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On the Linear Relation of Signals

机译：信号的线性关系

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摘要

In this paper two signals over finite and closed interval of time are considered. A linear relation of the two with shifting is defined. In general, when a signal is expressed linearly of another with shifting, this gives an error. The absolute error should be minimized to have the best approximation. For the case of discrete signals without shifting, minimizing this error is just the well-known least square method. Hence, the proposed relation of signals is a generalization of the least square method. Criterion of approximating a signal linearly of the other is based on the relative error. The relative error is defined by comparing the norm of absolute error with the norm of the normalized approximated signal. Geometric interpretation and the applications in finance are also discussed. Especially in finance, the relation of signals can be applied to predict the dynamics of stocks, exchange rates and commodity prices. Predicting such dynamics on itself is very difficult, especially for high frequency data.
机译：本文考虑了有限和封闭时间间隔内的两个信号。定义了两者之间具有线性关系的线性关系。通常，当一个信号线性移动表示另一个信号时，会产生误差。绝对误差应最小化以具有最佳近似值。对于不移位的离散信号，将误差最小化只是众所周知的最小二乘法。因此，建议的信号关系是最小二乘法的概括。线性近似另一个信号的标准基于相对误差。通过将绝对误差的范数与归一化的近似信号的范数进行比较来定义相对误差。还讨论了几何解释及其在金融中的应用。特别是在金融领域，信号的关系可以用来预测股票，汇率和商品价格的动态。很难自行预测这种动态，特别是对于高频数据。

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