Instrumental variables (eliminate the bias that afflicts least-squares identification of dynamical systems through noisy data, yet traditionally relies on external instruments that are seldom available for nonlinear time series data. We propose an IV estimator that synthesizes instruments from the data. We establish finite-sample $L^{p}$ consistency for all $p \ge 1$ in both discrete- and continuous-time models, recovering a nonparametric $\sqrt{n}$-convergence rate. On a forced Lorenz system our estimator reduces parameter bias by 200x (continuous-time) and 500x (discrete-time) relative to least squares and reduces RMSE by up to tenfold. Because the method only assumes that the model is linear in the unknown parameters, it is broadly applicable to modern sparsity-promoting dynamics learning models.

翻译：工具变量法能够消除因噪声数据导致动态系统最小二乘辨识中的偏差，但传统方法依赖外部工具变量，而这类变量在非线性时间序列数据中往往难以获取。我们提出一种从数据中合成工具变量的IV估计器。我们在离散时间与连续时间模型中均建立了针对所有$p \ge 1$的有限样本$L^{p}$一致性，并恢复了非参数化的$\sqrt{n}$收敛速率。在受迫洛伦兹系统上的实验表明，相较于最小二乘法，我们的估计器将参数偏差降低了200倍（连续时间）和500倍（离散时间），并将均方根误差降低至多十倍。由于该方法仅假设模型在未知参数上是线性的，因此可广泛应用于现代促进稀疏性的动力学学习模型。