a one factor multivariate time series model of metropolitan wage rates abstract the paper formulates and estimates a single factor multivariate time series model the model is a dynamic generalization of the multiple indicator or factor analysis model it is shown to be a special case of the general state space model and can be estimated by maximum likelihood methods using the kalman filter algorithm the model is used to obtain estimates of the unobserved metropolitan wage rate for los angeles based on observations of sectoral wages within the standard metropolitan statistical area hypothesis tests model diagnostics and out of sample forecasts are used to evaluate the model

four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences a class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed in each of the models the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix the component scores series reveal the latent data structure in the course of time the interpretation of the components is based on the loading matrix the simultaneous component models model not only intraindividual variability but interindividual variability as well the four models can be ordered hierarchically from weakly to severely constrained thus allowing for big to small interindividual differences in the model the use of the models is illustrated by an empirical example

meaningful design and contrast estimability in fmri optimising the efficiency of an experimental design is known to be of great importance however existing methods for calculating design rank deficiency and contrast estimability an important aspect of experimental design relate to computational precision rather than image noise and are therefore not very meaningful for example a contrast between two experimental conditions may be mathematically estimable while requiring a huge differential bold response for statistical significance to be reached in this paper we formulate standard efficiency equations in terms of required bold effect and use this to generate measures of rank estimability which are meaningful this takes into account the strength and smoothness of the timeseries noise and is applicable to complex contrasts we show how to re express several regressors and an associated contrast vector as a single equivalent regressor so that we can calculate the contrast s effective peak peak height unambiguously we also present some example results on typical designs and characterise noise results from a range of typical fmri acquisitions in order to allow experimenters to apply efficiency estimation in advance of acquiring data

the generalized dynamic factor model consistency and rates abstract a factor model generalizing those proposed by geweke in d j aigner and a s goldberger latent variables in socio economic models north holland amsterdam 1977 sargent and sims new methods in business research federal reserve bank of minneapolis minneapolis 1977 engle and watson j amer statist assoc 76 1981 774 and stock and watson j business econom statist 20 2002 147 has been introduced in forni et al rev econ statist 80 2000 540 where consistent as the number n of series and the number t of observations both tend to infinity along appropriate paths n t n estimation methods for the common component are proposed rates of convergence associated with these methods are obtained here as functions of the paths n t n along which n and t go to infinity these results show that under suitable assumptions consistency requires t n to be at least of the same order as n whereas an optimal rate of n is reached for t n of the order of n2 if convergence to the space of common components is considered consistency holds irrespective of the path t n thus can be arbitrarily slow the optimal rate is still n but only requires t n to be of the order of n

incomplete generalized l statistics given data x 1 x n and a kernel h with m arguments serfling introduced the class of generalized l statistics gl statistics which is defined by taking linear combinations of the ordered h x i1 x im where i 1 i m ranges over all n n m distinct m tuples of 1 n in this paper we derive a class of incomplete generalized l statistics igl statistics by taking linear combinations of the ordered elements from a subset of h x i1 x im with size n n a special case is the class of incomplete u statistics introduced by blom under very general conditions the igl statistic is asymptotically equivalent to the gl statistic as soon as n n n as n which makes the igl much more computationally feasible we also discuss various ways of selecting the subset of h x i x im several examples are discussed in particular some new estimates of the scale parameter in nonparametric regression are introduced it is shown that these estimates are asymptotically equivalent to an igl statistic some extensions for example functionals other than l and multivariate kernels are also addressed

detection power estimation efficiency and predictability in event related fmri experimental designs for event related functional magnetic resonance imaging can be characterized by both their detection power a measure of the ability to detect an activation and their estimation efficiency a measure of the ability to estimate the shape of the hemodynamic response randomized designs offer maximum estimation efficiency but poor detection power while block designs offer good detection power at the cost of minimum estimation efficiency periodic single trial designs are poor by both criteria we present here a theoretical model of the relation between estimation efficiency and detection power and show that the observed trade off between efficiency and power is fundamental using the model we explore the properties of semirandom designs that offer intermediate trade offs between efficiency and power these designs can simultaneously achieve the estimation efficiency of randomized designs and the detection power of block designs at the cost of increasing the length of an experiment by less than a factor of 2 experimental designs can also be characterized by their predictability a measure of the ability to circumvent confounds such as habituation and anticipation we examine the relation between detection power estimation efficiency and predictability and show that small increases in predictability can offer significant gains in detection power with only a minor decrease in estimation efficiency

efficient design of event related fmri experiments using m sequences rapid event related fmri erfmri allows estimation of the shape of hemodynamic responses hdr associated with transient brain activation evoked by various sensory motor and cognitive events choosing a sequence of events that maximizes efficiency of estimating the hdr is essential for conducting event related brain imaging experiments since increasing efficiency is essentially equivalent to reducing scanning time or increasing the strength of the principal magnetic field the efficiency of an erfmri design depends critically on the temporal arrangement of the sequence of events and the noise in the fmri signal we introduce to erfmri a simple method for generating efficient event sequences based on maximum length shift register sequences or m sequences we show that under the assumption of white uncorrelated mri noise efficiency of erfmri experimental designs that employ m sequences exceeds efficiency of the best randomly generated sequences this is true for single and multiple event type experiments which allow either parallel events overlapping events design or designs in which only one event occurs at a time nonoverlapping events design hdr estimation efficiency afforded by m sequences grows with the number of event types and is greatest when event sequences are relatively short albeit within commonly used scan times i e 63 255 total events per scan the improvement in efficiency however comes at a cost of constraints imposed by m sequence generation rules such as predetermined sequence lengths for nonoverlapping events design m sequence based designs are not available for all possible numbers of event types nevertheless designs that are available with m sequences cover a large subset of commonly used erfmri experimental designs under conditions of characteristic time correlated fmri noise randomly generated sequences may yield efficiencies that exceed those afforded by m sequences for single event type designs since in this case one can generate random sequences that partially decorrelate mri noise by chance our simulations suggest that for designs of realistic sequence lengths that use more than one event type m sequence based designs tend to outperform random designs thus making the knowledge of noise inessential finally within an r th order m sequence generated by a shift register of length r all possible combinations of subsequences of length r occur and thus these subsequences are exactly counterbalanced this property is essential for minimizing effects of psychological and neuronal adaptation and expectation

optimal design of multi subject blocked fmri experiments the design of a multi subject fmri experiment needs specification of the number of subjects and scanning time per subject for example for a blocked design with conditions a or b fixed block length and block order abn where n denotes a null block the optimal number of cycles of abn and the optimal number of subjects have to be determined this paper presents a method to determine the optimal number of subjects and optimal number of cycles for a blocked design based on the a optimality criterion and a linear cost function by which the number of cycles and the number of subjects are restricted estimation of individual stimulus effects and estimation of contrasts between stimulus effects are both considered the mixed effects model is applied and analytical results for the a optimal number of subjects and a optimal number of cycles are obtained under the assumption of uncorrelated errors for correlated errors with a first order autoregressive ar1 error structure numerical results are presented our results show how the optimal number of cycles and subjects depend on the within to between subject variance ratio our method is a new approach to determine the optimal scanning time and optimal number of subjects for a multi subject fmri experiment in contrast to previous results based on power analyses the optimal number of cycles and subjects can be described analytically and costs are considered

characterization of event related designs using bold and iron fmri abstract despite many desirable characteristics event related er stimulus designs for bold and iron suffer from low detection power relative to block designs because the hemodynamic impulse response function irf acts as a low pass filter on neural activation to attenuate the size of differential responses to alternate stimuli while the use of exogenous contrast agent iron technique provides an alternative fmri method in animal models to improve sensitivity and spatial localization the inherently slower hemodynamic irf causes iron detection efficiency to decrease faster than bold efficiency as the interstimulus interval isi is shortened using simulations based upon assumptions of stimulus response linearity and experimental data obtained in awake non human primates this study compared detection efficiencies for fixed random and semi random isi distributions for bold and iron techniques a larger relative gain in detection efficiency at short isi was obtained by randomized designs using iron contrast relative to bold contrast due to the slower irf of the iron method to quantify tradeoffs between detection efficiency and the predictability of stimulus presentation the shannon entropy was introduced as an objective measure of predictability small amounts of entropy can be traded for large gains in efficiency particularly for the iron method

efficiency power and entropy in event related fmri with multiple trial types part i theory experimental designs for functional magnetic resonance imaging fmri experiments can be characterized by their estimation efficiency which is a measure of the variance in the estimate of the hemodynamic response function hrf and their detection power which is a measure of the variance in the estimate of the amplitude of functional activity previous studies have shown that there exists a fundamental trade off between efficiency and power for experiments with a single trial type of interest this paper extends the prior work by presenting a theoretical model for the relation between detection power and estimation efficiency in experiments with multiple trial types it is shown that the trade off between efficiency and power present in multiple trial type experiments is identical in form to that observed for single trial type experiments departures from the predicted trade off due to the inclusion of basis function expansions and the assumption of correlated noise are examined finally conditional entropy is introduced as measure for the randomness of a design and an empirical relation between entropy and estimation efficiency is presented

improved detection of event related functional mri signals using probability functions selecting an optimal event distribution for experimental use in event related fmri studies can require the generation of large numbers of event sequences with characteristics hard to control the use of known probability distributions offers the possibility to control event timing and constrain the search space for finding optimal event sequences we investigated different probability distributions in terms of response estimation estimation efficiency detectability detection power parameter estimation efficiency sensitivity to true positives and false positive activation numerous simulated event sequences were generated selecting interevent intervals iei from the uniform uniform permuted latin square exponential binomial poisson chi 2 geometric and bimodal probability distributions and fixed iei event sequences from the bimodal distribution like block designs had the best performance for detection and the poorest for estimation while high estimation and detectability occurred for the long decay exponential distribution the uniform distribution also yielded high estimation efficiency but probability functions with a long tail toward higher iei such as the geometric and the chi 2 distributions had superior detectability the distributions with the best detection performance also had a relatively high incidence of false positives in contrast to the ordered distributions latin square and uniform permuted the predictions of improved sensitivities for distributions with long tails were confirmed with empirical data moreover the latin square design yielded detection of activated voxels similar to the chi 2 distribution these results indicate that high detection and suitable behavioral designs have compatibility for application of functional mri methods to experiments requiring complex designs

efficiency power and entropy in event related fmri with multiple trial types part ii design of experiments the performance of an experimental design for functional magnetic resonance imaging fmri can be characterized by its estimation efficiency which is the ability to make an estimate of the hemodynamic response its detection power which is the ability to detect an activation and its conditional entropy which is a measure of the randomness of the design in liu and frank neuroimage 21 2004 387 400 it is shown that there is a fundamental theoretical trade off between estimation efficiency and detection power for experiments with multiple trial types and that there is an empirical relation between estimation efficiency and conditional entropy this paper provides an intuitive interpretation of the theoretical results and examines the practical implications of these results for the optimal design of fmri experiments with multiple trial types the properties of block designs permuted block designs m sequence designs clustered m sequence designs and mixed designs are explored it is shown that these designs nearly achieve the theoretically predicted performance and can be used in practice to obtain advantageous trade offs among efficiency power and entropy

estimating efficiency a priori a comparison of blocked and randomized designs this technical note deals with a priori estimation of efficiency of functional magnetic resonance imaging fmri designs the efficiency of an estimator is a measure of how reliable it is and depends on error variance the variance not modeled by explanatory variables in the design matrix and the design variance a function of the explanatory variables and the contrast tested changes in the experimental design can induce changes in the variance of estimated responses this translates into changes in the standard error of the response estimate or equivalently into changes in efficiency one consequence is that statistics testing for the same activation in different contexts i e experimental designs can change substantially even if the activation and error variance are exactly the same we demonstrate this effect using an event related fmri study of single word reading during blocked and randomized trial presentations furthermore we show that the error variance can change with the experimental design this highlights a problem with a priori comparison of efficiency for two or more experimental designs which usually assumes identical error variance

signal processing applications of oblique projection operators oblique projection operators are used to project measurements onto a low rank subspace along a direction that is oblique to the subspace they may be used to enhance signals while nulling interferences in the paper the authors give several basic results for oblique projections including formulas for constructing oblique projections with desired range and null space they analyze the algebra and geometry of oblique projections in order to understand their properties they then show how oblique projections can be used to separate signals from structured noise such as impulse noise damped or undamped interfering sinusoids such as power line interference and narrow band noise in some of the problems addressed the oblique projection provides an alternative way to implement an already known solution expressing these solutions as oblique projections brings geometrical insight to the study of the solution the geometry of oblique projections enables one to compute performance in terms of angles between signal and noise subspaces as a special case of removing impulse noise the authors can use oblique projections to interpolate missing data samples in array processing oblique projections can be used to simultaneously steer beams and nulls in communications oblique projections can be used to remove intersymbol interference

robustness of optimal design of fmri experiments with application of a genetic algorithm in this paper we apply the genetic algorithm developed by kao et al 2009 to find designs which are robust against misspecification of the error autocorrelation two common optimality criteria the a optimality criterion and the d optimality criterion based upon a general linear model are employed to obtain locally optimal designs for a given value of the autocorrelation the maximin criterion is then used to obtain designs which are robust against misspecification of the autocorrelation furthermore robustness depending on the choice of optimality criterion is evaluated we show analytically and empirically that the a and d optimality criterion will result in different optimal designs e g with different stimulus frequencies optimal stimulus frequency for the a optimality criterion has been derived by liu et al 2004 whereas we derive here the optimal stimulus frequency for the d optimality criterion conclusions about the robustness of an optimal design against misspecification of model parameters and choice of optimality criterion are drawn based upon our results

an alternative approach towards assessing and accounting for individual motion in fmri timeseries abstract motion is a significant problem for the analysis of functional mri data this manuscript addresses the question of whether an individualized assessment of motion may be informative and whether it may be beneficial with regard to explaining motion related variance two independent datasets are used to explore and test this hypothesis from a total of 21 healthy children performing either no externally cued task resting state or an active listening paradigm beep story translations and rotations are combined into one single individual measure of total displacement which is demonstrated to be substantially different between brain regions as a function of their distance from the individual origin an increasing number of covariates leads to a loss of detection power but more so on the first than on the second level and more so in less powerful designs synthetic timeseries are calculated from which the direct effects of motion as well as motion b0 effects can be isolated allowing to extract individual timecourses which reflect both direct and indirect motion effects including three timecourses from such an individually derived motion fingerprint into first level statistical analyses explains variance to a similar degree as the commonly used approach of including the realignment parameters and performance is statistically equivalent to including the realignment parameters on the second level a more individualized approach to explaining motion related variance may therefore be beneficial depending on the scenario

matched subspace detectors we formulate a general class of problems for detecting subspace signals in subspace interference and broadband noise we derive the generalized likelihood ratio glr for each problem in the class we then establish the invariances for the glr and argue that these are the natural invariances for the problem in each case the glr is a maximal invariant statistic and the distribution of the maximal invariant statistic is monotone this means that the glr test glrt is the uniformly most powerful invariant detector we illustrate the utility of this finding by solving a number of problems for detecting subspace signals in subspace interference and broadband noise in each case we give the distribution for the detector and compute performance curves

the special sign indeterminacy of the direct fitting parafac2 model some implications cautions and recommendations for simultaneous component analysis parafac2 is the most flexible simultaneous component analysis sca model that produces an essentially unique solution in this paper we discuss how parafac2 s special sign indeterminacy affects applications of sca and we reveal how an external criterion variable can be used to ensure that estimated parafac2 weights are meaningfully signed across the levels of the nesting mode we present an example with real data from clinical psychology that illustrates the importance of parafac2 s special sign indeterminacy as well as the effectiveness of our proposed solution we also discuss the implications of our results for general applications of sca

estimating densities of functions of observations abstract density estimates such as histograms and more sophisticated versions are important in applied and theoretical statistics in applied statistics a density estimate gives the data analyst a graphical overview of the shape of the distribution this overview allows the data analyst to arrive immediately at a qualitative impression of the location scale and various aspects of the extremes of the distribution in theoretical statistics the shape of the density allows the researcher to link the data to families of curves perhaps indexed parametrically by estimating a density nonparametrically certain aspects of the data can be viewed without the restriction of a priori imposing limitations of a class of parametric curves in this article we introduce density estimation for functions of observations to motivate the study one type of function used is the interpoint distance between observations arising in spatial statistics from the fields of biometry and regional science a second type of func

multi objective optimal experimental designs for event related fmri studies in this article we propose an efficient approach to find optimal experimental designs for event related functional magnetic resonance imaging er fmri we consider multiple objectives including estimating the hemodynamic response function hrf detecting activation circumventing psychological confounds and fulfilling customized requirements taking into account these goals we formulate a family of multi objective design criteria and develop a genetic algorithm based technique to search for optimal designs our proposed technique incorporates existing knowledge about the performance of fmri designs and its usefulness is shown through simulations although our approach also works for other linear combinations of parameters we primarily focus on the case when the interest lies either in the individual stimulus effects or in pairwise contrasts between stimulus types under either of these popular cases our algorithm outperforms the previous approaches we also find designs yielding higher estimation efficiencies than m sequences when the underlying model is with white noise and a constant nuisance parameter the stimulus frequencies of the designs we obtained are in good agreement with the optimal stimulus frequencies derived by liu and frank 2004 neuroimage 21 387 400 in addition our approach is built upon a rigorous model formulation

analysis and design of perfusion based event related fmri experiments perfusion based functional magnetic resonance imaging fmri using arterial spin labeling asl methods has the potential to provide better localization of the functional signal to the sites of neural activity compared to blood oxygenation level dependent bold contrast fmri at present experiments using asl have been limited to simple block and periodic single trial designs we present here an adaptation of the general linear model to perfusion based fmri that enables the design and analysis of more complicated designs such as random and semirandom event related designs formulas for the least squares estimate of the perfusion response and the f statistic for the detection of a response are derived exact expressions and useful approximations for detection power and estimation efficiency are presented and it is shown that the trade off between power and efficiency for perfusion experiments is similar to that previously observed for bold experiments the least squares estimate is compared with an estimate formed from the running subtraction of tag and control images the running subtraction estimate is shown to be approximately equal to a temporally low pass filtered version of the least squares estimate numerical simulations and results from asl experiments are used to support the theoretical findings

glivenko cantelli properties of some generalized empirical df s and strong convergence of generalized l statistics we study a nonclassical form of empirical df hnwhich is of u statistic structure and extend to hnthe classical exponential probability inequalities and glivenko cantelli convergence properties known for the usual empirical df an important class of statistics is given byt hn where t is a generalized form of l functional for such statisticswe prove almost sure convergence using an approach which separates the functional analytic and stochastic components of the problem and handles the latter component by application of glivenko cantelli type properties classical results for u statistics and l statistics are obtained as special cases without addition of unnecessary restrictions many important new types of statistics of current interest are covered as well by our result

optimization of blocked designs in fmri studies blocked designs in functional magnetic resonance imaging fmri are useful to localize functional brain areas a blocked design consists of different blocks of trials of the same stimulus type and is characterized by three factors the length of blocks i e number of trials per blocks the ordering of task and rest blocks and the time between trials within one block optimal design theory was applied to find the optimal combination of these three design factors furthermore different error structures were used within a general linear model for the analysis of fmri data and the maximin criterion was applied to find designs which are robust against misspecification of model parameters

generalized l m and r statistics abstract a class of statisticss generalizing u statistics and l statistics and containing other varieties of statistics as well such as trimmed u statistics is studied using the differentiable statistical function approach differential approximations are obtained and the influence curves of these generalized l statistics are derived these results are employed to establish asymptotic normality for such statistics parallel generalizations of m and r statistics are noted strong convergence berry esseen rates and computational aspects are discussed

the impact of temporal regularization on estimates of the bold hemodynamic response function a comparative analysis in fmri data analysis it has been shown that for a wide range of situations the hemodynamic response function hrf can be reasonably characterized as the impulse response function of a linear and time invariant system an accurate and robust extraction of the hrf is essential to infer quantitative information about the relative timing of the neuronal events in different brain regions when no assumptions are made about the hrf shape it is most commonly estimated using time windowed averaging or a least squares estimated general linear model based on either fourier or delta basis functions recently regularization methods have been employed to increase the estimation efficiency of the hrf typically these methods produce more accurate hrf estimates than the least squares approach goutte c nielsen f a hansen l k 2000 modeling the haemodynamic response in fmri using smooth fir filters ieee trans med imag 19 12 1188 1201 here we use simulations to clarify the relative merit of temporal regularization based methods compared to the least squares methods with respect to the accuracy of estimating certain characteristics of the hrf such as time to peak ttp height hr and width w of the response we implemented a bayesian approach proposed by marrelec et al marrelec g benali h ciuciu p pelegrini issac m poline j b 2003 robust estimation of the hemodynamic response function in event related bold fmri using basic physiological information hum brain mapp 19 1 17 marrelec g benali h ciuciu p poline j b bayesian estimation of the hemodynamic of the hemodynamic response function in functional mri in r f editor 2001 melville p 229 247 and its deterministic counterpart based on a combination of tikhonov regularization tikhonov a n arsenin v y 1977 solution of ill posed problems washington dc w h winston and generalized cross validation gcv wahba g 1990 spline models for observational data philadelphia siam for selecting the regularization parameter the performance of both methods is compared with least square estimates as a function of temporal resolution color and strength of the noise and the type of stimulus sequences used in almost all situations under the considered assumptions e g linearity time invariance and smooth hrf the regularization based techniques more accurately characterize the hrf compared to the least squares method our results clarify the effects of temporal resolution noise color and experimental design on the accuracy of hrf estimation

raw data maximum likelihood estimation for common principal component models a state space approach the specifications of state space model for some principal component related models are described including the independent group common principal component cpc model the dependent group cpc model and principal component based multivariate analysis of variance some derivations are provided to show the equivalence of the state space approach and the existing wishart likelihood approach for each model a numeric example is used to illustrate the state space approach in addition a simulation study is conducted to evaluate the standard error estimates under the normality and nonnormality conditions in order to cope with the nonnormality conditions the robust standard errors are also computed finally other possible applications of the state space approach are discussed at the end

a comparison of methods for characterizing the event related bold timeseries in rapid fmri abstract information about the shape and temporal duration of the blood oxygenation level dependent bold response can inform both functional neuroanatomy and psychological theory however the bold response evolves over 20 s or more making it difficult to distinguish the unique characteristics of the response evoked by temporally adjacent stimuli fortunately event related bold signals can be extracted given that there is adequate variance in the distribution of inter stimulus intervals isi unfortunately the isi distribution that yields the highest statistical efficiency is not always optimal from a psychological perspective variability in the stimulus timing may complicate the interpretation of neuroimaging data in terms of underlying cognitive operations in the present paper monte carlo simulations are used to evaluate two techniques for estimating the event related bold timeseries event related averaging and deconvolution using the ordinary least squares estimate with respect to maintaining acceptable levels of statistical power and experimental validity while the unbiased deconvolution technique more robustly estimates the shape of the bold response functions both methods succeed in accurately re producing known differences between evoked bold responses when the stimulus ordering is randomized however the deconvolution method is more effective at preserving differences when there are sequential dependencies in the stimulus presentation order and restricted isi distributions are used particularly if the second of two sequentially dependent stimuli is omitted on some portion of the trials importantly the successful re production of the evoked bold response using restricted isi distributions often maximizes the ability to make psychologically valid experimental conclusions

regularized generalized canonical correlation analysis for multiblock or multigroup data analysis this paper presents an overview of methods for the analysis of data structured in blocks of variables or in groups of individuals more specifically regularized generalized canonical correlation analysis rgcca which is a unifying approach for multiblock data analysis is extended to be also a unifying tool for multigroup data analysis the versatility and usefulness of our approach is illustrated on two real datasets

modeling differences in the dimensionality of multiblock data by means of clusterwise simultaneous component analysis given multivariate multiblock data e g subjects nested in groups are measured on multiple variables one may be interested in the nature and number of dimensions that underlie the variables and in differences in dimensional structure across data blocks to this end clusterwise simultaneous component analysis sca was proposed which simultaneously clusters blocks with a similar structure and performs an sca per cluster however the number of components was restricted to be the same across clusters which is often unrealistic in this paper this restriction is removed the resulting challenges with respect to model estimation and selection are resolved