曲线的下界

Question

我有一条如下曲线：

可以看出，上面有两条上限和下限曲线（参见下图中的近似可视化！）：

我试图通过以下代码找到下限：

lower_bound = rbind()
for(i in 2:(length(data[,1])-1))
{
    if(data[i,2] < data[i+1,2] && data[i,2] < data[i-1,2])
    {
        lower_bound = rbind2(lower_bound, c(data[i,1], data[i,2]))
    }    
}

但是，问题是中间有一些局部最小值，所以结果不是我预期的（如下）。

因此问题是如何在R 中找到这样一条曲线的下界？是否有任何已知的算法来找到下限？

关于最小的可重复性示例，我在下面包含了500个数据（有意义）：

structure(c(3.54246575342466, 5.75616438356164, 6.54246575342466, 
9.73424657534247, 7.75616438356164, 9.75616438356164, 1.41917808219178, 
2.75342465753425, 3.16986301369863, 2.27123287671233, 3.74520547945205, 
5.74520547945205, 4.29315068493151, 3.37260273972603, 8.37534246575343, 
3.33698630136986, 5.04657534246575, 6.92328767123288, 1.33972602739726, 
6.34246575342466, 26.358904109589, 2.34794520547945, 4.35068493150685, 
7.35342465753425, 27.3671232876712, 1.15068493150685, 2.14794520547945, 
4.15342465753425, 6.15342465753425, 9.15616438356164, 19.1616438356164, 
29.1698630136986, 2.58904109589041, 4.5972602739726, 9.6, 29.613698630137, 
3.10684931506849, 8.11780821917808, 3.35068493150685, 8.37260273972603, 
5.11506849315068, 28.1315068493151, 5.36986301369863, 28.386301369863, 
4.87945205479452, 4.46027397260274, 5.35616438356164, 25.3698630136986, 
25.8684931506849, 5.85479452054795, 1.85205479452055, 4.37260273972603, 
6.37260273972603, 9.37534246575343, 19.3835616438356, 29.3890410958904, 
18.3808219178082, 1.36986301369863, 8.37534246575343, 3.37260273972603, 
5.85479452054795, 28.3890410958904, 2.87671232876712, 4.87945205479452, 
7.37808219178082, 8.17260273972603, 28.186301369863, 4.29315068493151, 
6.96164383561644, 26.3479452054795, 7.96438356164384, 1.53972602739726, 
1.58630136986301, 5.04657534246575, 16.6356164383562, 18.4684931506849, 
25.0602739726027, 15.6493150684932, 24.8547945205479, 25.3506849315068, 
26.1068493150685, 16.0547945205479, 2.16712328767123, 19.2657534246575, 
20.6849315068493, 1.04383561643836, 3.04383561643836, 23.0575342465753, 
3.29315068493151, 4.13150684931507, 2.04383561643836, 22.0575342465753, 
22.8904109589041, 2.87671232876712, 5.54246575342466, 25.5561643835616, 
1.04931506849315, 6.05205479452055, 26.0657534246575, 2.09041095890411, 
7.09315068493151, 27.1068493150685, 2.09041095890411, 4.09315068493151, 
6.09315068493151, 9.0958904109589, 19.1013698630137, 29.1095890410959, 
1.83287671232877, 1.42739726027397, 3.43013698630137, 9.3041095890411, 
2.72328767123288, 8.58082191780822, 2.8958904109589, 4.92054794520548, 
9.92328767123288, 6.92054794520548, 19.9315068493151, 29.9369863013699, 
20.9315068493151, 2.25205479452055, 22.2657534246575, 3.40821917808219, 
23.4219178082192, 8.21095890410959, 3.20821917808219, 5.20821917808219, 
18.8027397260274, 19.8054794520548, 21.7232876712329, 2.25205479452055, 
22.8054794520548, 24.4301369863014, 4.41643835616438, 25.2684931506849, 
5.25479452054795, 6.33698630136986, 26.3506849315068, 2.87671232876712, 
4.29315068493151, 2.78630136986301, 1.2958904109589, 1.75890410958904, 
2.29041095890411, 7.2958904109589, 3.44657534246575, 4.5041095890411, 
9.73424657534247, 8.21643835616438, 6.29315068493151, 7.50684931506849, 
3.75616438356164, 7.96438356164384, 9.96438356164384, 29.9780821917808, 
17.3452054794521, 26.8547945205479, 5.5041095890411, 7.17260273972603, 
6.13150684931507, 3.62739726027397, 5.79452054794521, 7.7972602739726, 
5.75616438356164, 3.09041095890411, 8.0958904109589, 9.63013698630137, 
4.93424657534247, 6.93424657534247, 9.93698630136986, 19.9452054794521, 
6.42191780821918, 8.96438356164384, 28.9780821917808, 15.7643835616438, 
18.7205479452055, 1.20547945205479, 4.16986301369863, 24.1835616438356, 
25.2520547945205, 26.3095890410959, 9.61917808219178, 12.3424657534247, 
14.2986301369863, 23.7698630136986, 3.71232876712329, 8.71506849315069, 
3.58082191780822, 1.25205479452055, 5.96164383561644, 3.46027397260274, 
8.46301369863014, 7.88219178082192, 5.54246575342466, 6.46027397260274, 
4.79452054794521, 3.20821917808219, 15.5506849315068, 9.5972602739726, 
20.6, 2.09041095890411, 1.87671232876712, 22.2191780821918, 5.67397260273973, 
25.7260273972603, 26.9753424657534, 7.2958904109589, 27.5205479452055, 
8.17260273972603, 28.3095890410959, 29.2246575342466, 3.87945205479452, 
9.13424657534247, 5.84109589041096, 3.96164383561644, 4.54246575342466, 
9.92602739726027, 29.9397260273973, 16.1397260273973, 16.1780821917808, 
16.1397260273973, 18.7643835616438, 18.7643835616438, 20.2191780821918, 
20.6383561643836, 20.8904109589041, 1.16712328767123, 21.2657534246575, 
1.33424657534247, 2.04383561643836, 2.20547945205479, 4.75616438356164, 
2.92054794520548, 23.2219178082192, 4.09315068493151, 4.24931506849315, 
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29.2246575342466, 9.75890410958904, 8.75890410958904, 28.2712328767123, 
8.42465753424658, 28.4383561643836, 2.37260273972603, 4.37534246575343, 
4.75616438356164, 1.67123287671233, 3.96164383561644, 2.37260273972603, 
5.04657534246575, 2.25205479452055, 5.62739726027397, 4.29315068493151, 
1.29041095890411, 3.5041095890411, 5.5041095890411, 5.09315068493151, 
3.83561643835616, 8.46301369863014, 3.42191780821918, 1.33424657534247, 
4.5041095890411, 4.79452054794521, 6.37534246575343, 7.21095890410959, 
9.37808219178082, 29.3534246575342, 3.54246575342466, 28.6904109589041, 
8.2958904109589, 8.54246575342466, 4.62739726027397, 12.4082191780822, 
19.4712328767123, 20.972602739726, 3.67397260273973, 4.42191780821918, 
4.46027397260274, 25.3890410958904, 5.87945205479452, 25.8931506849315, 
2.20547945205479, 27.2246575342466, 7.88219178082192, 1.43561643835616, 
2.43561643835616, 4.46027397260274, 6.46027397260274, 9.46301369863014, 
19.4712328767123, 29.4767123287671, 2.45753424657534, 9.63013698630137, 
7.88219178082192, 23.641095890411, 5.13150684931507, 6.46027397260274, 
9.7972602739726, 29.8109589041096, 3.21095890410959, 8.21643835616438, 
4.92328767123288, 3.25479452054795, 2.62465753424658, 3.71232876712329, 
5.75616438356164, 25.7698630136986, 6.20821917808219, 1.20547945205479, 
4.96164383561644, 2.09041095890411, 4.13150684931507, 2.45753424657534, 
27.4767123287671, 7.46301369863014, 3.2, 18.2164383561644, 1.1972602739726, 
5.20821917808219, 8.21095890410959, 28.2246575342466, 19.972602739726, 
7.50684931506849, 9.92602739726027, 3.75616438356164, 1.38904109589041, 
3.39178082191781, 19.4082191780822, 2.04383561643836, 24.0602739726027, 
5.75616438356164, 26.4356164383562, 6.42191780821918, 28.0630136986301, 
8.21095890410959, 18.2164383561644, 6.37534246575343, 3.96164383561644, 
6.87945205479452, 7.75890410958904, 5.54246575342466, 7.75890410958904, 
2.19452054794521, 27.213698630137, 29.227397260274, 8.2, 5.87945205479452, 
1.12876712328767, 3.12876712328767, 7.54520547945206, 2.74246575342466, 
4.74520547945205, 6.74520547945205, 9.74794520547945, 29.7616438356164, 
3.58904109589041, 8.53150684931507, 18.5369863013699, 28.5452054794521, 
4.96164383561644, 19.3479452054795, 1.04657534246575, 21.386301369863, 
3.96164383561644, 5.42191780821918, 3.62739726027397, 7.13150684931507, 
2.54246575342466, 27.5616438356164, 18.0547945205479, 4.16986301369863, 
9.25753424657534, 4.76986301369863, 15.2164383561644, 6.37534246575343, 
8.17808219178082, 3.73424657534247, 1.26849315068493, 6.27123287671233, 
6.2986301369863, 19.3945205479452, 17.1397260273973, 1.20547945205479, 
6.20821917808219, 3.37534246575342, 5.25479452054795, 4.75616438356164, 
6.54246575342466, 3.16986301369863, 2.04383561643836, 4.25479452054795, 
1.04383561643836, 7.71506849315069, 5.09315068493151, 4.42191780821918, 
6.71232876712329, 3.04383561643836, 1.67945205479452, 3.30958904109589, 
3.96164383561644, 2.92054794520548, 7.92602739726027, 27.9397260273973, 
2.83835616438356, 7.46301369863014, 8.21095890410959, 7.23835616438356, 
27.2520547945205, 2.53972602739726, 6.58904109589041, 9.96438356164384, 
29.9780821917808, 16.4301369863014, 18.3671232876712, 19.2657534246575, 
20.386301369863, 2.37534246575342, 6.46027397260274, 26.4739726027397, 
6.62739726027397, 4.75616438356164, 6.71232876712329, 28.8958904109589, 
15.5123287671233, 14.8465753424658, 14.8465753424658, 2.0027397260274, 
3.20821917808219, 3.71232876712329, 1.6958904109589, 2.07671232876712, 
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8.25205479452055, 3.24931506849315, 1.38904109589041, 5.89315068493151, 
6.60821917808219, 1.5013698630137, 3.46027397260274, 6.5041095890411, 
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9.37808219178082, 7.96438356164384, 11.0493150684932, 12.213698630137, 
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4.21614538082358, 4.46897200885451, 4.78330432863798, 2.44945159376164, 
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3.24894671597498, 3.81640051029768, 4.29811641034397, 2.41235239871336, 
4.16381972504066, 4.82320604897032, 2.85924829032196, 3.60102245001335, 
4.38947242891732, 4.83225858506487, 2.32276057802484, 2.77466777269801, 
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3.19775298791517, 5.05733248819791, 2.34500731502677, 3.86347792664764, 
4.55194984744145, 4.8536532162969, 4.68580090262939, 3.8560422018531, 
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Answer 1

对于相对简单的实现，您可以尝试如下：

考虑一个运行的数据切片，具有恒定的宽度。宽度必须大于最大间隙。

使用Monotone Chain算法构建点的上部和下部船体（无需排序，您的点已经通过增加横坐标）。 https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain

对于切片中的中间横坐标，估计凸包上的两个纵坐标（通过识别该特定absicssa处的船体边缘）。通过扫描，这将创建所需的曲线。在范围的两端，使用第一个和最后一个半切片中的所有横坐标。

曲率很高和/或窗口很大会有一个小的向下偏差，但没有人会完美。

总运行时间将与点数和窗口宽度的乘积成正比。但我想增量版本可以做得更好。

Answer 2

有一种线性编程方法，您可能会发现R包将为您进行线性编程。

找到平滑曲线的空间，例如样条曲线，其中平滑曲线是基函数的线性组合。

现在解决线性程序，以最小化该点上的值的所有点的总和减去该点处基函数的线性组合的值，但受基函数的线性组合必须的约束不得大于那时的价值。