Static Covariance Research Articles

Representation learning with unconditional denoising diffusion models for dynamical systems

Abstract. We propose denoising diffusion models for data-driven representation learning of dynamical systems. In this type of generative deep learning, a neural network is trained to denoise and reverse a diffusion process, where Gaussian noise is added to states from the attractor of a dynamical system. Iteratively applied, the neural network can then map samples from isotropic Gaussian noise to the state distribution. We showcase the potential of such neural networks in proof-of-concept experiments with the Lorenz 1963 system. Trained for state generation, the neural network can produce samples that are almost indistinguishable from those on the attractor. The model has thereby learned an internal representation of the system, applicable for different tasks other than state generation. As a first task, we fine-tune the pre-trained neural network for surrogate modelling by retraining its last layer and keeping the remaining network as a fixed feature extractor. In these low-dimensional settings, such fine-tuned models perform similarly to deep neural networks trained from scratch. As a second task, we apply the pre-trained model to generate an ensemble out of a deterministic run. Diffusing the run, and then iteratively applying the neural network, conditions the state generation, which allows us to sample from the attractor in the run's neighbouring region. To control the resulting ensemble spread and Gaussianity, we tune the diffusion time and, thus, the sampled portion of the attractor. While easier to tune, this proposed ensemble sampler can outperform tuned static covariances in ensemble optimal interpolation. Therefore, these two applications show that denoising diffusion models are a promising way towards representation learning for dynamical systems.

Characterization of the Ionospheric Vertical Error Correlation Lengths Based on Global Ionosonde Observations

AbstractData assimilation is one of the most important approaches to monitoring the variations of ionospheric electron densities. The construction of the background error covariance matrix is an important component of ionospheric data assimilations. To construct the background error covariance matrix, the information about the spatial ionospheric correlations is required. We present a statistical analysis on the ionospheric vertical error correlation length (VCL) based on a global network of ionosondes and the Neustrelitz Electron Density Model. We show that the locally derived VCL is well‐defined and the VCL does not show a considerable dependency on the geographical seasons while local time dependencies of the VCL are shown to be present. A novel VCL model is also established based on the ionospheric scale heights. We show that the ionospheric VCL can be characterized by the variance ratio between the ionosphere model and ionospheric measurements. The altitudinal variations of VCLs are controlled by the interactions between the inherent VCLs of the ionosphere model and the measurements. Two experiments are conducted at two different latitudes based on the proposed model. The results show that the proposed model is stable and well‐correlated with the observed VCLs, which implies a potential to be generalized for a global correlation model. The proposed model can be used in the temporal evolution of error covariance matrices in the ionospheric 4D‐Variational (4D‐Var) assimilations, which may overcome the main drawbacks of the static error covariance specifications in the ionospheric 4D‐Var assimilations.

A Hybrid Gain Analog Offline EnKF for Paleoclimate Data Assimilation

AbstractFor Paleoclimate data assimilation (PDA), a hybrid gain analog offline ensemble Kalman filter (HGAOEnKF) is proposed. It keeps the benefits of the analog offline ensemble Kalman filter (AOEnKF) that constructs analog ensembles from existing climate simulations with joint information of the proxies. The analog ensembles can provide more accurate prior ensemble mean and “flow‐dependent” error covariances than randomly sampled ensembles. HGAOEnKF further incorporates the benefits of static prior error covariances that better capture large‐scale error correlations and mitigate sampling errors than the sample prior error covariances, through a hybrid gain approach within an ensemble framework. Observing system simulation experiments are conducted for various data assimilation methods, using ensemble simulations from the Community Earth System Model‐Last Millennium Ensemble Project. Results show that using the static prior error covariances estimated from a sufficiently large sample set is beneficial for the traditional offline ensemble Kalman filter (OEnKF) and AOEnKF. HGAOEnKF method is superior to the OEnKF and AOEnKF with and without static prior error covariances, especially for the reconstruction of extreme events. The advantages of HGAOEnKF over OEnKF and AOEnKF with and without static prior error covariances are persistent with varying sample sizes and presence of model errors.

Representing uncertainty in limited-area data assimilating ocean models

A limited-area ocean prediction system is developed to acquire forecast error covariances related to uncertainty in atmospheric forcing and turbulent mixing using perturbed model parameters within an Ensemble Kalman Filter (EnKF). The system performs sequential data assimilation delivering realistic ocean state estimation and forecasts. It is initialised to observations using the EnKF, the hybrid-EnKF and Ensemble Optimal Interpolation (EnOI). It has higher resolution than the parent global model, includes tides and assimilates altimetric sea level anomaly to constrain the offshore mesoscale circulation. Dynamic ensemble spread introduced by parameter uncertainty shows agreement with error estimates obtained from forecast innovation statistics. Hybrid EnKF offers improvements to both EnKF and EnOI, having smaller analysis increments and forecast errors. The ensemble mean of the hybrid-EnKF is more accurate than any member due to the introduction of parameterised error which behaves as additive inflation in the hybrid EnKF system. This indicates the parameterisation acts as a non-linear filter for model forecast error growth. To demonstrate consistency, experiments are carried out for various regions that differ in their oceanographic situations. The hybrid EnKF data assimilation system enables practical use of dynamic ensembles that capture the flow dependent errors which are mixed with static or climatological covariances for situations where model error is systematically under-estimated.

Bridge weigh-in-motion considering dynamic response in observation noise with application to multiple driving conditions

In addition to causing a static response when traversing a bridge, moving vehicles also produce a dynamic response, which is thought to be one of the factors adversely affecting the accuracy of bridge weigh-in-motion approaches. This paper proposes a novel method for improving the accuracy of such approaches. Rather than modeling the dynamic response directly, the proposed model considers it as noise with autocorrelation. A Bayesian inference approach is considered as it allows a rational way of assimilating data into the model. Here, the dynamic response is considered in the form of a covariance matrix of observation noise, composed of the residuals of the measured and calculated responses, in a Bayesian framework. In a validation study using actual measured data, it was confirmed that the estimation of axle weights in the presence of multiple vehicles was significantly improved. From the result, it can be concluded that proper modeling of the dynamic response in observation data leads to more accurate weigh-in-motion estimates of axle weights and GVW, particularly when complicated vehicle travel is involved. A comparison of static and dynamic influence lines is also discussed using actual measured data. It is shown that the weights estimated by the dynamic influence line are generally more accurate than those by the static influence line when correlation of noise is not considered. The dynamic component in influence line can be interpreted as a regularization term. The proposed method, which uses static influence line and covariance matrix considering correlation of noise, is more accurate than the method with dynamic influence line and also, it does not cause any bias.

Improving Odometric Model Performance Based on LSTM Networks

This paper presents a localization system for an autonomous wheelchair that includes several sensors, such as odometers, LIDARs, and an IMU. It focuses on improving the odometric localization accuracy using an LSTM neural network. Improved odometry will improve the result of the localization algorithm, obtaining a more accurate pose. The localization system is composed by a neural network designed to estimate the current pose using the odometric encoder information as input. The training is carried out by analyzing multiple random paths and defining the velodyne sensor data as training ground truth. During wheelchair navigation, the localization system retrains the network in real time to adjust any change or systematic error that occurs with respect to the initial conditions. Furthermore, another network manages to avoid certain random errors by using the relationship between the power consumed by the motors and the actual wheel speeds. The experimental results show several examples that demonstrate the ability to self-correct against variations over time, and to detect non-systematic errors in different situations using this relation. The final robot localization is improved with the designed odometric model compared to the classic robot localization based on sensor fusion using a static covariance.

Influences of 1DVAR Background Covariances and Observation Operators on Retrieving Tropical Cyclone Thermal Structures

Spaceborne passive microwave sounding instruments are important for monitoring tropical cyclones (TCs) over oceans. However, previous studies have found large retrieval errors at TCs’ inner region at the lower troposphere where heavy precipitation occurs. In this study, the background error covariance matrix used in the variational retrieval algorithm is designed to vary with atmospheric conditions. It is found that the errors of retrieved temperature and humidity profiles are significantly reduced under the TC conditions, when they are compared with those from using a static covariance matrix. The retrieval errors of temperature and humidity are about 1.5 K and 10–20%, respectively, in the troposphere. Moreover, the influence of different observation operators on the retrievals are also investigated. It is shown that ARMS (Advanced Radiative Transfer Modeling System) used as an observation operator can produce a higher retrieval accuracy, compared to CRTM (Community Radiative Transfer Model). For the relative humidity profile, the error can be reduced by up to 5% from ARMS. The reason may be attributed to the more comprehensive handling of the scattering from various hydrometeors in ARMS, which results in a higher retrieval accuracy under cloudy conditions.

Coverage hole detection using social spider optimized Gaussian Mixture Model

Wireless sensors are backed by their battery life. The battery depletion creates a communication hole in the network. The replacement of certain energy-depleted sensor nodes necessitates their detection. The work in this article presents a novel solution for the detection of coverage hole in the WSN. An undirected graph was employed in this study to divide the WSN region into two parts: the outer and inner nodes. The Gaussian Mixture Model (GMM) clustering detects the outer/edge nodes at the coverage hole area. Node degree, closeness, betweenness, and page rank are the four graph metrics that GMM uses for clustering in this work. GMM’s clustering performance is governed by the static mean and covariance. A novel combination of social spider optimization and GMM is then leveraged to fine-tune the cluster to decrease false detection rates of holes. Distinct holes, such as round, E-shape, and rectangular, are used to conduct the analysis. Using the SSO-EM GMM approach, the comparison shows that it outperforms the current methods of coverage hole identification.

Development of Convective-Scale Static Background Error Covariance within GSI-Based Hybrid EnVar System for Direct Radar Reflectivity Data Assimilation

AbstractA convective-scale static background-error covariance (BEC) matrix is further developed to include the capability of direct reflectivity assimilation and evaluated within the GSI-based three-dimensional variational (3DVar) and hybrid ensemble–variational (EnVar) methods. Specific developments are summarized as follows: 1) Control variables (CVs) are extended to include reflectivity, vertical velocity, and all hydrometeor types. Various horizontal momentum and moisture CV options are included. 2) Cross correlations between all CVs are established. 3) A storm intensity-dependent binning method is adopted to separately calculate static error matrices for clear-air and storms with varying intensities. The resultant static BEC matrices are simultaneously applied at proper locations guided by the observed reflectivity. 4) The EnVar is extended to adaptively incorporate static BECs based on the quality of ensemble covariances. Evaluation and examination of the new static BECs are first performed on the 8 May 2003 Oklahoma City supercell. Detailed diagnostics and 3DVar examinations suggest zonal/meridional winds and pseudo–relative humidity are selected as horizontal momentum and moisture CVs for direct reflectivity assimilation, respectively; inclusion of cross correlations favors spin up and maintains the analyzed storms; application of binning improves characteristics and persistence of the simulated storm. Relative to an experiment using the full ensemble BECs (Exp-PureEnVar), incorporating static BECs in hybrid EnVar reduces spinup time and better analyzes reflectivity distributions while the background ensemble is deficient in sampling errors. Compared to both pure 3DVar and Exp-PureEnVar, hybrid EnVar better predicts reflectivity distributions and better maintains a strong mesocyclone. Further examination through the 20 May 2013 Oklahoma supercells confirms these results and additionally demonstrates the effectiveness of adaptive hybridization.

The effect of static pressure-wind covariance on vertical carbon dioxide exchange at a windy subalpine forest site

Accounting for turbulent temperature and water vapor fluctuations on the vertical flux of CO2 measured by an open-path infrared gas analyzer are commonly known as the Webb–Pearman–Leuning (WPL) corrections. Static pressure fluctuations also affect air sample density, but the magnitude and effect of these changes on CO2 flux is not usually considered. In our study, the turbulent static pressure p and the vertical wind component w covariance w′p′¯ from just-above a Rocky Mountain subalpine forest are examined. The magnitude of w′p′¯ was highly correlated to the mean horizontal wind speed U with similar characteristics during the daytime and nighttime. The pressure term was calculated from w′p′¯ and compared to other terms in the equation for the vertical net ecosystem exchange of CO2 (NEE). We found the following: (1) for U≲6 m s−1, the pressure term was small, (2) as U increased beyond 6 m s−1, the pressure term became more and more important reaching a magnitude of ≈ 2 μmol m−2 s−1 at U of 12 m s−1 (for leaf area index LAI ≈ 2.6 m2 m−2), (3) for a more dense forest (LAI ≈ 4.8 m2 m−2), the magnitude of the pressure term at U of 12 m s−1 was ≈ 1 μmol m−2 s−1, or about half the open forest value, and (4) based on 14 years of measurements, the interannual mean and standard deviation of the yearly cumulative pressure and NEE terms were −32.8± 24.5 g C m−2 year−1 and −147.4± 231.7 g C m−2 year−1, respectively. Therefore, on average, including the pressure term in the NEE calculation reduces the annual carbon uptake by the GLEES forest from 147.4 g C m−2 year−1 to 114.6 g C m−2 year−1. This implies that carbon uptake by forests in windy locations that ignore the pressure term can be overestimated by as much as 20%. However, the year-to-year variability of both NEE and the pressure term was large due to changes in the forest structure and LAI from beetle attack and tree die-off. Finally, we present results from sensor-manipulation experiments examining the effect of tilting the quad-disk pressure ports on the turbulent static pressure.

Empirical determination of the covariance of forecast errors: An empirical justification and reformulation of hybrid covariance models

AbstractDuring the last decade, the replacement of static climatological forecast error covariance models with hybrid error covariance models that linearly combine localised ensemble covariances with static climatological error covariances has led to significant forecast improvements at several major forecasting centres. Here, a deeper understanding of why the hybrid's superficially ad hoc mix of ensemble‐based and climatological covariances yields such significant improvements is pursued. In practice, ensemble covariances are not equal to the true flow‐dependent forecast error covariance matrix. Here, the relationship between actual forecast error covariance and the corresponding ensemble covariance is empirically demonstrated. Using a simplified global circulation model and the local ensemble transform Kalman filter (LETKF), the covariance of the set of actual forecast errors corresponding to ensemble covariances close to a fixed target value is computed. By doing this for differing target values, an estimate of the actual forecast error covariance as a function of ensemble covariance is obtained. A demonstration that the hybrid is a much better approximation to this estimate than either the static climatological covariance or the localised ensemble covariance is given. The empirical estimate has two features that current hybrid error covariance models fail to represent: (i) The weight given to the static covariance matrix is an increasing function of the horizontal separation distance of the covarying model variables, and (ii) for small ensemble sizes and ensemble covariances near zero but negative, the actual forecast error covariance is a decreasing function of increasing ensemble covariance. While the first finding has been anticipated by other authors, the second finding has not been anticipated, as far as the authors are aware. Here, (ii) is hypothesised to be a consequence of spurious sample correlations and variances associated with reduced ensembles. Consistent with this hypothesis, the non‐monotonicity of this relationship is almost eliminated by quadrupling the ensemble size.

Spatially resolved atomic-scale friction: Theory and simulation

We analyze the friction force exerted on a small probe particle sliding over an atomic-scale surface by means of a Green-Kubo relation and classical Molecular Dynamics simulations. We find that, on the atomic scale, the friction tensor can drastically vary as a function of position and sliding direction. The Green-Kubo relation yields this positional and directional dependence from equilibrium simulations of the time dependent covariance of force acting on the probe. We find, unexpectedly, that the positional and directional dependence of energy dissipation is related to the (much simpler) static force covariance, especially in the limit where the probe only mildly perturbs the surface particles. In contrast, the (free) energy landscape experienced by the probe is in general not a good indicator of local dissipation. We also discuss optimization strategies making use of the locally and directionally resolved friction tensor. This enables us to find optimal sliding paths and velocity protocols, e.g., minimizing energy dissipation, between two points on the surface in a given time.

Improving Robot Localization Using Doppler-Based Variable Sensor Covariance Calculation.

This paper describes a localization module for an autonomous wheelchair. This module includes a combination of various sensors such as odometers, laser scanners, IMU and Doppler speed sensors. Every sensor used in the module features variable covariance estimation in order to yield a final accurate localization. The main problem of a localization module composed of different sensors is the accuracy estimation of each sensor. Average static values are normally used, but these can lead to failure in some situations. In this paper, all the sensors have a variable covariance estimation that depends on the data quality. A Doppler speed sensor is used to estimate the covariance of the encoder odometric localization. Lidar is also used as a scan matching localization algorithm, comparing the difference between two consecutive scans to obtain the change in position. Matching quality gives the accuracy of the scan matcher localization. This structure yields a better position than a traditional odometric static covariance method. This is tested in a real prototype and compared to a standard fusion technique.

The development of a hybrid EnSRF-En3DVar system for convective-scale data assimilation

A coupled hybrid ensemble square root filter and three-dimensional ensemble-variational (EnSRF-En3DVar) radar data assimilation system, developed for the Weather Research and Forecasting model, was applied to a mesoscale convective system (MCS) that occurred over southeastern China on 5 June 2009. This hybrid system uses the extended control variable method to combine the static and ensemble flow-dependent forecast error covariance. The potential of the hybrid EnSRF-En3DVar system was first explored by comparing the derived results with those obtained using 3DVar and EnSRF approaches alone. Assimilation results showed the hybrid EnSRF-En3DVar system reduces the root mean square innovations for reflectivity and radial velocity, and better represents the pattern of the MCS than either 3DVar or EnSRF. This successful analysis improved quantitative reflectivity and precipitation forecasting skills, and helped forecast the MCS more realistically with regard to location, structure and intensity. Moreover, the root mean square error of the forecast wind, temperature and water vapor mixing ratio were found reduced by the hybrid EnSRF-En3DVar system when compared with the other methods. Diagnoses of the forecast fields showed the hybrid EnSRF-En3DVar system increases low-level cooling and mid-level warming within the convective area. Furthermore, sensitivity experiments testing the ensemble covariance weighting factor effect in the hybrid EnSRF-En3DVar method suggested that a stronger ensemble covariance weighting value led to improved precipitation forecast results.

Impact of non‐stationarity on hybrid ensemble filters: A study with a doubly stochastic advection‐diffusion‐decay model

Effects of non‐stationarity on the performance of hybrid ensemble filters is studied. (By hybrid filters we mean those which blend ensemble covariances with some other regularizing covariances.) To isolate effects of non‐stationarity from effects due to nonlinearity (and the non‐Gaussianity it causes), a new doubly stochastic advection‐diffusion‐decay model (DSADM) is proposed. The model is hierarchical: it is a linear stochastic partial differential equation whose coefficients are random fields defined through their own stochastic partial differential equations. DSADM generates conditionally Gaussian spatiotemporal random fields with a tunable degree of non‐stationarity in space and time. DSADM allows the use of the exact Kalman filter as a baseline benchmark.In numerical experiments with DSADM as the “model of truth”, the relative importance of the three kinds of covariance blending is studied: with static, time‐smoothed, and space‐smoothed covariances. It is shown that the stronger the non‐stationarity, the less useful the static covariance matrix becomes and the more beneficial the time‐smoothed covariances are. Time‐smoothing of background‐error covariances proved to be systematically more useful than their space‐smoothing. Under non‐stationarity, a filter that extends the (previously proposed by the authors) Hierarchical Bayes Ensemble Filter and accommodates the three covariance‐blending techniques is shown to outperform all other configurations of the filters tested.

The Role of Correlation Dynamics in Equity Factor Allocation

The five equity factors from Fama and French (2015) and the momentum factor are pervasive in academia and practice. However, not much is known about their joint distribution and dynamics. We find strong evidence of time-varying volatilities and correlations, but no signs of asymmetric comovements in the tails. Based on our findings, we assess the economic value of volatility and correlation forecasts for mean-variance portfolio optimization. Our analysis shows that, in particular, volatility timing leads to significant utility gains over a static covariance strategy. Correlation timing can enhance this results for highly risk-averse investors.

The added value of the multi-system spread information for ocean heat content and steric sea level investigations in the CMEMS GREP ensemble reanalysis product

Since 2016, the Copernicus Marine Environment Monitoring Service (CMEMS) has produced and disseminated an ensemble of four global ocean reanalyses produced at eddy-permitting resolution for the period from 1993 to present, called GREP (Global ocean Reanalysis Ensemble Product). This dataset offers the possibility to investigate the potential benefits of a multi-system approach for ocean reanalyses, since the four reanalyses span by construction the same spatial and temporal scales. In particular, our investigations focus on the added value of the information on the ensemble spread, implicitly contained in the GREP ensemble, for temperature, salinity, and steric sea level studies. It is shown that in spite of the small ensemble size, the spread is capable of estimating the flow-dependent uncertainty in the ensemble mean, although proper re-scaling is needed to achieve reliability. The GREP members also exhibit larger consistency (smaller spread) than their predecessors, suggesting advancement with time of the reanalysis vintage. The uncertainty information is crucial for monitoring the climate of the ocean, even at regional level, as GREP shows consistency with CMEMS high-resolution regional products and complement the regional estimates with uncertainty estimates. Further applications of the spread include the monitoring of the impact of changes in ocean observing networks; the use of multi-model ensemble anomalies in hybrid ensemble-variational retrospective analysis systems, which outperform static covariances and represent a promising application of GREP. Overall, the spread information of the GREP product is found to significantly contribute to the crucial requirement of uncertainty estimates for climatic datasets.

A comparison of hybrid variational data assimilation methods for global NWP

Variational data assimilation methods are reviewed and compared in the Met Office global numerical weather prediction system. This supports hybrid background‐error covariances which are a weighted combination of modelled static “climatological” covariances with covariances calculated from a current ensemble of forecasts, in both three‐ and four‐dimensional methods. For the latter, we compare the use of linear and adjoint models (hybrid‐4DVar) with the direct use of ensemble forecast trajectories (hybrid‐4DEnVar). Earlier studies had shown that hybrid‐4DVar outperforms hybrid‐4DEnVar, and 4DVar outperforms 3DVar. Improvements in the processing of ensemble covariances and computer enhancements mean we are now able to explore these comparisons for the full range of hybrid weights. We find that, using our operational 44‐member ensemble, the static covariance is still beneficial in hybrid‐4DVar, so that it significantly outperforms hybrid‐4DEnVar. In schemes not using linear and adjoint models, the static covariance is less beneficial. It is shown that the time‐propagated static covariance is the main cause of the better performance of 4DVar; when using pure ensemble covariances, 4DVar and 4DEnVar show similar skill. These results are consistent with nonlinear dynamics theory about assimilation in the unstable subspace.

Development of a Hybrid En3DVar Data Assimilation System and Comparisons with 3DVar and EnKF for Radar Data Assimilation with Observing System Simulation Experiments

Abstract A hybrid ensemble–3DVar (En3DVar) system is developed and compared with 3DVar, EnKF, “deterministic forecast” EnKF (DfEnKF), and pure En3DVar for assimilating radar data through perfect-model observing system simulation experiments (OSSEs). DfEnKF uses a deterministic forecast as the background and is therefore parallel to pure En3DVar. Different results are found between DfEnKF and pure En3DVar: 1) the serial versus global nature and 2) the variational minimization versus direct filter updating nature of the two algorithms are identified as the main causes for the differences. For 3DVar (EnKF/DfEnKF and En3DVar), optimal decorrelation scales (localization radii) for static (ensemble) background error covariances are obtained and used in hybrid En3DVar. The sensitivity of hybrid En3DVar to covariance weights and ensemble size is examined. On average, when ensemble size is 20 or larger, a 5%–10% static covariance gives the best results, while for smaller ensembles, more static covariance is beneficial. Using an ensemble size of 40, EnKF and DfEnKF perform similarly, and both are better than pure and hybrid En3DVar overall. Using 5% static error covariance, hybrid En3DVar outperforms pure En3DVar for most state variables but underperforms for hydrometeor variables, and the improvement (degradation) is most notable for water vapor mixing ratio qυ (snow mixing ratio qs). Overall, EnKF/DfEnKF performs the best, 3DVar performs the worst, and static covariance only helps slightly via hybrid En3DVar.

On the choice of covariance specifications for portfolio selection problems

Two crucial aspects to the problem of portfolio selection are the specification of the model for expected returns and their covariances, as well as the choice of the investment policy to be adopted. A common trade-off is to consider dynamic covariance specifications vis-a-vis static models such as those based on shrinkage methods. This work empirically shows that these two aspects are intrinsically attached to the impact of transaction costs. To address this question, we implement a broad range of covariance specifications to generate a set of 16 portfolio selection policies in a high dimensional sample composed by the 50 most traded stocks of the S\&P100 index. We find that GARCH-type dynamic covariances yield portfolios with superior risk-adjusted performance only in the absence of transaction costs. In more realistic scenarios involving alternative levels of transaction costs, portfolios based on static covariance models outperform. In particular, we find that a risk-averse investor with quadratic utility function is willing to pay an annualized fee of 368 basis points (bp) on average in order to switch from the dynamic covariance models to a static covariance specification when the level of transaction costs is 20 bp. Finally, portfolio policies that seek to alleviate estimation error by ignoring off-diagonal covariance elements as those proposed in \citet{kirby2012s} are more robust specially in scenarios with higher transaction costs.