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PurposeThe paper aims to identify the effect of ignorance of correlatedness among process observations and to implement new sampling schemes; skip and mixed sampling, in order to reduce the effect of autocorrelation on process capability index (PCI) Cpm.Design/methodology/approachAutocorrelated observations are generated using autoregressive process of order two (AR (2)) using Monte Carlo simulations. The PCI is computed based on these observations assuming the independence. The skip and mixed sampling schemes are then used to form sub-groups among correlated observations. The PCI obtained using sub-groups from skip and mixed sampling schemes are assessed using sample mean and sample standard deviation.FindingsThe paper provides empirical insights into how the effect of autocorrelation decreases in the estimated value of PCI Cpm. The use of new sampling schemes, skip and mixed sampling, reduces the effect of autocorrelation on estimates of PCI Cpm.Originality/valueThis paper fulfills an identified need to study how to reduce the effect of autocorrelation on PCI Cpm.

We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\\ge0}$ with parameters $a_1,\\ldots,a_p \\in \\mathbb{R}$ , $p\\in\\mathbb{N}$ , and $\\lambda \\ge 0$ , such that, for all $n\\ge p$ , conditioned on $X_0,\\ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \\cdots + a_p X_{n-p} + \\lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \\lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.

Although the impact of serial correlation (autocorrelation) in residuals of general linear models for fMRI time-series has been studied extensively, the effect of autocorrelation on functional connectivity studies has been largely neglected until recently. Some recent studies based on results from economics have questioned the conventional estimation of functional connectivity and argue that not correcting for autocorrelation in fMRI time-series results in “spurious” correlation coefficients. In this paper, first we assess the effect of autocorrelation on Pearson correlation coefficient through theoretical approximation and simulation. Then we present this effect on real fMRI data. To our knowledge this is the first work comprehensively investigating the effect of autocorrelation on functional connectivity estimates. Our results show that although FC values are altered, even following correction for autocorrelation, results of hypothesis testing on FC values remain very similar to those before correction. In real data we show this is true for main effects and also for group difference testing between healthy controls and schizophrenia patients. We further discuss model order selection in the context of autoregressive processes, effects of frequency filtering and propose a preprocessing pipeline for connectivity studies. •The effect of autocorrelation on functional connectivity (FC) has been investigated.•We study autocorrelation in theory, simulations and real fMRI data.•Model order selection and proper preprocessing for FC studies have been discussed.

The natural generalization of the XLindley distribution is proposed. The mathematical properties of the generalized XLindley distribution are derived. The importance of the proposed model is evaluated on the first-order autoregressive process, and compared with its counterparts. Extensive simulation studies are carried out to demonstrate the suitability of the estimation methods. Empirical findings reveal that the first-order autoregressive process with generalized XLindley innovations produces better forecasting results than those of the gamma, weighted Lindley, and normal innovations. Additionally, a web-tool application of the proposed model is developed and deployed on a free server that is accessible for practitioners.

Let $\\{X_n\\}_{n\\in{\\mathbb{N}}}$ be an ${\\mathbb{X}}$ -valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \\in {\\mathbb{X}}$ and for $n\\geq 1$ , $X_n\\;:\\!=\\;F(X_{n-1},\\vartheta_n)$ , where $\\{\\vartheta_n\\}_{n \\ge 1}$ are independent and identically distributed random variables with common probability distribution $\\mathfrak{p}$ , $F(\\cdot,\\cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of $\\{\\vartheta_n\\}_{n \\ge 1}$ . Under parametric perturbation of both F and $\\mathfrak{p}$ , we are interested in the robustness of the V-geometrical ergodicity property of $\\{X_n\\}_{n\\in{\\mathbb{N}}}$ , of its invariant probability measure, and finally of the probability distribution of $X_n$ . Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

In this study, we apply the Fredholm-type integral equation method to derive the explicit formulas of the average run length (ARL) for an autoregressive moving average process with explanatory variables (ARMAX(p,q,r)) with exponential white noise running on a modified exponentially weighted moving average (EWMA) control chart. As a performance measure, we compared the computational times of calculating the ARL based on explicit formulas and the classical numerical integral equation (NIE) method. We found that although the ARLs using both methods were very close with an absolute percentage difference of less than 0.00001%, their calculational times were less than 0.01 and 10 seconds, respectively. Furthermore, the comparison of the performances of the ARL methods for ARMAX(p,q,r) processes with exponential white noise by practical application for time series data comprising exchange rates and the price of energy running on modified and standard EWMA and cumulative sum (CUSUM) control charts using the relative mean index (RMI) criteria. The results show that the explicit formulas method for the ARL of the process on the modified EWMA control chart is more powerful than the CUSUM and standard EWMA control charts.

Some large language models (LLMs) are open source and are therefore fully open for scientific study. However, many LLMs are proprietary, and their internals are hidden, which hinders the ability of the research community to study their behavior under controlled conditions. For instance, the token input embedding specifies an internal vector representation of each token used by the model. If the token input embedding is hidden, latent semantic information about the set of tokens is unavailable to researchers. This article presents a general and flexible method for prompting an LLM to reveal its token input embedding, even if this information is not published with the model. Moreover, this article provides strong theoretical justification—a mathematical proof for generic LLMs—for why this method should be expected to work. If the LLM can be prompted systematically and certain benign conditions about the quantity of data collected from the responses are met, the topology of the token embedding is recovered. With this method in hand, we demonstrate its effectiveness by recovering the token subspace of the Llemma-7BLLM. We demonstrate the flexibility of this method by performing the recovery at three different times, each using the same algorithm applied to different information collected from the responses. While the prompting can be a performance bottleneck depending on the size and complexity of the LLM, the recovery runs within a few hours on a typical workstation. The results of this paper apply not only to LLMs but also to general nonlinear autoregressive processes.

The EUREF Permanent GNSS Network (EPN) provides a unique atmospheric dataset over Europe in the form of Zenith Total Delay (ZTD) time series. These ZTD time series are estimated independently by different analysis centers, but a combined solution is also provided. Previous studies showed that changes in the processing strategy do not affect trends and seasonal amplitudes. However, its effect on the temporal and spatial variations of the stochastic component of ZTD time series has not yet been investigated. This study analyses the temporal and spatial correlations of the ZTD residuals obtained from four different datasets: one solution provided by ASI (Agenzia Spaziale Italiana Centro di Geodesia Spaziale, Italy), two solutions provided by GOP (Geodetic Observatory Pecny, Czech Republic), and one combined solution resulting from the EPN’s second reprocessing campaign. We find that the ZTD residuals obtained from the three individual solutions can be modeled using a first-order autoregressive stochastic process, which is less significant and must be completed by an additional white noise process in the combined solution. Although the combination procedure changes the temporal correlation in the ZTD residuals, it neither affects its spatial correlation structure nor its time-variability, for which an annual modulation is observed for stations up to 1,000 km apart. The main spatial patterns in the ZTD residuals also remain identical. Finally, we compare two GOP solutions, one of which only differs in the modeling of non-tidal atmospheric loading at the observation level, and conclude that its modeling has a negligible effect on ZTD values.

The Markov property is shared by several popular models for time series such as autoregressive or integer-valued autoregressive processes as well as integer-valued ARCH processes. A natural assumption which is fulfilled by corresponding parametric versions of these models is that the random variable at time  t gets stochastically greater conditioned on the past, as the value of the random variable at time  t-1increases. Then the associated family of conditional distribution functions has a certain monotonicity property which allows us to employ a nonparametric antitonic estimator. This estimator does not involve any tuning parameter which controls the degree of smoothing and is therefore easy to apply. Nevertheless, it is shown that it attains a rate of convergence which is known to be optimal in similar cases. This estimator forms the basis for a new method of bootstrapping Markov chains which inherits the properties of simplicity and consistency from the underlying estimator of the conditional distribution function.