Recently, we introduced the notion of a random walk based on a discrete sequence of data samples (data walk) and discovered a surprising link between ordinary least squares (OLS) fits to evenly sampled data and random walks. Here we generalize earlier results by showing that the slope of a linear fit to data which annuls the net area under a residual data walk equals that found by OLS for irregularly spaced data sequence. We also discover a deep connection with the orthogonality principle of estimation theory, leading to interpretation of suitably defined scalar products of data vectors as areas under data walks. The results are extended to weighted and generalized least squares (GLS). The new approach is illustrated on cosmic ray arrival time series.

We present a theoretical framework, derived from Maxwell’s equations for conducting media, to describe transient current decay in circuits with finite time constants. The temporal behaviour of the current is shown to be governed by a single dimensionless parameter β, defined as the ratio of the circuit time constant to the magnetic diffusion time. For finite values of β, numerical solutions of the resulting Volterra equation show a systematic deviation from the conventional exponential decay predicted by the lumped-element model. In RC circuits, the model predicts a crossover from monotonic decay to damped oscillatory behaviour below a critical value βcr≈0.4, as obtained numerically and supported by a controlled single-mode approximation. The number of such oscillations is finite, and they turn out to be irregular on the time scale. As the β-parameter decreases, this number increases. A simple physical interpretation of these results is also provided.