Sequential Quantiles via Hermite Series Density Estimation

Sequential quantile estimation refers to incorporating observations into quantile estimates in an incremental fashion thus furnishing an online estimate of one or more quantiles at any given point in time. Sequential quantile estimation is also known as online quantile estimation. This area is relevant to the analysis of data streams and to the one-pass analysis of massive data sets. Applications include network traffic and latency analysis, real time fraud detection and high frequency trading. We introduce new techniques for online quantile estimation based on Hermite series estimators in the settings of static quantile estimation and dynamic quantile estimation. In the static quantile estimation setting we apply the existing Gauss-Hermite expansion in a novel manner. In particular, we exploit the fact that Gauss-Hermite coefficients can be updated in a sequential manner. To treat dynamic quantile estimation we introduce a novel expansion with an exponentially weighted estimator for the Gauss-Hermite coefficients which we term the Exponentially Weighted Gauss-Hermite (EWGH) expansion. These algorithms go beyond existing sequential quantile estimation algorithms in that they allow arbitrary quantiles (as opposed to pre-specified quantiles) to be estimated at any point in time. In doing so we provide a solution to online distribution function and online quantile function estimation on data streams. In particular we derive an analytical expression for the CDF and prove consistency results for the CDF under certain conditions. In addition we analyse the associated quantile estimator. Simulation studies and tests on real data reveal the Gauss-Hermite based algorithms to be competitive with a leading existing algorithm.Comment: 43 pages, 9 figures. Improved version incorporating referee comments, as appears in Electronic Journal of Statistic

The Parabolic variance (PVAR), a wavelet variance based on least-square fit

This article introduces the Parabolic Variance (PVAR), a wavelet variance similar to the Allan variance, based on the Linear Regression (LR) of phase data. The companion article arXiv:1506.05009 [physics.ins-det] details the $\Omega$ frequency counter, which implements the LR estimate. The PVAR combines the advantages of AVAR and MVAR. PVAR is good for long-term analysis because the wavelet spans over $2 \tau$, the same of the AVAR wavelet; and good for short-term analysis because the response to white and flicker PM is $1/\tau^3$ and $1/\tau^2$, same as the MVAR. After setting the theoretical framework, we study the degrees of freedom and the confidence interval for the most common noise types. Then, we focus on the detection of a weak noise process at the transition - or corner - where a faster process rolls off. This new perspective raises the question of which variance detects the weak process with the shortest data record. Our simulations show that PVAR is a fortunate tradeoff. PVAR is superior to MVAR in all cases, exhibits the best ability to divide between fast noise phenomena (up to flicker FM), and is almost as good as AVAR for the detection of random walk and drift

Assessing the performance of ultrafast vector flow imaging in the neonatal heart via multiphysics modeling and In vitro experiments

Ultrafast vector flow imaging would benefit newborn patients with congenital heart disorders, but still requires thorough validation before translation to clinical practice. This paper investigates 2-D speckle tracking (ST) of intraventricular blood flow in neonates when transmitting diverging waves at ultrafast frame rate. Computational and in vitro studies enabled us to quantify the performance and identify artifacts related to the flow and the imaging sequence. First, synthetic ultrasound images of a neonate's left ventricular flow pattern were obtained with the ultrasound simulator Field II by propagating point scatterers according to 3-D intraventricular flow fields obtained with computational fluid dynamics (CFD). Noncompounded diverging waves (opening angle of 60 degrees) were transmitted at a pulse repetition frequency of 9 kHz. ST of the B-mode data provided 2-D flow estimates at 180 Hz, which were compared with the CFD flow field. We demonstrated that the diastolic inflow jet showed a strong bias in the lateral velocity estimates at the edges of the jet, as confirmed by additional in vitro tests on a jet flow phantom. Furthermore, ST performance was highly dependent on the cardiac phase with low flows (< 5 cm/s), high spatial flow gradients, and out-of-plane flow as deteriorating factors. Despite the observed artifacts, a good overall performance of 2-D ST was obtained with a median magnitude underestimation and angular deviation of, respectively, 28% and 13.5 degrees during systole and 16% and 10.5 degrees during diastole

Low Cost 3D Flow Estimation in Medical Ultrasound

abstract: Medical ultrasound imaging is widely used today because of it being non-invasive and cost-effective. Flow estimation helps in accurate diagnosis of vascular diseases and adds an important dimension to medical ultrasound imaging. Traditionally flow estimation is done using Doppler-based methods which only estimate velocity in the beam direction. Thus when blood vessels are close to being orthogonal to the beam direction, there are large errors in the estimation results. In this dissertation, a low cost blood flow estimation method that does not have the angle dependency of Doppler-based methods, is presented. First, a velocity estimator based on speckle tracking and synthetic lateral phase is proposed for clutter-free blood flow. Speckle tracking is based on kernel matching and does not have any angle dependency. While velocity estimation in axial dimension is accurate, lateral velocity estimation is challenging due to reduced resolution and lack of phase information. This work presents a two tiered method which estimates the pixel level movement using sum-of-absolute difference, and then estimates the sub-pixel level using synthetic phase information in the lateral dimension. Such a method achieves highly accurate velocity estimation with reduced complexity compared to a cross correlation based method. The average bias of the proposed estimation method is less than 2% for plug flow and less than 7% for parabolic flow. Blood is always accompanied by clutter which originates from vessel wall and surrounding tissues. As magnitude of the blood signal is usually 40-60 dB lower than magnitude of the clutter signal, clutter filtering is necessary before blood flow estimation. Clutter filters utilize the high magnitude and low frequency features of clutter signal to effectively remove them from the compound (blood + clutter) signal. Instead of low complexity FIR filter or high complexity SVD-based filters, here a power/subspace iteration based method is proposed for clutter filtering. Excellent clutter filtering performance is achieved for both slow and fast moving clutters with lower complexity compared to SVD-based filters. For instance, use of the proposed method results in the bias being less than 8% and standard deviation being less than 12% for fast moving clutter when the beam-to-flow-angle is $90^o$. Third, a flow rate estimation method based on kernel power weighting is proposed. As the velocity estimator is a kernel-based method, the estimation accuracy degrades near the vessel boundary. In order to account for kernels that are not fully inside the vessel, fractional weights are given to these kernels based on their signal power. The proposed method achieves excellent flow rate estimation results with less than 8% bias for both slow and fast moving clutters. The performance of the velocity estimator is also evaluated for challenging models. A 2D version of our two-tiered method is able to accurately estimate velocity vectors in a spinning disk as well as in a carotid bifurcation model, both of which are part of the synthetic aperture vector flow imaging (SA-VFI) challenge of 2018. In fact, the proposed method ranked 3rd in the challenge for testing dataset with carotid bifurcation. The flow estimation method is also evaluated for blood flow in vessels with stenosis. Simulation results show that the proposed method is able to estimate the flow rate with less than 9% bias.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201