Constrained monotone regression of ROC curves and histograms using splines and polynomials
Constrained monotone regression of ROC curves and histograms using splines and polynomials
复制标题
使用样条曲线和多项式对 ROC 曲线和直方图进行约束单调回归
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
R. Haralick
中科院分区：
文献类型：
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作者：
T. Kanungo;D. Gay;R. Haralick
受试者工作特征（ROC）曲线具有从（0，1）开始到（1，0）结束且单调递减的特性。此外，曲线的参数表示更自然，因为ROC不需要是单值函数：它们可以从无限斜率开始。我们展示了如何将参数样条和多项式拟合到具有端点和单调性约束的ROC数据。样条和多项式表示为我们提供了一种计算ROC曲线各个位置导数的方法，这对于找到最佳工作点是必要的。密度函数不是单调的，但累积密度函数是单调的。因此，为了将样条拟合到密度函数，我们将单调样条拟合到累积密度函数，然后对拟合的样条函数求导。就像ROC有端点约束一样，密度函数也有端点约束。此外，样条函数的导数是样条函数，并且可以以封闭形式计算。因此，平滑直方图也可以被视为一个约束单调回归问题。算法的实现在一个数学编程语言称为AMPL和样本数据集上的结果。
Receiver operating characteristics (ROC) curves have the property that they start at (0,1) and end at (1,0) and are monotonically decreasing. Furthermore, a parametric representation for the curves is more natural, since ROCs need not be single valued functions: they can start with infinite slope. We show how to fit parametric splines and polynomials to ROC data with the end-point and monotonicity constraints. Spline and polynomial representations provide us a way of computing derivatives at various locations of the ROC curve, which are necessary in order to find the optimal operating points. Density functions are not monotonic but the cumulative density functions are. Thus in order to fit a spline to a density function, we fit a monotonic spline to the cumulative density function and then take the derivative of the fitted spline function. Just as ROCs have end-point constraints, the density functions have end-point constraints. Furthermore, derivatives of splines are spline functions and can be computed in closed form. Thus smoothing of histograms can also be treated as a constrained monotone regression problem. The algorithms were implementation in a mathematical programming language called AMPL and results on sample data sets are given.