Fig. 12
From: Quasi-differentiation and its applications to noisy time series data from complex systems
(a) The daily closing prices and (b) daily returns of GOOG and MSFT, between 2005 and 2023. (c) The heat map of the quasi-derivative of the cross correlations \(C_{\text {GOOG}, \text {MSFT}}(t)\) between GOOG and MSFT, over 2005 to 2023, obtained using window sizes \(20 \le w \le 250\). (d) The integrated quasi-derivative of \(C_{\text {GOOG}, \text {MSFT}}(t)\) for \(w = 150\) showing complex oscillations between 2005 and 2023. To test how well the method of integrated quasi-differentiation can recover a time-dependent cross correlation, (e) we sampled the artificial stochastic time series \(x_1(t)\) and \(x_2(t)\) with no cross correlations between them, and created an artificial stochastic time series \(x_3(t) = \rho (t) x_1(t) + \sqrt{1 - \rho ^2(t)} x_2(t)\), where \(\rho (t) = 0.4\cos (t/400)\). (f) The integrated quasi-derivative of \(\rho (t)\) (blue) is compared against the exact value of \(\rho (t)\) (red).
