many real - world brain - computer interface ( bci ) applications rely on single - trial classification of event - related potentials ( erps ) in eeg signals .for example , in a rapid serial visual presentation ( rsvp ) based bci system , a sequence of images are shown to the subject rapidly ( e.g. 2 - 10 hz ) , and the subject needs to detect some target images in them .the target images are much less frequent than the non - target ones , so that they can elicit p300 erps in the oddball paradigm .the p300 erps can be detected by a bci system , and the corresponding images are then triaged for further inspection .research has shown that these bci systems enable the subject to detect targets in large aerial photographs faster and more accurately than traditional standard searches .unfortunately , because different subjects have different neural responses to even the same visual stimulus , it is very difficult , if not impossible , to build a generic erp classifier whose parameters fit all subjects .so , we need to calibrate the classifier for each individual subject , using some labeled subject - specific data . reducing this calibration effort ,i.e. , minimizing the amount of labeled subject - specific data required in the calibration , would greatly increase the utility of the bci system .this is the research problem tackled in this paper .more specifically , we distinguish between two types of calibration in bci :  1 .  _ offline calibration _ , in which a pool of unlabeled eeg epochs have been obtained_ a priori _ , and a subject is queried to label some of these epochs , which are then used to train a classifier to label the remaining epochs in that pool .a potential application of offline calibration is personalized automatic game highlight detection , e.g. , a subject s eeg signals are recorded continuously while watching a football game ; after the game , the subject manually labels a few highlights , which are then used to train an erp classifier to detect more highlights . 2 ._ online calibration _ ,in which some labeled eeg epochs are obtained on - the - fly , and then a classifier is trained from them to classify future eeg epochs .a potential application of online calibration is the afore - mentioned rsvp image tagging problem : at the beginning of the task , the subject is asked to explicitly indicate it ( e.g. , press a button ) every time he / she detects a target image , and that information is used to train a p300 erp classifier . after a certain number of calibration epochs ,the performance of the classifier can become reliable enough so that it can label further images using eeg epochs only .one major difference between offline calibration and online calibration is that , in offline calibration , the unlabeled eeg epochs can be used to help design the erp classifier , whereas in online calibration there are no unlabeled eeg epochs .additionally , in offline calibration we can query any epoch in the pool for the label ( an optimal query strategy can hence be designed by using machine learning methods such as active learning ) , but in online calibration usually the sequence of the epochs is pre - determined and the subject has no control over which epoch he / she will see next .many signal processing and machine learning approaches have been proposed to reduce the bci calibration effort .they may be grouped into five categories :  1 .  _regularization _ , which is a very effective machine learning approach for constructing robust models , especially when the training data size is small . a popular regularization approach in bci calibration is shrinkage , which gives a regularized estimate of the covariance matrices ._ transfer / multi - task learning _ , which uses relevant data from other subjects to help the current subject .the transfer learning ( tl ) based approaches are particularly popular , because in many bci applications we can easily find legacy data from the same subject in the same task or similar tasks , or legacy data from different subjects in the same task or similar tasks . these data , which will be called auxiliary data in this paper , can be used to improve the learning performance of a new subject , or for a new task .adaptive learning _ , which refines the machine learning model as new ( labeled or unlabeled ) subject - specific data are available .the main approach in this category is semi - supervised learning , which is often used for offline bci calibration where unlabeled data are available .semi - supervised learning first constructs an initial model from the labeled training data and then applies it to the unlabeled test data .the newly labeled test data are then integrated with the groundtruth training data to retrain the model , and hence to improve it iteratively .active learning _ , which optimally selects the most informative unlabeled samples to label .there are many criteria to determine which unlabeled samples are the most informative .the most popular , and probably also the simplest , approach for classification is to select the samples that are closest to the current decision boundary , because the classifier is most uncertain about them .active learning has been mainly used for offline bci calibration , where unlabeled samples are available . however, a closely related technique , active class selection , can be used for online bci calibration .its idea is to optimize the classes from which the new training samples are generated ._ _ a priori _ physiological information _ , which can be used to construct the most useful eeg features . for example , prior information on which eeg channels are the most likely to be useful was used in as a regularizer to optimize spatial filters , and beamforming has been used in to find relevant features from prior regions of interest to reduce the calibration data requirement .interestingly , these five categories of approaches are not mutually exclusive : in fact they can be freely combined to further reduce the amount of subject - specific calibration data .an optimal spatial filters was designed in for efficient subject - to - subject transfer by combining regularization and _ a prior _ information on which channels are the most likely to be useful .a collaborative filtering approach was developed in , which combined tl and active class selection to minimize the online calibration effort .an active tl approach was proposed in for offline calibration , which combined tl and active learning to minimize the offline calibration effort .an active weighted adaptation regularization approach , which combines active learning , tl , regularization and semi - supervised learning , was proposed in to facilitate the switching between different eeg headsets .all these approaches are for bci classification problems .however , recently researchers have also started to apply these techniques for regression problems in bci calibration .for example , a domain adaptation with model fusion approach , which combines regularization and tl , was developed in to estimate driver s drowsiness online continuously .this paper presents a comprehensive overview and comparison of the offline and online weighted adaptation regularization with source domain selection ( warsds ) algorithms , which we proposed recently .the offline warsds algorithm , which combined tl , regularization , and semi - supervised learning , was first developed in for offline single - trial classification of erps in a visually - evoked potential ( vep ) oddball task .it was later extended to online calibration in a rsvp task , which still includes tl and regularization but not semi - supervised learning because unlabeled samples are not available in online calibration . in this paperwe use a vep oddball task and three different eeg headsets to show that they have consistently good performance across subjects and headsets .we also compare the performances of the offline and online warsds algorithms in identical experimental settings to investigate the effect of semi - supervised learning , and show that it can indeed help improve the calibration performance .the remainder of the paper is organized as follows : section [ sect : warsds ] introduces the details of the offline warsds algorithm .section [ sect : owarsds ] introduces the online warsds ( owarsds ) algorithm .section [ sect : experiments ] describes the experiment setup that is used to evaluate the performances of different algorithms .section [ sect : offline ] presents performance comparison of different offline calibration algorithms .section [ sect : online ] presents performance comparison of different online calibration algorithms .section [ sect : onoff ] compares the performances of offline and online algorithms .finally , section [ sect : conclusions ] draws conclusions .this section describes the offline warsds algorithm , which originates from the adaptation regularization regularized least squares ( arrls ) algorithm in .we made several major enhancements to arrls to better handle class - imbalance and multiple source domains , and also to make use of labeled samples in the target domain .warsds first uses source domain selection ( sds ) to select the closest source domains for a given target domain , then uses weighted adaptation regularization ( war ) for each selected source domain to build individual classifiers , and finally performs model fusion . for simplicity , we only consider 2-class classification .a _ domain _  in tl consists of a multi - dimensional feature space and a marginal probability distribution , i.e. , , where .two domains and are different if , and/or .  a _ task _  in tl consists of a label space and a conditional probability distribution . two tasks and are different if , or .  given a source domain with labeled samples , , and a target domain with labeled samples and unlabeled samples , _ domain adaptation _ ( da ) tl learns a target prediction function with low expected error on , under the assumptions , , , and .  for example , in single - trial classification of veps , eeg epochs from the new subject are in the target domain , while eeg epochs from an existing subject ( usually different from the new subject ) are in the source domain . when there are multiple source domains , we perform da for each of them separately and then aggregate the classifiers .a sample consists of the feature vector for an eeg epoch from a subject in either domain , collected as a response to a specific visual stimulus .though usually the source and target domains employ the same feature extraction method , generally their marginal and conditional probability distributions are different , i.e. , and , because the two subjects usually have different neural responses to the same visual stimulus . as a result, the auxiliary data from a source domain can not represent the primary data in the target domain accurately , and must be integrated with some labeled target domain data to induce an accurate target domain classifier .because to use the source domain data in the target domain , we need to make sure is close to , is close to , and is also close to .however , in this paper we focus only on the first two requirements by assuming all subjects conduct similar vep tasks [ so and are intrinsically close ]. our future research will consider the more general case that and are different .let the classifier be , where is the classifier parameters , and is the feature mapping function that projects the original feature vector to a hilbert space . as in , the learning framework of war is formulated as : \label{eq : f}\end{aligned}\ ] ] where is the loss function , is the kernel matrix with , and and are non - negative regularization parameters . is the overall weight for target domain samples , which should be larger than 1 so that more emphasis is given to target domain samples than source domain samples .however , eventually the learned classifier will be applied to the target domain .so , the target domain should be emphasized .we choose so that the labeled target domain samples are less overwhelmed by the source domain samples . ] . and are the weight for the sample in the source domain and target domain , respectively , i.e. , in which is the set of samples in class of the source domain , is the set of samples in class of the target domain , is the number of elements in , and is the number of elements in .the goal of ( ) is to balance the number of samples from difference classes in the source ( target ) domain .this is very important , because class imbalance is intrinsic to many applications , particularly bci applications . in many casesthe minority class is the one of particular interest ( e.g. , the vep experiment presented in this paper ) , but it can be easily overwhelmed by the majority class if not properly weighted .of course , there are many other approaches for handling class imbalance .we used the weighting approach for its simplicity .briefly speaking , the 1st term in ( [ eq : f ] ) minimizes the loss on fitting the labeled samples in the source domain , the 2nd term minimizes the loss on fitting the labeled samples in the target domain , the 3rd term minimizes the structural risk of the classifier , and the 4th term minimizes the distance between the marginal probability distributions and , and also the distance between the conditional probability distributions and .according to the representer theorem , the solution of ( [ eq : f ] ) can be expressed as : where ^t \label{eq : x}\end{aligned}\ ] ] and ^t ] are coefficients to be computed .it has been shown that the closed - form solution for is : ^{-1}e^o\mathbf{y}^o \label{eq : alphaonline}\end{aligned}\ ] ] next we briefly introduce how the various terms in ( [ eq : alphaonline ] ) are derived .define ^t \label{eq : yonline}\end{aligned}\ ] ] where are known labels in the source domain , and are known labels in the target domain . define also as a diagonal matrix with then , following the derivation in ( [ eq : l3 ] ) , we now have ^o(\mathbf{y}^o - k^o\boldsymbol{\alpha}^o ) \label{eq : l3online}\end{aligned}\ ] ]   again , we define the structural risk as the squared norm of in , i.e. ,    we compute using the projected mmd between the source and target domains : ^ 2 \nonumber \\ & = ( \boldsymbol{\alpha}^o)^tk^om_0^ok^o\boldsymbol{\alpha}^o \label{eq : dfkponline}\end{aligned}\ ] ] where is the mmd matrix :    in offline calibration , to minimize the discrepancy between the conditional probability distributions in the source and target domains , we need to first compute the pseudo - labels for the unlabeled target domain samples . in online calibration , because there are no unlabeled target domain samples , this step is skipped . following the derivation of ( [ eq : dfkq2 ] ) , we still have : where is still computed by ( [ eq : m ] ) , but using only the source domain samples and target domain samples .the sds procedure in owarsds is almost identical to that in warsds .the only difference is that the latter also makes use of the unlabeled target domain samples in computing in ( [ eq : dst ] ) , whereas the former only uses the labeled target domain samples , because there are no unlabeled target domain samples in online calibration .the pseudo code for the complete owarsds algorithm is described in algorithm 2 .it first uses sds to select the closest source domains , then performs owar for each of them separately to build individual classifiers , and finally aggregates them using a weighted average , where the weights are the corresponding training accuracies .observe that owarsds is very similar to warsds , the major difference being that no unlabeled target domain samples are available for use in owarsds .this section describes the setup of the vep oddball experiment , which is used in the following three sections to evaluate the performances of different algorithms .a two - stimulus vep oddball task was used . in this task , participants were seated in a sound - attenuated recording chamber , and image stimuli were presented to them at a rate of 0.5 hz ( one image every two seconds ) . the images ( 152 pixels ) , presented for 150 ms at the center of a 24 inch dell p2410 monitor at a distance of approximately 70 cm , were either an enemy combatant [ _ target _ ; an example is shown in fig .[ fig : t ] ] or a u.s . soldier [ _ non - target _ ; an example is shown in fig . [fig : nt ] ] .the subjects were instructed to maintain fixation on the center of the screen and identify each image as being target or non - target with a unique button press as quickly and accurately as possible .a total of 270 images were presented to each subject , among which 34 were targets .the experiments were approved by u.s .army research laboratory ( arl ) institutional review board .the voluntary , fully informed consent of the persons used in this research was obtained as required by federal and army regulations .the investigator has adhered to army policies for the protection of human subjects .18 subjects participated the experiments , which lasted on average 15 minutes .signals for each subject were recorded with three different eeg headsets , including a 64-channel 512hz biosemi activetwo system , a 9-channel 256hz advanced brain monitoring ( abm ) x10 system , and a 14-channel 128hz emotiv epoc headset . however , due to some exceptions at the experiment , data were correctly recorded for only 16 subjects for abm , 15 subjects for biosemi , and 15 subjects for emotiv . there were 14 subjects whose data were correctly recorded for all three headsets , so we used only these 14 subjects in this study .the preprocessing and feature extraction method for all three headsets was the same , except that for abm and emotiv headsets we used all the channels , but for the biosemi headset we only used 21 channels ( cz , fz , p1 , p3 , p5 , p7 , p9 , po7 , po3 , o1 , oz , poz , pz , p2 , p4 , p6 , p8 , p10 , po8 , po4 , o2 ) mainly in the parietal and occipital areas , as in .eeglab was used for eeg signal preprocessing and feature extraction . for each headset ,we first band - passed the eeg signals to [ 1 , 50 ] hz , then downsampled them to 64 hz , performed average reference , and next epoched them to the  ] .let and be the true number of epochs from the target and non - target class , respectively .let and be the number of epochs that are correctly classified by an algorithm as target and non - target , respectively .then , we compute where is the classification accuracy on the target class , and is the classification accuracy on the non - target class .  the following balanced classification accuracy ( bca ) was then used as the performance measure in this paper : selection presents performance comparison of warsds with several other offline calibration algorithms .   although we knew the labels of all eeg epochs for all 14 subjects in the vep experiment , we simulated a realistic offline calibration scenario : we had labeled eeg epochs from 13 subjects , and also all epochs from the 14th subject , but initially none of them was labeled .our goal was to iteratively label epochs from the 14th subject and build a classifier so that his / her remaining unlabeled epochs can be reliably classified .the flowchart for the simulated offline calibration scenario is shown in fig .[ fig : offlinec ] .assume the 14th subject has sequential epochs in the vep experiment , and we want to label epochs in each iteration , starting from zero .we first generate a random number  ] .a larger aupc value indicates a better overall classification performance .first , we checked the normality of our data to see if parametric anova tests can be used .the histograms of the aupcs for each of the six algorithms on the three headsets are shown in fig .[ fig : hist ] . observe that most of them are not even close to normal .so , parametric anova tests can not be applied .    as a result, we used friedman s test , a two - way non - parametric anova where column effects are tested for significant differences after adjusting for possible row effects .we treated the algorithm type ( bl2 , tl , tlsds , arrls , war , warsds ) as the column effects , with subjects as the row effects .each combination of algorithm and subject had 30 values corresponding to 30 runs performed .friedman s test showed statistically significant differences among the six algorithms for each headset ( , ) .then , non - parametric multiple comparison tests using dunn s procedure was used to determine if the difference between any pair of algorithms was statistically significant , with a -value correction using the false discovery rate method .the results showed that the performances of war and warsds were statistically significantly different from bl2 , tl , tlsds and arrls for each headset ( for all cases , except for abm war vs bl2 , and for abm warsds vs bl2 ) .there was no statistically significant performance difference between war and warsds ( for biosemi , for abm , and for emotiv ) .  in summary, we have demonstrated that given the same number of labeled subject - specific training samples , war and warsds can significantly improve the offline calibration performance . in other words ,given a desired classification accuracy , war and warsds can significantly reduce the number of labeled subject - specific training samples .for example , in fig .[ fig : biosemi_offline_avgs ] , the average bca of bl2 is , given 100 labeled subject - specific training samples .however , to achieve that bca , on average war and warsds only need 5 samples , corresponding to saving of the labeling effort . moreover , fig .[ fig : biosemi_offline_avgs ] also shows that , without using any labeled subject - specific samples , war and warsds can achieve similar performance as bl2 which uses 65 samples .similar observations can also be made for the abm and emotiv headsets .   in this subsectionwe study the sensitivity of war and warsds to parameters and ( ) . to save space, we only show the bca results for the biosemi headset .similar results were obtained from the other two headsets .the average bcas of war and warsds for different ( and were fixed at 10 ) are shown in fig .[ fig : sigma ] , and for different and and identical value because they are conceptually close . ] ( was fixed at 0.1 ) are shown in fig .[ fig : lambda ] . observe from fig .[ fig : sigma ] that both war and warsds achieved good bcas for  ] and  ] , representing the starting position in the vep sequence .then , in the first iteration , we use all labeled epochs from the other 13 subjects to build different classifiers , and compute their bcas on the unlabeled epochs . in the second iteration ,we generated labeled epochs , , ... , from the 14th subject , build different classifiers , and compute their bcas on the remaining unlabeled epochs .we iterate until the maximum number of iterations is reached . to obtain statistically meaningful results, the above process was repeated 30 times for each subject , each time with a random starting point .the whole process was repeated 14 times so that each subject had a chance to be the 14th " subject .we compared the performances of owarsds with five other algorithms :  1 .  _bl1 _ in section [ sect : aoffline ] .bl2 _ in section [ sect : aoffline ] , using different pca features .tl _ in section [ sect : aoffline ] , using different pca features .tlsds _ , which is the above tl algorithm with sds ._ owar _ , which uses all existing subjects , instead of performing sds .again , weighted libsvm with rbf kernel was used as the classifier in bl1 , bl2 , tl and tlsds .we chose , , and .note that the online algorithms still used pca features , but they were computed differently from those in offline calibration . in offline calibration we had access to the labeled samples plus the unlabeled samples ,so the pca bases can be pre - computed from all samples and kept fixed in each iteration .however , in online calibration , we only had access to the labeled samples , so the pca bases were computed from the samples only , and we updated them in each iteration as changed .the bcas of the six algorithms , averaged over the 30 runs and across the 14 subjects , are shown in fig .[ fig : online ] for different eeg headsets .the observations made in section [ sect : results ] for offline calibration still hold here , except that arrls was not included in online calibration .particularly , both owar and owarsds achieved much better performance than bl2 , tl and tlsds .however , instead of using 13 auxiliary subjects in owar , owarsds only used on average 6.51 subjects for biosemi , 6.01 subjects for abm , and 7.09 subjects for emotiv , corresponding to , and computational cost saving , respectively .friedman s test showed statistically significant performance differences among the five algorithms ( excluding bl1 , which is not iterative ) for each headset ( , ) .dunn s procedure showed that the bcas of owar and owarsds are statistically significantly different from bl2 , tl , and tlsds for each headset ( for abm owarsds vs bl2 , for abm owarsds vs tlsds , and in all other cases ) .there was no statistically significant performance difference between owar and owarsds ( for biosemi , for abm , and for emotiv ) .  in summary, we have demonstrated that given the same number of labeled subject - specific training samples , owar and owarsds can significantly improve the online calibration performance .in other words , given a desired classification accuracy , owar and owarsds can significantly reduce the number of labeled subject - specific samples . for example , in fig .[ fig : biosemi_online_avgs ] , the average bca of bl2 was , given 100 labeled subject - specific training samples .however , to achieve that bca , on average owar only needed 15 samples , and owarsds only needed 20 samples , corresponding to 85% and 80% saving of labeling effort , respectively . moreover , fig .[ fig : biosemi_online_avgs ] also shows that , without using any labeled subject - specific samples , owar and owarsds can achieve similar bca as bl2 which used 60 labeled subject - specific samples .similar observations can also be made for the abm and emotiv headsets .this section compares the performances of warsds and owarsds ( war and owar ) .intuitively , we expect the performances of the offline calibration algorithms to be better than their online counterparts , because : 1 ) offline calibration uses all eeg epochs to compute the pca bases , whereas online calibration only uses epochs , so the pca bases in offline calibration should be more representative ; and , 2 ) offline calibration also uses the unlabeled epochs in the optimization , whereas online calibration does not , so offline calibration makes use of more information . in other words ,offline calibration makes use of semi - supervised learning whereas online calibration does not .the average performances of war , warsds , owar and owarsds across the 14 subjects are shown in fig .[ fig : onoff ] . observe that the results were consistent with our expectation : for all three headsets , the offline algorithms ( war and warsds ) achieved better bcas than their online counterparts ( owar and owarsds ) .additionally , fig .[ fig : onoff ] shows that the algorithms had best performance using the biosemi headset , and worst performance using the abm headset .this is not surprising , as biosemi used the most number of channels , and it was wired , which means better signal quality .the abm headset had the least number of channels , and was wireless. moreover , epochs with incorrect button presses were filtered out for biosemi and emotiv headsets , but not for most subjects for the abm headset .so , the epochs in abm were noisier .we also performed statistical tests to check if the performance differences among the four algorithms were statistically significant .friedman s test showed statistically significant differences among the four algorithms for biosemi ( , ) and emotiv ( , ) , but not abm ( , ) .dunn s procedure showed that for biosemi the bcas of war and owar were statistically significantly different ( ) , so were bcas of warsds and owarsds ( ) . for abm andemotiv the performance differences between online and offline algorithms were not statistically significant ( for abm war vs owar , for abm warsds vs owarsds , for emotiv war vs owar , and for emotiv warsds vs owarsds ) .  in conclusion ,we have shown that generally the offline war and warsds algorithms , which include a semi - supervised learning component , can achieve better calibration performance than the corresponding online owar and owarsds algorithms , i.e. , semi - supervised learning is effective .single - trial classification of erps in eeg signals is used in many bci applications . however , because different subjects have different neural responses to even the same stimulus , it is very difficult to build a generic erp classifier whose parameters fit all subjects .so , the classifier needs to be calibrated for each individual subject , using some labeled subject - specific data . reducing this calibration effort ,i.e. , minimizing the number of labeled subject - specific data required in the calibration , would greatly increase the utility of a bci system .this paper introduced both online and offline war algorithms for this purpose .we have demonstrated using a vep oddball task and three different eeg headsets that both algorithms can cope well with the class - imbalance problem , which is intrinsic in many real - world bci applications , and they also significantly outperformed several other algorithms .we also compared the performances of the online and offline war algorithms in identical experimental settings and showed that the offline war algorithm , which includes an extra semi - supervised component than the online war algorithm , can achieve better calibration performance , i.e. , semi - supervised learning is effective in bci calibration .moreover , we further proposed a source domain selection approach , which can reduce the computational cost of both online and offline war algorithms by about .we expect our algorithms to find broad applications in various bci calibration scenarios , and beyond .the most intuitive bci calibration scenario , as described in this paper , is to reduce the subject - specific calibration data requirement by making use of relevant data from other subjects .another scenario is to make use of the same subject s data from previous usages to facilitate a new calibration .for example , the subject may need to work on the same bci task at different locations using different eeg headsets ( office , home , etc . ) , or may upgrade a bci game with a new eeg headset . in such applications , war can be used to make use of the data obtained from a previous eeg headset to facilitate the calibration for the new headset , as introduced in .of course , the above two scenarios can also be combined : auxiliary data from other subjects and from the subject himself / herself can be integrated to expedite the calibration .furthermore , because of human - machine mutual adaptation and non - stationarity , a well - calibrated bci system may degrade gradually .the proposed war algorithms can be used to re - calibrate it from time to time .additionally , eegs , together with many other body signals ( facial expressions , speech , gesture , galvanic skin response , etc . ) , are also frequently used in affective computing , _ computing that relates to , arises from , or deliberately influences emotion or other affective phenomena . _" the war algorithms can also be used to handle individual differences and non - stationarity in such applications .  finally , we need to point out that the current war algorithms still have some limitations , which will be improved in our future research .first , we will develop incremental updating rules to reduce their computational cost .second , we will develop criteria to determine when a negative transfer may occur , and hence use subject - only calibration data in such cases .third , although war can map the features to a new kernel space to make them more consistent across the source and target domains , it still relies on good initial features .simple pca features were used in this paper . in our future researchwe will consider more sophisticated and robust features , e.g. , the information geometry .the author would like to thank scott kerick , jean vettel , anthony ries , and david w. hairston at the u.s .army research laboratory ( arl ) for designing the experiment and collecting the data , and brent j. lance and vernon j. lawhern from the arl for helpful discussions ._ research was sponsored by the u.s .army research laboratory and was accomplished under cooperative agreement numbers w911nf-10 - 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Abstract: many real - world brain - computer interface ( bci ) applications rely on single - trial classification of event - related potentials ( erps ) in eeg signals . however , because different subjects have different neural responses to even the same stimulus , it is very difficult to build a generic erp classifier whose parameters fit all subjects . the classifier needs to be calibrated for each individual subject , using some labeled subject - specific data . this paper proposes both online and offline weighted adaptation regularization ( war ) algorithms to reduce this calibration effort , i.e. , to minimize the amount of labeled subject - specific eeg data required in bci calibration , and hence to increase the utility of the bci system . we demonstrate using a visually - evoked potential oddball task and three different eeg headsets that both online and offline war algorithms significantly outperform several other algorithms . moreover , through source domain selection , we can reduce their computational cost by about , making them more suitable for real - time applications .   brain - computer interface , event - related potential , eeg , domain adaptation , transfer learning


intermittency , the inherent inhomogeneity of turbulence , causes energy dissipation to be highly localized in space and in time .spatial intermittency is manifest by coherent structures , while temporal intermittency is characterized by irregular , bursty events .both of these phenomena are related , play key roles in the turbulent dynamics , and have important observational consequences . as with any dynamical physics problem , a complete understanding of intermittency can only be claimed when the solution is described in both the spatial and the temporal dimensions .intermittency is a challenging theoretical problem that impedes our progress toward a complete theory of turbulence .many of the methods used to study turbulence , such as energy spectra and correlation functions , are insensitive to intermittency .the methods conventionally employed to measure intermittency include structure functions , scale - dependent kurtosis , topological methods , and statistics of discontinuities ; however , these often give incomplete information or are difficult to measure accurately .therefore , in order to better guide theoretical models of turbulence , it is essential to develop new tools for studying intermittency in a robust and informative manner .the statistical analysis of coherent structures is a possible route forward in this direction .it is convenient , for both practical and theoretical purposes , to treat coherent structures as discrete objects due to their localized nature and their central role in the dynamics .the statistical properties of these structures , including their intensities and morphologies , give insight into the underlying dynamics from which they formed .this information reveals the anisotropy , inhomogeneity , and characteristic scales of the dynamics .furthermore , such an analysis can be extended beyond simulations and theory , being well suited for a variety of experimental and observational applications , including solar flares , the solar photosphere , the interstellar medium , instabilities in fusion devices , and radiative signatures in optically thin astrophysical plasmas , e.g. , in black - hole accretion disk coronae , hot accretion flows , jets , pulsar wind nebulae ( e.g. * ? ? ?* ; * ? ? ?* ) , and hot gas in galaxy clusters .the statistical analysis of spatial structures has been a fruitful approach to studying the intermittency of turbulence in numerical simulations .the methods for identification of structures vary between different studies , but one of the simplest and most common approaches is to identify structures as regions in space bounded by an isosurface of the relevant field . for example , this was applied in hydrodynamic turbulence to study dissipative vorticity filaments , which revealed that the radii of intense filaments scale with the kolmogorov microscale while lengths occupy large scales .more recently , similar methods were applied to study current sheets and other dissipative structures in simulations of plasma turbulence , notably in 2d magnetohydrodynamic ( mhd ) turbulence , 3d mhd turbulence , the kinematic dynamo , ambipolar diffusion mhd , boundary - driven mhd , and decaying kinetic turbulence .these previous studies succeeded in describing the morphology and scaling properties of intermittent structures , but are incomplete in the sense that they give no information on how the structures evolve in time . for this reason , very little can be said about the dynamics , even if one assumes statistical stationarity .important temporal information about intermittent structures includes their characteristic timescales , stability , motion , interactions , and impulsiveness .if one is interested in understanding these temporal aspects of intermittency , then a broader framework must be developed .temporal intermittency was investigated to a limited extent in hydrodynamic turbulence .the temporal statistics of vortices in 2d turbulence and decaying charney - isotropic geostrophic turbulence gave insight into global changes in the population size and morphology of vortices .however , the structures were not entirely treated as temporal objects in these studies , which requires an algorithm for tracking the structures through time .this can be much more difficult to design and implement than the algorithms used to identify and characterize structures in individual fixed - time snapshots .to the best of our knowledge , apart from our precursor paper , no systematic studies of spatiotemporal structures were previously undertaken for hydrodynamic or mhd turbulence in 3d , and only limited studies were performed in 1d and 2d systems ( e.g. , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?the objective of this paper is to extend the framework previously used for the statistical analysis of spatial dissipative structures in numerical simulations of mhd turbulence into the temporal realm .we therefore consider the statistical properties of 4d spatiotemporal objects which represent structures evolving through time .these objects are dissipative processes , analogous to flares in astrophysical systems , which , in general , may involve many interacting coherent structures .we consider the distributions and scaling relations for process characteristics including the total dissipated energy , peak energy dissipation rate , duration , and geometric scales ; measures of complexity such as the number of interacting structures and types of interactions ( mergers and divisions ) ; and the temporal evolution of individual processes .this novel methodology is applied to study intermittency in numerical simulations of driven incompressible mhd turbulence ; however , it can also be applied to hydrodynamic turbulence , kinetic plasma turbulence , and other complex dynamical systems exhibiting spatiotemporal self - organization .the analysis presented in this paper addresses several fundamental questions regarding the intermittency of mhd turbulence .one key question is whether , in the limit of high reynolds number , the overall energy dissipation is dominated by a few intense , long - lasting events residing at large scales , or by many weak , short - lived events residing near the dissipation scale .another question is whether there is an inherent relationship between spatial intermittency and temporal intermittency , e.g. , whether larger structures better retain their coherency in time .a third question is whether the dissipative events show any characteristic temporal asymmetry , e.g. , impulsive onset followed by a slow decay .the primary questions addressed in this paper for incompressible mhd turbulence are also fundamentally important for the solar corona . in fact, our approach has many similarities with observational studies of solar flares and stellar flares . in these studies , the time - series of extreme uv , soft x - ray , and hard x - ray emissions from the sun are used to characterize solar flares .measured quantities include the size , duration , peak intensity , and fluence of the flares , from which the dissipated energy is inferred . in particular , the probability distribution for dissipated energy is of central importance due to its role in assessing the feasibility of the nanoflare model for coronal heating .this distribution is observed to obey a power law across eight orders of magnitude , with an index generally close to  , although the precise value of the index varies significantly between different studies depending on the time period , region , type of emission , and methods used to identify the flares .most importantly , this index is shallower than the critical value of , suggesting that nanoflares do not dominate the overall heating of the solar corona ., the critical index is derived by noting that the total energy dissipation , , scales with the lower bound if and with the upper bound if . ]this analogy with the analysis of solar flares suggests a practical application for the statistical analysis of spatiotemporal structures : to assess whether turbulence plays a role in the intermittent energy dissipation of the solar corona .it is presently unknown whether the complex dynamics of the solar corona arise from self - organized criticality , turbulence , or some other phenomena .this problem can eventually be addressed by a better comparison between numerical simulations of the various models with observations .our methodology is ideal for making this comparison . as a first step in this direction, we compare the results in this paper to the observed statistical properties of solar flares , collected from a number of previous studies . we find several nontrivial similarities in both cases , despite the anticipated differences between driven incompressible mhd turbulence and the essentially force - free dynamics of the solar corona .this suggests that mhd turbulence may play a role in the energetics of the corona , a possibility which should be investigated more carefully in future studies .the outline of this paper is as follows .we first describe the basic concepts and subtleties associated with the temporal analysis of structures in section [ sec2 ] .we then outline the technical aspects of our procedure , based on extending our previous framework for the statistical analysis of spatial structures , including detailed discussion of the algorithms involved , in section [ sec3 ] . in section [ sec4 ], we introduce the measurements that are made on spatiotemporal dissipative structures .the results of our analysis on the combined spatial and temporal intermittency of energy dissipation in 3d mhd turbulence are described in section [ sec5 ] .we discuss the implications of our results in section [ sec6 ] , which includes a comparison of our results with the observed statistical properties of solar flares .we summarize our conclusions in section [ sec7 ] .mhd describes the macroscopic dynamics of an electrically conducting fluid , such as a plasma ( see , e.g. ) .given a uniform background magnetic field that is strong relative to turbulent fluctuations ( estimated by the rms value , ) , and assuming that gradients along the background field are small relative to those in the perpendicular direction , the incompressible mhd equations can be written in a reduced form as where are the elssser variables ( with directions strictly perpendicular to ) , is the fluctuating plasma velocity , is the fluctuating magnetic field ( in units of the alfvn velocity , , where is plasma density ) , is the total pressure , is the gradient in the directions while is the gradient in the direction , and is the large - scale external forcing . for simplicity ,the fluid viscosity is taken to be equal to the magnetic diffusivity , and both are assumed to be uniform in space . in the reduced mhd approximation , only the -component of current density and vorticity are retained ; these two scalar fields contain complete information to describe the dynamics as well as energetics .energy loss from the system is governed by resistive dissipation and viscous dissipation , with respective energy dissipation rates per unit volume given by and .since the current density and vorticity are in direct correspondence with energy dissipation rates , they are logical dynamical fields in which to study the intermittency of dissipation . in this work ,we focus on resistive dissipation only for simplicity ; it is straightforward to apply the methods to obtain similar statistical results for the viscous dissipation .intermittency in strong mhd turbulence is characterized by the appearance of thin , quasi-2d ribbon - like structures in the current density , known as ( electric ) current sheets , with similar structures in the vorticity field .these current sheets are anisotropic in three directions , with the largest scale coinciding with the direction of the local mean field , which is the direction of the guide field to good approximation .their lengths and widths are distributed mainly in the inertial range , while the thicknesses are strongly localized inside the dissipation range .the current sheets are notable for their role in heating , particle acceleration , and magnetic reconnection .   in this paper , we aim to study the temporal properties of intense , spatially - coherent structures produced by intermittency in 3d turbulence . for concreteness , we consider current sheets in mhd turbulence . despite their prominence ,intermittent current sheets occupy a small volume that represents the tail of the probability distribution for current density . to systematically study these structures ,we employ a framework based on identifying and characterizing the regions in space with current densities exceeding some fixed threshold value , , which is the only parameter inherent to the methodology .each structure is then represented as a 3d volume bounded by an isosurface of .it is straightforward in principle to extend this procedure into the temporal realm by applying the same threshold criterion to the 4d spatiotemporal field , thereby obtaining 4d spatiotemporal structures .however , although simple in principle , it is challenging in practice to analyze a high - resolution 4d data set in this way . assuming that the analysis is applied to post - processed data from well - resolved simulations stored on a computer system , the data storage conditions limit the total number of available snapshots , negatively affecting the possible time cadence of snapshots and size of the overall time interval .hence , the time resolution of post - processed data must generally be worse than the internal time resolution of the simulation . this is a serious issuesince the data must be well - resolved in all dimensions in order to properly resolve and track structures across a wide range of scales .a related problem is that the primary memory limits the amount of data that can be loaded by the analysis program at any given time , requiring the numerical procedures to work with small pieces of the overall data set at a time .therefore , a feasible temporal analysis should be based on first identifying the structures in the spatial dimensions of a given snapshot , and then tracking them through time , from their formation to their destruction .the algorithms required to perform this task are rather complex , and must be designed in a robust and efficient manner .in particular , there needs to be an algorithm that accurately associates structures in one snapshot with their time - evolved counterparts in subsequent snapshots .a fundamental challenge here lies in the fact that there may not be a unique correspondence between a structure in one snapshot and a structure in the adjacent snapshot , due to mergers and divisions .  the interactions between structurescause the notion of a coherent time - evolving structure to be ambiguous . instead , the objects of central importance are the _ processes involving structures .these processes can be variously thought of as sets of interacting structures , as dissipative events , or as flares ( if we equate the dissipated energy with outgoing radiation ) .more generally , they can be thought of as branched spatiotemporal structures ._    diagrams of some simple processes , where formation is represented by o , interaction by a vertex , and destruction by x. an isolated structure is a process with no vertices .division and merger processes are the next simplest case , with a single vertex each .higher - order processes such as loops and scatterings have a larger number of vertices or vertices with more paths.,width=302 ]  in order to build an intuition and facilitate the subsequent discussion of processes , we now describe a convenient classification scheme for processes .this allows processes to be visualized diagrammatically , which has some superficial similarities to feynman diagrams for quantum mechanical scattering processes ; however , none of the mathematical symmetries characterized by the feynman rules carry over , since , to the best of our knowledge , isosurfaces in the current density described by the mhd equations contain no conserved quantities .regardless , the following classification scheme is a simple way to describe processes and their complexity .we first introduce some terminology .we define a _ state to be an individual spatial structure at fixed time , which represents the basic building block of processes .we assume that the states are given at times spaced by an infinitesimal increment .we also assume that there exists a map between all states at any time to other states at time and , which represents the instantaneous temporal evolution of structures from one state to another state .we define a _ path segment to be a bijective ( i.e. , one - to - one ) sequence of states under this map , which represents the coherent temporal evolution of an individual structure while it does not interact with any other structures .we then define a _ path to be a path segment with bijectivity breaking down only at the initial and final states in the sequence .a process is then described as a set of paths that are ( non - bijectively ) connected at their endpoints , which represents a set of interacting structures . _ _ _  diagrams of some simple conceivable processes are shown in fig .[ fig : interactions ] . here ,we represent paths schematically as lines with an arrow marking the direction of time .if the path begins by spontaneous formation , i.e. , from a peak that grows to exceed the detection threshold , then we mark the beginning of the path with an o. if the path ends by spontaneous destruction , i.e. , from a structure that recedes below the detection threshold , then we mark the end of the path with an x. the third possibility is for the path to start or end with an interaction .interactions between structures are represented by_ vertices connecting sets of three ( or more ) paths .a process schematically consists of a set of paths connected by a set of vertices . _the simplest process is the evolution of an _ isolated structure , i.e. , a structure that is formed and then destroyed without interactions .isolated structures are described by a single path and have well - defined histories with well - defined properties .other structures undergo at least one interaction .the two simplest processes involving an interaction are the _ division of one structure into two structures and the _ merger of two structures into one structure .since an interaction is non - bijective , the structures in these processes do not have completely well - defined histories , so many of the quantities used to describe isolated structures are ambiguous .however , we will see that a meaningful set of more general characteristics can be introduced ._ _ _  we claim that the most logical approach for a temporal analysis is to study processes rather than individual spatially - coherent structures , i.e. , paths , which lose their identity upon interacting .this is also the most conservative approach , as it requires no fundamental changes to the methodology used for the statistical analysis of spatial structures at fixed time , and requires no ad - hoc assumptions to treat the interactions .we also find that the statistical trends are more robust for processes rather than paths .a schematic of structures evolving in time , highlighted in green ( shown in one - dimensional space for clarity ) .structures from the initial and final states are marked in red.,width=604 ]  in this section , we describe our algorithms for the temporal analysis . since the procedure in its entirety is rather complicated , we include a brief outline in this subsection .a more detailed description is presented in the rest of this section .the procedure rests on a hierarchy of steps : first , we find sets of contiguous points ( above the threshold ) in each snapshot to obtain the spatial structures ; next , we find sets of bijectively - connected states to obtain the paths ; finally , we find sets of connected paths to obtain the processes .a schematic of the final result is shown in fig .[ fig : scheme ] , where processes ( colored in green ) of varying complexity are identified on the space - time lattice .  1 .* identify all states ( i.e. , spatial structures ) in each snapshot , and represent temporal connectivity by constructing a map between states in adjacent snapshots . * 1 .load initial snapshot and determine states ( by using threshold algorithm ) .* store constituent points of each state in a temporary array ; * perform measurements on states and store in a permanent array ; 2 .  for , do the following : * load snapshot , determine states , and store constituent points and measurements ; * construct the evolution map , which associates states in snapshot with temporally - connected states in snapshot , and vice - versa ( do this by comparing constituent points of structures in both snapshots ) ; remove temporary array for snapshot ; 2 .  * identify paths . * 1 .  obtain paths from states that form a bijective sequence under the evolution mapsperform measurements on paths by referencing the constituent states .3 .  for each path , construct ( non - bijective ) map that identifies other paths connected to it .3 .  * identify processes . * 1 .obtain processes from sets of connected paths .perform measurements on processes by referencing the constituent paths .treat processes that exist during the initial snapshot or final snapshot as special cases ( incomplete processes ) .we first describe our algorithm to identify the states on each snapshot , i.e. , the spatial structures at fixed times .we define states as spatially - connected sets ( i.e. , clusters ) of points with current density magnitudes larger than a fixed threshold , .two points on the lattice are considered spatially connected if the distance between the points is strictly less than two lattice spacings , i.e. , one is contained in the other s 26 nearest neighbors .we note that this is not the only conceivable way to define a state ; one can , for example , use a variable threshold based on local field quantities such as the local peak current density . however , a fixed threshold appears to be the simplest approach , having only one free parameter ( ) and no need for ad - hoc treatment of special cases ( e.g. , mergers ) .the main drawback of using the fixed threshold is that it generally gives a large number of unresolved structures due to peaks near the threshold ; these do not exhibit physically meaningful scaling relations and must be carefully ignored .fortunately , they are only manifest as noise in the low - intensity , small - scale regime of parameter space and generally have a negligible contribution to the total energy dissipation and volume of the structures .to some extent , unresolved structures are inevitable regardless of how they are defined , since there will always be a population of small , short - lived processes representing structures that barely cross the detection criteria .filtering procedures can be applied to remove the population of unresolved , noisy structures residing near the threshold , but for simplicity we do not apply any filtering in our analysis .the definition of a state can affect how a _ given process appears ; it may determine , for example , whether the close approach of two structures is registered as a merger or not .the threshold value may likewise have a similar effect .however , we would like to emphasize that the _ statistical conclusions should be , and in our experience are found to be , broadly consistent regardless of the method or threshold ._ _  our algorithm identifies states by scanning the lattice for points with current densities above the threshold . for each such pointfound , the neighboring points satisfying are identified , then neighbors of those neighboring points satisfying , and so on , until no more points remain .the coordinates of the constituent points of the state are stored in an array for later use , and then the lattice is scanned for any additional states ( while ignoring the points already found to belong to a state ) .once every state in a snapshot is identified , measurements are performed on the states ( see section [ sec4 ] ) and stored for later use .   in this subsection, we consider the connectivity of states in the temporal dimension .we describe our algorithm for identifying the time - evolved counterparts of a state , i.e. , the states in the adjacent snapshots that represent the evolved structure .the result is a ( non - bijective ) map associating the set of states in one snapshot with the set of states in an adjacent snapshot , representing the instantaneous temporal evolution of structures .consider a given state in the snapshot . to find the future counterparts of the given state , we iterate through all states in the subsequent snapshot and determine which ones have any constituent points that are spatially - connected to the given state .in other words , we look for states with points that coincide with or neighbor any of the given state s points ( using the 26 nearest neighbor criterion ) .any such states are identified as future counterparts of the given state .let the states in each snapshot be denoted by an index , where is the number of states in the snapshot .we construct an array that stores the indices of the corresponding future states in the snapshot , which we call the _ forward evolution map where .thus , if the state in snapshot is associated with the state in snapshot , then .if the state has no future counterparts , then we assign , representing the null state , which corresponds to the destruction of the structure .if the state is associated to multiple future states ( i.e. , it divides ) , then we assign multiple values containing all of these future states ._  we perform a similar procedure to find the past counterparts of the given state . in this case, we iterate through all states in the preceding snapshot to determine which ones have points that are spatially - connected to the given state .any such states are identified as past counterparts of the given state .likewise , we construct a _ backward evolution map , , which identifies state in the snapshot with its corresponding past counterparts in the snapshot .if the state has no past counterpart , then we assign , which corresponds to the formation of the structure .if the state is associated to multiple past states ( i.e. , it results from a merger ) , then we assign multiple values containing all of these past states . _  the entire set of evolution maps where is constructed by iterating through the snapshots and applying the above procedure immediately after identifying the states in pairs of consecutive snapshots .after is constructed for all states in a given snapshot , the array containing the constituent points of the states is deleted to free up memory before loading the next snapshot .schematic of structure evolution , shown in 2d space for clarity .the procedure stores the constituent points of the present state , checks the future snapshot for any states with points that are spatially - connected to the present state s points , and then identifies these states as future counterparts of the present state . in the left panel ,a structure evolves without interacting . in the right panel ,a structure divides , having multiple future counterparts .this procedure is reversed in time to determine past counterparts , with a merger occuring for multiple past counterparts.,width=453 ]   in this subsection , we describe our algorithm for identifying paths across multiple snapshots , which is an important intermediate step before processes themselves can be identified .a path is abstractly defined as a sequence of states that is bijectively - connected under the evolution maps ( see sec .[ class ] ) . from the algorithms in the preceding subsections, we have a sequence of snapshots ( denoted by index ) , each with a set of states , along with the evolution maps associating the states in each snapshot with their time - evolved counterparts in adjacent snapshots . to identify paths, we must find the bijective sequences in .consider the state in snapshot .bijectivity of the forward evolution map is satisfied if there exists a unique state in snapshot such that and .any two states satisfying this condition form a path segment . if this condition fails , however , then state is the endpoint of a path .this happens either when the structure is destroyed ( ) or when it interacts ( either or is multi - valued ) .likewise , if there exists a unique state in snapshot such that and , then these two states form a path segment ; otherwise state is the beginning of a path due to either formation ( ) or an interaction ( either or is multi - valued ) . using these conditions, we can track the state of any given structure along a path through a sequence of snapshots , until an interaction is encountered .we iterate through the states in all snapshots and use the above tracking procedure to identify the associated path and its constituent states , marking those states so that they are ignored in the remaining iterations . as a result , we obtain a set of paths . for each path , we construct an array which contains the indices of the constituent states , which is referenced to perform measurements on the path .we also construct an array containing the indices of the _ predecessors , which are the other paths connecting to it from the beginning of the path .the predecessors are determined by operating with on the first state of the path to obtain all of the past states , and finding the paths that contain these past states . in a separate array ,we store the indices of _ successors , which are the other paths connecting to the end of the path , obtained by operating with on the final state of the path .the predecessors and successors of paths characterize the vertices between paths .note that if the number of predecessors is zero , then the path is formed spontaneously .if the number of successors is zero , then the path is destroyed spontaneously . _ _we finally describe how to identify processes from the set of paths and their predecessors and successors . recall that processes are described as sets of connected paths .therefore , we first iterate through the set of paths . for each path ,iterate through the predecessors and successors of the path , and then through the predecessors and successors of those , and so on , until no new paths can be obtained .the set of paths acquired in this way constitute a single process , and their indices are stored in an array corresponding to that process .the paths that have already been identified as belonging to a process are ignored in the remaining iterations of paths .some processes will contain states from the initial snapshot or in the final snapshot of the dataset , which we call _ initial processes or _final processes , respectively .these processes must be treated as special cases , since our information about them is incomplete .the simplest treatments of these processes are either to ignore them or to treat them as normal processes undergoing formation or destruction in the initial or final snapshots . for most of our analysis, we will ignore initial and final processes .however , they are included in the probability distributions for better statistics at long durations . due to the relatively long interval of time in our simulations ,the initial and final processes are a very minor contribution to the statistics , unless low thresholds are used . _ _  this concludes our discussion of the algorithms used to identify processes .we now have a sample of processes , each including an array of constituent paths .the paths contain all of the information necessary to perform measurements on the processes .these measurements are described in the next section .in this subsection , we describe measurements made on states , i.e. , spatial structures at fixed time , which will be used in the next subsections to construct similar , more general quantities for paths and processes .the volume of the structure is immediately obtained by counting the number of constituent points of the state and multiplying by the lattice volume element . the ohmic energy dissipation rate is given by where integration is performed across all constituent points of the given state .  to characterize the geometry of each state , we measure the linear scales in three orthogonal directions . for length ,we take the maximum distance between any two points of the structure . for width , we consider the plane orthogonal to the length and coinciding with the point of peak current density .we then take the maximum distance between any two constituent points in this plane to be the width .the direction for thickness is then fixed by the orthogonality condition .we take the thickness to be the distance across the structure in this direction through the point of peak current density .since typical thicknesses tend to be comparable to the lattice spacing , we use a linear interpolation scheme to obtain finer measurements .note that this method can cause the thickness to be over - estimated in some cases with complex morphologies , for example , in structures with s - shaped cross - sections , but these appear to be a relatively small population manifest as spurious high - value measurements .these definitions automatically satisfy .we measure these scales in units of the system size in the direction perpendicular to the guide field .see for more detailed information on this method applied to high - resolution simulations of mhd turbulence , as well as a discussion on alternative methods based on the minkowski functionals .states can also be characterized by some discrete properties .one example is the direction of the current flow , i.e. , orientation .another example is the presence / absence of topological features such as x - points or o - points [ see and ] .a final example is the euler characteristic , i.e. , genus .these characteristics will be relegated to a future paper .   in this subsection, we describe measurements for paths , which are conceptually simpler than those for processes due to the bijectivity condition .these measurements will be used and generalized in the next subsection to characterize processes .the evolution of a path can be described by the time - series , , , , and of instantaneous characteristics defined in the previous section for the constituent states .consider a path given by a sequence of states at times , where and is the number of constituent states . assuming a fixed cadence of snapshots , the states are separated by a fixed time interval .the state has characteristics denoted by , , , , and .one of the most basic properties of a path is its duration ( or lifetime ) , defined as the energy dissipation rate of states generalizes to the dissipated energy , where time integration is performed over the duration of the path .we also define the peak volume , peak energy dissipation rate , maximum length , maximum width , and maximum thickness as note that since these quantities are local rather than time - integrated , they may be sensitive to chaotic fluctuations . as an alternative, we can consider time - averaged quantities , which however are less easily generalized for processes .   we now describe how to generalize the quantities defined for paths in the previous section to processes , i.e. , sets of interacting structures .we characterize each process by the number of constituent paths , .processes with a single path , , are isolated structures. processes with three paths , , are division or merger processes. processes consisting of more than three paths , , are higher - order processes , containing either more than one vertex or vertices joining more than three paths .other related measures of the complexity of a process include the number of vertices , the number of constituent states , the number of internal paths ( i.e. paths that begin and end in vertices ) , and the number of incoming paths or outgoing paths  .consider a process with constituent paths enumerated by index , each extending from initial times to final times .let , , , , , and be the characteristics of the path as defined in the previous subsection .we define the process duration by which is simply the time interval from the formation of the first existing constituent path to the destruction of the last existing constituent path .the total dissipated energy of a process is defined by integrating across the enclosed 4d spacetime region .this is found in a straightforward manner from likewise , we can define the peak volume and peak energy dissipation rate as the maximum of the peaks corresponding to the constituent paths , note that an alternative definition for ( ) can be based on the maximum of the volume ( energy dissipation rate ) summed for all states belonging to the process at any given time .these two definitions may differ for processes with a large number of paths , but will otherwise give similar results ; we use the first definition only for simplicity . the most difficult quantities to generalize for a process are the characteristic spatial scales .there appears to be no universally satisfactory way to obtain _ average characteristic scales for a general configuration with many paths .if we apply an average across all paths ( or all states ) constituting the process , then the result may be skewed toward unphysical short - lived paths ( or states ) .a simpler alternative is to take the maximum scale corresponding to any path in the process , one alternative definition for the spatial scales , which gives results consistent with the definition that we apply , is to take the spatial scales from the largest state at the moment of peak energy dissipation rate . _before discussing the results of our temporal analysis of mhd turbulence , we first describe the numerical simulations .we consider simulations of strong , incompressible mhd turbulence driven at large scales .these simulations solve the reduced mhd equations , eq .[ rmhd - elsasser ] , using a fully dealiased 3d pseudo - spectral algorithm ( see for specific details on simulations ) .the ratio of magnetic guide field to rms fluctuations is set to .the periodic box is elongated in ( the direction of the guide field ) by a factor of , where is the perpendicular size of the domain in simulation units .turbulence is driven at the largest scales by colliding alfvn modes , generated from statistically independent random forces in fourier space at low wave - numbers , . the fourier coefficients of are gaussian random numbers with amplitudes chosen so that .the forcing is solenoidal in the perpendicular plane and has no component along .the random values of the different fourier components of the forces are refreshed independently on average about times per eddy turnover time .the reynolds number is defined by ; we set so that the magnetic reynolds number , , is equal to . durations ( and other timescales ) are measured in terms of large - scale eddy turnover times of the turbulence , given by .the analysis is performed on time intervals of durations , all of which begin after the simulations reach statistical steady state .     if we suppose that all processes are single paths that evolve with identical ( but rescaled ) functional forms for the energy dissipation rate , then we can derive nontrivial constraints between the indices of the various distributions given in sec .[ sec : prob ] .specifically , for simplicity , assume that all processes of duration consist of a single path with the energy dissipation rate for , where the universal shape function satisfies , and for .if we assume power - law distributions for all quantities , then they are given by with the constraint the corresponding scaling relations are given by normalizable distributions require .for the measured value of , is required , so that .an example set of parameters satisfying this constraint are , , and , along with scalings and , all of which are consistent with the results in this paper .the derivation is as follows .we first relate the distribution of instantaneous energy dissipation rates , , measured from states at random , to the distribution of peak energy dissipation rates , , measured from processes . assuming that one samples a random value from with uniform time sampling , so , we obtain the distribution of energy dissipation rates from a process of duration , where ( ) are the roots of , and we have defined the function .the total distribution of energy dissipation rates is then where is defined such that .the lower bound of the integral is required since processes with durations do not reach high enough energy dissipation rates to contribute to the distribution . now assume that .then eq .[ magic ] becomes where .therefore , assuming the integral in eq .[ magic2 ] converges , has the same index as . to relate this to other indices ,note that the dissipated energy per process is given by where , and the integral evaluates to a constant of order unity ; hence , is exactly satisfied . assuming and , we can find the exponent for , hence , and therefore we have , as required .storlie , c. , davis , c. , hoar , t. , lee , t. , nychka , d. , weiss , j. , & whitcher , b. 2004 , in signals , systems and computers , 2004 .conference record of the thirty - eighth asilomar conference on , vol . 2 , ieee , 17001704

Abstract: energy dissipation is highly intermittent in turbulent plasmas , being localized in coherent structures such as current sheets . the statistical analysis of spatial dissipative structures is an effective approach to studying turbulence . in this paper , we generalize this methodology to investigate four - dimensional spatiotemporal structures , i.e. , dissipative processes representing sets of interacting coherent structures , which correspond to flares in astrophysical systems . we develop methods for identifying and characterizing these processes , and then perform a statistical analysis of dissipative processes in numerical simulations of driven magnetohydrodynamic turbulence . we find that processes are often highly complex , long - lived , and weakly asymmetric in time . they exhibit robust power - law probability distributions and scaling relations , including a distribution of dissipated energy with power - law index near , indicating that intense dissipative events dominate the overall energy dissipation . we compare our results with the previously observed statistical properties of solar flares .


the intention of this talk is to summarize the application of linearized gravity , in the specific form of the theory of black hole perturbations , to the study of the collision of black holes .most of the results are already present in the literature , and the rest of the work is still in progress so i present here only a brief survey .the motivation for studying black hole collisions is quite clear . in the next few decadesgravitational wave detectors will come online that will require `` templates '' of possible waveforms from different sources .the collision of black holes is one of the main candidates for observable sources of gravitational radiation .although the initial and advanced ligo detectors will not quite have the frequency range to detect the waves produced in the final moments of the most common collisions , it is expected that future detectors will , and knowing the waveform for the final moments can also lead to insights into the waveforms emitted earlier on .the presence of this strong motivation from the experimental side has led to the formation of an alliance of numerical relativity groups ( the `` binary black hole grand challenge collaboration '' ) with the goal of numerically simulating the collision of two black holes using supercomputers .the degree of difficulty of this project is reflected in the fact that several established numerical relativity groups have decided to team efforts in order to tackle it .  herewe will like to offer a much more modest approach , which is based on a simple idea : when a collision of two black holes starts with the holes so close to each other that they are surrounded by a common horizon , the problem looks from the point of view of an external observer as a single distorted black hole .it can therefore be treated with perturbation theory .although one expects this approach to only yield results in a small range of initial separation , it provides at least for that range a benchmark against which one can calibrate numerical codes of the fully numerical approach . in reference an explicit calculation was carried out using this idea .we took the initial data for the head - on collision of two black holes given by the misner solution and re - wrote it in such a way that in the case that the two black holes are close to each other it explicitly looks like schwarzschild plus something small " .we took the something small " and evolved it using the equations of linearized gravity ( the zerilli equation ) and computed the radiated energy .the results are shown in figure 1 , where we plot the energy radiated in the collision as a function of the initial separation and compare with the results of the ncsa group using a numerical integration of the full einstein equations .we see that the close approximation works very well until the holes are no longer surrounded by a common apparent horizon ( ) and works within the correct order of magnitude up to when the holes are no longer surrounded by an event horizon ( ) .also shown is a `` far approximation '' based on a particle - membrane paradigm .comparisons of waveforms have also been performed and they also show very good agreement between the linearized theory and the full numerical simulations .all this shows that the use of linearized gravity in the close limit can be a valuable aid to full numerical evolutions of the two black hole problem .it is therefore quite tempting to apply the linearized treatment to more interesting situations , specifically the in - spiraling collision of two black holes with angular momentum .there are two main obstacles to doing this computation and we will detail them in the next two sections .assuming initial data for a black hole collision is given , we can rather easily evolve and compute energies in linearized theory .why therefore not do it for the in - spiraling collision ?the main reason is that for that case there are no numerical results with which to compare and the linearized formalism does not have a measure of error in it : it therefore has little predictive power .there is no consistent way to say when the close approximation breaks down .in fact , this example teaches us a valuable lesson about perturbation theory : when is linearized perturbation valid ?the obvious answer `` when perturbations are small '' is clearly naive . to begin with , ``small '' should be characterized in a coordinate invariant way . moreover , as this example shows , perturbations can be `` large '' and perturbation theory can still be valid : it just needs to happen that the perturbations be large in regions of spacetime that do not contribute in a significant way to the physics of interest . in the two black hole example , such a region is the interior of the horizon and regions close to it , in which perturbations mostly fall into the black hole .  how is one to characterize when to trust the approximation ? the answer is simple : work out the second order perturbations , compute the physical quantities of interest and use how much the first and second order calculation differ as a measure of the accuracy of the first order results .the advantage of this answer is that it is phrased in terms of what one is exactly interested in : the physical quantities . in the case of the collision of two black holesthese are the radiated waveforms and energies .the formalism for second order perturbations of black holes has not been worked out in the past .it can be studied in detail as we do in reference . herei just sketch some of the outstanding points .it turns out that all the information can be coded into a single variable , exactly as in the first order perturbation case and that that variable satisfies a `` zerilli equation '' , where and the zerilli function is a coordinate invariant combination of the perturbed metric coefficients .this equation is exactly the same as the one satisfied by the first order perturbations ( including the `` potential '' v(r ) , which can be seen in reference ) .however , there is an important difference : the right - hand side is not zero but a `` source '' term s , which is listed explicitly in reference and which is a complicated function quadratic in the first order perturbations and their derivatives .the way in which we derived this equation is to compute a particular combination of the einstein equations , writing the perturbed metric in a particular coordinate system , the so called `` regge - wheeler '' gauge .this , in turn is a way of deriving the original zerilli equation .the expression we get for is therefore a representation in that gauge of a gauge invariant quantity .the explicitly gauge invariant form of can also be computed .we therefore are in a position to evolve to second order the problem of black hole collisions and therefore to endow the first order predictions with `` error bars ''. this will be crucial for the inspiralling case , where numerical results are not expected for some time .in the head - on collision case we were lucky to have an exact solution to the initial value problem that we could evolve .for the more realistic cases there are no exact solutions available at present and it is unlikely that they will be easily found in the future . there is an immediate alternative at hand . there exist already well tested numerical codes for solving the initial value problem in general relativity in the context of black hole collisions .one could simply take these initial data evaluated for the case in which the black holes are close and `` read off '' from them the departures from schwarzschild to be evolved using the linearized theory .this is certainly possible and has already been illustrated for brill - lindquist - type initial data by abrahams and price .apart from the possibility of using numerical initial data for realistic collisions it is interesting to notice that one can , up to a certain extent , solve the initial value problem analytically if one is only interested in initial data for the close approximation .the idea is simple : in the close approximation the initial data for a black hole collision departs a small amount from the initial data for a schwarzschild spacetime for a single black hole with mass equal to the sum of the masses of the colliding holes . therefore one can develop an approximation technique for the initial data starting from the initial data of schwarzschild and adding small corrections proportional to the separation of the holes .we illustrate here only the zeroth order results , details will be given in a forthcoming paper in collaboration with john baker .the initial value problem of general relativity can be conveniently cast in the conformal formalism .one is interested in solving the momentum and hamiltonian constraints where is the spatial metric , is the extrinsic curvature and is the scalar curvature of the three metric .one proposes a three metric that is conformally flat , with the conformal factor and a decomposition of the extrinsic curvature .the constraints become , where is a derivative with respect to the flat spacetime . since the momentum constraint is linear, one can propose as a solution for it for the case of two black holes the sum of the solutions for the case of individual holes with momentum , \ ] ] where is a unit normal in the direction of and all vector fields are defined in the flat background spacetime .one now can put this solution in the hamiltonian constraint and one is left with an elliptic , highly non - linear equation for .this is the equation that is usually solved numerically .there exist situations , however , where one can make some progress analytically .consider the case in which the momenta of the holes is small . in that caseone can neglect the right - hand side of the hamiltonian constraint and one only needs to solve a vacuum laplace equation for .the solution can therefore be very simply found , the difficulty depending on the boundary conditions one chooses for the problem ( typically a `` symmetrized '' boundary condition is imposed , which complicates calculation quite a bit in certain cases , see for details ) .another situation in which one can obtain an approximate solution is in the `` close approximation '' . in that case onehas two black holes of momenta equal and opposite , and since the black holes are close , the unit normals appearing in the form for the extrinsic curvature for each hole are approximately equal .that implies that the extrinsic curvature for the problem is approximately zero ( as it should , since in the close limit the problem looks like a schwarzschild black hole at rest . )therefore one can again neglect the right - hand side of the hamiltonian constraint and one is again left with a laplace equation .let us compare this approximation with the full numerical results .in order to do this we will compare the adm energy of initial data for a collision of two holes of momentum .the adm energy in the conformal formalism is given by    and we notice that it does not depend explicitly on the extrinsic curvature ( it does implicitly via the constraints ) .therefore at the approximation we are working , in which the constraints do not couple the conformal factor and the extrinsic curvature , the energy is independent of the extrinsic curvature and therefore independent of the momenta of the holes .we compare this prediction with the full numerical results of cook in figure [ cook ] .an interesting aspect is that one can advance this approximation one step further .one can input the extrinsic curvature and the conformal factor found as a fixed `` source '' in the equation determining the conformal factor and one can obtain a correction through the integration of a poisson equation .comparison of this approximation with the numerical data is currently in progress .details are complicated by the particular boundary conditions that are usually chosen in the numerical computations .it is evident that the `` close approximation '' can work in many other cases , apart from the head - on , equal momenta holes we considered here .the only changes will be that the solution one obtains in the `` close limit '' rather than being a slice of schwarzschild will be a slice of kerr or boosted schwarzschild if the net result of the collision has angular momentum or linear momentum .we have seen that the use of the `` close approximation '' can be a valuable aid to full numerical computations of the collision of two black holes . with the introduction of a second order scheme we are now in a position of offering reliable estimates of energies and waveforms that we expect people working on the full numerical simulations will find of use to calibrate codes and design strategies for better integrating the einstein equations in this problem of great current physical interest .the work described here is in collaboration with richard price , john baker , reinaldo gleiser and oscar nicasio .i acknowledge support of nsf through grants phy94 - 06269 phy93 - 96246 , funds of the pennsylvania state university , its office for minority faculty development , and the eberly family research fund .r. price , j. pullin , phys .rev . lett . * 72 * , 3297 ( 1994 ) . c. misner , phys. rev . * 118 * , 1110 ( 1960 ) .p. anninos , e. seidel , l. smarr , w .- m .suen , phys .lett . * 71 * , 2851 ( 1993 ) .p. anninos , r. price , j. pullin , e. seidel , w .- m .* d * ( in press ) .o. nicasio , r. gleiser , r. price , j. pullin , in preparation .a. abrahams , r. price , in preparation .g. cook , phys . rev . * d44 * , 2983 ( 1991 ) ; * 50 * , 5025 ( 1994 ) . see for instance j. bowen , j. york phys .* d21 * , 2047 ( 1980 ) and references therein .

Abstract: we give a brief summary of results and ongoing research in the application of linearized theory to the study of black hole collisions in the limit in which the holes start close to each other . this approximation can be a valuable tool for comparison and code - checking of full numerical relativity computations . the approximation works quite well for the head - on case and this is motivation to pursue its use in other more interesting contexts . we summarize current efforts towards establishing the domain of validity of the approximation and its use in generation and evolution of initial data for more interesting physical cases .   epsf  cgpg-95/8 - 2 + gr - qc/9508026 +