# 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.

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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:
$$
...

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### Ornstein-Uhlenbeck proc. Max Likelihood estimators [closed]

I'm currently working with OUP. I've already applied the MLE found by Tim Leung from Optimal Mean Reversion Trading to my normal time series. If I detrend my ts, my theta becomes negative and i can't ...

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### 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 +...

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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 $X = \{ X(t), t \in \mathbb{Z}\}$ ...

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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 ...

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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 $(X,Y)$ ...

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### 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 ...

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### Poisson random measure according a Lévy Process

$\DeclareMathOperator\Poisson{Poisson}$Let $(\Omega, \mathcal{F}, P)$ and $(\Theta, \mathcal{B}, \rho)$. A Poisson random measure (PRM) with intensity $\rho$ is a kernel $\mathcal{N}: \Omega \times \...

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### 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 $N$ ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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### 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 ...

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169
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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&...

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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 ...

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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 $m$-...

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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 ...

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### 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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### 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 ...

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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 ...

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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 ...

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### 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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### 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 ...