time series, I don't understand the description below. The white noise series $w_t$ has $E(w_t) = 0$ and $$\gamma_w(s, t) = \operatorname{cov}(w_s, w_t)= \begin{cases} \sigma_w^2 & ...
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148 views

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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57 views

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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43 views

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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87 views

Add Noise to a Dataset

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 \\ \...
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175 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 (...
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14 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 ...
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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 ...
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207 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 ...
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110 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 ...
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162 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 ...
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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 ...
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32 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 ...
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45 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 ...
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585 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 does summing up a bunch of white noises related to predicting your particular time series data? Suppose I have a MA(q) ...
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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 ...
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207 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 ...
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“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 ...
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516 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 ...
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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 ...
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404 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?
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65 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|\...
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Least Squares fit of model - R

The data file (X in code thread below) contains the record of monthly data X[t] over a twenty year period. The data can be modelled by X[12j+i] = Mu + s[i] + Y[12j+i] where (i=1,...,12; j=1,...,k) ...
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301 views

Create Gaussian noise for artificial dataset with different noise levels

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 ...
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587 views

Why is the variance of ACF of white noise 1/T

In many books, articles and comments on this website I read that the variance of the autocorrelation of a white noise process is $\frac{1}{T}$ when T is sufficiently large. Often this characteristic ...
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5k views

Does white noise imply wide-sense stationary?

White noise has the ACF: $R_{WW}[\kappa] = c_0 \delta [\kappa]$ and zero mean $m_W[\kappa] = 0$. The first and second order moments of a WSS process depend only upon the time difference $\kappa$. ...
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342 views

White noise test taking into account homoscedasticity

I try to test a time series for white noise. The ultimate goal is to show that scaling volatility from daily to longer time periods by the square-root of time rule is justified. Fore white noise I ...
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1answer
58 views

Choosing right Model exchange rate

I have obtained these two plots using R, I have to fit a model and the trouble is choosing between an ARMA(0,0), and an AR3. The main issue is the autocorrelation at lag 3, is it enough significant to ...