Python Data Frame如何在2D数组中查找局部最大值

Question

我有两列的数据框，即x，y。我想在x，y图中找到局部最大值，如所附图的图1所示。我按照这种方式进行：将数据帧的每一列转换为两个单独的矩阵数组。步骤1：我的代码首先确定Y中局部最大值的索引位置。步骤2：将标识与那些索引位置相对应的x值。而已。结果，我只能找到两个局部最大值。但是，存在三个局部最大值。我的方法无法识别它。我的问题：有没有办法可以直接从2D数组识别局部最大值？

我现在的代码：

x = my_dataframe.iloc[:,0].values # conversion of Data frame column into an array
y = my_dataframe.iloc[:,2].values # conversion of Data frame column into an array        

# Step 1: for local maximum in y list
local_y_index = argrelextrema(y, np.greater)
print("Index position of local maximum in y = ",local_y_index[0])

# Step 2: Below code is for identifying the value of x at local maximum
local_x = x[local_mpp_index[0]]
print("value of x corresponding to local maximum in y = ",local_x)

输出为：

Index position of local maximum in y =  [105 197]
value of x corresponding to local maximum in y =  [149.21 281.06]

我的问题：如图1所示，我的上述方法仅识别了两个局部峰。但是有三个高峰。是否有更好的方法直接从x和y的2D数组中识别出局部最大值？

x = [1.0330e-01, 1.0380e-01, 1.0430e-01, 1.0680e-01, 1.1932e-01, 1.8192e-01,
 3.6365e-01, 5.4539e-01, 7.9191e-01, 1.0384e+00, 1.3626e+00, 1.6869e+00,
 1.7438e+00, 2.0286e+00, 2.4825e+00, 2.9363e+00, 3.4787e+00, 4.0212e+00,
 4.7129e+00, 5.2137e+00, 6.0460e+00, 6.9486e+00, 7.8511e+00, 8.6835e+00,
 1.0092e+01, 1.0418e+01, 1.2153e+01, 1.3888e+01, 1.5623e+01, 1.7358e+01,
 1.9093e+01, 2.0828e+01, 2.2563e+01, 2.4298e+01, 2.6033e+01, 2.7768e+01,
 2.9503e+01, 3.1237e+01, 3.2972e+01, 3.4707e+01, 3.6442e+01, 3.8177e+01,
 3.9912e+01, 4.1647e+01, 4.3382e+01, 4.5117e+01, 4.6852e+01, 4.8587e+01,
 5.0322e+01, 5.2056e+01, 5.3791e+01, 5.5526e+01, 5.7261e+01, 5.8996e+01,
 6.0731e+01, 6.2466e+01, 6.4201e+01, 6.5936e+01, 6.7671e+01, 6.9406e+01,
 7.1141e+01, 7.2875e+01, 7.4610e+01, 7.6345e+01, 7.8080e+01, 7.9815e+01,
 8.1550e+01, 8.3285e+01, 8.5020e+01, 8.6755e+01, 8.8490e+01, 9.0225e+01,
 9.1960e+01, 9.3694e+01, 9.5429e+01, 9.7164e+01, 9.8899e+01, 1.0063e+02,
 1.0237e+02, 1.0410e+02, 1.0584e+02, 1.0757e+02, 1.0931e+02, 1.1104e+02,
 1.1278e+02, 1.1451e+02, 1.1625e+02, 1.1798e+02, 1.1972e+02, 1.2145e+02,
 1.2319e+02, 1.2492e+02, 1.2666e+02, 1.2839e+02, 1.3013e+02, 1.3186e+02,
 1.3360e+02, 1.3533e+02, 1.3707e+02, 1.3880e+02, 1.4054e+02, 1.4227e+02,
 1.4401e+02, 1.4574e+02, 1.4748e+02, 1.4921e+02, 1.5095e+02, 1.5268e+02,
 1.5442e+02, 1.5615e+02, 1.5684e+02, 1.5753e+02, 1.5789e+02, 1.5861e+02,
 1.5934e+02, 1.5962e+02, 1.6056e+02, 1.6136e+02, 1.6256e+02, 1.6309e+02,
 1.6482e+02, 1.6656e+02, 1.6829e+02, 1.7003e+02, 1.7176e+02, 1.7350e+02,
 1.7523e+02, 1.7697e+02, 1.7870e+02, 1.8044e+02, 1.8217e+02, 1.8391e+02,
 1.8564e+02, 1.8738e+02, 1.8911e+02, 1.9085e+02, 1.9258e+02, 1.9432e+02,
 1.9605e+02, 1.9779e+02, 1.9952e+02, 2.0126e+02, 2.0299e+02, 2.0473e+02,
 2.0646e+02, 2.0820e+02, 2.0993e+02, 2.1167e+02, 2.1340e+02, 2.1514e+02,
 2.1687e+02, 2.1861e+02, 2.1927e+02, 2.1993e+02, 2.2034e+02, 2.2103e+02,
 2.2172e+02, 2.2208e+02, 2.2296e+02, 2.2381e+02, 2.2493e+02, 2.2555e+02,
 2.2700e+02, 2.2728e+02, 2.2871e+02, 2.2902e+02, 2.3057e+02, 2.3075e+02,
 2.3164e+02, 2.3249e+02, 2.3422e+02, 2.3596e+02, 2.3769e+02, 2.3943e+02,
 2.4116e+02, 2.4290e+02, 2.4463e+02, 2.4637e+02, 2.4810e+02, 2.4984e+02,
 2.5157e+02, 2.5331e+02, 2.5504e+02, 2.5678e+02, 2.5851e+02, 2.6025e+02,
 2.6198e+02, 2.6371e+02, 2.6545e+02, 2.6718e+02, 2.6892e+02, 2.7065e+02,
 2.7239e+02, 2.7412e+02, 2.7586e+02, 2.7759e+02, 2.7933e+02, 2.8106e+02,
 2.8280e+02, 2.8453e+02, 2.8627e+02, 2.8800e+02, 2.8974e+02, 2.9147e+02,
 2.9321e+02, 2.9494e+02, 2.9668e+02, 2.9841e+02, 3.0015e+02, 3.0188e+02,
 3.0362e+02, 3.0535e+02, 3.0709e+02, 3.0882e+02, 3.1056e+02, 3.1229e+02,
 3.1403e+02, 3.1576e+02, 3.1749e+02, 3.1923e+02, 3.2096e+02, 3.2270e+02,
 3.2443e+02, 3.2617e+02, 3.2790e+02, 3.2964e+02, 3.3137e+02, 3.3311e+02,
 3.3484e+02, 3.3658e+02, 3.4686e+02, 3.4686e+02, 3.4686e+02, 3.4686e+02,
 3.4686e+02, 3.4686e+02, 3.4686e+02, 3.4686e+02, 3.4687e+02]

y = [4.2014e-01, 4.2237e-01, 4.2460e-01, 4.3574e-01, 4.9146e-01, 7.7004e-01,
     1.5788e+00, 2.3874e+00, 3.4842e+00, 4.5808e+00, 6.0228e+00, 7.4647e+00,
     7.7180e+00, 8.9843e+00, 1.1002e+01, 1.3020e+01, 1.5431e+01, 1.7842e+01,
     2.0916e+01, 2.3141e+01, 2.6839e+01, 3.0848e+01, 3.4856e+01, 3.8552e+01,
     4.4807e+01, 4.6254e+01, 5.3953e+01, 6.1650e+01, 6.9344e+01, 7.7035e+01,
     8.4723e+01, 9.2409e+01, 1.0009e+02, 1.0777e+02, 1.1545e+02, 1.2312e+02,
     1.3079e+02, 1.3846e+02, 1.4613e+02, 1.5379e+02, 1.6145e+02, 1.6911e+02,
     1.7677e+02, 1.8442e+02, 1.9207e+02, 1.9971e+02, 2.0735e+02, 2.1499e+02,
     2.2263e+02, 2.3027e+02, 2.3790e+02, 2.4552e+02, 2.5315e+02, 2.6077e+02,
     2.6839e+02, 2.7600e+02, 2.8361e+02, 2.9122e+02, 2.9882e+02, 3.0642e+02,
     3.1401e+02, 3.2160e+02, 3.2918e+02, 3.3676e+02, 3.4433e+02, 3.5190e+02,
     3.5946e+02, 3.6701e+02, 3.7455e+02, 3.8209e+02, 3.8961e+02, 3.9712e+02,
     4.0462e+02, 4.1211e+02, 4.1958e+02, 4.2703e+02, 4.3447e+02, 4.4188e+02,
     4.4926e+02, 4.5661e+02, 4.6393e+02, 4.7122e+02, 4.7846e+02, 4.8565e+02,
     4.9278e+02, 4.9985e+02, 5.0685e+02, 5.1376e+02, 5.2057e+02, 5.2728e+02,
     5.3386e+02, 5.4029e+02, 5.4656e+02, 5.5265e+02, 5.5852e+02, 5.6415e+02,
     5.6950e+02, 5.7453e+02, 5.7920e+02, 5.8347e+02, 5.8727e+02, 5.9056e+02,
     5.9325e+02, 5.9527e+02, 5.9654e+02, 5.9697e+02, 5.9646e+02, 5.9490e+02,
     5.9217e+02, 5.9175e+02, 5.9419e+02, 5.9665e+02, 5.9790e+02, 6.0049e+02,
     6.0309e+02, 6.0410e+02, 6.0748e+02, 6.1034e+02, 6.1467e+02, 6.1658e+02,
     6.2282e+02, 6.2905e+02, 6.3528e+02, 6.4151e+02, 6.4772e+02, 6.5393e+02,
     6.6013e+02, 6.6632e+02, 6.7251e+02, 6.7868e+02, 6.8484e+02, 6.9099e+02,
     6.9712e+02, 7.0323e+02, 7.0931e+02, 7.1536e+02, 7.2137e+02, 7.2732e+02,
     7.3320e+02, 7.3899e+02, 7.4464e+02, 7.5013e+02, 7.5540e+02, 7.6039e+02,
     7.6502e+02, 7.6922e+02, 7.7287e+02, 7.7589e+02, 7.7817e+02, 7.7962e+02,
     7.8014e+02, 7.8039e+02, 7.8250e+02, 7.8464e+02, 7.8598e+02, 7.8823e+02,
     7.9050e+02, 7.9166e+02, 7.9458e+02, 7.9739e+02, 8.0109e+02, 8.0313e+02,
     8.0793e+02, 8.0888e+02, 8.1359e+02, 8.1462e+02, 8.1978e+02, 8.2036e+02,
     8.2330e+02, 8.2610e+02, 8.3183e+02, 8.3755e+02, 8.4326e+02, 8.4897e+02,
     8.5466e+02, 8.6035e+02, 8.6602e+02, 8.7168e+02, 8.7732e+02, 8.8295e+02,
     8.8855e+02, 8.9412e+02, 8.9965e+02, 9.0513e+02, 9.1055e+02, 9.1588e+02,
     9.2110e+02, 9.2618e+02, 9.3108e+02, 9.3576e+02, 9.4015e+02, 9.4420e+02,
     9.4784e+02, 9.5100e+02, 9.5362e+02, 9.5563e+02, 9.5698e+02, 9.5761e+02,
     9.5746e+02, 9.5650e+02, 9.5468e+02, 9.5195e+02, 9.4828e+02, 9.4363e+02,
     9.3796e+02, 9.3122e+02, 9.2337e+02, 9.1437e+02, 9.0418e+02, 8.9275e+02,
     8.8004e+02, 8.6600e+02, 8.5059e+02, 8.3376e+02, 8.1546e+02, 7.9566e+02,
     7.7430e+02, 7.5134e+02, 7.2674e+02, 7.0046e+02, 6.7244e+02, 6.4266e+02,
     6.1108e+02, 5.7765e+02, 5.4234e+02, 5.0512e+02, 4.6596e+02, 4.2483e+02,
     3.8170e+02, 3.3654e+02, 6.8800e-05, 5.1500e-05, 4.8000e-05, 4.7300e-05,
     4.7200e-05, 4.7200e-05, 4.7200e-05, 4.7200e-05, 1.5520e-04]

Answer 1

任何极值使得极值处的导数为零。由于我们没有数据的解析表达式，因此我们可以做的下一件最好的事情是近似导数。这本质上与采取1步差异并寻找“小”值相同。

以下对我来说很好，

def find_extrema(frame, tolerance=0.5):
    diff = frame.diff()

    extrema = diff[np.abs(diff) < tolerance]

    return extrema[~np.isnan(extrema.y)]


df = pd.DataFrame(dict(y=y), index=x)

candidates = find_extrema(df)

print(candidates)

我发现，

                      y
0.10380    2.230000e-03
0.10430    2.230000e-03
0.10680    1.114000e-02
0.11932    5.572000e-02
0.18192    2.785800e-01
1.74380    2.533000e-01
149.21000  4.300000e-01
156.15000 -4.200000e-01
218.61000  2.500000e-01
282.80000 -1.500000e-01
346.86000 -1.730000e-05
346.86000 -3.500000e-06
346.86000 -7.000000e-07
346.86000 -1.000000e-07
346.86000  0.000000e+00
346.86000  0.000000e+00
346.86000  0.000000e+00
346.87000  1.080000e-04

这将需要进行一些清洁（大部分在边缘），但是一般的想法应该对您很清楚。

以下图是用

绘制的

tolerance = 0.75

diff = df.diff()

ax = diff[np.abs(diff) < tolerance].y.plot(
     title="Derivative approximation for tolerance = {0}".format(tolerance))

ax.set_xlabel("x")
ax.set_ylabel("y[x] - y[x - 1]")

plt.show()

（请注意较大的公差，因此我们实际上可以观察到一些线而不仅仅是点）

Answer 2

您还可以使用np.gradient函数并查看渐变符号的位置：

z = np.gradient(y, x)
i = 0
while i < len(x)-2:
if (z[i]*z[i+2]<=0 and z[i]>0): #gradient changes sign > optima, and point previous to optima has a positive slope
        print(i+1, x[i+1], y[i+1])
        i = i+1
    i+=1

plt.ylim(-1, 1)
plt.plot(x, z)

查看该图，似乎在210处的点不是最大值（梯度未达到零）。您可以通过将if语句替换为以下if (y[i+1]>y[i] and y[i+1]>y[i+2]):

来进行检查

Answer 3

这是我幼稚的方法：

步骤1：查找包含斜率的列表，如果两个连续的+1值正在增加，则为y，如果减小则为-1，而{{1 }}是否相同：

0

现在，如果您打印import numpy as np slope = [np.sign(y[i]-y[i-1]) for i in range(1, len(y))] ，它将只是slope，它说明每两个连续的0,1,-1点之间的斜率。

第2步：为了找到y和minimas，我编写了这段代码来评估斜率是否发生变化。如果它从maximas变为1，则索引将另存为-1，否则另存为maxima。

minima

，如果您打印结果：

x_prev = slope[0]
optima_dic={'minima':[], 'maxima':[]}
for i in range(1, len(slope)):
    if slope[i]*x_prev==-1: #slope changed
        if x_prev==1: # slope changed from 1 to -1
            optima_dic['maxima'].append(i)
        else: # slope changed from -1 to 1
            optima_dic['minima'].append(i)
        x_prev=-x_prev

输出：

print(optima_dic)

又快又脏：）