Error Covariance Structure Research Articles

Comparison of ensemble‐based data assimilation methods for sparse oceanographic data

AbstractProbabilistic forecasts in oceanographic applications, such as drift trajectory forecasts for search‐and‐rescue operations, face challenges due to high‐dimensional complex models and sparse spatial observations. We discuss localisation strategies for assimilating sparse point observations and compare the implicit equal‐weights particle filter and a localised version of the ensemble‐transform Kalman filter. First, we verify these methods thoroughly against the analytic Kalman filter solution for a linear advection diffusion model. We then use a nonlinear simplified ocean model to do state estimation and drift prediction. The methods are rigorously compared using a wide range of metrics and skill scores. Our findings indicate that both methods succeed in approximating the Kalman filter reference for linear models of moderate dimensions, even for small ensemble sizes. However, in high‐dimensional settings with a nonlinear model, we discover that the outcomes are significantly influenced by the dependence of the ensemble Kalman filter on relaxation and the particle filter's sensitivity to the chosen model error covariance structure. Upon proper relaxation and localisation parametrisation, the ensemble Kalman filter version outperforms the particle filter in our experiments.

Design of Hotelling T2 control chart using various covariance structures

The objective of this research is to evaluate different serial correlated structures in Hotelling T2 and Shewhart control charts in monitoring multivariate statistical process control. For this purpose, the false alarm rate and shift detection performance were calculated by considering different autocorrelations in the presence of three covariance matrices for the errors. Both simulated data and real data are used for validating the analysis. The result of the simulation shows that the false alarm rate decreases with the increase in the magnitude of autocorrelation in all three low, moderate, and high error covariance structures. In real data, two quality variables of the production process are used. By ignoring autocorrelation in real data, it is concluded that ARL0 shows the process in controlling on the average for nominal value. The performance of Hotelling T 2 in ARL 1 varies as the shift changes only in x1 otherwise ARL1 have small values as compared to the individual Shewhart chart by increasing shifts in both variables.

A XCO2 Retrieval Algorithm Coupled Spatial Correlation for the Aerosol and Carbon Detection Lidar

We describe the approach to estimating the atmospheric carbon dioxide (CO2) for the Aerosol and Carbon Detection Lidar (ACDL) onboard the Atmospheric Environment Monitoring Satellite (AEMS). The method estimates the optimal state vector by maximizing the measurement posterior probability under a given prior state vector probability distribution. A priori constraint considering the spatial correlation is used as regularization to solve the ill-posed problem. We ran a series of observing system simulation experiments to demonstrate the critical outcome and character percentage uncertainty reduction. The results show that the state vector uncertainty can be reduced by ∼ 10% near the surface for the single sounding. The CO2 column-averaged dry air mole fraction (XCO2) derived by this algorithm is more stable than that obtained by the conventional algorithm and enables the monitoring of concentration changes for the multiple soundings. Similar to the Total Carbon Column Observing Network (TCCON), the averaging kernel is also provided for the subsequent flux inversion. Our simulation experiments demonstrate that the structure of the prior error covariance plays an important role in revealing vertical information from observations. In addition, we applied this algorithm to an airborne ACDL experiment for the retrieval of atmospheric CO2 over Bohai Bay on March 14, 2019. AEMS’s observations with a small footprint will yield important information on the carbon cycle, especially for small but strong emission sources.

How economic weights translate into genetic and phenotypic progress, and vice versa

BackgroundThis paper highlights the relationships between economic weights, genetic progress, and phenotypic progress in genomic breeding programs that aim at generating genetic progress in complex, i.e., multi-trait, breeding objectives via a combination of estimated breeding values for different trait complexes.ResultsBased on classical selection index theory in combination with quantitative genetic models, we provide a methodological framework for calculating expected genetic and phenotypic progress for all components of a complex breeding objective. We further provide an approach to study the sensitivity of the system to modifications, e.g. to changes in the economic weights. We propose a novel approach to derive the covariance structure of the stochastic errors of estimated breeding values from the observed correlations of estimated breeding values. We define ‘realized economic weights’ as those weights that would coincide with the observed composition of the genetic trend and show, how they can be calculated. The suggested methodology is illustrated with an index that aims at achieving a breeding goal composed of six trait complexes, that was applied in German Holstein cattle breeding until 2021.ConclusionsBased on the presented results, the main conclusions are (i) the composition of the observed genetic progress matches the expectations well, with predictions being slightly better when the covariance of estimation errors is taken into account; (ii) the composition of the expected phenotypic trend deviates significantly from the expected genetic trend due to the differences in trait heritabilities; and (iii) the realized economic weights derived from the observed genetic trend deviate substantially from the predefined ones, in one case even with a reversed sign. Further results highlight the implications of the change to a modified breeding goal based on the example of a new index comprising eight, partly new, trait complexes, which is used since 2021 in the German Holstein breeding program. The proposed framework and the analytical tools and software provided will be useful to define more rational and generally accepted breeding objectives in the future.

Seemingly Unrelated Regression Model to Measure the Profitability of Some Iraqi Private Commercial Banks with Presence of Outliers

A seemingly uncorrelated regression (SUR) model is a special case of multivariate models, in which the error terms in these equations are contemporaneously related. The method estimator (GLS) is efficient because it takes into account the covariance structure of errors, but it is also very sensitive to outliers. The robust SUR estimator can dealing outliers. We propose two robust methods for calculating the estimator, which are (S-Estimations, and FastSUR). We find that it significantly improved the quality of SUR model estimates. In addition, the results gave the FastSUR method superiority over the S method in dealing with outliers contained in the data set, as it has lower (MSE and RMSE) and higher (R-Squared and R-Square Adjusted) values.

Mixed Multinomial Probit Model Accommodating Flexible Covariance Structure and Random Taste Variation: An Application to Commute Mode Choice Behavior

This paper developed a mixed multinomial probit (MMNP) model with alternative error specification and random coefficients (for both generic variables and personal attributes) to accommodate flexible covariance structure and taste variation. The MMNP model can be efficiently estimated with analytic approximations of multivariate normal cumulative distribution functions, which avoid defects of simulation-based integration in the mixed multinomial logit (MMNL) model. The integral dimension of the MMNL model increases as random coefficients increase, but it only depends on the number of available alternatives in the MMNP model. Simulation experiments and empirical analysis of Shanghai commuters’ mode choice behavior were undertaken to examine the performance of MMNP models. Both simulation results and empirical results show that MMNP models can well accommodate flexible covariance structures and taste variation reflected through random coefficients being associated with both generic and personal variables. Empirical results indicate that the MMNP model performs better than traditional discrete choice models, such as the multinomial logit, the cross-nested logit, MMNL, and multinomial probit models. Random coefficients of “in-vehicle time of car” and “number of companions” indicate taste heterogeneity and the identifiability of random coefficients associated with both generic and personal attributes. Pairwise positive correlations between car/taxi, bus/metro, and bus/bus and metro are to be expected. However, the positive correlation between the car and metro modes may be unique to the Chinese city, Shanghai, because of the developed metro system. Unequal error variances reflect heterogeneities in unspecified factors in commute modes’ utilities. The MMNP model will offer an alternative efficient way to accommodate taste heterogeneity and flexible error covariance structure in discrete choice models. Compared with the MMNL model, the MMNP model can accommodate more random coefficients without increasing computational complexity.

The impact of hybrid oceanic data assimilation in a coupled model: A case study of a tropical cyclone

AbstractTropical cyclones tend to result in distinctive spatial and temporal characteristics in the upper ocean, which suggests that traditional, parametrisation‐based background‐error covariances in oceanic data assimilation (DA) may not be suitable. Using the case study of Cyclone Titli, which affected the Bay of Bengal in October 2018, we explore hybrid methods that combine the traditional covariance modelling strategy used in variational methods with flow‐dependent estimates of the ocean's error covariance structures based on a short‐range ensemble forecast. This hybrid approach is investigated in the UK Met Office's state‐of‐the‐art system. Single‐observation experiments in the ocean reveal that the hybrid approach is capable of producing analysis increments that are time‐varying, more anisotropic and vertically less uniform. When the hybrid oceanic covariances are incorporated into a weakly coupled DA system, the sea‐surface temperature (SST) in the path of the cyclone is changed, not only through the different specifications of background‐error covariances used in the SST assimilation, but also through the propagation of subsurface temperature differences to the surface as a result of vertical mixing associated with the cyclone's strong winds. The coupling with the atmosphere then leads to a discrepancy in the cyclone's central pressure, which brings forth further SST differences due to the different representations of the cyclone's emerging cold wake.

A Variational Method for Analyzing Vortex Flows in Radar-Scanned Tornadic Mesocyclones. Part III: Sensitivities to Vortex Center Location Errors

Abstract When the vortex center location is estimated from a radar-scanned tornadic mesocyclone, the estimated location is not error-free. This raises an important issue concerning the sensitivities of analyzed vortex flow (VF) fields by the VF-Var (formulated in Part I of this paper series and tested in Part II) to vortex center location errors, denoted by Δxc. Numerical experiments are performed to address this issue with the following findings: The increase of |Δxc| from zero to a half of vortex core radius causes large analysis error increases in the vortex core but the increased analysis errors decrease rapidly away from the vortex core especially for dual-Doppler analyses. The increased horizontal-velocity errors in the vortex core are mainly in the Δxc-normal component, because this component varies much more rapidly than the other component along the Δxc direction in the vortex core. The vertical variations of Δxc distort the vertical correlation structure of Δxc-dislocated VF-dependent background error covariance, which can increase the analysis errors in the vortex core. The dual-Doppler analyses have adequate accuracies outside the vortex core even when |Δxc| increases to a half of vortex core radius, while single-Doppler analyses can also have adequate accuracies outside the vortex core mainly for the single-Doppler-observed velocity component. The sensitivities to Δxc are largely unaffected by the vortex slanting. The above findings are important and useful for assessing the accuracies of analyzed VFs for real radar-observed tornadic mesocyclones. Significance Statement When the vortex center location is estimated from a radar-scanned tornadic mesocyclone, the estimated location is not error-free. This raises an issue concerning the sensitivity of analyzed vortex flow (VF) by the VF-Var (formulated in Part I of this paper series and tested with simulated radar observations in Part II) to vortex center location error. This issue and its required investigations are very important for the VF-Var to be applied to real radar-observed tornadic mesocyclones, especially in an operational setting with the WSR-88Ds. Numerical experiments are performed to address this issue. The findings from these experiments are important and useful for assessing the accuracies of VF-Var analyzed VF fields for real radar-observed tornadic mesocyclones.

Generalised non-negative matrix factorisation for air pollution source apportionment

Source Apportionment (SA) techniques are widely used for identifying key sources of air pollution, thereby providing critical inputs for policy measures. Positive Matrix Factorisation (PMF) (Paatero and Tapper, 1994) is a widely used SA technique. PMF uses the speciated concentration data (X) collected over several days and factorises it into source contribution (G) and source profile (F) matrices, albeit under positivity constraint. Towards this end, it involves solving an optimisation problem where the elements of X are weighted by the inverse of the standard deviations of the corresponding errors introduced during the sampling and chemical analysis process. Thus, PMF implicitly assumes that the errors in different elements of the X matrix are uncorrelated. This assumption may not hold since the sampling, and chemical analysis steps deployed in any data-collection campaign will inevitably lead to correlated errors. While there are other existing Non-Negative Matrix Factorisation (NMF) methods in literature that can be potentially used for SA, these also make various restrictive assumptions about the error covariance structure. In this work, we propose a new method called Generalised Non-Negative Matrix Factorisation (GNMF) to fill this gap. In particular, the proposed method is able to incorporate any error covariance matrix without making any restrictive assumptions on its structure. Towards this end, we integrate the full error covariance matrix in the objective function to be minimised to obtain F and G matrices. We derive the corresponding update rules for obtaining these matrices iteratively. To ensure non-negativity, we extend the multiplicative and projected gradient-based ideas available in NMF literature to the proposed GNMF approach. The proposed method subsumes various NMF methods available in literature as special cases. The utility of the proposed approach is demonstrated by comparing its performance with other methods on an SA problem using a dataset derived from field measurements.

Satellite Constraints on the Latitudinal Distribution and Temperature Sensitivity of Wetland Methane Emissions

AbstractWetland methane (CH4) emissions comprise about one‐third of the global CH4 source. The latitudinal distribution and climate sensitivity of wetland CH4 fluxes are the key determinants of the global CH4‐climate feedback. However, large differences exist between bottom‐up estimates, informed by ground‐based flux measurements, and top‐down estimates derived from spaceborne total column CH4. Despite the extensive coverage of satellite CH4 concentration observations, challenges remain with using top‐down estimates to test bottom‐up models, mainly because of the uncertainties in the satellite retrievals, the model representation errors, the variable prior emissions, and the confounding role of the posterior error covariance structures. Here, we use satellite‐based top‐down CH4 flux estimates (2010–2012) to test and refine 42 bottom‐up estimates of wetland emissions that use a range of hypothesized wetland extents and process controls. Our comparison between bottom‐up models and satellite‐based fluxes innovatively accounts for cross‐correlations and spatial uncertainties typically found in top‐down inverse estimates, such that only the information from satellite observations and the atmospheric transport model is kept as a constraint. We present a satellite‐constrained wetland CH4 ensemble product derived from assembling the highest‐performance bottom‐up models, which estimates global wetland CH4 emissions of 148 (117–189, 5th–95th percentile) Tg CH4 yr−1. We find that tropical wetland emissions contribute 72% (63%–85%) to the global wetland total. We also find that a lower‐than‐expected temperature sensitivity agrees better with atmospheric CH4 measurements. Overall, our approach demonstrates the potential for using satellites to quantitatively refine bottom‐up wetland CH4 emission estimates, their latitudinal distributions, and their sensitivity to climate.

144 Temporal and Bidirectional Associations between Objectively Measured Physical Activity and Sleep in Preschoolers

Abstract Introduction Physical activity (PA) and sleep contribute to overall health in early childhood. To explore the interactive relationships of these behaviors in older children and adults, previous studies have examined temporal between- and within-person associations through micro-longitudinal designs. However, such analyses have not been conducted in early childhood, when behaviors are guided by adult caregivers. The purpose of this analysis was to examine temporal and bidirectional associations between SED, PA, and sleep in preschool children. Methods Wake (activity counts/min and percent time in SED, light PA [LPA], and moderate-to-vigorous PA [MVPA]) and overnight sleep (sleep duration, sleep efficiency [SE], mid-sleep point [MSP]) were assessed via wrist-based actigraphy (mean = 10.4 days and 9.8 nights) and recorded as repeated (daily) measures. Multilevel models with lagged effects and AR(1) error covariance structure were used to examine the temporal associations between wake and sleep measures and adjusted for age, sex, socioeconomic status, and nap frequency. Results With PA measures as predictors, between-person associations were positive between activity counts and SE (p=0.004), SED and SE (p=0.004), LPA and sleep duration (p=0.005), and negative between LPA and MSP (p=0.039) and MVPA and SE (p=0.003). Within-person associations were positive between activity counts and sleep duration (p=0.010), activity counts and SE (p=0.018), MVPA and sleep duration (p=0.003), MVPA and SE (p=0.004), and negative between SED and SE (p=0.034) and LPA and sleep duration (p=0.045). With sleep measures as predictors, associations were positive between sleep duration and LPA (p<0.001) and SE and SED (p=0.008), and negative between MSP and LPA (p=0.009), SE and activity counts (p=0.001), and SE and MVPA (p=0.003). Within-person associations were positive between SE and activity counts (p=0.001) and SE and MVPA (p=0.001), and negative between sleep duration and LPA (p=0.001) and SE and SED (p=0.012). Conclusion Generally, days with higher levels of activity or sleep were not associated with greater subsequent sleep or PA. Conversely, when participants obtained greater PA or sleep compared to their individual average, some beneficial associations were evident. These findings demonstrate some evidence of temporal associations between PA and sleep, although the bidirectional nature was not conclusive. Support (if any) NIH R01 HL111695

Functional Linear Regression: Dependence and Error Contamination

Functional linear regression is an important topic in functional data analysis. It is commonly assumed that samples of the functional predictor are independent realizations of an underlying stochastic process, and are observed over a grid of points contaminated by iid measurement errors. In practice, however, the dynamical dependence across different curves may exist and the parametric assumption on the error covariance structure could be unrealistic. In this article, we consider functional linear regression with serially dependent observations of the functional predictor, when the contamination of the predictor by the white noise is genuinely functional with fully nonparametric covariance structure. Inspired by the fact that the autocovariance function of observed functional predictors automatically filters out the impact from the unobservable noise term, we propose a novel autocovariance-based generalized method-of-moments estimate of the slope function. We also develop a nonparametric smoothing approach to handle the scenario of partially observed functional predictors. The asymptotic properties of the resulting estimators under different scenarios are established. Finally, we demonstrate that our proposed method significantly outperforms possible competing methods through an extensive set of simulations and an analysis of a public financial dataset.

Analysis of color differences in stained contemporary esthetic dental materials

Statement of problemAlthough contemporary analytical methods are available for application to data which exhibit a lack of equality of variances or a lack of normality in the error distribution, little guidance is provided for selecting the methods of data handling and analysis which best fit color difference data for stained esthetic materials. PurposeThe purposes of this in vitro study were to apply information criteria of analysis of variance (ANOVA) methods of differing error distributions and covariance structures when analyzing color differences to determine the degree of alienation among 3 Commission Internationale de l’Eclairage (CIE) color difference formulae to assess the linearity of relationships among these formulae and to independently assess differences among various computer-aided design and computer-aided manufacture (CAD-CAM) materials in any color change after common forms of staining over time. Material and methodsHybrid ceramic, resin nanoceramic, feldspathic-ceramic, and lithium-disilicate ceramic specimens (N=128) were subjected to staining from water, tea, coffee, and red wine over 1, 7, and 30 days, with color differences calculated from baseline. Akaike information criteria (AIC) and Bayesian information criteria (BIC) values were determined for Gaussian and lognormal error distributions at covariance structures of standard variance components and compound-symmetry. The analysis of variance used to analyze any significant effects on these color differences was the one with the lowest AIC and BIC values. Then, for each solution, day, and CIE color difference formula, any significant difference in the color differences between all pairs of materials was found by Bonferroni-corrected Student t tests. Those statistically significant pairwise comparisons where the larger of the color differences met or exceeded the acceptability threshold were labeled as statistically and visually noteworthy. ResultsFor this color difference data set, the lognormal error distribution and the covariance structure of compound symmetry provided the best AIC and BIC. Because the interaction between material, solution, and day was statistically significant (P<.001), pairwise comparisons were made between all pairs of materials for each level of solution and day studied. Noteworthy differences were identified, where hybrid ceramic and resin nanoceramic each had color changes after staining in coffee and red wine that were greater than each of feldspathic-ceramic and lithium-disilicate ceramic. ConclusionsAIC and BIC values evaluate distinctively the Gaussian and lognormal error distributions when analyzing highly varying color differences. Although there is a high linear correlation between the 3 color difference formulae studied, each formula is unique, and each represents a different assessment of the perceived color difference. CAD-CAM materials, staining liquids, and time points affected the notable color changes.

Modeling speed profiles on mountainous freeways using high resolution data

Operating speed profiles represent drivers’ responses to roadway geometry and are widely used to evaluate safety performance of roadway design. To predict operating speed profile, the majority of early research followed a two-step modeling procedure: (1) estimate speeds at start, middle, and end points of road segments, and (2) fill the profile between the points with assumed driver behavior. This sparse-spot-based modeling strategy has been shown to be inadequate for capturing the complex speed changes resulting from the overlapping horizontal and vertical curves on mountainous roads. This paper proposes a high-resolution modeling approach for operating speeds measured in a dense series of equidistant spots along a road. This type of model is more conducive to analysis of mountainous freeway alignments as operating speeds are predicted along the entire roadway. The high-resolution data were obtained, using the Tongji University Driving Simulator, from a simulated section of mountainous freeway. The estimated linear mixed model includes geometric variables representing the road upstream and downstream of each data collection spot. To determine the suitable lengths of the upstream and downstream segments, the data were extracted from several alternative segment lengths, including fixed lengths and varying downstream length accordingly to sight distances. The model with a spherical structure of error covariance, using geometric data extracted from 300-meter upstream and downstream segments, performed the best. An out-of-sample evaluation of the model has the mean absolute error of 3.2 km/h and the root mean square error of 4.2 km/h, which indicates a promising prediction ability of the proposed model.

Evaluating Ensemble Kalman Filter Analyses of Severe Hailstorms on 8 May 2017 in Colorado: Effects of State Variable Updating and Multimoment Microphysics Schemes on State Variable Cross Covariances

Abstract Ensemble Kalman filter (EnKF) analyses of the storms associated with the 8 May 2017 Colorado severe hail event using either the Milbrandt and Yau (MY) or the NSSL double-moment bulk microphysics scheme in the forecast model are evaluated. With each scheme, two experiments are conducted in which the reflectivity (Z) observations update in addition to dynamic and thermodynamic variables: 1) only the hydrometeor mixing ratios or 2) all microphysical variables. With fewer microphysical variables directly constrained by the Z observations, only updating hydrometeor mixing ratios causes the forecast error covariance structure to become unreliable, and results in larger errors in the analysis. Experiments that update all microphysical variables produce analyses with the lowest Z root-mean-square innovations; however, comparing the estimated hail size against hydrometeor classification algorithm output suggests that further constraint from observations is needed to more accurately estimate surface hail size. Ensemble correlation analyses are performed to determine the impact of hail growth assumptions in the MY and NSSL schemes on the forecast error covariance between microphysical and thermodynamic variables. In the MY scheme, Z is negatively correlated with updraft intensity because the strong updrafts produce abundant small hail aloft. The NSSL scheme predicts the growth of large hail aloft; consequently, Z is positively correlated with storm updraft intensity and hail state variables. Hail production processes are also shown to alter the background error covariance for liquid and frozen hydrometeor species. Results in this study suggest that EnKF analyses are sensitive to the choice of MP scheme (e.g., the treatment of hail growth processes).

Prediction in polynomial errors-in-variables models

A multivariate errors-in-variables (EIV) model with an intercept term, and a polynomial EIV model are considered. Focus is made on a structural homoskedastic case, where vectors of covariates are i.i.d. and measurement errors are i.i.d. as well. The covariates contaminated with errors are normally distributed and the corresponding classical errors are also assumed normal. In both models, it is shown that (inconsistent) ordinary least squares estimators of regression parameters yield an a.s. approximation to the best prediction of response given the values of observable covariates. Thus, not only in the linear EIV, but in the polynomial EIV models as well, consistent estimators of regression parameters are useless in the prediction problem, provided the size and covariance structure of observation errors for the predicted subject do not differ from those in the data used for the model fitting.

Real-time Forecasting of the Australian Macroeconomy Using Flexible Bayesian VARs

This paper evaluates the real-time forecast performance of alternative Bayesian Vector Autoregressive (VAR) models for the Australian macroeconomy. To this end, we construct an updated vintage database and estimate a set of model specifications with different covariance structures. The results suggest that a large VAR model with 20 variables tends to outperform a small VAR model when forecasting GDP growth, CPI inflation and unemployment rate. We find consistent evidence that the models with more flexible error covariance structures forecast GDP growth and inflation better than the standard VAR, while the standard VAR does better than its counterparts for unemployment rate. The results are robust under alternative priors and when the data includes the early stage of the COVID-19 crisis.

Comparisons of statistical models for growth curves from 90-day rat feeding studies

The objective of this work was to compare several models of body weight data from 90-day rodent feeding trials. Polynomial and nonlinear functions relating time and weight were examined as were the use of Toeplitz error covariance structures and random coefficients. The models were evaluated by fitting them to five publicly available datasets from rat feeding studies. Model performance was assessed in terms of their ability to capture the complexity of the growth patterns, validity of necessary assumptions, and information criteria scores. The results demonstrated the importance of selecting a curve function that effectively reflects the mean response. Toeplitz error covariance structures resulted in superior model fit, while failing to address deviations from model assumptions. Models using the Richards function and random coefficients were generally superior to the other models evaluated and dramatically improved upon linear models with complex error structures.

Inference of stochastic parametrizations for model error treatment using nested ensemble Kalman filters

Stochastic parametrizations are increasingly used to represent the uncertainty associated with model errors in ensemble forecasting and data assimilation. One of the challenges associated with the use of these parametrizations is the characterization of the statistical properties of the stochastic processes within their formulation. In this work, a hierarchical Bayesian approach based on two nested ensemble Kalman filters is proposed for inferring parameters associated with stochastic parametrizations. The proposed technique is based on the Rao‐Blackwellization of the parameter estimation problem. It consists of an ensemble of ensemble Kalman filters, each of them using a different set of stochastic parameter values. We show the ability of the technique to infer parameters related to the covariance of stochastic representations of model error in the Lorenz‐96 dynamical system. The evaluation is conducted with stochastic twin experiments and with imperfect model experiments with unresolved physics in the forecast model. The technique performs successfully under different model error covariance structures. The technique is conceived to be applied offline as part of an apriori optimization of the data assimilation system and could, in principle, be extended to the estimation of other hyperparameters of the data assimilation system.

COMPLEX-VALUED TIME SERIES MODELING FOR IMPROVED ACTIVATION DETECTION IN FMRI STUDIES.

A complex-valued data-based model with pth order autoregressive errors and general real/imaginary error covariance structure is proposed as an alternative to the commonly-used magnitude-only data-based autoregressive model for fMRI time series. Likelihood-ratio-test-based activation statistics are derived for both models and compared for experimental and simulated data. For a dataset from a right-hand finger-tapping experiment, the activation map obtained using complex-valued modeling more clearly identifies the primary activation region (left functional central sulcus) than the magnitude-only model. Such improved accuracy in mapping the left functional central sulcus has important implications in neurosurgical planning for tumor and epilepsy patients. Additionally, we develop magnitude and phase detrending procedures for complex-valued time series and examine the effect of spatial smoothing. These methods improve the power of complex-valued data-based activation statistics. Our results advocate for the use of the complex-valued data and the modeling of its dependence structures as a more efficient and reliable tool in fMRI experiments over the current practice of using only magnitude-valued datasets.