In this paper we propose three distinct points for improving scale sensitive DFA based multiple linear regression inference: the scale dependent standardized Regression Coefficients ${\hat{\beta }}_{std}^{DFA}\left(s\right)$ as a measure of dependent variables relative importance, scale dependent effect size $f^2\left(s\right)$, and Intersection-Union hypothesis test, which can handle the composite hypothesis of no cross-correlation, and of having no advantage over the standard OLS linear regression model. We applied this framework on a model for acoustic impedance of well log data, as a function of neutron effective porosity, shale volume, and resistivity. We find that the neutron effective porosity is more important for modeling acoustic impedance than resistivity and, that both are more important than the shale volume, for all scales. Results from Intersection-Union test suggest the rejection of compound null hypothesis for neutron effective porosity and resistivity for scales between 40 and 500 ft, suggesting a robust model for acoustic impedance based on neutron effective porosity and resistivity on these scales. This approach represents a novel framework for scale sensitive regression models, and we believe that it can be useful for time series studies in many diverse areas.

在本文中，我们提出了三个不同的点来改进基于尺度敏感的 DFA 多元线性回归推理：尺度相关的标准化回归系数 ${\widehat{\beta }}_{秒吨d}^{DF\mathrm{一种}}\left(秒\right)$ 作为因变量相对重要性的度量，尺度依赖效应大小 $F^2\left(秒\right)$，以及 Intersection-Union 假设检验，它可以处理没有互相关的复合假设，并且与标准的 OLS 线性回归模型相比没有优势。我们将此框架应用于测井数据声阻抗模型，作为中子有效孔隙度、页岩体积和电阻率的函数。我们发现中子有效孔隙度对于模拟声阻抗比电阻率更重要，并且对于所有尺度，两者都比页岩体积更重要。Intersection-Union 检验的结果表明，对于 40 到 500 英尺之间的尺度，拒绝中子有效孔隙率和电阻率的复合零假设，这表明基于这些尺度上的中子有效孔隙率和电阻率的声阻抗模型是稳健的。