# 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.

30
questions

0
votes

1
answer

67
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 ...

0
votes

0
answers

53
views

### 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

36
views

### Stationarity of ARMA-like time series

It is well known that for $X_t \sim ARMA(p,q)$ where $\phi(B)X_t = \theta(B)Z_t, Z_t\sim WN(0, \sigma^2)$, if $\phi(z)\neq0$ in the unit circle, $\{X_t\}$ is stationary.
Now assume $\{Y_t, t=0, \pm1, ....

0
votes

0
answers

48
views

### Multivariate continuous wavelet transform

I have used the continuous wavelet transform (CWT) and cross-wavelet transform (bivariate; XWT) several times while researching geophysical systems.
I'm trying to understand how the XWT can be ...

1
vote

0
answers

76
views

### 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 ...

0
votes

0
answers

20
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_{...

0
votes

0
answers

29
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, ...

0
votes

1
answer

633
views

### 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
vote

0
answers

39
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 +...

3
votes

0
answers

144
views

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

4
votes

1
answer

500
views

### 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 ...

2
votes

1
answer

113
views

### 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 $(X,Y)$ ...

1
vote

2
answers

146
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 ...

1
vote

1
answer

527
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 $N$ ...

0
votes

0
answers

228
views

### 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 ...

2
votes

0
answers

75
views

### 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 ...

3
votes

0
answers

90
views

### 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 ...

1
vote

0
answers

29
views

### 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 ...

3
votes

1
answer

224
views

### 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 ...

2
votes

0
answers

344
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 ...

1
vote

2
answers

214
views

### 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&...

-4
votes

1
answer

285
views

### 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 ...

5
votes

0
answers

155
views

### 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 $m$-...

4
votes

1
answer

332
views

### 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 ...

1
vote

1
answer

357
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 ...

1
vote

0
answers

88
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 ...

2
votes

1
answer

112
views

### 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 ...

2
votes

1
answer

100
views

### 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 ...

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 ...

9
votes

1
answer

1k
views

### 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 ...