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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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31 views

Interpretation of Complex Roots of a Linear Difference equation.

I solved this linear difference equation $$x_{k+2} = c_1 x_{k+1} + c_2 x_k$$ to get the general solution \begin{align} x_k = \lambda \left(\frac{c_1 + \sqrt{c_1^2 + 4 c_2}}{2} \right)^k + \mu \left(\...
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15 views

Is it possible to represent two time series as a common and different component?

First of all, thank you for reading my question. I would like to know if it is possible to rearrange two correlated time series as the sum of two individual components plus a a third common component? ...
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19 views

Moving Average for Speed and Acceleration

I am tracking a person using a set of ultrasound tracking device. The ultrasound system provides (x,y) location data at 8Hz frequency, and there are 131 data points in total. When calculating the ...
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22 views

Solution to minimization problem when parameter goes to infinity

I have a time series with $T$ periods, $\{y_1, y_2, ..., y_T\}$. I need to minimize the following expression: $\min_{\{\tau_t\}} \sum_{t=1}^T (y_t - \tau_t)^2 + \lambda \sum_{t=2}^{T-1} ((\tau_{t+1} - ...
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35 views

First and second moments of a piecewise time series

Let $x=\{x_1,x_2,...\}$ be a first-order autoregressive process, $AR(1)$. Since $x$ is an $AR(1)$ process, it is defined as follows: $x_t=\mu+\phi x_{t-1}+\epsilon_t$ where $\mu$ and $\phi$ are ...
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33 views

Find asymptotic variance $w_{hh}$ of $\rho_{n.X}(h)$

I have to consider Bartlett’s formula for a causal AR(1) process. And then I have to find the asymptotic variance $w_{hh}$ of $\rho_{n.X}(h)$ and find the limit of $w_{hh}$ as $h \rightarrow \infty$....
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1answer
35 views

Analytical derivative of a signal $y(t)$ wrt to a signal $x(t)$

I am running a sensitivity study on the model $y(t) = x(t - \tau)$ where $y(t)$ and $x(t)$ are 2 time signals and $\tau$ a time lag. Basically I want to study the sensitivity of $y$ to a change in $x$....
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35 views

Does the following function have an integral?

Can the following function be integrated? $X_t = c + \phi_1X_{t-1} + \phi_2X_{t-2} + \epsilon_t$ Where: $X_t$ is the variable of interest; $X_{t-1}$ is the value of $X_t$ at one time step earlier and ...
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1answer
36 views

Standard deviation of multiplicative stochastic process

I am trying to build the mathematical background for an stochastic simulation. Right now I have the mathematical model for: $$y_t = N(\mu_t, \sigma_t)$$ So that $\mu_t$ and $\sigma_t$ are known. These ...
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11 views

Time serie model(find Yule-Walker estimators)

I got the time serie model $X_t=0.9X_{t-1}+Z_t$ with (Z_t) iid studentt-distributed with 10 degrees of freedom. And then I have to find the Yule-Walker estimators $\hat{\phi}$ and $\hat{\sigma^2}$. I ...
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12 views

Time series of web hit trend data

Each product on our website has its view hits logged. I have the data grouped by month, for the last 3 months. Data ...
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14 views

Why does this expansion hold in ARMA model?

I am studying time series analysis with R.S. Tsay's text "Analysis of Financial Time Seres". As you know, $ARMA(p, q)$ model is in the form $$ (1-\phi_1B - \cdots - \phi_pB^p)r_t = \phi_0+(1-...
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1answer
26 views

Why we use the expectation conditional on signal one period before as the estimation of state variable in Kalman filter?

Consider a simple state space model $$x_{t+1}=Ax_t+C\omega_t\\ y_t=Gx_t+v_t $$ Besides the orthogonal assumption for $w_{t+1}$ and $v_t$, we assume that $w_{t+1}\sim N(0,I)$ and $v_t\sim N(0,R)$. We ...
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30 views

$Cov(X_t , X_{t-2})$ in $AR$ model.

What is $Cov(X_t , X_{t-2})$ based on $Var({X_t})$ in model $AR(1)$. I tried using $X_t - \mu = a(X_{t-1} - \mu) + Z_t$ to find $X_{t-2}$ based on $X_t$ as fallow $$ X_{t-2} = \frac{X_t - \mu - aZ_{t-...
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61 views

Is an $\mathrm{ARMA}(1,1)$ process with an $\mathrm{ARCH}(1)$ innovation strictly stationary?

During some self studies of time series and ARCH processes, I thought of the following example. Given an $\mathrm{ARCH}(1)$ process $Z_t$, \begin{align*} Z_t &= e_t\sqrt{h_t}\\ h_t&=\alpha_0+\...
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22 views

Annual Probability Over Time When Not Exactly Independent

I haven't found an answer to this specific question. Say that I have a 1% chance of my appendix bursting in any given year. To calculate the probability over twenty years, if these were considered ...
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13 views

Hilbert Huang Transform until $t-1$

I want to decompose a signal using the HHT trying to avoid the end effect at $t_0$ and $t$ analyzing the signal only between $t_1$ and $t-1$. If I have a signal x(t) and I want to decompose it for $...
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13 views

How to smooth a projected value obtained with a multiplicator in a time series?

I have a time series chart where each bar interval = 600 seconds. Each bar accumulates a number of transactions in a variable called 'volume'. The goal is, while the current 10 mn bar is forming in ...
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6 views

Impulse response function of the unit root model

Trying to calculate the impulse response function of the unit root model. IRF = enter image description here Where Y = enter image description here Not sure how to derivate that.
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1answer
26 views

Autocovariance of the process $Y_t = X_t - X_{t-1}$

Let $ \{X_t, t=1,2,...\} $ be a stationary process. Obtain the autocovariance function of $ Y_t = X_t - X_{t-1} $.\ Solution. Since $ X_t $ is stationary, then $ E(X_t) $ is constant and $ Cov(X_t, ...
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1answer
27 views

A statistic to capture the degree of mean reversion

Given a realization of a stochastic process, $x_{t_1}, x_{t_2}, \ldots, x_{t_n}$, is there a simple statistic that captures the degree to which the stochastic process is mean reverting? For example, ...
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9 views

Reference Request: Measure convergence for spectrum of Stationary time series

Let $\{Y_t\}_{t\in \mathbb{Z}_{>=0}}$ be a centered, stationary random process, with autocovariance $R(t)=Cov(Y_0Y_t)$. The spectral density $S$ is defined as the Fourier transform of $R$, $S(\...
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15 views

Constrained interpolation/smoothing of multi-dimensional time series

Consider an $N$ dimensional time series $x_i(t),~i\in\{0,1,\cdots, N-1\}$ where $x_i(t)$ is smooth. It turns out that for all $t$: $x_i(t)>x_{i-1}(t)$. The multi-dimensional series is sampled at ...
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1answer
41 views

Does Portmanteau Test need?

I am studying time series with Ruey S. Tsay Financial time Series(3rd ed). In his text (page 32), he introduced Box & Pierce Portmanteau statistics $$Q^*(m) = T\sum_{l=1}^m\hat{\rho}_l^2$$ for ...
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50 views

Is a ratio of numerical derivatives, $\frac{dx}{dt} / \frac{dy}{dt}$, considered a "partial derivative"?

I am reading a paper on genetic programming by Lipson and Schmidt called "Distilling Free-Form Natural Laws from Experimental Data" at the following link: ...
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8 views

Multinomial model: starting value for time parameter

I’m a beginner data science student and for a course we need to replicate a paper which models multi-party elections in a Bayesian framework. Basically, we understand the model: poll respondents $y$ ...
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20 views

Predicting complex periodic signals without "enough" samples

I was recently reasoning about the analysis of complex periodic signals and about the prediction guarantees that one can achieve without sampling the whole period. Say I have a signal that I know can ...
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1answer
33 views

Multivariate time series and machine learning.

I have a question relate to apply machine learning algorithm to time series data. Because time series data has the impact of "order time or sequence" (I am meaning that time indexing, for ...
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13 views

Spectral representation of real/valued stationary process

It is known that a wide-sense stationary process (for the purpose of time series analysis, let's say we are in discrete time) has a spectral representation as a stochastic integral $$ X_t = \int_{(-\...
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5 views

How to specify this model's space-state equation?

I am using this python package called CausalImpact (http://google.github.io/CausalImpact/) as one of the statistical methods in my research paper. I explained it in my methodology section as much as ...
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32 views

Does an ARMA-GARCH better estimates AR parameters than an ARMA in presence of heteroskedasticity?

Working on a time series with time varying volatility, if one was to apply an ARMA-GARCH and an ARMA on the said series, would these Auto Regressive parameters vary much ? My understanding is the ...
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45 views

Bayes' factors with related sets of observations

I have a set of time series observations that come in threes, say $\textbf{y}= (y_{1,1}, y_{2,1}, y_{3,1}... y_{1,T}, y_{2,T}, y_{3,T}$). It's important to note that temporally, $y_{1,t}$ still occurs ...
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18 views

Understanding and Computing the Discrete Correlation Function of two time series

I am attempting to compute the discrete correlation function (DCF) that is defined in Edelson & Krolik for two time series, but I do not understand the final step. Let my two data sets $(a_i, t_{...
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1answer
33 views

Showing a certain time series is stationary

I have been trying to solve the following problem, but have not been successful yet. I was hoping anyone could nudge me in the correct direction. Let $\{ w_t; t = 0,1,\ldots \}$ be a white noice ...
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25 views

What does autocorrelation means for the following data?

Let's suppose we have a time series is a=[1,1,1,3,3,3,1,1,1,3,3,3] then the autocorrelation figure for this time series is enter image description here The lag here ...
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4 views

Comparing two classifier g-mean result for statistical significance

I want to compare two G-mean values at 0.9923 and 0.9884 from the two classifiers. What is the recommended test on how to accomplish this? I will be very grateful for your time Best wishes
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10 views

How to calculate process correlation with sparse unequal observations at different times?

Suppose I have two processes: $dx_t = \sigma_x dW_t$ $dy_t = \sigma_y dV_t$ $dW_t dV_t = \rho dt$ where $W_t,V_t$ are two correlated standard Brownian motions and $\sigma_x,\sigma_y,\rho$ are all ...
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20 views

How to compare Poisson Point Process, ARIMA and LSTM?

I'm trying to compare three forecasting techniques: A stationary stochastic Poisson-GEV : where the rate of occurrence of the events is given by a Poisson process and, it's intensity is given by a ...
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16 views

Rotating radar time domain data

I want to simulate the time domain data for a rotating radar. I assume that the space around the radar is filled up with a very big extended object and it moves with a constant speed in one direction. ...
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40 views

Numerical Computation of Chapman-Kolmogorov Condition for a Dataset

Good day every one, I am trying to solve a particular problem relating to probabilities: I have a time series dataset, where the price at time $t_i$ can be given as $x(t_i)$ We can then define a ...
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16 views

Integral of a function of the form $\exp(1j A \cos(Bt) t - C t)$

I want to see how the Fourier transform looks like when the time domain signal phase is modulated with a cosine function that varies with time. Let's start with a monochromatic signal first and find ...
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1answer
46 views

Closed-form expression for h-step forecast for AR(2) process

I'm trying to derive $\mathbb E[y_{t+h} \mid y_t]$ where $y_t$ is an AR(2) process $$ y_t = \phi_1 y_{t-1} + \phi_2y_{t-2} + c + \varepsilon_t $$ As per usual I'm recursively applying $\mathbb E[y_{t+...
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1answer
21 views

conversion from per year to per decade for time-series data

I was wondering if this logic makes sense. I have a time series where the slope is -0.19 over 21 years, so the rate would be -0.19 per year. Now if I want a rate per decade, would it logically make ...
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1answer
20 views

Is it possible to generate multi-step forecasts with the Kalman Filter (KF)? [closed]

I wanted to double check with the community but I think I already have the answer to this question. Can the KF (or its variants such as the ensemble KF) produce meaningful multi-step forecasts? My ...
2
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1answer
37 views

Variance of random walk

Consider the Random Walk $$X_t = X_{t-1}+\epsilon_t$$ with $\epsilon_t \sim N(\mu,\sigma^2)$ and $X_0=0$. We can write $$X_t=X_0+\sum_{n=1}^t\epsilon_t.$$ Using the equation above we have $$\mathbb{E}[...
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1answer
100 views

Approximating the spectral density of white noise by a moving average process

Suppose that $X_t$ is a weakly stationary process; then, its autocovariance function can be represented as: $$\gamma_X(h) = \int_{(-\pi , \pi ]} e^{i h v} dF(v)$$ The function $F$ is called the ...
2
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1answer
141 views

Order of a non-linear autoregressive exogenous (NARX) model

Suppose I have a non-linear autoregressive exogenous (NARX) model of the kind $$y(k+1)=f(y(k), y(k-1), ..., y(k-s), u(k), u(k-1),..., u(k-t)) $$ where $y$ and $u$ represent respectively the output and ...
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1answer
49 views

Bounded AR(1) Process [closed]

I am trying to estimate an AR(1) model for a stochastic process Xt that is bounded in some interval (a,b]. Is there a simple transformation I can do on Xt that ensures my AR(1) model does not violate ...
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1answer
25 views

How to judge second-order stationary?

Given that a series {$x_t$} where $x_t=\sin(t+U)$, $U$ has uniform distrubution on $(0,2\pi)$. Is {$x_t$} second-order stationary? I know here we need to show $E(x_t)$ is constant and $cov(x_{t+s},x_{...
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1answer
38 views

Identifying stochastic process through its autocovariance function

I am unable to understand how to correctly identify time series processes through their autocovariance functions (acvf). For example I have an acvf $$\gamma(h) = \begin{cases} 4-|h|, & |h|\leq4,\\ ...

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