Exploring time-delay-based numerical differentiation using principal component analysis

Natural systems, including the dynamics of engineered structures, are often considered complex; hence, engineers employ different statistical methods to understand these systems better. Analyzing these systems usually require accurate derivative estimations for better understanding, i.e. a measured displacement can be used to estimate the forces on a cylindrical structure in water by using its velocity, and acceleration estimations. In this study, we use a nonlinear method based on embedding theory and consider the time-delay coordinates of a signal with a fixed lag time. We propose a new method for estimating the derivatives of the signal via redefining the delay matrix. That is, the original signal is updated with the second principal component of the delay matrix in each derivation. We apply this simple method to both linear and nonlinear systems and show that derivatives of both clean and/or noisy signals can be estimated with sufficient accuracy. By optimizing the required embedding dimension for the best derivative approximation, we find a constant value for the embedding dimension, which illustrates the simplicity of the proposed method. Lastly, we compare the method with some common differentiation techniques.