Walter Köhler

A Flexible and Fast Method for Automatic Smoothing

Abstract The choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators. The optimal choice depends on the data at hand. A data-driven bandwidth selection, close to the optimal one, would make these curve estimators objective, more reliable, and easier to use. Minimizing residual mean squared error criteria, such as cross-validation, have been frequently proposed to estimate the optimal smoothing parameter. Empirical and theoretical evidence indicates that cross-validation rules and related methods lead to rather variable estimated optimal bandwidths. The method presented here builds on estimating the asymptotically optimal bandwidth from the data. Since estimators for the residual variance and for an asymptotic expression for the bias are plugged into the asymptotic formula, such selection rules are called “plug-in” estimators. The functional that quantifies bias is approximated by the integrated squared second derivative of the regre...

Velocity and acceleration of height growth using kernel estimation

A method is introduced for estimating acceleration, velocity and distance of longitudinal growth curves and it is illustrated by analysing human height growth. This approach, called kernel estimation, belongs to the class of smoothing methods and does not assume an a priori fixed functional model, and not even that one and the same model is applicable for all children. The examples presented show that acceleration curves might allow a better quantification of the mid-growth spurt (MS) and a more differentiated analysis of the pubertal spurt (PS). Accelerations are prone to follow random variations present in the data, and parameters defined in terms of acceleration are, therefore, validated by a comparison with parameters defined in terms of velocity. Our non-parametric-curve-fitting approach is also compared with parametric fitting via a model suggested by Preece and Baines (1978).

An analysis of the mid-growth and adolescent spurts of height based on acceleration

Height growth between four weeks and 20 years of 45 boys and 45 girls from the Zürich Longitudinal Growth Study (1955-1976) was analysed using kernel estimates. Timings of the mid-growth spurt (MS) and of the pubertal spurt (PS) were determined in an automatic way from the individual acceleration curves, together with height, percentage of height, velocity and acceleration at these ages. The small mid-growth spurt is a consistent phenomenon, peaking at 6.4 years (M,F) in acceleration and at 7.7 years (M) and 7.5 years (F) in velocity. There are no significant sex differences in its intensity. In girls, the PS follows in close succession to the MS; in boys there is a substantial period in between. In addition to the age of peak height velocity, ages of onset, maximal acceleration and end of the PS are defined. Sex differences in timing and size of the pubertal peak previously established were again verified. New results relate to the asymmetry of the PS, which is more pronounced in girls, and to sex differences in intensity and duration of the first rising phase of the PS. After this phase, boys and girls do not differ in timing but only in the intensity of deceleration.

Does filtering and smoothing of average evoked potentials really pay? a statistical comparison

Averaging of sweeps to obtain evoked potentials provides an unsatisfactory reduction of the background activity for a small number of stimuli. A posteriori Wiener filtering, time varying filtering, and smoothing of the average EP have been proposed to meet this problem. As to a posteriori Wiener filtering, a controversy regarding its merits has been going on for several years. The present paper gives a statistical comparison of the above methods, based on real data of two groups of subjects (flash evoked potentials in 41 subjects, pattern reversal evoked potentials in 9 subjects). It is shown that most of the improvement of the filtering approaches was due to an attenuation effect, without any improvement in smoothness of the potentials. The strength of the attenuation introduced by the filtering approaches depended on the specific underlying signal-to-noise ratio. This effect led to an artificially enhanced interindividual variability and could intraindividually lead to biased topographical distribution, when several electrodes are considered. The smoothing method did not show this undesired feature, but, when applying strong smoothing, this method also rendered sizable distortions of the potentials.

Human height growth: correlational and multivariate structure of velocity and acceleration

In this paper the correlations between parameters quantifying the mid-growth spurt (MS) and the pubertal spurt (PS) of height are explored. These parameters are defined in individual acceleration, velocity, and distance curves, which are obtained via kernel estimates as previously introduced. The MS proves to be a phenomenon largely independent of the PS. Intensity, timing, and duration of the PS are also independent mechanisms of growth and neither of them influences adult height in an appreciable way. Adult height depends mostly on pre-adolescent velocity. Short-, medium- and long-term regulatory mechanisms for explaining height growth are considered. For short-term mechanisms, acceleration plays a crucial role.

Trial-to-Trial Variability of Single Potentials: Methodological Concepts and Results

Variability of single visual evoked potentials was investigated by means of three statistical tests sensitive to amplitude variations, gradual changes, and latency jitter, respectively. In a sample of (n = 78) normal children, a considerable number of inhomogeneous responders was found, and most prominent were gradual potential changes and latency jitter. Removal of latency jitter demonstrated that the gradual changes are not of latency type and only partly of the amplitude type. As found from empirical densities, there is strong indication that there were subpopulations differing in their response style. On the whole, however, it was concluded that there was no clear, interindividually stable type of response variation in these data.

The Analysis of Noisy Signals by Nonparametric Smoothing and Differentiation

While smoothing methods are commonly applied to noisy signals, this is not true for differentiation. Derivatives are often of intrinsic interest when analyzing biological dynamics, and as will be illustrated, they are useful for determining characteristic points (local extrema, inflection, and saddle points) in the curve, in the presence of noise. There are inherent difficulties in computing derivatives which might have inhibited wider usage. Kernel estimation is a statistical approach to nonparametric regression (i.e., without specifying a functional model for the signal), which allows detertining the signal itself and its derivatives from noisy data. This method is presented, together with its properties. The influence and the choice of the weight function (kernel) of the smoothing parameter and the treatment of boundary points deserve particular attention.

VISUAL EVOKED POTENTIALS OF MILDLY MENTALLY RETARDED AND CONTROL CHILDREN

Visual evoked potentials (VEPs) were recorded from 25 10‐ to 13–year‐old mildly mentally retarded children and compared with those from 31 control children of the same age‐range. Correlations of VEPs with age were weak, but a relationship between VEPs and IQ was demonstrated for the control group. The retarded group had significantly longer latencies and higher amplitude peaks than the control group, with the differences occurring primarily over non‐specific cortex and for secondary components. Analysis also showed that the retarded group were neurophysiologically heterogeneous. Since the same children had been analyzed earlier by quantitative EEG methods, comparisons are made with respect to these two methods of investigating brain function.

Performance and Measures of Performance for Estimators of Brain Potentials Using Real Data

A number of filtering and smoothing procedures have been proposed for estimating evoked brain potentials. Their common goal is to reduce noise further than averaging. A statistical method is proposed for comparing these estimators for real data, thereby avoiding the use of simulated data, which usually are not representative of the shapes encountered for signal and noise and might also rely on some artificial assumptions. This approach is applied to visual evoked potential (both flash and pattern reversal). Filtering methods offer substantial gains in mean square error, but most of the gain is obtained trivially by attenuating the average. This points out the need for also using other loss functions. In terms of these, adaptive filtering brings at best gains equivalent to smoothing.

The intra-individual reproducibility of flash-evoked potentials in a sample of children

Visual evoked potentials (VEPs) to flash stimuli were recorded twice from 26 children aged 10-13 years, with an intersession interval of about 10 months. Test-retest reliability was poor for recordings taken from scalp locations overlying non-specific cortex and somewhat better for specific cortex. The size of consistency coefficients (i.e. correlations within session) showed that noise and artefacts were not the decisive factors which lower reliability. A comparison with retest correlations of broad band parameters of the EEG at rest for the same sample showed, to our surprise, smaller retest reliability for VEP parameters. Variability of the VEP in children over time seems to be a substantial as its well-known inter-individual variability.