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The piecewise monotonic approximation problem makes the least changes to n univariate noisy data so that the piecewise linear interpolant to the new values is composed of at most k monotonic sections. The term "least changes" is defined in the sense of a global sum of strictly convex functions of changes. The main difficulty in this calculation is that the extrema of the interpolant have to be found automatically, but the number of all possible combinations of extrema can be O (n k - 1) , which makes not practicable to test each one separately. It is known that the case k = 1 is straightforward, and that the case k > 1 reduces to partitioning the data into at most k disjoint sets of adjacent data and solving a k = 1 problem for each set. Some ordering relations of the extrema are studied that establish three quite efficient algorithms by using a binary search method for partitioning the data. In the least squares case the total work is only O (n σ + k σ log 2 σ) computer operations when k ≥ 3 and is only O (n) when k = 1 or 2, where σ - 2 is the number of sign changes in the sequence of the first differences of the data. Fortran software has been written for this case and the numerical results indicate superior performance to existing algorithms. Some examples with real data illustrate the method. Many applications of the method arise from bioinformatics, energy, geophysics, medical imaging, and peak finding in spectroscopy, for instance. [ABSTRACT FROM AUTHOR]

There are many methods used in resolving data clustering. One of them is the Fuzzy C-Means (FCM) method, which is a reliable method to solve clustering problems in the East Java region. This study aims to determine the optimum cluster in the East Java region which can help the government to identify problems and assist policymaking in regencies/cities in East Java province. The research process uses data from the central statistical agency, namely the unemployment rate and poverty rate from 2010 to 2015. In this study, the Davies Bouldin Index (DBI) is used as a cluster validation test for determining the optimum cluster. Unemployment rate and poverty rate data were analyzed using RStudio. From the calculation of the FCM method and also the determination of the optimum cluster results obtained in 2 clusters with a DBI value of 1.2759, 3 clusters with a DBI value of 0.9937, 4 clusters with a DBI value of 0.8737. The optimum cluster is in 4 clusters with a minimum DBI value.

High survey nonresponse in unemployment duration studies may have a strong effect on inference if the so called causal mechanism is present. A robust method of testing the causal nonresponse is proposed for data sets where survey information can be combined with complete administrative records. It is assumed that population distribution follows approximately the Cox regression model. Formal justification of the method and a comparative simulation study are included. [ABSTRACT FROM AUTHOR]

. Leading economic indicators are often used to anticipate changes in key economic variables. Understanding the dynamics of these indicators is of primary interest for policy-making objectives and for sustainable economic welfare. We are concerned with the problem of setting a dynamic threshold above which the value of leading indicators would be considered as extreme. We propose a dynamic threshold modelling approach based on fractionally integrated processes where a semiparametric method is used to determine the amount of differencing that is required to obtain a weakly stationary process-to which standard methods of statistics of extremes apply. Given that our approach is linked to the Box-Jenkins method, we refer to the procedure proposed and applied herein as the Box-Jenkins-Pareto procedure. We use our approach to analyse the weekly number of unemployment insurance claims in the USA and explore the connection between its threshold exceedances and the US business cycle. [ABSTRACT FROM AUTHOR]

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