Fig. 4
From: Quasi-differentiation and its applications to noisy time series data from complex systems
(a) A noisy time series f(t) (red) obtained by adding Gaussian noise with \(\sigma = 0.2\) to the function \(\tanh (t)\), and its quasi-derivative (blue) obtained by averaging f(t) over a pair of sliding windows with width \(w=1\). As we can see, the quasi-derivative compares favorably with the analytical derivative \(\operatorname {sech}^2(t)\) (green). (b) A noisy time series f(t) (red) obtained by adding Gaussian noise with \(\sigma = 0.2\) to the signum function \(\operatorname {sgn}(t)\), and its quasi-derivative (blue) obtained by averaging f(t) over a pair of sliding windows with width \(w=1\). We also show the quasi-derivative of the noise-free \(\operatorname {sgn}(t)\) in green. As we can see, the noisy quasi-derivative is almost identical to the noise-free quasi-derivative, for this level of noise in the time series. (c) A noisy time series f(t) (red) consisting of Gaussian noise with \(\sigma _1 = 0.5\) for \(t \le 0\) and \(\sigma _2 = 2\) for \(t>0\). For a pair of sliding windows with width \(w=1\), the quasi-derivatives of its mean and variance are shown in green and blue. We also show the quasi-derivative of its variance for \(w=2\) (magenta).
