最小一乘法在岩土地基沉降中的应用

最小一乘法是一个既古典又新颖的方法，随着最小一乘解的实现问题近年来有大的突破，一些最小二乘法所不具有的优良特性，如直观性、稳健性、零误差性、可预测性、广义性等逐渐显现。最小一乘逼近是最小绝对值误差极小化的逼近，也称为“极小极小”逼近，由于极小极小逼近的最佳结果一定是零，所以零误差原理是最小一乘法的基本原理。最小一乘解是通过“代表式”的数据处理方式来实现的，由于排除了大误差数据的干扰，使最小一乘法具有较好的稳健性。而代表数据可按不同的应用而选择并确定，使应用具有了广泛性。对于预测而言，将端点数据设定为零误差数据，使数据的权重不再相等，而是往端点方向倾斜，端点数据具有最大的权重，且建立在无误差的基点上，这使得预测理论与模式变得合理，使预测的准确性得到保证。文中通过3个工程实例，介绍了最小一乘法在探索岩石或软土地基在沉降过程中的应用，其结果与最小二乘法的进行了比较，通过分析后给出如下主要结论：（1）最小一乘法的数据处理稳定性较好，波动幅度较小，预测结果较准；（2）一般不会出现最小二乘法数据处理中的矛盾及不合理的现象；（3）虽然时间t→∞的极限下沉量具有不可验证性，但最小一乘法的预测是建立在无误差的基点上，比最小二乘法的预测建立在有误差的基点上合理，加上有较好的稳健性，其结果更具参考性。

Application of least absolute deviation method to settlement of rock and soil foundations

The least absolute deviation (LAD) method has some favorable properties, including intuitiveness, robustness, zero-error, predictability and generalization, compared to the least square (LS) method. The LAD approximation is a "mini-mini approximation", which is minimization of the minimum absolute value of error. Because the best result of the mini-mini approximation must be zero, the zero-error principle is indeed the basic principle of LAD method. The realization of the LAD solution is achieved by a "representative" data processing mode. Because the disturbance induced by the data with big error is removed, the LAD method has a good robustness. Since the representative data are selected according to different applications, the LAD method is widely applicable. When the LAD method is applied to prediction, the endpoint data can be set to be zero-error data, yielding the unequal weights of the data which are closer to the end point. As a result, the endpoint data have the greatest weight, and have a basis without errors, making the proposed procedure more reasonable and more accurate. Based on three engineering cases, the LAD method is applied to predicting the settlement process of rock and soil foundation. By comparing with LS method, it is concluded that: 1) The stability of data processing of LDA method is better, the fluctuation range is smaller, and the forecasted result is more accurate; 2) Ccntradictory and unreasonable phenomenon no longer appear as in the data processing of LS method; 3) when t→∞, although the limit settlement values cannot be verified, the prediction of LAD method is more reasonable than those of LS method, since the LAD method is built on the basis without errors.