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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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Relation between Vector Auto-regressive models and correlation matrix

I am generating a multivariate time series using Vector Autoregressive Models- $$X(t) = AX(t-1) + \epsilon$$ where $X \in R^{n \times 1}$, $A \in R^{n \times n}$ and $\epsilon \in R^{n \times 1}$ is a ...
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Approximating the tangent vector in a phase space (or state space) reconstruction

I am investigating an application of differential geometry in experimental dynamical systems. Given a 1D time series (e.g., one that has been experimentally obtained), $x(t)$, I am considering the ...
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Statistics Time series model problem

$X_t = 2X_{t−1} − 1.7X_{t−2} + 0.7X_{t−3} + w_t − 0.1w_{t−1}$ Find the characteristic equations, ARIMA order, and $E[X_t]$ Also for $E [\nabla X_t]$ and determine if $\nabla X_t$ is stationary and ...
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1answer
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Statistic ARMA model

$Xt = 5+0.8(X_{t−1} −5)+0.8(X_{t−2} −5)−0.6(X_{t−3} −5)−0.5e_{t−1} + 0.25e_{t−2} +e_t$ Write the process using characteristic equations then determine ARIMA order and E[Xt]. Finally, find the values ...
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28 views

Statistics--Time Series Problem

$X_t$ = $3/2X_{t-1} - 1/2X_{t-2} + 1/2e_t - e_{t-1}$ Write the model into an ARMA form, determine if it is stationary if it is invertible and determine p and q. If it can be reduced, write the ...
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21 views

How to convert values from one range to another with in-equal slopes and real time data?

There are two series , Sa and Sb. Sa ranges always within 0-1 . Sb ranges variably sometimes from 130-145, 2017-2077 and many more etc.. The data points are real time in nature. For each second ...
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Is there any theoretical result on how to stabilize a polynomial by changing its coefficients?

The stability of a general $n$ order polynomial is associated with the following statement: if all the roots of the following equation falls in unit circle on the complex plane, then the system is ...
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Conditional densities and known past values

Let there be a random variable with the following properties $Y_{t} = \mu + \beta Y_{t-1} + \epsilon_{t} \quad\text{such that} \quad \epsilon_{t}\sim \mathcal{N}(0,\sigma^{2})$ Estimate the ...
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Error in time series forecasting

If I have hourly input data and I want to produce an output forecast of half hourly granularity, I must interpolate the hourly input data - but how can I calculate the average error attributed to my ...
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128 views

How to transform maximum distances between two graphs into cross-over and vice-versa?

Considering there are have two series v1,v2 over time which ranges between 0 and 1. Two graphs are plotted below in colour Blue and Green and the pattern between the graphs is shown in the diagram.How ...
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Bias adjustment for the Box-Cox back-transformation

I'm learning time series analysis and I don't understand why the back-transform of Box-Cox transformation outputs the median instead of the mean of the forecast distribution. The family of Box-Cox ...
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How to assess the convergence of a time series towards a periodic signal and assess the sufficiency of the sample size?

I have a sampled temporal signal, result of a transient fluid simulation. At the beginning the flow is being established, so the first few seconds of signal should be ignored (question : is there a ...
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upper bound of an inequality

Let $X_t = \theta_1\epsilon_{t-1} + ... + \theta_q\epsilon_{t-q} + \epsilon_t$ where $\epsilon_t$ is a white noise with zero mean variance $\sigma^2$. $X_t$ is said to be invertible if $\epsilon_t = ...
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Calculating Coefficients for MA and AR representations of ARMA Models

Was working on a problem - how to find the coefficients for a MA and AR representation of given ARMA models. Specifications in the image. Thanks! Image of Problem
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1answer
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Calculating Mean, Variance and Autocovariance Functions of Time Series

I'm having trouble finding the mean, variance and autocovariance functions of a time series function. Looked around and couldn't find a problem like this. Image attached Vsubt = 1/q * summation Xsub(...
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Classifying Customers by Purchase Periodicity

I want to classify retail customers into groups reflecting the periodicity of their purchasing behaviors: e.g. weekly, monthly, seasonal, annual, etc. I thought this question was trivial, but now find ...
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Time Series' Question

enter image description here Here is a question regarding a linear trend process: $yt = b1 + b2t + \epsilon t,$ where $\epsilon t ~ N(0, \sigma^2), b_1\neq 0,b_2\neq 0$ It is known that DES model ...
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Sample autocorrelation function as $n\rightarrow\infty$

Let $\{X_1,\cdots,X_n\}$ be observed values of a time series at times $1,\cdots,n$, and let $\hat{\rho}(h)$ be the sample autocorrelation function at lag $h$ which is defined as \begin{align*} \hat{\...
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6 views

Simplified expression for autocovariance function.

Given $X_t=\beta_1+\beta_2t+Z_t$ where $\beta_1$ and $\beta_2$ are known constants and $Z_t$ is a white noise process with variance $\sigma^2$. I'm trying to find a simplified expression of the for ...
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2answers
40 views

What is the baby-step to derive this?

Could somebody help me with this one quickly? My midterm is coming in 3 hours. I wonder how to go from the first line to the second line? How does (Z of t)^2 suddenly become (Z of t-1) ? Thank you ...
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Decomposition of time series into weighted (weights known) stretched exponentials with unknown offset function

Looking for solution to decomposing function $f(t)$ into stretched exponential functions all with same meta-exponent and decay constants, but with different amplitudes $p$ (known) and offsets $\tau$ (...
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1answer
26 views

When is the following process stationary?

Let $Y$ be a random variable with mean zero and variance $\sigma^2$, and let $c$ be a constant. Let $$X_t = Y\cos(ct)$$ When is the process $X_t$ stationary? I find $E(X_t) = \cos(ct)E(Y) = 0$ $E(...
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How to verify/classify a stochastic process

Let say I have a time series $X_t$. I then calculate $Y_t = X_{t+1}-X_t$ and find that $Y_t$ are normally distributed from Kolmogorov-Smirnov test. Is this sufficient to infer $X_t$ is a Wiener ...
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1answer
25 views

Finding expected value and variance of time series

Consider a time series generated by the following model: $y_t = 0.8y_{t-1} + \epsilon_t$ $E(\epsilon_t) = 0$ and $Var(\epsilon_t) = 1$. Note that for any s < t, $y_s$ is independent with $...
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Determine if AR(p) model is causal stationary or invertible

I was going through these problems and think I figured out most of them both, but am having some troubles at one of the last steps. The question is for each of the following models: Express them ...
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1answer
69 views

ARMA (1,2) model - Auto covariance function

I am struggling with finding the Autocovariance function $\gamma(k)$, of the following ARMA(1,2) model: $x_t-0.9x_{t-1}=e_t+2e_{i-1}+0.5e_{t-2}$. I have already found this model to be stationary, ...
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Questions about ARIMA modelling

I am estimating this model: But I want to do some analysis of the variables before. In particular, I am interested in fitting some ARIMA models. First, I am doing it for the inflation rate in Mexico. ...
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1answer
37 views

How can I solve this quadratic problem?

For φ2> 0, show that the inequalities for $φ_1$ and $φ_2$ which ensure the roots of $φ(z) = 1 − φ_1(z) − φ_2(z^2)$ are greater than $1$ are given by: $φ_1 + φ_2 < 1$, $φ_2 − φ_1 < 1$, and $|...
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1answer
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Representing the “surprisingness” or “acyclicness” of events given binary time-series

I might just be missing a term or a concept, as Googling things like "measure acyclic events" mostly brings up non-relevant results (like DAGs). I'd be happy to continue researching on my own given ...
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Online estimation of drifting discrete probability

I recently come across (in a practical setting) to the following problem. Suppose I receive items from a finite set ,one at a time . At each moment one item is drawn independently from an unknown ...
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Find likelihood for time series

There is time series defined with the equation (ARIMAX model) $$X_t - 1.5X_{t-1} + 0.7X_{t-2} = u_{t-1} + 0.5u_{t-2} + \epsilon_{t} - \epsilon_{t-1} + 0.2\epsilon_{t-2}$$ where $\epsilon_{t}$ is ...
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Difficulty solving a system of differential equations

I have difficulties solving a system of differential equations. I am in the process of estimating several arbitrage-free Nelson-Siegel functions and I want to investigate whether it is possible to ...
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1answer
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How to detect inflection points without function expression in a live time series scenario?

Most of the learnings and examples out there talks about finding inflection point given there is a function expression. What happens when I do-not know the function expression? The movement of the ...
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1answer
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Wavelets for signal modulation

How does the Continuous Wavelet Transform handle signal modulation? for instance if an external 8-year period influenced the amplitude of an annual period. i.e. every 8 years the amplitude of the ...
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Question on Levinson's proposed discrete form of Wiener filter: the stationarity assumption

The whole foundation of Levinson's discrete version of Wiener filter is based on the assumption of stationarity of a time series, and aims to predict a value based on the past observed values. Now, if ...
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Reduced order model from time series to generate a new time series

I am working on simulation and created a model made of many equations. The simulation takes very long time to simulate and make results. So my aim is I want to be able to speed up the process using ...
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What is an unconditional model for a time series variable?

If I am being asked to do an unconditional analysis of a time series variable, lets say GDP starting in 2000, what model am I supposed to estimate?
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Decrease Beta bewteen to variables

I have two time series of return data. One is global equity and one is a seperate stock. I would like to know if there is a theoretical way of decreasing the EQ Beta of the stock? To be more clear: ...
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What's the entropy of a time-series?

Entropy is defined as $H(X)=\sum\limits_{x \in X} p(x)log p(x)$ and is usually used for measuring the uncertainty of a system. I wonder if the entropy concept can be applied to time series ? If so ...
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Method for deriving constrained displacement field from noisy and incomplete multitemporal point ovservations

I have some incomplete and noisy point observations (x,y coordinate) of particles moving in a spatially and temporally dynamic flow in 2D space. The observations are at uneven time steps, and points ...
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80 views

Create a new vector from a constant

I have pair of decimal values with the corresponding monotonically increasing vectors (integer). ...
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55 views

Is an AR(p)-process a martingale?

Is an AR(p)-process a martingale? I think it is not, but I don't know how to explain this. The expected value of the martingale must be zero. In the case of an AR(p)-process it isn't but in the case ...
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A quantification of variation for a time series, aggregated over various time intervals.

This question has been bugging me for quite a while, and I did not manage to find an answer online. I have a time series of integer flow data (a flow of cyclists, per second, average 0.3 bicycles/s), ...
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1answer
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Word for cyclic, non-periodic function

The additive decomposition of a time series can be written as $$ Y_t = T_t + C_t + S_t + I_t $$ where $T_t$ is the trend component, $C_t$ is the cyclic component, $S_t$ is the seasonal component, ...
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Velocity and acceleration of currency

I have been reading this Thesis. And I faced some difficulties to understand the features engineering section. The author is calculating some features from a time series of transactions. Some of this ...
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Stationary solution of AR(p) [closed]

If $W_t$ is a white noise then, how can I show that $X_t-\phi_1X_{t-1} - \phi_2X_{t-2} - … - \phi_pX_{t-p} = W_t$ has a stationary solution when $\phi(z) = 1-\phi_1z - \phi_2z^2 - … - \phi_pz^p \neq ...
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Filtered Historical Simulation FHS for VAR

If I wish to run a FHS for VaR model, first I estimate the GARCH model on the historical returns $r_{t}$, then I obtain the historical innovation time series as $z_{t}=\frac{r_{t}}{\sigma_{t}}$, where ...
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ARMA-GARCH versus GARCH, questions about skewness and kurtosis.

I am currently researching the following statement: The ARMA-GARCH model better captures the skewness and leptokurtosis of financial time series than a GARCH model would. What I know so far: For a ...
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35 views

explanation of beta random variable

I've been reading a paper for self study on the beta-egarch model here: https://core.ac.uk/download/pdf/42337476.pdf One of the things I don't understand is how the following: $$ b_{t}=\frac{(y_{t}-\...
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Is there a closed form solution to compute expectation of this time series / stochastic process?

$ \textrm{Suppose you have a time series } Y_{t} \textrm{ which moves via normal random variable }\boldsymbol{l}_{t}\\\\ y_{t+1} = \left\{\begin{matrix} 0, y_{t}\leq 0 \\ \max[0, y_{t}\cdot (1+r)+...