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Background and Exploration in Time Series
Published in Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos, Statistical and Econometric Methods for Transportation Data Analysis, 2020
Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos
A first step in investigating stationarity is to use either the ACF for the series (Figure 7.7), or to use a formal statistic to examine whether autocorrelations of Xt are jointly zero. The Ljung–Box (1978) statistic is used for testing the null hypothesis H0:ρ1=…=ρm=0against the alternative that H1:ρi≠0 for some i∈{1,…,m} as follows (the Ljung–Box statistic is a modification of the well-known Portmanteau test to increase the power of the test in finite samples; Tsay, 2002): () Q(m)=T(T+2)∑L=1mρL2T−L
A Monitoring and Diagnostic Approach for Stochastic Textured Surfaces
Published in Technometrics, 2018
A B-P (aka portmanteau) test (Box and Pierce 1970) is widely used for testing the existence of autocorrelations in time series. Likewise, a B-P type statistic can be used to detect spatial correlations in our stochastic textured surface images. Because local defects in the stochastic textured surfaces are likely to result in local spatial correlations in the residuals, the B-P type statistic is intuitively appealing for our objective. We define the B-P type SMS for the ith pixel as where is some local estimate of the covariance between the residual ri at the ith pixel and another residual rk within the moving window of n pixels surrounding the ith pixel (e.g., the moving window in Figure 3). Note that is included in Ti in (5). To estimate , we use a kernel weighted window centered at the ith pixel. For ease of illustration, let i1 and i2 be the row and column indices of the ith pixel, and let k1 and k2 be the row and column indices of the kth pixel. Then, where K(h, m) is the Epanechnikov quadratic kernel:
A multivariate statistical representation of railway track irregularities using ARMA models
Published in Vehicle System Dynamics, 2022
J. N. Costa, J. Ambrósio, D. Frey, A. R. Andrade
VARMA models are also estimated to check whether adding an MA component improves the fit of the model. The order of a VARMA model can be estimated with a multivariate Portmanteau test statistic of a transformed process [34] and summarised on a two-way table of p-values [34]. The intuition behind the test is that the cross-correlation of a VMA process is zero after lag q. Thus, the iterated VAR is transformed into a VMA model, and its cross-correlation evaluated. The null hypothesis of the Portmanteau test is that the correlation matrices of the transformed process are zero, and consequently, rejecting the null hypothesis identifies the order q. Setting a maximum order for p and q and evaluating the test statistic for every combination results in a table of p-values. The order q is identified in the table by looking for the upper left entry above a certain significance level. Note that the explanation of the test has been oversimplified and is only meant to help the reader navigate the results of the two-way table. The reader is referred to [34] for more details on the iterated VAR, extended correlation matrix, the Portmanteau test statistic, and the two-way table. The order of VARMA models is estimated for = 20 and = 5 using the MTS package [37], and the results are summarised in Table 1. The table suggests that the VARMA(7,1) or the VARMA(5,2) are suitable to model the track irregularities. Another interesting note is that, unlike the information criteria that favoured higher-order models, this test suggests that a VAR(7) might also be a suitable candidate for modelling the irregularities. The VARMA models are estimated using the marima package [38], a fast estimation method for VARMA models [39]. Insignificant parameters at the 0.05 significance level are removed using stepwise linear regression.