Change-point Problem and Regression: An Annotated Bibliography

The problems of identifying changes at unknown times and of estimating the location of changes in stochastic processes are referred to as the change-point problem or, in the Eastern literature, as disorder . The change-point problem, first introduced in the quality control context, has since developed into a fundamental problem in the areas of statistical control theory, stationarity of a stochastic process, estimation of the current position of a time series, testing and estimation of change in the patterns of a regression model, and most recently in the comparison and matching of DNA sequences in microarray data analysis. Numerous methodological approaches have been implemented in examining change-point models. Maximum-likelihood estimation, Bayesian estimation, isotonic regression, piecewise regression, quasi-likelihood and non-parametric regression are among the methods which have been applied to resolving challenges in change-point problems. Grid-searching approaches have also been used to examine the change-point problem. Statistical analysis of change-point problems depends on the method of data collection. If the data collection is ongoing until some random time, then the appropriate statistical procedure is called sequential. If, however, a large finite set of data is collected with the purpose of determining if at least one change-point occurred, then this may be referred to as non-sequential. Not surprisingly, both the former and the latter have a rich literature with much of the earlier work focusing on sequential methods inspired by applications in quality control for industrial processes. In the regression literature, the change-point model is also referred to as two- or multiple-phase regression, switching regression, segmented regression, two-stage least squares (Shaban, 1980), or broken-line regression. The area of the change-point problem has been the subject of intensive research in the past half-century. The subject has evolved considerably and found applications in many different areas. It seems rather impossible to summarize all of the research carried out over the past 50 years on the change-point problem. We have therefore confined ourselves to those articles on change-point problems which pertain to regression. The important branch of sequential procedures in change-point problems has been left out entirely. We refer the readers to the seminal review papers by Lai (1995, 2001). The so called structural change models, which occupy a considerable portion of the research in the area of change-point, particularly among econometricians, have not been fully considered. We refer the reader to Perron (2005) for an updated review in this area. Articles on change-point in time series are considered only if the methodologies presented in the paper pertain to regression analysis

Evaluating the mobility and safety benefits of adaptive signal control technology (ASCT)

The Adaptive Signal Control Technology (ASCT) is a traffic management strategy that optimizes signal timing based on real-time traffic demand. This thesis proposes a comprehensive methodology of quantifying the mobility and safety benefits of the ASCT deployed in the state of Florida. A Bayesian switch-point regression model was proposed to evaluate the mobility benefits of ASCT. The analysis was based on a 3.3-mile corridor along Mayport Road from Atlantic Boulevard to Wonderwood Drive in Jacksonville, Florida. The proposed analysis was used to estimate the possible dates that separate the two operating characteristics, i.e., with and without ASCT. Also, the posterior estimated distributions were used for the Bayesian hypothesis test to investigate if there is a significant difference in the operating characteristics for two scenarios - with and without ASCT. The results revealed that ASCT increases travel speeds by 4% in typical days of the week (Tuesday, Wednesday and Thursday) in the northbound direction. However, the implementation of ASCT did not yield a significant increase in travel speed in the southbound direction. In addition, ASCT exhibited more benefits in AM peak in the northbound direction indicating a 7% increase in travel speeds. A Bayesian hypothesis test revealed that there is a significant difference in the operating characteristics between scenarios with and without ASCT. Moreover, an observational before-after Empirical Bayes (EB) with a comparison-group approach was adopted to develop the Crash Modification Factors (CMFs) for certain crash types (total and rear-end crashes) and crash severity levels (fatalities and injury crashes). The CMFs developed were used to quantify the safety benefits of the ASCT. The analysis was based on 42 treatment intersections with ASCT and their corresponding 47 comparison intersections without ASCT. Florida-specific Safety Performance Functions (SPFs) for total and rear-end crashes and for fatal plus injury crashes were also developed. The deployment of ASCT was found to reduce total crashes and rear-end crashes by 5.2% (CMF = 0.948) and 10.6% (CMF = 0.894), respectively. On the other hand, fatal plus injury crashes and PDO crashes were reduced by 6.1% (CMF = 0.939) and 5.4% (CMF = 0.946), respectively, after the ASCT deployment. The CMFs for total crashes and rear-end crashes, and for fatal plus injury crashes and PDO crashes were found to be statistically significant at 95% confidence level. These findings provide researchers and practitioners with an effective means for quantifying the mobility and safety benefits of ASCT, economic appraisal of the ASCT as well as a key consideration to transportation agencies for future ASCT deployment in the state

Coherent frequentism

By representing the range of fair betting odds according to a pair of confidence set estimators, dual probability measures on parameter space called frequentist posteriors secure the coherence of subjective inference without any prior distribution. The closure of the set of expected losses corresponding to the dual frequentist posteriors constrains decisions without arbitrarily forcing optimization under all circumstances. This decision theory reduces to those that maximize expected utility when the pair of frequentist posteriors is induced by an exact or approximate confidence set estimator or when an automatic reduction rule is applied to the pair. In such cases, the resulting frequentist posterior is coherent in the sense that, as a probability distribution of the parameter of interest, it satisfies the axioms of the decision-theoretic and logic-theoretic systems typically cited in support of the Bayesian posterior. Unlike the p-value, the confidence level of an interval hypothesis derived from such a measure is suitable as an estimator of the indicator of hypothesis truth since it converges in sample-space probability to 1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly extended to vector parameters of interest. The derivation of upper and lower confidence levels from valid and nonconservative set estimators is formalize

Generalized shrinkage F-like statistics for testing an interaction term in gene expression analysis in the presence of heteroscedasticity

<p>Abstract</p> <p>Background</p> <p>Many analyses of gene expression data involve hypothesis tests of an interaction term between two fixed effects, typically tested using a residual variance. In expression studies, the issue of variance heteroscedasticity has received much attention, and previous work has focused on either between-gene or within-gene heteroscedasticity. However, in a single experiment, heteroscedasticity may exist both within and between genes. Here we develop flexible shrinkage error estimators considering both between-gene and within-gene heteroscedasticity and use them to construct <it>F</it>-like test statistics for testing interactions, with cutoff values obtained by permutation. These permutation tests are complicated, and several permutation tests are investigated here.</p> <p>Results</p> <p>Our proposed test statistics are compared with other existing shrinkage-type test statistics through extensive simulation studies and a real data example. The results show that the choice of permutation procedures has dramatically more influence on detection power than the choice of <it>F </it>or <it>F</it>-like test statistics. When both types of gene heteroscedasticity exist, our proposed test statistics can control preselected type-I errors and are more powerful. Raw data permutation is not valid in this setting. Whether unrestricted or restricted residual permutation should be used depends on the specific type of test statistic.</p> <p>Conclusions</p> <p>The <it>F</it>-like test statistic that uses the proposed flexible shrinkage error estimator considering both types of gene heteroscedasticity and unrestricted residual permutation can provide a statistically valid and powerful test. Therefore, we recommended that it should always applied in the analysis of real gene expression data analysis to test an interaction term.</p