Estimating the Gait Speed of Older Adults in Smart Home Environments

Mobility is one of the key performance indicators of the health condition of older adults. One important parameter is the gait speed. The mobility is usually assessed under the supervision of a professional by standardised geriatric assessments. Using sensors in smart home environments for continuous monitoring of the gait speed enables physicians to detect early stages of functional decline and to initiate appropriate interventions. This in combination with a floor plan smart home sensors were used to calculate the distance that a person walked in the apartment and the inertial measurement unit data for estimating the actual walking time. A Gaussian kernel density estimator was applied to the computed values and the maximum of the kernel density estimator was considered as the gait speed. The proposed method was evaluated on a real-world dataset and the estimations of the gait speed had a deviation smaller than $$0.10 \, \frac{\mathrm{m}}{\mathrm{s}}$$ 0.10 m s , which is smaller than the minimal clinically important difference, compared to a baseline from a standardised geriatrics assessment.

A Fast-Converging Kernel Density Estimator for Dispersion in Horizontally Homogeneous Meteorological Conditions

Kernel smoothers are often used in Lagrangian particle dispersion simulations to estimate the concentration distribution of tracer gasses, pollutants etc. Their main disadvantage is that they suffer from the curse of dimensionality, i.e., they converge at a rate of 4/(d+4) with d the number of dimensions. Under the assumption of horizontally homogeneous meteorological conditions, we present a kernel density estimator that estimates a 3D concentration field with the faster convergence rate of a 1D kernel smoother, i.e., 4/5. This density estimator has been derived from the Langevin equation using path integral theory and simply consists of the product between a Gaussian kernel and a 1D kernel smoother. Its numerical convergence rate and efficiency are compared with that of a 3D kernel smoother. The convergence study shows that the path integral-based estimator has a superior convergence rate with efficiency, in mean integrated squared error sense, comparable with the one of the optimal 3D Epanechnikov kernel. Horizontally homogeneous meteorological conditions are often assumed in near-field range dispersion studies. Therefore, we illustrate the performance of our method by simulating experiments from the Project Prairie Grass data set.

An Improved Variable Kernel Density Estimator Based on L2 Regularization

The nature of the kernel density estimator (KDE) is to find the underlying probability density function (p.d.f) for a given dataset. The key to training the KDE is to determine the optimal bandwidth or Parzen window. All the data points share a fixed bandwidth (scalar for univariate KDE and vector for multivariate KDE) in the fixed KDE (FKDE). In this paper, we propose an improved variable KDE (IVKDE) which determines the optimal bandwidth for each data point in the given dataset based on the integrated squared error (ISE) criterion with the L2 regularization term. An effective optimization algorithm is developed to solve the improved objective function. We compare the estimation performance of IVKDE with FKDE and VKDE based on ISE criterion without L2 regularization on four univariate and four multivariate probability distributions. The experimental results show that IVKDE obtains lower estimation errors and thus demonstrate the effectiveness of IVKDE.