# 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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### 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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1 vote
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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 $$x_t = \frac{u_t}{\sum_{s=1}^{t-1} u_s}$$ where $u_t \overset{i.i.d.}{\sim} \mathcal{N}(0,\sigma^2)$, $\sigma^2<\infty$, and ...
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### 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 $X_t=ϵ_t+ϵ_{t−1}$...
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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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### 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?