On Consistency of Signature Using Lasso
This work addresses feature selection for time series analysis, but it is incremental as it applies an existing method (Lasso) to a known technique (signatures) with specific theoretical extensions.
The paper tackles the problem of feature selection in time series data analysis by studying the consistency of signature methods using Lasso regression, establishing theoretical conditions for consistency and showing that Lasso is more consistent with Itô signatures for Brownian-like processes and with Stratonovich signatures for mean-reverting processes, with demonstrations achieving high accuracy in learning nonlinear functions and option prices.
Signatures are iterated path integrals of continuous and discrete-time processes, and their universal nonlinearity linearizes the problem of feature selection in time series data analysis. This paper studies the consistency of signature using Lasso regression, both theoretically and numerically. We establish conditions under which the Lasso regression is consistent both asymptotically and in finite sample. Furthermore, we show that the Lasso regression is more consistent with the Itô signature for time series and processes that are closer to the Brownian motion and with weaker inter-dimensional correlations, while it is more consistent with the Stratonovich signature for mean-reverting time series and processes. We demonstrate that signature can be applied to learn nonlinear functions and option prices with high accuracy, and the performance depends on properties of the underlying process and the choice of the signature.