Questions tagged [time-series]
The analysis and inference about data observed over a general(continuous or discrete) time space. Usually related to stochastic processes and will probably receive better response under that tag.
33 questions
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Fourth moments of Gaussian processes
I am working on a topic outside my main research area, so I am afraid I am reproving obvious results, so I would like to ask for a reference. Google didn't help, mostly because I am looking for ...
- 20.2k
5
votes
0
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197
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Theoretical justification of time-series forecasting using Takens' embedding
This is a cross-posting
where I couldn't get an answer. In the meantime I have tried to improve the original logic:
As in Takens original paper about his embedding theorem, consider a compact $m$-...
- 51
4
votes
1
answer
622
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Why is every Gaussian process a linear process?
In Section 4.2.4 of [1], the authors write
In this section we consider a causal linear process
$$
X_t = \sum_{j = 0}^\infty a_j \varepsilon_{t - j}, \quad t \in \mathbb{N},
$$
where, without loss of ...
- 299
4
votes
1
answer
337
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Support of bivariate joint distribution of stationary and ergodic sequence
Let $\{X_t\}_{t\in \mathbb{N}}$ be a strictly stationary and ergodic sequence of real valued random variables and let the support of $X_1$ equal $[-1,1]$. Can the support of $(X_1,X_2)$ equal the unit ...
- 43
3
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1
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Temporal generalization of graphs: density vs $n$ and $m$?
In short: we generalize graphs to the temporal case, but fail to fully preserve the usual relation between density, number of vertices, and number of edges; how to make better?
Context.
We propose a ...
- 1,444
3
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0
answers
199
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An infinite moving average is stationary iff its innovations are stationary
Let $(a_j)_{j \in \mathbb{N}_0}$ be a real-valued sequence such that $\sum_{j = 0}^\infty a_j^2 < \infty$.
Further, define an infinite moving average time series $X = \{ X(t), t \in \mathbb{Z}\}$ ...
- 299
3
votes
0
answers
98
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Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix
In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
- 279
2
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1
answer
120
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Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes
Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...
- 135
2
votes
1
answer
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On the stationarity of Gaussian processes
I am trying to understand and prove the statement:
The normal (or Gaussian) process is stationary in the wide sense if and only if it is strictly stationary.
I know the following:
A strictly ...
2
votes
1
answer
117
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Quantifying an increasing spacing between data points
Is there a measure or statistic that could quantify a steady increase in the spacing between data points in a time series?
For instance, in the figure, the points are clustered and dense near 0, but ...
- 167
2
votes
1
answer
107
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Relation between invertibility and strong mixing of a time series
Setup:
I have a sequence of stationary ergodic random variables $(\epsilon_t)_{t\in\mathbb{Z}}$ and a function $\phi:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$. Define the sequence of random ...
- 479
2
votes
0
answers
81
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Convergence of random operators
I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
- 121
2
votes
0
answers
431
views
Cointegration of (multiple) time series
Some time ago I stumbled upon the notion of cointegration of time series (see the wiki for some basic fact).
Unfortunately, my knowledge of time series is a bit sketchy, and moreover I was able to ...
- 7,853
1
vote
2
answers
160
views
Expected value of long memory moving average
Let $X$ be an infinite moving average time series, i.e.
$$
X(t) = \sum_{k = -\infty}^\infty a_j \varepsilon_{t-j}, \quad t \in \mathbb{Z},
$$
where $\varepsilon_{j}$ are uncorrelated zero mean, finite ...
- 299
1
vote
2
answers
235
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Counterexample for absolute summability of autocovariances of strictly stationary strongly mixing sequence
Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strictly stationary, strongly (i.e. $\alpha-$)mixing sequence of real random variables. If we have $\mathbb{E}[|X_1|^{2+\epsilon}]<\infty$ for some $\epsilon&...
- 203
1
vote
1
answer
79
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Does the "random Krylov-Bogolyubov theorem" hold in a non-skew-product setting?
Informal description.
Suppose I have a dynamical system $f$ defined on the product of a compact space $X$ representing the state space of an "experimentally visible" variable and a compact ...
- 698
1
vote
1
answer
77
views
Why shocks are independent with weighted sum of normal process
I am doing a problem and got stuck by the definition of "normal process". The problem is stated as follows:
Suppose $e_t = \sum_{j}^{\infty}\theta^j Y_{t - j} $ and assume that $Y_t$ is a ...
- 119
1
vote
1
answer
371
views
Calculate Average and Correlation of WSS Random Processes
Given two stochastic processes, $X[n]$ and $Y[n]$, both being WSS (wide state stationary) and independents. What would be the Average and Autocorrelation function of $Z[n] = Y[n] X[n]$?
Is the ...
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vote
0
answers
81
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Urn model with delayed replacement
Suppose I have x red and y blue balls. At each timestep I draw a ball with probability $$P(\text{red ball}) = (x/(x+y))^z, P(\text{blue ball}) = 1-P(\text{red ball})$$ where z is fixed.
Each ball is ...
- 11
1
vote
0
answers
42
views
Subordinated non-deterministic Gaussian process is non-deterministic
Let $X = \{ X(k), k \in \mathbb{Z} \}$ be a strictly stationary, Gaussian time series whose spectral density $f_X$ exists.
Furthermore, let $X$ be non-deterministic, i.e.
$$
\mathbb{E}\big[ \vert X(n +...
- 299
1
vote
1
answer
737
views
How do the singular values of a Hankel matrix, generated by some data time series, change when we add/remove rows and columns?
Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ ...
- 11
1
vote
0
answers
29
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Deriving periodical processes from a finite time series
Suppose we have a finite time series of real-world events measured at $(t_k), k \in \mathbb{N}$ with $(t_{k-1} < t_k)$. The content of the actual events is irrelevant.
I would like an automated ...
- 111
1
vote
0
answers
92
views
Calculating right values of Periodogram using Fourier Analysis
In the book, Economic Cycles: There Law and Cause By Henry Ludwell Moore, he plots Periodogram of rainfall of Ohio valley. He uses 72 years data (1839-1910) and tries to find the most dominant cycle ...
- 11
0
votes
1
answer
1k
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The derivation of Reynolds-averaged Navier-Stokes equations
The following procedure is used to derive the Reynolds-averaged Navier-Stokes equations (Wikipedia: RANS equations)
When we talk about turbulent flows we can represent the velocity of the fluid as:
$$
...
- 1
0
votes
1
answer
126
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Stationary distribution of AR(1) processes and Lyapunov central limit theorem
Let $X_t$ follow the following AR(1) process:
$$
X_t=\rho X_{t-1}+e_t
$$
in which $|\rho|<1$ and $e_t$ is iid noise term with density $f$, mean $0$ and finite moments up to a certain order.
I am ...
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0
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71
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How to analyze a nonlinear time series dataset?
I have a time series that appears chaotic that I would like to analyze with Python. To draw its logistic map, I must use the logistic equation: $$x_{t+1}=rx_{t}(1-x_{t})$$
I have the data in a text ...
- 1
0
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0
answers
60
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Given a set of time-series data, how would I determine another time-series is a linear combination of the set?
In other words, determine if sum linear combination of existing time-series could result in the desired time-series. I'm unsure if assumptions about the time-series may clarify the problem better, so ...
0
votes
0
answers
24
views
How to calculate the 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_{...
- 109
0
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0
answers
30
views
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, ...
- 23
0
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0
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233
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A new method for processing music scores?
I have developed a method and python script:
https://github.com/githubuser1983/algorithmic_python_music
which allows the user to input a midi file and then chose a few numbers as parameters, and the ...
- 3,032
0
votes
1
answer
66
views
Estimating operators of functional linear processes
Let $H= L^2[0,1]$ be the space of measurable and square integrable functions from $[0,1]$ to $\mathbb{R},$ let $(\varepsilon_k)_{k \in \mathbb{Z}}$ denote the iid (or strict stationary) $H$-valued ...
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Bayesian Inference for Parameters Estimation in ARMA Model [closed]
In the usual sense, Maximum Likelihood Estimation is the common method for Parameter estimation in ARMA(p,q) model.
If I am looking to estimate parameters for ARMA(p,q) with Bayesian Inference, how ...
- 1
-4
votes
1
answer
303
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Reference request in optimal stopping [closed]
I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
