subsampling and block - based bootstrap methods have been widely used in inference problems for time series ; see politis et al .( 1999a ) and lahiri ( 2003 ) for book - length treatments of these important resampling methods . to accommodate the unknown time series dependence nonparametrically, these methods introduce a bandwidth parameter , such as the block size in the block - based bootstrap and the subsampling window width in subsampling .the bandwidth plays an important role in the finite sample performance of subsampling or block bootstrap based inference .intuitively , if the bandwidth ( or block size ) is too small , it may not capture the dependence in a time series sufficiently , whereas if it is too large , the number of blocks for subsampling / resampling is too small to lead to a good approximation of finite sample distribution .statistically speaking , the bandwidth is a smoothing parameter as it usually leads to a bias - variance tradeoff in variance estimation or size - power tradeoff in testing on the basis of subsampling and block bootstrap . in the traditional asymptotic theory, goes to infinity as sample size goes to infinity and the fraction goes to zero , which is a necessary condition for the general consistency of subsampling and block - based bootstrap methods without additional assumptions .therefore , the role of ( or ) does not show up in the conventional first order asymptotics , although in practice the choice of does affect the subsampling / block bootstrap distribution estimator and related operating characteristics . in this paper, we aim to offer a new perspective on the use of these smoothing parameter dependent resampling methods based on the so - called fixed- approach , which was first proposed by kiefer and vogelsang ( 2005 ) in the context of heteroscedasticity - autocorrelation robust ( har ) testing .it was found that the asymptotic distribution obtained under the fixed- framework ( i.e.  ] ) under the small- asymptotics , but is dependent upon and differs from under the fixed- asymptotics . for a scalar parameter, we calibrate the nominal coverage level on the basis of the pivotal limiting null distribution of the p - value under the fixed- framework , and modify the small- based confidence interval by inverting the corresponding test .thus the impact of the bandwidth parameter on the subsampling / block bootstrap distribution approximation is captured to the first order using a p - value based adjustment .simulation studies are conducted to demonstrate that the fixed- approach delivers confidence intervals of better coverage in most situations and that the fixed- based intervals are slightly wider than the small- counterparts , consistent with early findings associated with the fixed - b approach ; see e.g. kiefer and vogelsang ( 2005 ) .so far the use of the fixed- approach has been restricted to the inference of a finite dimensional parameter . since the subsampling and moving block bootstrap have also been used in the inference of infinite dimensional parameters , such as marginal distribution function and ( normalized )spectral distribution function of a stationary time series , we explore an extension of the fixed- idea to construct confidence bands for these infinite dimensional parameters .unlike the case of a scalar parameter , the limiting null distribution of the subsampling - based p - value is not pivotal under the fixed- asymptotics and it depends on the unknown dependence structure of the underlying process .to alleviate the problem , we apply the subsampling method to approximate the sampling distribution of the p - value so inference becomes feasible .this double subsampling approach is also used in constructing the confidence region for a vector parameter .the remainder of the article is organized as follows .section [ sec : mean ] introduces an extension of the fixed- approach to subsampling and the moving block bootstrap in the mean case .section [ sec : general ] lays out a general framework and describes the fixed- based confidence interval ( region ) for a finite dimensional parameter and some implementational issues .section [ sec : infinite ] presents an extension of the fixed- approach to confidence band construction for the marginal distribution function and normalized spectral distribution function .simulation results are reported in section [ sec : simulation ] .section [ sec : conclusion ] concludes and section [ sec : appendix ] contains some technical details .to help the reader understand the essence of the fixed- approach , we shall focus on the simple problem : inference for the mean of a stationary time series .suppose we want to test versus based on the observations from a univariate stationary time series with . under suitable moment and weak dependence conditions , we have where is the long run variance with and `` '' denotes convergence in distribution. the scale parameter can be consistently estimated by the so - called lag window estimator , where is a bandwidth parameter , is a kernel function and is the sample autocovariance at lag .a natural test statistic is . to ensure the consistency of as an estimator of , the bandwidth parameter , where  ] . in the testing context , if the alternative hypothesis is , then we reject the null hypothesis at the significance level if ; if the alternative hypothesis is , then the null is rejected provided that  ] .the above inference is based on the traditional small- based asymptotic theory , under which , where is the distribution function for the standard normal distribution , and , i.e. , the subsampling method provides a consistent approximation to the sampling distribution of and its limiting distribution .note that we are using the data - centered subsampling distribution for testing as recommended by berg et al .( 2010 ) .  under the fixed- framework , does not converge to in distribution .instead , lahiri ( 2001 ) showed that the limit of is dt,\ ] ] where is a standard brownian motion . since converges to a random measure, the subsampling - based inference is asymptotically invalid under the fixed- framework . to alleviate the problem, we shall modify the traditional subsampling - based inference procedure by considering the subsampling - based p - value and its limiting null distribution .for the one - sided alternative hypothesis , we define the p - value as under the small- asymptotics , it can be shown that converges to  ] and another one - sided confidence interval  ] under the small- asymptotic theory .the p - value is defined as under the fixed- asymptotics , the limiting null distribution of is the distribution of , where let denote the quantile of the distribution . then the fixed- based symmetric confidence interval is , i.e. , .\end{aligned}\ ] ]   in this subsection , we shall consider the approximation of the sampling distribution of by the moving block bootstrap ( mbb ) .denote the mbb sample by with the dependence on the block size being explicit .then we approximate the sampling distribution of by the conditional distribution of or  ] , where .let .the corresponding fixed- based two sided equal tailed confidence interval for is then \ ] ] and the one - sided confidence intervals can be formed analogous to those developed for the subsampling method .the details are omitted .the above discussion is based on the assumption that is an integer .when is not an integer , we use a fraction of the last resampled block to make the bootstrap sample size equal to original sample size. then the p - value is and its limiting null distribution can be derived similarly .below we shall focus our discussion on the case is an integer for simplicity .  in a similar fashion , if we want to construct an mbb - based symmetric confidence interval for , we consider the approximation of the sampling distribution of by the conditional distribution of given the data .the corresponding p - value is and it converges in distribution to under the null .define and .then the fixed- based symmetric confidence interval for is  ] be the space of functions on  ] or more generally in the -valued function space  ] , \times [ 0,\pi]) ] .   following politis et al .( 1999a ) , we assume that the parameter of interest is , where is the joint probability law that governs the -dimensional stationary sequence .let be an estimator of based on the observations .further we define the subsampling estimator of by on the basis of the subsample , .let be a norm in .the subsampling - based distribution estimator of is denoted as . in the testing context( say versus ) , we define the subsampling based p - value as where we do not distinguish and for notational convenience because they are the same under the null .  to obtain the limiting null distribution of the p - value under the fixed- framework, we further assume , where is the marginal distribution of , and is a functional that takes value in .then a natural estimator of is , where is the empirical distribution . here stands for the point mass at .similarly , , where .assume that there is an expansion of in the neighborhood of , i.e. , where is the influence function of ( hampel , ronchetti , rousseeuw and stahel , 1986 ) defined by and is the remainder term .similarly , .below are the two key assumptions we need .( a.1 ) assume that and , where is a positive definite matrix and denotes the -th dimensional vector of independent brownian motions .( a.2 ) assume that and .note that ( a.1 ) is assumption 1 in shao ( 2010a ) and its verification has been discussed in remark 1 therein .the assumption ( a.2 ) is to ensure the negligibility of remainder terms . in the samplemean case , and the remainder terms vanish , so ( a.2 ) is automatically satisfied and ( a.1 ) reduces to a functional central limit theorem for the partial sum process of .[ th : finite ] suppose the assumptions ( a.1 ) and ( a.2 ) hold and  ].  please insert table [ tb : cv ] about here !  for a vector parameter ( i.e. ) ,the limiting null distribution of the p - value depends on the unknown long run variance matrix , so is not pivotal in general .one way out is to approximate the limiting null distribution by subsampling .denote by the subsampling width at the second stage .let and .for each subsample , we define the subsampling counterpart of as for .denote the empirical distribution function of by , which can be used to approximate the sampling distribution or the limiting null distribution of .let .then the calibrated subsampling - based confidence region for is whereas the traditional subsampling - based confidence region is .[ th : theoremregion ] assume that and  ] in section [ subsec : mdf ] and  ] is a two - parameter mean zero gaussian process with where . to construct a confidence band for , we note that by the continuous mapping theorem , ( [ eq : weakconv ] ) implies that since is a gaussian process with mean zero and unknown covariance , direct inference of is difficult . to circumvent the difficulty , both the moving block bootstrap and subsampling have been proposed to approximate the limiting distribution consistently ; see bhlmann ( 1994 ) , naik - nimbalkar and rajarshi ( 1994 ) , and politis et al .( 1999b ) .below we shall focus our discussion on the subsampling method , and a similar argument applies to the moving block bootstrap approach in view of the argument in section [ subsec : mbb ] .let , be the subsampling counterpart of , where .assuming and other regularity conditions , politis et al .( 1999b ) showed that the subsampling approximation based on is consistent in certain function space .this implies that the sampling distribution of ( or the distribution of ) can be consistently approximated by the empirical distribution of .the above result is obtained under the small- asymptotics . to introduce our calibration method, we again start with the p - value and study its limiting null distribution under the fixed- asymptotics . for notational simplicity, we do not distinguish the true marginal distribution function and the hypothesized function , because they are identical under the null hypothesis .define the p - value let . under the fixed- asymptotics ,the limiting null distribution of the p - value is the distribution of , where note that the distribution of is not pivotal for a given , because it depends upon the gaussian process , whose covariance structure is tied to the unknown dependence structure of .so subsampling at the first stage is insufficient under the fixed- asymptotic framework .it is worth noting that in the iid setting , the quantity is pivotal provided that is continuous , so the inferential difficulty is mainly caused by the presence of unknown weak dependence .  to make the inference feasible, we propose to approximate the sampling distribution of the p - value or its limiting null distribution by subsampling ; see section [ subsec : subsampling ] . let be the subsampling window size at the second stage , and . for each subsample , the subsampling counterpart of is defined as for .then we can approximate the sampling distribution of or its limit null distribution by the empirical distribution associated with , denoted as .for a given , the traditional subsampling - based confidence band for is and the calibrated confidence band is where .the following theorem states the consistency of subsampling at the second stage , which implies that the coverage for the calibrated confidence band is asymptotically correct .let   [ th : theorem2 ] assume that , ( [ eq : weakconv ] ) and  ] .( b ) suppose that the process is -mixing , is a continuous random variable and for every  ] , where is the spectral density function of .let be the periodogram .a commonly used estimator for is or its discretized version .it has been shown that the two versions are asymptotically equivalent [ dahlhaus ( 1985a ) ] and we shall use the discrete version for the computational convenience . under certain regularity conditions , we have that , where is a mean zero gaussian process with covariance here is the fourth order cumulant spectrum . for various sets of conditions for this weak convergence to hold ,see brillinger ( 1975 ) , dahlhaus ( 1985b ) and anderson ( 1993 ) . applying the continuous mapping theorem, we get } |g(\lambda)|.\end{aligned}\ ] ] since the covariance of depends on unknown second order and fourth order spectrum , the distribution of }|g(\lambda)| ] , which does not depend on the marginal variance of , is of more practical relevance .politis et al . ( 1999b ) mentioned that the subsampling method is still consistent in the approximation of the sampling distribution or the limiting distribution of , where .  to introduce our calibration method, we need to define the estimate of based on the subsample for .in particular , we define the periodogram on the basis of the subsample as , where , , and .the subsampling method approximates the sampling distribution of by the empirical distribution generated from , and the corresponding p - value is under the small - b asymptotics , the limiting null distribution of is , but under the fixed- asymptotics , the limiting null distribution is expected to depend on and the intricate second and fourth order dependence structure of .the derivation of the limiting distribution of the p - value relies on the functional central limit theorem for , \times [ 0,\pi] ] is held fixed and as .in this section , we conduct simulation studies to evaluate the accuracy of the asymptotic approximations provided by both small- and fixed- approaches to the finite sample distribution .specifically , we examine the empirical coverage probabilities and the volumes of confidence sets to see if the fixed- approach corresponds to smaller coverage errors .   in this subsection, we consider a univariate stationary time series model with various types of dependence structure . to be specific , we let we consider ( i ) ar(1)- error : , and ; ( ii ) ma(1)- error : ; and their corresponding ar(1)-exp(1 ) and ma(1)-exp(1 ) models , where has mean zero , unit variance but with an asymmetric distribution .following the suggestion of a referee , we also include two nonlinear time series models : nonlinear 1 , , where .this model was used in the simulation work of paparoditis and politis ( 2001 ) and shao ( 2010c ) ; nonlinear 2 ( threshold autoregressive model of order 1 ) , , where .sample size and the number of bootstrap replications is .the bandwidth parameter varies from to , i.e. .we examine the empirical coverages and average widths of symmetric confidence intervals for and the trimmed mean based on 10000 replications .nominal coverage is set to be .for the models with normally distributed errors , the results for the mean case are depicted in figure [ fig : mean1 ] , in which the left panel shows the empirical coverages and the right panel shows the corresponding ratios of average interval widths ( fixed- over small- ) .the symbols ss " and bb " stand for subsampling and the moving block bootstrap , respectively .for both subsampling and the moving block bootstrap , the undercoverage occurs and it gets more severe as the dependence strengthens .the empirical coverages for the fixed- approach are closer to the nominal level than those for the small- approach , with the difference between two empirical coverages increasing as gets large . on the other hand ,the fixed- based interval is slightly wider than its small- counterpart , with the ratio of widths increasing with respect to in general .these findings are consistent with the intuition that the larger is , the more accurate the fixed- based approximation provides relative to its small- counterpart .the intervals constructed based on the moving block bootstrap have noticeably better coverage than the ones based on subsampling , especially for large . for the ma(1 ) model with , there is an overcoverage for the fixed- based interval , which is usually slightly more conservative than the small- counterpart .the overcoverage in the case of negatively correlated time series corresponds to the underrejection for kiefer and vogelsang s ( 2005 ) studentized statistic when using normal approximation ( i.e. small- approach ) and is small ( see figure 1 therein ) , so our results are in a sense consistent with those in kiefer and vogelsang ( 2005 ) . practically speaking ,the overcoverage is less harmful to the practitioner than the undercoverage , so are less concerned in practice .the results for the models with exponentially distributed errors as presented in figure [ fig : meanexp1 ] are very similar to the ones for normally distributed errors , suggesting that the asymmetric shape of exponentially distributed errors has little impact on the finite sample performance .figures [ fig : trmean1 ] and [ fig : trmeanexp1 ] present the results for the trimmed mean case , which are fairly similar to the results in the mean case .additionally , the results for the nonlinear models in figure [ fig : nonlinear ] resemble those for ar(1)-n(0,1 ) models with in both the mean and the trimmed mean case , indicating that nonlinearity does not affect our results much .  due to the duality of confidence interval construction and hypothesis testing, we would expect that the fixed - b approach leads to better size ( i.e. size closer to the nominal one ) in all the models except ma(1 ) with , at the sacrifice of ( raw ) power .the power loss is expected to be moderate because the ratio of fixed- based interval width over the small- based interval width is quite close to .overall , the simulation results demonstrate that the fixed- approach delivers more accurate inference for both subsampling and the moving block bootstrap in most situations owing to its more accurate approximation to the finite sample distribution .of course , we only show the improved accuracy of the fixed- approximation for a specific , which is also what kiefer and vogelsang did .we also tried and qualitatively similar results are obtained .it would be interesting to provide some theoretical justifications on the order of the error rejection probability .for the subsampling method , this boils down to the order of } |p(\widetilde{pval}_{n , l}^{sub}\le \alpha)-p(\widetilde{g}(l / n)\le \alpha)| ] which is expected to be larger . a formal theoretical proof seems quite challenging and is left for future research .   in this subsection, we examine the coverage probabilities of confidence regions for the vector parameter of mean and median , and confidence bands for the marginal distribution function and the normalized spectral distribution function .let be generated from the ar model : , where , or .sample size and number of replications is 1000 .we use the euclidean norm in the confidence region construction . for both confidence regions and confidence bands , we compared the following three schemes : ( 1 ) traditional subsampling - based confidence region ( band ) ; ( 2 ) calibrated subsampling - based confidence region ( band ) with a fixed , where for confidence region construction and for confidence band construction ; ( 3 ) calibrated subsampling - based confidence region ( band ) with chosen in a data driven fashion . herewe employ a variant of a block size selection procedure proposed in bickel and sakov ( 2008 ) for the out of bootstrap ( also see gtze and ra ( 2001 ) ) , which is closely related to the subsampling method .the use of bickel and sakov s automatic bandwidth selection in the subsampling context has been explored in jach et al .( 2011 ) recently .the procedure consists of the following steps ( in the confidence band case ) :  1 .  for a predetermined interval ] .then . let for any .then as . on , we have that .\ ] ] so the expression is bounded by ,\ ] ] which , by the continuous mapping theorem , converges in distribution to , where .\ ] ] it is not hard to see that for each  ] , the fact ( [ eq : fact1 ] ) can be easily shown by noticing that the joint distribution of is multivariate normal with a positive definite covariance matrix .so holds and the use of the continuous mapping theorem is justified .the proof is thus complete .proof of theorem [ th : theorem2 ] : ( a ) the proof follows from the use of the continuous mapping theorem . here the mapping \times [ 0,1])\rightarrow { { \mathbb{r}}} ] , then the mapping is -continuous , i.e. , the probability that the gaussian process falls into the discontinuity set of is zero .this completes the proof .\(b ) in view of the continuity assumption of and the monotonicity of , it suffices to show for each .let for and .note that for any , let and then as .on , we have that for each , , where by stationarity , we have that the above convergence follows from theorem 3 of ferguson ( 1996 ) and the fact that , where again the continuous mapping theorem is invoked to derive the weak convergence of and following the argument in the proof of theorem [ th : finite ] , its use can be justified under the assumption that for each  ] .thus for large enough . furthermore , by the markov inequality , . using the same argument in the proof of theorem 3.2.1 . of politis et al .( 1999a ) , we can show that , which follows from the stationarity and strong mixing properties of and the boundness of .the conclusion then follows from an elementary argument .2.3em1 gtze , f. and ra , a. ( 2001 ) adaptive choice of bootstrap sample sizes . in _ state of the art in probability and statistics _ , ims lecture notes monogr .aad van der vaart mathisca de gunst chris klaassen ) , pp .286 - 309 .cambridge university press .2.3em1 hashimzade , n. and vogelsang , t. j. ( 2008 ) fixed - b asymptotic approximation of the sampling behavior of nonparametric spectral density estimators ._ journal of time series analysis _, 29 , 142 - 162 .2.3em1 liu , r. y. and singh , k. ( 1992 ) . movingblocks jackknife and bootstrap capture weak dependence . in_ exploring the limits of bootstrap _ , ( ed .r. lepage and l.billard ) , 225 - 248 .john wiley , new york .2.3em1 politis , d. n. and romano , j. p. ( 1992 ) a circular block - resampling procedure for stationary data ._ exploring the limits of bootstrap _ , ( raoul lepage and lynne billard , eds . ) , john wiley , p263 - 270 .2.3em1 politis , d. n. , romano , j. p. and you , l. ( 1993 ) uniform confidence bands for the spectrum based on subsamples , in computing science and statistics , proceedings of the 25th symposium on the interface , san diego , california .( m. tarter and m. lock , eds . ) , the interface foundation of north america , 346 - 351 .2.3em1 politis , d. n. , romano , j. p. and wolf , m. ( 1999b) weak convergence of dependent empirical measures with application to subsampling in function spaces ._ journal of statistical planning and inference _ , 79 , 179 - 190 .2.3em1 sayginsoy , o. and vogelsang , t. j. ( 2011 ) testing for a shift in trend at an unknown date : a fixed - b analysis of heteroskedasticity autocorrelation robust ols based tests ._ econometric theory _ , 27 , 992 - 1025 .2.3em1 shao , x. ( 2010a ) a self - normalized approach to confidence interval construction in time series ._ journal of the royal statistical society , series b _ , 72(3 ) , 343 - 366 .corrigendum : 2010 , 72(5 ) , 695 - 696 .2.3em1 shao , x. and politis , d. ( 2011 ) fixed- subsampling and block bootstrap : improved confidence sets based on p - value calibration .technical report , department of statistics , university of illinois at urbana - champaign .available at arxiv   .simulated values of , , and when fitted with a quadratic polynomial , $ ] .the simulated values are based on and 50000 replications . in the moving block bootstrap case ,we use bootstrap replications . [ cols="^,^,^,^,^",options="header " , ]   over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 . ,title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 . ,title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ] [ fig : mean1 ]   over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 . ,title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ] [ fig : meanexp1 ]   over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 . , title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with normally distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ] [ fig : trmean1 ]   over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 . , title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 . ,title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the trimmed mean and for the models with exponentially distributed errors .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ] [ fig : trmeanexp1 ]   over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models . sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models .sample size and number of replications is 10000 . , title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the mean ( top two plots ) and the trimmed mean ( bottom two plots ) for the two nonlinear models .sample size and number of replications is 10000 ., title="fig:",width=170,height=302 ] [ fig : nonlinear ]   over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 . , title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with normally distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ] [ fig : region ]   over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 . ,title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the vector parameter and for the models with exponentially distributed errors .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ] [ fig : regionexp ]   over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the marginal distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ] [ fig : band1 ]   over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 . ,title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ]  over traditional small- ) ( right panel ) for the normalized spectral distribution function .sample size and number of replications is 1000 ., title="fig:",width=170,height=302 ] [ fig : band2 ]

Abstract: subsampling and block - based bootstrap methods have been used in a wide range of inference problems for time series . to accommodate the dependence , these resampling methods involve a bandwidth parameter , such as subsampling window width and block size in the block - based bootstrap . in empirical work , using different bandwidth parameters could lead to different inference results , but the traditional first order asymptotic theory does not capture the choice of the bandwidth . in this article , we propose to adopt the fixed- approach , as advocated by kiefer and vogelsang ( 2005 ) in the heteroscedasticity - autocorrelation robust testing context , to account for the influence of the bandwidth on the inference . under the fixed- asymptotic framework , we derive the asymptotic null distribution of the p - values for subsampling and the moving block bootstrap , and further propose a calibration of the traditional small- based confidence intervals ( regions , bands ) and tests . our treatment is fairly general as it includes both finite dimensional parameters and infinite dimensional parameters , such as marginal distribution function and normalized spectral distribution function . simulation results show that the fixed- approach is more accurate than the traditional small- approach in terms of approximating the finite sample distribution , and that the calibrated confidence sets tend to have smaller coverage errors than the uncalibrated counterparts .   _ keywords _ : block bootstrap , calibration , iterative bootstrap , prepivoting , subsampling .   fixed- subsampling and block bootstrap : improved confidence sets based on p - value calibration  by xiaofeng shao and dimitris n. politis


process mining is a recently emerging discipline that links data analysis with process management .one of the central tasks of process mining is * process discovery * , where knowledge is extracted from event logs ( readily available in today s information systems ) in order to discover real processes .several tools , techniques , and algorithms have been proposed to discover business process models from event logs .the quality of discovered models and the efficiency of discovery techniques vary according to several criteria including : fitness , precision , generalization and simplicity .this document is our submission to the process discovery contest organized by the ieee cis task force on process mining , and which is collocated with the bpm conference in rio de janeiro in september 2016 .the contest is dedicated to the assessment of tools and techniques that discover business process models from event logs .the objective is to compare the efficiency of techniques to discover process models that provide a proper balance between overfitting " and underfitting " .a process model is overfitting ( the event log ) if it is too restrictive , disallowing behavior which is part of the underlying process .conversely , it is underfitting ( the reality ) if it is not restrictive enough , allowing behavior which is not part of the underlying process . within the process - discovery contest ,10 event logs are provided to the contestants as a training set .these event logs are generated from business process models that show different behavioral characteristics .the process models are not disclosed to the contestants .the contestants should use the training event logs to discover process models that are the closest to the original process models , in term of balancing between overfitting " and underfitting " . to assess this balance, the contest has a classification perspective where a test event log will be used .the test event log contains traces representing real process behavior and traces representing behavior not related to the process .a model is as good in balancing overfitting " and underfitting " as it is able to correctly classify the traces in the test " event log :  * given a trace representing real process behavior , the model should classify it as allowed . *given a trace representing a behavior not related to the process , the model should classify it as disallowed .  with a classification view, the winner is the contestant who can classify correct the largest number of traces in all the test event logs .moreover , in order to help the contestants tune their discovered models , two validation " sets are made available in april and may 2016 .  in our submission to the contest , we discovered process models using two techniques : inductive miner and decomposition based discovery .we chose these techniques because they guarantee rediscoverability ; that is , models discovered using those techniques _( generate ) all the traces in their original training event logs .we used prom tool ( http://www.promtools.org/ ) to apply those techniques in discovering our submitted models .  in the reminder of this report, we first show in section [ sec:2 ] how do we use two discovery techniques , namely : _ inductive miner _ and _ decomposition _ , to discover process models from the training set using prom tool .second , we present in section [ sec:3 ] how we _ replay _ an event log on a process model . then , we show how we use replay results :  1 .  to check the rediscoverability of models ( section [ sec:31 ] ) , and 2 .  to classify unseen traces ( in validation and/or test logs ) as fitting or not ( section [ sec : class ] )section [ sec : discussion ] is dedicated for a discussion about the classification results of april and may validation logs , the complexity of discovered models , and how these two points affect our selection of models for submission . in the end of the report , we show the pictures of our submitted models .we used two well known discovery techniques : inductive miner and decomposed process mining .we used those techniques to discover process models for each one of the 10 training event logs .however , since only one process model should be submitted for every event log , we decided to include in our submission a mix of those models .rationally , we have chosen those techniques because they guarantee a level of _ * rediscoverability * _ , namely the ability of discovering again the models used to generate the traces of the event log .we consider that * rediscoverability * property is essential to accomplish the task of correctly classify new unseen traces .  the models for processes 1 , 2 , 4 , 8 and 9 are discovered using * inductive miner*. we used prom lite for this group of models .other models - for processes 3 , 5 , 6 , 7 , and 10 - are discovered using * decomposition * technique .we used prom 6.5.1 for this group of models .the justification of this selection of process models is given in section [ sec : discussion ] .  in the reminder of this section, we describe the step - by - step procedure to discover process models using the above mentioned techniques.   inductive miner is a discovery approach to construct a process tree for a given log .the main advantages of this approach are :  * all discovered models correspond to sound , block - structured workflow ( wf ) net systems .* the model always fits , i.e. , the model can generate the traces in the log .the approach works recursively with _ divide and conquer _ strategy : split log , construct part of process tree , and proceed with handling split parts of log separately .a process tree can be directly transformed into a petri net . however , prom has a plugin that can mine a petri net directly using inductive miner technique .  to discover a model ( petri net ) from an event log with inductive miner technique , we used prom lite and followed the following steps :  1 .first , the log ( in * xes * format ) is loaded onto prom and used .then , in _ actions_ window , the plugin * mine petri net with inductive miner * " by leemans is selected ( see figure [ fig : discv_induc_1 ] ) .a settings page appears then ( figure [ fig : discv_induc_2 ] ) . for * variant * , we select _ inductive miner_. for * event classifier * , we select _ event name_. +  3 .  finally , we get a petri net discovered using inductive miner .the discovered model can be visualized using _graphviz petri net visualization _( figure [ fig : discv_induc_3 ] ) . +   decomposition is a process mining technique that discovers an accepting petri net from a log using decomposition .first , the log will be split into a number of sublogs .second , a subnet will be discovered ( using the given discovery algorithm ) for every sublog .third , these subnets will be merged into a single accepting petri net .fourth , known reduction techniques will be applied on this accepting petri net , and the result is returned .  to discover a model from an event log using decomposition technique , we used prom 6.5.1 ( this technique is not available in prom lite ) .  1 .  first, the log ( in * xes * format ) is loaded onto prom and used .then , in _ actions _ window , the plugin * discovery using decomposition * " is selected ( see figure [ fig : discv_decomp_1 ] ) .actually , there are two available plugins in prom 6.5.1 for discovery using decomposition : an * interactive * plugin that allows for custom settings , and a * batch * one that uses some _ default _ settings .we used the * batch * plugin , hence default settings are used ; for instance , the underlying discovery algorithm is ilp miner .we get an accepting petri net discovered using decomposition .it can be visualized using different visualizations such as _ visualize accepting petri net ( dot ) _( figure [ fig : discv_decomp_2 ] ) .+  3 .  finally, we convert the discovered accepting petri net into a ` regular ' petri net using * unpack accepting petri net * plugin .our models are all regular petri nets that are exported as pnml files .to replay an event log on a process model , we used the prom plugin * replay a log on petri net for conformance analysis * ( available in both prom lite and prom 6.5.1 , with identical results obtained ) .we need * replay * for two tasks :  1 .verify that the discovered model can generate its originating event log ( rediscoverability ) .2 .  classify new unseen traces ( in validation and test datasets ) as fitting or not - fitting with respect to the discovered models .  to replay a log on a process model :  1 .first , in _ workspace _ page , both the log and the model ( petri net ) are selected and used .2 .  second , in _ actions _ page , the plugin * replay a log on petri net for conformance analysis * is selected ( figure [ fig : replay_1 ] ) .then , a dialog appears that allow for mapping between the transitions of the petri net and the event classes of the log ( figure [ fig : replay_2 ] ) .we need only to choose * event name * as classifier ; thus each transition is matched with the event class having the same name .the next dialog allows us to select a replay algorithm .we use the following settings ( as shown in figure [ fig : replay_4 ] ) : * * the purpose of replay * measuring fitness . * * penalize improper completion * : yes . * * suggested algorithm * a * cost - based fitness express , assuming at most 32767 tokens in each place .the last page of the dialog is for selecting the costs of particular deviation ( figure [ fig : replay_5 ] ) .again , we keep the defaults : the cost is always 1 except for _ tau _ transitions where the cost is 0. +   those are the required steps to replay a log on a process model .as mentioned above , we use those steps for rediscoverability checking and for classifying unseen traces .our choice of discovery techniques is mainly based on the rediscoverability property .that is , we chose inductive miner and decomposition techniques because they both guarantee this property .therefore , once we discover a model using one of those techniques , as discussed in section [ sec:2 ] , we replay the originating / training event log on the discovered model to make sure that all the traces in the log * fit * properly in the model .  to do so, we follow the replay steps mentioned above using the training log and the discovered model .the result of the replay can be visualized in different ways .for instance , using * model projected with alignments * choice , the model can be visualized with alignments on top of it , as shown in figure [ fig : replay_6 ] .    in this visualization, we can take a look at the * _ global statistics _ * within the inspector window . among the displayed statistics ,we are mainly interested in three statistics : * trace * fitness , * move - model * fitness , and * move - log * fitness .the values of those statistics give an indicator about the rediscoverability property .when those global statistics are all equal to 1 , this means that all the traces fit in the model ; thus the model is able to rediscover the event log .otherwise , if anyone of those statistics has a value less than 1 , then there are some traces in the log that do nt fit in the model .  in our submission , all the models perfectly satisfy the rediscoverability property .   in the context of the process discovery contest, process discovery is turned into a classification task with a training set and a test set .moreover , two validation sets are made available to the contestants to allow them tune their discovered models .a process model needs to decide whether traces are fitting or not .  for this classification task, a test or validation log is replayed on the discovered model .again , we used the prom plugin * replay a log on petri net for conformance analysis * according to the steps mentioned earlier .when the replay results are obtained , we visualize them using * project alignment to log * choice .this will display log - model alignments for each case / trace , as shown in figure [ fig : replay_results ] .for each trace , a table of statistics is shown to the left ; and a colored picture of the trace alignment is shown to the right .the meaning of colors is illustrated in a legend shown on the right - top of the screen ( figure [ fig : legend ] ) .  among the displayed statistics , we are interested in : trace fitness , move - log fitness , and move - model fitness ; and we use them in the classification task .we classify each trace as _ fitting _ if , and only if , those three values are perfect ( equal to 1 ) .otherwise , when anyone of those values is less than 1 , then we classify the trace as _ not - fitting_.  for example , table [ tab : replay1 ] shows those values for replaying log-1 of april validation dataset over our model-1 ( discovered using inductive miner ) .1|l * 20|y| case i d & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 + move - model & 1 & 1 & .73 & 1 & .85 & .64 & .83 & 1 & .75 & .92 & 1 & .67 & .82 & 1 & .92 & 1 & 1 & .79 & 1 & 1 + trace & 1 & 1 & .73 & 1 & .87 & .67 & .83 & 1 & .74 & .79 & 1 & .65 & .82 & 1 & .88 & 1 & 1 & .63 & 1 & 1 + move - log & 1 & 1 & .73 & 1 & .92 & .70 & .83 & 1 & .75 & .71 & 1 & .67 & .82 & 1 & .85 & 1 & 1 & .58 & 1 & 1 + classification & & & & & & & & & & & & & & & & & & & & +  in the same sense , the colored picture of the trace alignment can be also used to determine fitness .if the picture consists of * green * and * gray * alignments * only * , then the trace is classified as _fitting_. otherwise ,if the picture also contains other colors ( like pink and/or yellow ) , then the trace is classified as _ not - fitting_.in our solution to the process discovery contest , we used inductive miner and decomposition techniques - as presented in section [ sec:2 ] - to discover process models from the 10 _ training _ logs .both groups of models are then used to replay again the training logs - as presented in section [ sec:31]- to ensure _rediscoverability_. we found that all our process models _ satisfy the rediscoverability _ property .then , we replayed april and may _ validation _ event logs over both groups of process models - processes discovered using inductive miner and processes discovered using decomposition .the replay results are then used to classify the traces of the validation logs as fitting or non - fitting - as presented in section [ sec : class ] .we communicated the obtained classifications with the contest organizers , who replied by indicating how many traces have been correctly classified for each one of the 10 logs .table [ tab : april_results ] shows the classification results of april validation logs using the two groups of process models : inductive miner and decomposition .analogue classification results of may logs are shown in table [ tab : may_results ] .l3.4cm|yyyyyyyyyy|y log & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & +  + true positive & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & + false positive & 0 & 0 & 8 & 0 & 1 & 3 & 0 & 5 & 0 & 0 & + true negative & 10 & 10 & 2 & 10 & 9 & 7 & 10 & 5 & 10 & 10 & + false negative & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & + correctly classified & 20 & 20 & 12 & 20 & 19 & 17 & 20 & 15 & 20 & 20 & * 183 * +  + true positive & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & + false positive & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 6 & 0 & 0 & + true negative & 10 & 10 & 9 & 10 & 10 & 9 & 10 & 4 & 10 & 10 & + false negative & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & + correctly classified & 20 & 20 & 19 & 20 & 20 & 19 & 20 & 14 & 20 & 20 & * 192 * +   l3.4cm|yyyyyyyyyy|y log & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & +  + true positive & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & + false positive & 0 & 0 & 9 & 0 & 1 & 4 & 3 & 2 & 0 & 2 & + true negative & 10 & 10 & 1 & 10 & 9 & 6 & 7 & 8 & 10 & 8 & + false negative & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & + correctly classified & 20 & 20 & 11 & 20 & 19 & 16 & 17 & 18 & 20 & 18 & * 179 * +  + true positive & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & + false positive & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 4 & 0 & 0 & + true negative & 10 & 10 & 10 & 10 & 9 & 10 & 9 & 6 & 10 & 10 & + false negative & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & + correctly classified & 20 & 20 & 20 & 20 & 19 & 20 & 19 & 16 & 20 & 20 & * 194 * +  a quick look at these classification results tells that decomposition models classify better than inductive miner models .that is , out of 200 traces in april logs , 192 are correctly classified using decomposition models versus 183 using inductive miner models .similarly , in may logs , 194 are correctly classified using decomposition models versus 179 using inductive miner models  however , with a more careful examination of the classification results , we made the following observations :  * logs 1 , 2 , 4 , and 9 are always classified correctly : 20 out of 20 traces are classified correctly .* log 8 has a better classification using inductive miner model than using decomposition model ( april : , may : ) . * other logs - namely 3 , 5 , 6 , 7 , and 10 - has a better classification using decomposition models than using inductive miner models .now , based on these observations , we should decide which models shall we include in our submission to the contest . from the second and third observation , we can easily decide to consider inductive miner model for log : 8 , and decomposition models for logs : 3 , 5 , 6 , 7 and 10 .however , the first observation tells us that neither one of the two groups of process models has a preference over the other with respect to the classification of logs : 1 , 2 , 4 and 9 .therefore , we have to consider another quality criteria of process models , namely : _simplicity_.  actually , the inductive miner models are generally more _ simple _ than decomposition ones .one can clearly see this by looking at the pictures of both groups of models ( see appendix [ apx : models ] ) .however , in order to prove this claim we computed different complexity metrics of process models . table[ tab : metrics ] shows the complexity metrics for both inductive miner and decomposition process models .these metrics are : _ ecam _ : extended cardoso metric , _ ecym _ : extended cyclomatic metric , and : number of edges and nodes in the reachability graph , , and : number of arcs , places , and transitions in the net , and _ sm _ : structuredness metric .  based on the values listed in table [ tab : metrics ]we can see that the inductive miner models are , in general , _ simpler _ than decomposition ones .for instance , the average structuredness of decomposition models is roughly one order of magnitude higher than for inductive miner models .  c|c|c|c|cc|ccc|c model & density & _ ecam _ & _ ecym _ & & & & & & _ sm _ +  + 1 & 0.0387 & 35 & 177 & 185 & 78 & 72 & 31 & 30 & 293 + 2 & 0.0589 & 20 & 67 & 85 & 27 & 70 & 18 & 33 & 6180 + 3 & 0.0795 & 18 & 51 & 65 & 20 & 62 & 13 & 30 & 1550 + 4 & 0.0376 & 31 & 174 & 174 & 77 & 70 & 31 & 30 & 212 + 5 & 0.0499 & 24 & 215 & 247 & 50 & 90 & 22 & 41 & 29205 + 6 & 0.1038 & 12 & 23 & 31 & 10 & 54 & 10 & 26 & + 7 & 0.0586 & 22 & 3730 & 4082 & 482 & 82 & 20 & 35 & 4160 + 8 & 0.0536 & 24 & 159 & 183 & 56 & 72 & 21 & 32 & 740 + 9 & 0.034 & 35 & 142 & 142 & 72 & 74 & 34 & 32 & 242 + 10 & 0.0322 & 45 & 9418 & 10001 & 1852 & 98 & 39 & 39 & 16435 + avg . &0.0547 & 26.6 & 1415.6 & 1519.5 & 272.4 & 74.4 & 23.9 & 32.8 & 6557.44 +  + 1 & 0.0722 & 39 & 129 & 133 & 57 & 78 & 27 & 20 & 32000 + 2 & 0.2325 & 120 & && & 266 & 26 & 22 & 72270 + 3 & 0.2167 & 156 & && & 312 & 30 & 24 & 93240 + 4 & 0.0745 & 40 & 862 & 900 & 282 & 93 & 26 & 24 & 79200 + 5 & 0.088 & 80 & && & 169 & 32 & 30 & 48600 + 6 & 0.262 & 133 & && & 283 & 27 & 20 & 23700 + 7 & 0.0821 & 58 & 9501 & 10003 & 2140 & 128 & 26 & 30 & 65700 + 8 & 0.5132 & 149 & && & 349 & 17 & 20 & 31300 + 9 & 0.0741 & 41 & 403 & 418 & 154 & 88 & 27 & 22 & 13455 + 10 & 0.1705 & 94 & && & 180 & 24 & 22 & 89100 + avg . &0.1786 & 91 & 2723.75 & 2863.5 & 658.25 & 194.6 & 26.2 & 23.4 & 54856.5 +  now , given that for logs : 1 , 2 , 4 and 9 , there is no preference of a particular group of models based on classification results ; and given that inductive miner models are simpler than decomposition models , we decided to consider inductive miner models for these logs , based on _occam s razor _ principle ,  in summary , our submitted models to the contest are : models 1 , 2 , 4 , 8 and 9 of inductive miner models , and models 3 , 5 , 6 , 7 and 10 of decomposition models .the pictures of these process models are depicted in appendix [ apx : models ]  based on this selection of process models , the contest organizers replayed the _ june _ test logs .the reported classifications are shown in table [ tab : juneresults ] , whereas the classification results are shown in table [ tab : juneresults2 ] .our selection of models correctly classified 192 out of 200 traces .the organizers confirmed later that this submission gave us the * second * place in the contest among 14 participant teams .l1cm|yy|yyyyyyyyyyyyyyyyyyyy & & & + log & + & - & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 + 1 & 10 & 10 & & - & + & + & - & + & - & - & + & + & - & + & + & - & + & - & + & - & - & - + 2 & 12 & 8 & - & - & + & + & - & + & + & + & - & + & + & - & + & + & - & - & + & + & + & - + 3 & 9 & 11 & + & - & + & - & - & - & + & + & + & + & + & - & - & - & - & - & + & - & + & - + 4 & 10 & 10 & - & + & - & + & + & + & - & + & - & - & + & + & - & - & - & + & + & - & + & - + 5 & 10 & 10 & - & + & - & - & - & - & - & - & + & - & + & + & + & - & + & + & + & + & - & + + 6 & 12 & 8 & - & + & - & + & + & + & + & + & - & - & - & + & + & - & + & - & + & + & + & - + 7 & 10 & 10 & + & + & - & - & - & + & - & + & + & + & - & + & - & - & - & - & + & + & + & - + 8 & 13 & 7 & + & - & + & + & + & - & + & + & + & + & + & + & - & + & - & + & - & - & + & - + 9 & 10 & 10 & + & + & + & + & + & - & - & + & - & + & - & + & - & - & - & + & - & - & - & + + 10 & 10 & 10 & + & + & - & - & - & + & - & + & + & + & + & - & + & + & - & - & + & - & - & - +   l3.4cm|yyyyyyyyyy|y log & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & + true positive & 10 & 10 & 9 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & + false positive & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 3 & 0 & 0 & + true negative & 10 & 8 & 10 & 10 & 10 & 8 & 10 & 7 & 10 & 10 & + false negative & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & + correctly classified & 20 & 18 & 19 & 20 & 20 & 18 & 20 & 17 & 20 & 20 & * 192 * +  5  carmona , j. , de leoni , m. , depair , b. , jouck , t. : process discovery contest @ bpm 2016 , https://www.win.tue.nl/ieeetfpm/doku.php?id=shared:process_discovery_contest  van der aalst , w.m.p . : .process mining : discovery , conformance and enhancement of business processes .springer - verlag , berlin , 2011 .leemans , s. , fahland , d. , van der aalst , w. : discovering block - structured process models from event logs - a constructive approach . in : petri nets 2013 .lncs , vol . 7927 , pp .springer , 2013 .hompes , b.f.a . : on decomposed process discovery , master thesis .eindhoven , 2014 .verbeek , h.m.w ., van der aalst , w.m.p . :decomposed process mining : the ilp case . in : bpi 2014 workshop , haifa , israel , 2014 .verbeek , h.m.w . : decomposed process mining with divideandconquer .proceedings of the bpm demo sessions 2014 , 15 , 2014 .werf , j.m.e.m.v.d . ,dongen , b.f.v . ,hurkens , c.a.j . ,serebrenik , a. : process discovery using integer linear programming .fundamenta informaticae , volume 94 issue 3 - 4 , pp . 387412 , 2009 .lassen , k.b . , van der aalst , w.m.p . : complexity metrics for workflow nets .information and software technology , volume 51 , issue 3 , pp . 610626 , 2009 .blum , f. rojas : metrics in process discovery .technical report tr / dcc-2015 - 6 , computer science department , university of chile , 2015 .

Abstract: this report presents a submission to the process discovery contest . the contest is dedicated to the assessment of tools and techniques that discover business process models from event logs . the objective is to compare the efficiency of techniques to discover process models that provide a proper balance between overfitting " and underfitting " . in the context of the process discovery contest , process discovery is turned into a classification task with a training set and a test set ; where a process model needs to decide whether traces are fitting or not .  in this report , we first show how we use two discovery techniques , namely : _ inductive miner _ and _ decomposition _ , to discover process models from the training set using prom tool . second , we show how we use replay results to 1 ) check the rediscoverability of models , and to 2 ) classify unseen traces ( in test logs ) as fitting or not . then , we discuss the classification results of validation logs , and the complexity of discovered models , and their impact on the selection of models for submission . the report ends with the pictures of the submitted process models .