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".

Filter by
Sorted by
Tagged with
0
votes
0answers
10 views

Calculating covariance and correlation of a moving average time series

I'm learning about MA(1) time series models and I have an example with the answers but not the worked through solutions. Let {$ϵ_t$} be a zero mean white noise with variance $σ^2$ and $X_t=ϵ_t+ϵ_{t−1}$...
1
vote
0answers
21 views

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, ...
16
votes
3answers
2k views

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 ...
1
vote
0answers
88 views

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-...
1
vote
0answers
15 views

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 (...
0
votes
1answer
20 views

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 ...
0
votes
0answers
18 views

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 ...
0
votes
0answers
8 views

How to understand the statistical noise level of wavelet bicoherence?

Wavelet bicoherence was given by Van Milligen1995, which used to analyze turbulence. And the normalized squared wavelet bicoherence (usually called wavelet bicoherence) is shown below. $$ WBC(a_1,a_2)=...
1
vote
0answers
20 views

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: ...
0
votes
1answer
58 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 ...
0
votes
0answers
24 views

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 ...
1
vote
1answer
213 views

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 ...
0
votes
1answer
39 views

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 $\sigma^2_a = 0.5$. ...
0
votes
0answers
10 views

Adding noise to ARIMA forecasts

I'm learning about ARIMA forecasts and I applied it on one of the example data I have. When I used model on the train set, it is giving me peaks, valleys and other irregularities. But on the test data,...
3
votes
1answer
45 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 $X_t \perp P_{H_{t-3}}X_t$ ...
0
votes
0answers
24 views

Over Differencing a Time Series

Consider the simple white noise process, $Z_t = a_t$. Discuss the consequences of over-differencing by examining the ACF and AR representation of the differenced series, $W_t = Z_t - Z_{t-1}$. Answer: ...
4
votes
1answer
38 views

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:...
0
votes
1answer
37 views

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}...
1
vote
1answer
14 views

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}]$?
1
vote
0answers
54 views

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 $t$ ...
1
vote
0answers
202 views

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 ...
1
vote
1answer
109 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 ...
4
votes
1answer
666 views

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 ...
1
vote
2answers
1k views

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) ...
1
vote
0answers
16 views

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-...
1
vote
1answer
157 views

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,...
0
votes
1answer
56 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 $y_{t}=w_{t}-\theta w_{t-1}+u_{t}$ ...
1
vote
1answer
109 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 ...
0
votes
1answer
279 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:...
2
votes
1answer
37 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 ...
0
votes
2answers
354 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?
0
votes
1answer
107 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.
1
vote
2answers
227 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 ...
1
vote
1answer
245 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} ...
1
vote
0answers
99 views

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 ...
0
votes
0answers
263 views

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?
0
votes
0answers
76 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 ...
2
votes
1answer
398 views

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 ...
0
votes
1answer
94 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 ...
1
vote
1answer
196 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 ...
1
vote
1answer
181 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 ...
0
votes
1answer
651 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?
1
vote
1answer
2k 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 & ...
0
votes
1answer
293 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 ...
1
vote
1answer
141 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 ...
0
votes
1answer
72 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 ...
0
votes
1answer
542 views

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 ...
1
vote
1answer
57 views

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 ...
0
votes
0answers
467 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 \\ \...
1
vote
1answer
295 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 (...