Questions tagged [time-series]
This tag is used for question related to time series models such as AR, ARMA, ARCH, GARCH and their properties and techniques used for inference.
932
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Estimating the current reliability of a flawed detector
I have a detector that tries to evaluate if a given measurement in our system indicated urgent intervention is required.
I have quite a bit of historical data. This can tell me the historical ...
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12
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How to find the current time when a vinyl is dragged up
I am trying to create a DJ Mixer, i have this so far:
When the mouse moves up how do i calculate the new current time of the vinyl rewinding.
This is my code so far:
...
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28
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Show that a certain stochastic process is not linear
Let $Y=(Y_t)_{t \in \mathbb Z}$ be a stochastic process defined on $(\Omega, \mathcal{F}, P)$. We say that $Y$ is a linear process if:
\begin{equation}\label{I}\tag{I}
Y_t = \sum_{j=0}^\infty \psi_j \...
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How to determine numerically that a fluctuating curve (which depends on random variables) is stationary, i.e. has no upward or downward tendency?
I need to determine if and when curves like the following are stationary:
My idea so far is to perform linear fits in different parts of the curve (see example)
When the modulus of the slope of a ...
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Testing predictability of a predictor of expected weekely returns on the stock market
say I have a T daily observations for the last ten years on a new predictor $x_t$ which I think is a predictor of the expected weekly return on the stock market, $r_{t,t+5} = r_{t+1}+...+r_{t+5}$, ...
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Inequality involving changing order of limits and probability
I read this paper, in Corollary 1 the author claims that $$\underset{\pi \in [0, 1]}{\sup}\ W_T(\pi) \overset{p}{\to} \infty$$ as $T \to \infty$. Where $W_T(\pi)$ is Wald statistics but I think it ...
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What is the definition of "periodicity of the data"?
I'm reading a research paper where it says. Here $m$ is the periodicity of the data (e.g., 12 for monthly series).
I understand:
There are twelve months in a year. So I suspect periodicity relates to ...
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Can you use combinatorics to determine the average time needed for something?
So im building an graffitiprinter and I use a matrix so could it be possible if I know the time of one spray to calculate the avarege time to complete a picture. If I also knew the total amoents of ...
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Derivation of Autocovariance Function of First-Order Autoregressive Process
In my textbook, the autocovariance of the AR(1) model is derived as such:
$$Y_t=\phi Y_{t-1}+e_t$$
After multiplying both sides by $Y_{t-k}(k=1,2,...)$ and take expected values, you get:
$$E(Y_{t-k}...
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Dickey Fuller Test coefficient distribution and appropriate functional form
I have two basic questions regarding the Dickey-Fuller test
Why does the asymptotic distribution of the coefficient in the Dickey-Fuller test not converge to a normal distribution?
How does one ...
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Moving averages notation
Suppose I'm fitting a cubic to best approximate sets of seven points. Then solving for the coefficient $a_0$ I get $$a_0 = \frac{1}{21}(-2, 3, 6, 7, 6, 3, -2)$$
Then expressing the formula for the ...
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White noise or IID noise
Let $Z_t \sim^{iid} N(0, 1)$ and
$$X_t = \begin{pmatrix}Z_t, \text{if t is even} & (Z_{t-1}^2-1)/\sqrt{2}, \text{if t is odd} \end{pmatrix}$$
Show that $X_t$ is WN(0, 1) and not IID(0, 1) noise, ...
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Prove $E(E(Z|X, Y )|Y ) = E(Z|Y )$.
Let $(X,Y)$ be a random vector.
Prove $E(E(Z|X, Y )|Y ) = E(Z|Y )$.
I'm having a hard time understanding how to get this set up. I'm not sure how to set up the integrals for this. This proof was done ...
- 463
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Kolmogorov’s theorem formulation
I have the following definition and theorem in my lecture notes :
Let $\mathcal F$ be the set of all vectors $\{t = (t_1, \dots ,t_n) \in T^n: t_1 < t_2 < \dots < t_n, n = 1, 2, \dots \}$. ...
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How to tell is time series is Gaussian
For the Gaussian process, this is the definition provided in class: A process is Gaussian if all the finite-dimensional distributions have a normal distribution.
This means that $[X_{t_1}, . . . , X_{...
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Brockwell & Davis, Time Series Theory and Methods, problem 4.4
If $\{X_t\}$ is the process defined by $$X_t=\sum_{j=1}^n A(\lambda_j)e^{it\lambda_j}$$ in which $-\pi<\lambda_1<\lambda_2<...<\lambda_n=\pi$, and $A(\lambda_1), ..., A(\lambda_n)$ are ...
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convergence of series
Suppose that $$\lim_{n\to\infty}\sum_{k=0}^nk|x_k|=L<\infty.$$
Is it possible to show that
$$\lim_{n\to\infty}\sum_{k=0}^n|x_k|=K<\infty.$$
In words, does the absolute summability of $\{nx_n\}_{...
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Inverse of a Lag Operator
I am having trouble understanding how the lag operator works. My understanding is that an operator is a mapping from one vector space to another. If I am working with a time series $\{X_t\}_{t=1}^{\...
- 1,244
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conditional covariance for a weakly stationary process
Given a stochastic process $\{X_k\}_{k \geq 1}$, we say it is weakly stationary if $\mathbb{E}X_k$ is a constant, and $\mathrm{Cov}(X_k, X_l)$ only depends on $|k - l|$.
My question is, for a weakly ...
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How to interpret a significant intercept estimate for segmented regression analysis in interrupted time series design?
I am using the following GLS model (Wagner et al. 2002) for my analyses.
model link
beta 0 stands for the estimate of the outcome at the baseline level. What does this mean when it has a p-value of &...
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How to compute the sample autocovariance with multiple i.i.d. samples.
This question is copied of https://stats.stackexchange.com/q/451404.
It has not been answered yet, however, I put the question here.
Consider two (discrete) samples of any stationary stochastic ...
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29
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Probability of two events occurring in same time period across time-series data
I haven't studied probability mathematics for a long time. However, some time-series data came my way and I'm trying to ascertain the likelihood of two events happening during the same time intervals. ...
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How is regression different from time series?
When you have data $(x_1,y_1), \cdots (x_n,y_n)$, then in regression one tries to find a function $f: X \rightarrow Y$ which explains the dependance between $x_i$ and $y_i$. Here it is assumed that ...
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I need an explanation on why the variance of the distribution of noise term is considered as such
AR: $\epsilon_t=\rho\epsilon_{t-1}+\eta_t$ with $\eta_t $ i.i.d. $N(0, \sigma^2), t=1,...,n$ and $\epsilon_0=0$ with $\eta_0\sim N(0,\frac{\sigma^2}{1-\rho^2}),\lvert\rho\rvert<1;$
Why is the ...
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Power Transformation of a Spectral Density Function
There is a problem I have been trying to solve for a while. Let $X_t$ be a stationary (univariate) time series. The spectral density of the moving average process $$X_t=\sum^{\infty}_{j=-\infty}a_je_{...
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Distribution of bivariate vectors for strictly stationary processes
Consider a strictly stationary process $X_t$, $t\in\mathbb{Z}_{\geq 1}$. Could you help me to disprove the following statement:
"For $t, s > 0$, the bivariate vectors $(X_s, X_t)$ and $(X_t, ...
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Definition of ergodicity and ergodicity for second moments of a stochastic process
According to this topic, we understand well the relation between the ergodicity for dynamical system and the mean ergodicity of a stochastic process. More exactly, we have the Ergodic Theorem (See ...
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What kind of convergence actually takes place in the definition of ergodicity?
We say that the stationary time series $(X_t)_{t \in \mathbb Z}$, , with mean $\mu$, is mean ergodic if the following converge in probability holds:
\begin{equation}\label{a}\tag{M-E}
\hat{\mu}_T := \...
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What properties does a mean-ergodic but non ergodic for second moments stochastic process have?
The first and the most trivial example of a mean-ergodic stochastic process is the i.i.d. case:
$$(u_t)_{t \in \mathbb Z} \sim N(0,1)$$
The first intuition we have of ergodicity is that the process $(...
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Can I use an identity to go from $\sin(2\pi(\omega(t + h) + \beta))\sin(2\pi(\omega t + \beta))$ to $\frac{1}{2}\cos(2\pi \omega h)$?
In my time series homework I had to compute the periodogram of a periodic series and in the solutions we have a step that goes from $\sum_{t=1}^{n-h} \sin(2\pi(\omega(t + h) + \beta))\sin(2\pi(\omega ...
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Does type of moving average for an irregular time series exist already?
Context
I have an irregular time series where data points occur at irregular intervals of time. As a way to observe the behavior of this data over time, I want to use some type of moving average.
The ...
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Proof of the stationarity condition of ARMA model
In the book Introduction to Time Series and Forecasting by Peter J. Brockwell and Richard A. Davis, at page 75, there is the Existence and Uniqueness of stationary solution of an ARMA process:
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Segregate/Decompose a prediction for a time period into smaller sub-periods
I have a dataset of Electric vehicle demand every 5 mins at every station in a cluster.
However, this data is sparse so I cannot train a model and extract the underlying patterns.
Therefore, I do some ...
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How to understand the cointegration test with the rank function in Rstudio
I'm learning the cointegration test for time series now and I ran this test with the rank function from 'tsDyn' package in Rstudio. Attached here is my results.
I have a problem in understanding the ...
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An inequality for first order autocorrelation.
Consider a discrete stationary process $\{X_n\}$ with autocorrelation $\rho_k=0,\forall k\ge2$, prove that $|\rho_1|\le\frac12$. WLOG assume that $EX_n=0,Var(X_n)=1$.
I have tried to use Cauchy's ...
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Why is the fourier transform of the autocovariance function only integrated from $-\frac{1}{2}$ to $\frac{1}{2}$?
in several time series texts like Shumway, the Fourier transform of the autocovariance function is integrated only over frequencies from $-\frac{1}{2}$ to $\frac{1}{2}$.
I do not understand why it ...
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Cumulative sum of trigonometric functions
I have two known time series $ Y, 𝜭 $
$Y = \left\{ Y_i \right\}_N \hspace {0.2 in} Y_i > 0 \hspace {0.2 in} -1 \leq Y_{i+1} - Y_i \leq 1 $ for all $i$
$𝜭 = \left\{ 𝜭_i \right\}_N $
Let ...
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What is the acf function of ARFIMA(0, d, 0) model
This should be a really easy question, but I cannot find any resources on the internet. So suppose $X_t$ follows a fractional ARIMA model with $|d| \leq \frac{1}{2}$. I am wondering what is the ...
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Fuzzy Entropy, what is the correct exponential term?
When computing the Fuzzy Entropy measure, most works use an Exponential fuzzy function. Yet, there are two deviations between the works. For the exponential membership function, some use the term
$\...
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Is the product of independent white noise also white noise?
Assume that the random vector $(u_t, v_t)$ is sampled iid over time and that $E[u_t v_t] = 0$. We also assume that $E[u_t] = E[v_t] = 0$ and that $E[u_t^2] = \sigma^2_u$, $E[v_t^2] = \sigma^2_v$.
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Comparing sample mean of a month X to the distribution of points during month X last year
Let's say $M$ is a metric of interest in the context of time series.
We are interested to see if the average $A$ of $M$ computed during month $M$ this year makes sense.I had many ideas:
compare $A$ ...
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Why do the bivariate vectors $(X_t, X_s)$ and $(X_s, X_t)$ from strictly stationary $X_t$ not have the same distribution?
from homework I got some weeks ago, we were asked if the bivariate vectors of $(X_t, X_s)$ and $(X_s, X_t)$ from strictly stationary Time Series$X_t$ do not have the same distribution.
I thought they ...
- 665
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Fourier transform of time series by diagonalising matrix
Does there exist a case where the Fourier transform of a time series is found via diagonalising a matrix? Ideally, I am looking for cases where the eigenvalues correspond to frequencies.
Any leads are ...
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6
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How to determine whether a reduced-form VAR is covariance-stationary or not?
I know how to determine whether a vector is covariance-stationary or not but I do not know how to determine whether a reduced-form VAR is stationary or not. For example, the expression is shown below,
...
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1
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What is the limit of autocorrelation of ARIMA(1,1,0) series
Consider the causal AR(1) time-series $y_t = \phi y_{t-1} + w_t$ such that $x_{t} - x_{t-1} = y_t$. I want to compute the limit $\lim_{t \to \infty} corr(x_t, x_{2t})$.
It is easy so see that the ACF ...
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32
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How to plot autocorrelation vs time instead of autocorrelation vs number of lags?
If I consider a stochastic process given by the monthly sunspots and I calculate the autocorrelation by using this Python code:
...
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10
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Autoregressive Time Series in Continuous Time?
I have seen that AR(1) process in discrete time corresponds to Ornstein–Uhlenbeck process in continuous time, is there a general analog of AR(n) process in continuous time?
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15
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Finding drops in timeseries - excluding spikes
I am trying to find drops that do not happen after spikes in a time series. This needs to be done without code (the platform supports min and max though).
I can detect drops already using this ...
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8
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Calculating rates of change of a curves on a graph in a time interval
I have a situation where I have continuous glucose response measurements and I would like to know the rate of change of the curve once it begins to change (i.e., how fast glucose rises and falls).
AUC ...
- 101
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29
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What does the superscript 'T' mean in this formula? [duplicate]
Not being a mathematician, I still need to understand this equation. I apologise in advance for the possibly trivial nature of this question. Here is the equation:
$$
\mathbf t_i = (t_{i, 1}, \ldots, ...
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