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.

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32 views

Show that all series satisfy this equation $\sum_{\mid h\mid < n } \hat{\gamma}(h)=0$

I need to show that the sample autocovariance of the series {$x_1,...,x_n$} satisfies $\sum_{h<\mid n \mid}\hat{\gamma}(h)=0$ I start by defining $\bar{X}= \frac{1}{n}\sum_{t=1}^{n}X_{t}$ and that ...
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41 views

How to estimate this (unusual?) model for a time series of covariance matrices?

I am wondering if anyone can point me to econometrics or statistics literature on estimating the covariance matrices of models of the following form. Or, if someone with enough domain knowledge would ...
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9 views

Measuring variation of mean and variance in time series

I don't get how the variation of mean and $\sigma$ are measured for instance in the following picture from here: The red line is the mean of the data which changes. It is said that the standard ...
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9 views

Time series dataset - averaged over five years [closed]

I have a dataset on daily accidents in a particular province from the years 2001-2005. I need to create a time series data set of daily accidents, averaged over five years. The way I'm thinking about ...
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10 views

Analytical Wavelets

I'm trying to understand the difference between a Time-frequency analysis done with standard wavelet, and another done with analytical wavelet. If I take as a signal a function $f \in L^2(\mathbb{R})$ ...
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21 views

Geometric Brownian motion: Correlation in the increments when looking at the average value over some period?

We have a Geometric Brownian Motion $X_t$ sampled at some time interval $\delta t$. Increments in the Brownian Motion are uncorrelated, i.e. we have $\text{corr}\left[(X_{t+\delta t} - X_t), (X_{s+\...
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34 views

minimising the square of the expected error in Gaussian time series model

I'm trying to derive the result that $\mathbb{E}(X_{n+h} - m(X_n))^2 = \sigma^2(1-\rho(h)^2)$ where $m(X_n) = \mu + \rho(h)(X_n - \mu)$ but i can't get the same result. I will post my derivation below....
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26 views

Can you give me some clue on proofing that squared log-returns of a GARCH(1,1) time series is ARMA(1,1)?

I am working on an exercise from René Carmona's Statistical Analysis of Financial Data in R and I am stuck at this exercise. The problem I have a series $\{Y_t\}_t$ of log-returns from an asset that ...
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19 views

Moving average and using calculus

I have a time series of data. Let’s say date on the X axis and Y is revenue. I calculate the moving average over, say, the past 5 data points and plot this (another time series). I calculate the ...
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Is there a formula for a cross-correlation function for more than two time series?

For discrete functions, the cross-correlation is defined as: $$c[n]=\sum _{m=-\infty }^{\infty }{\overline {f[m]}}g[m+n]$$ (see Wikipedia). This is valid for two time series, $f$ and $g$. Is there a ...
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39 views

Modeling differential equation

I am trying to model some differential equation which can explain buying some assets that can increase itself like a virus. So, let's assume my money is y(t), and number of assets is n(t). y(t) to ...
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36 views

Understanding measurement error and model consistency in a state space VAR model with Kalman Filtered coefficients.

I wish to create a VAR model of any order and dimensionality and find its prediction coefficients by a Kalman Filter, in order to avoid window size selection issues in time-series analysis. I have ...
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44 views

A beginner’s explanation for PCA on a multivariate time series

This is very much a beginner’s question. Say you have a 10 dimensional vector for every day in a time series of 100 days. I was reading about using PCA to reduce this to a one dimensional time series....
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What does this convolution result means?

I tried to predict a time series (for anecdotal purposes, it is a serie of daily detected coronavirus infected cases). For that purpose, I created a complex number for each day, were the real part is ...
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1answer
32 views

If I split a stationary ARMA process into two parts, are they also stationary?

Considering an Auto-Regressive Moving Average (ARMA) model, \begin{equation*} y_k = \phi_0 + \sum_{j=1}^{p} \phi_j y_{k-j} + \sum_{l=1}^{q} \theta_l \varepsilon_{k-l}+ \varepsilon_k, \qquad \text{for}...
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7 views

auto-covariance gamma function

I'm learning time series analysis and I'm a little confused as to how the auto-covariance of a time series dataset works. We define the auto-cov function of a time series as: $\gamma(h) = E[(Y_t - ...
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13 views

Equations of a trendcpt model

After fitting a Trend cpt changepoint model to certain data using the EnvCpt library in R, the estimated parameters are: ...
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10 views

Stationarity in seasonal time series

Given a time-series difference equation, how can I tell whether it's stationary or not? e.g. let's take $\phi_p(B)\Phi_P(B^s)=(1-0.2B^2)(1-6B^{12}+12B^{24})$ where seasonality=12, for the AR and ...
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Ways to obtain networks from multivariate time-series

I recently became aware of a bridge between (dynamical) properties of time-series and (topological) features of an associated network representation. A variety of methods exist to embed the time-...
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21 views

Unevenly spaced data

I'm working on a project that is made up of X-values (Julian dates) and Y-values (measurement data). My current problem is that the X-values have some dates that are too close together (let's say ...
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36 views

Summability conditions for coefficients of the reciprocal of a power series

I wonder if there is some reference for materials like the following. Let $\frac{1}{\beta(z)}$ be well-defined for all z with $|z|\leq1$, where $\beta(z)=\sum_{k=0}^{\infty} \beta_kz^k$. Then, we ...
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27 views

Does high standard error and high r-square imply spurious regression?

Does a regression passed on time series data with one independent variable and one dependent variable which yields parameters with very high standard errors (t-values) and also a high r-squared imply ...
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15 views

Distribution of maximum of a moving average sequence

Let $Z_n \sim WN(0,\sigma^2)$ and $a \in \mathbb{R}$, $$X_t = Z_t + aZ_{t-1} \qquad t = 0, \pm1, \pm 2,...$$ defines a $MA(1)$ sequence. I need to prove that this sequence has an independent extremal ...
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13 views

Which covariance matrix should I use for treating heteroskedasticity in my panel data?

I have a data set with panel structure (panel data) with 78 individuals observed over 5 three-year periods. I have 10 dependent variables an 1 independent variable. I applied logarithmic ...
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18 views

Interpretation of graphs of ACF and PACF

I plotted graphs of ACF and PACF (in R), but I do not know, how to find out, which orders of differences are statistically significant. Could somebody explain it to me? I found on the Internet that "...
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32 views

Time Series - ARMA$(1,1)$ Model

Consider the autocorrelation function of the ARMA$(1,1)$ process is given by $Y_t= \alpha Y_{t -1} +Z_t + \beta Z_{t-1}$. Show that $\text{Var}(Y_t) = \frac{1+\beta^2+2\alpha\beta}{1-\alpha^2}$. ...
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1answer
13 views

Cauchy Schwarz and Absolute Values

I am having trouble with the following: I know that by CS inequality I have $$ |E(xy)|\leq \sqrt{ E(x^2)E(y^2)} $$ However, say we have a process like $z_{t}= e_{t}e_{t-1}$ with $e_t$ ~ $iid(0,\...
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26 views

Stock prediction

I'm working on a stock prediction algorithm using LSTM network. However my test data are obviously from different distribution than my train set 1, hence my prediction looks like the one on the figure ...
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Maximum Likelihood for Uniformally Distributed Errors (MA process)

I couldn't find similar questions to mine so here it goes. Consider the MA(1) model given by $$ y_t = e_t +be_{t-1} $$ with $e_{t}$ distributed uniformally over the unit interval [0,1]. Simulate ...
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How do I create a MA model in R?

I'm running analysis on a certain dataset, and while looking over the data, I noticed that the PACF have no values (besides the first of course) that are significant, while the ACF's values are all ...
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10 views

How to determine whether the differences between two time series are significant (in R)?

There are two time series, that are generated in an experiment, where the intensity of signals of chemical bounds are measured along a wavelength interval, but the second series are generated by ...
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7 views

Can R's IRG function be used with ndvi data derived from MOD09Q1?

I am trying to use the IRG function to calculate instantaneous rate of green-up from NDVI values derived from MOD09Q1 bands 1 and 2. I have been following this tutorial but realized it requires a data ...
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What are the benefits of working in the frequency domain instead of time domain?

I have been conducting research in time-series analysis for quite some time and I am familiar with the studies performed within territory of the time domain. However, there are also numerous books and ...
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17 views

Regression Analysis of Time Series Data

My model is in the form of: $Y_t=\beta_0+\beta_1 Y_{t-1} + \beta_2 X_t + \beta_3Z_t$. I have only learned regression analysis of cross-section data. I'm wondering which function in R should I use to ...
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10 views

Usage of empirical auto-covariance matrices of time series

Let $x=(x_t)_{t\in \mathbb N}$ be a multivariate stochastic process that takes values $x_t\in\mathbb R^n$. Suppose we have a sample path of $x_t$ as $\bf X$, where $\mathbf X=(\mathbf x_1,\mathbf x_2,...
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1answer
20 views

Forecasting using LSTM network

I have a time series data of size 150. I trained 80% (120 data points) and tested the remaining 20% (30 data points) of the data set by LSTM network. So I got the predicted values of the series from ...
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1answer
65 views

Finding a sufficient statistic for $\theta$ that takes values in a subset of $\mathbb R^3$

This is the problem that I'm working on, for reference: For some $n\ge 3$, let $\varepsilon_1, \ldots , \varepsilon_n$ be i.i.d $N(0, 1)$. Set $X_1 = \varepsilon_1$ and $X_i=\theta X_{i-1}+(1-\...
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8 views

Basic time series representation

So, I am about to start researching models for efficient time series storage and management. But before I start I want to study basic Time Series Bibliography. Most "basic" books I have seen so far ...
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77 views

Probability of failure at year i vs Probability of failure up to year i

Assume you have a system with unknown resistance $(R)$ at the design phase, therefore that could be modelled as a random variable, but time-invariant (i.e. the value of the random variable generated ...
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8 views

If two random walk patterns follow each other, is it still considered a random walk?

I am wondering if these two lines, F1 and F2, representing time series, would still be considered "random walk", once the relationship between the two was discovered? Could this relationship ever even ...
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12 views

Variance change in a moving average of infinite order?

Does the variance or auto covariance change in the moving average of infinite order, of a weakly second order stationary time series process. Because if I find the variance of the time series ...
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22 views

Locally stationary processes: Can a tvAR(1) process by approximated by an AR(1) process?

I am trying to understand the asymptotic behaviour of stochastic processes of the form $\{X_{t,T}\}_{t=1, \dotsc,T}$ where the time is rescaled to $[0, 1]$, i.e. with $T \rightarrow \infty$ we get ...
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4 views

Classify a series of data to good series and noisy series data

I have time-series data for N commodities. I want to segregate them to two classes, one with commodities having good data without much noise & variance in the series, then the other with high ...
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9 views

How can I Wold Decompose a Strict White Noise Process

According to Wold Decomposition theorem, any covariance stationary process can be decomposed in deterministic and stochastic part. We know that strict white noise is covariance stationary. So, I ...
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13 views

How to assign single numeric value to a time series data, which denotes trend in a time-series data?

I am working with time-series data and I need to determine whether data is upward sloping or downward sloping. The value of 1 would indicate data is upward sloping and the value of -1 would indicate ...
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1answer
18 views

Finding conditions for an inequality, positive semi definite

I want to prove that the sequence: $$ ( 2 + \psi^2 ; \psi ) $$ is positive semi definite for $|\psi | < 1$, and ideally find conditions on $\psi$ about when it isn't. Recall that for a sequence, ...
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5 views

Calculate error in velocity from error given in position data

I am working on gps time series data and have 4 year daily position data of a gps station. Along with daily gps cordinates I have error in measurement (in position) associated with each entry. Now, I ...
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11 views

Positive semidefinite sequence, estimators of the acvs

definition of positive semidefinite for a sequence $s_n$: $\forall t_1, \cdots, t_n \in \mathbb T, \forall a_1, \cdots, a_n \in \mathbb R^*\colon$ $$ \sum_1^n \sum_k^n s_{t_j-t_k} a_j a_k \geq 0 $$ ...
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4 views

Estimates from Yule Walker Equations for AR(2) Model

Manually construct the Yule-Walker estimates $$\hat{\phi_1},\hat{\phi_2},\hat{\sigma_Z^2} $$ for an AR(2) Model. I know the Yule Walker equations for AR(2) model and that we can estimate $$\hat{\phi}$...
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1answer
17 views

How to normalize time series data

Forgive me if I am using the wrong terminology. I am trying to graph how productive a machine is over time with incomplete data. I am polling the machine at a random interval and getting the total ...

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