Frechet distance概念

弗雷彻距离(Frechet distance)定义了一种考虑位置和点的次序的计算两条曲线相似度的方法，常用于时间序列相似度度量和轨迹序列相似度度量。该指标的算法可用the walking dog problem来描述：想象一个人用牵引带遛狗的同时，穿过一条有限弯曲的路径，而狗则穿过一条单独的有限弯曲路径。每个人都可以改变自己的速度来放松牵引带，但都不能向后移动。则两条路径之间的Fréchet距离是牵引带的最短长度，足以让两条弯路从头到尾穿过各自的路径。
严格的数学定义为：
曲线A，B为度量空间S内的两条曲线，即$A:[0,1]\to S$,$B:[0,1]\to S$。$\alpha$和$\beta$是单位区间的两个重采样函数，$\alpha ：[0,1]\to [0,1]$。弗雷彻距离为：$$F(A,B)=\underset{\alpha ,\beta }{\inf }\underset{t\in [0,1]}{\max }(d(A(\alpha (t)),B(\beta (t))))$$
通俗的数学定义可以简化为：

代码实现

基于动态规划，求两条轨迹的弗雷彻距离，具体的，将轨迹A和B每步分为三类可能的路径状态，1）A和B都向前移动一步；2）A停留原地，B向前移动一步；3）A向前移动一步，B停留原地；

import numpy as np
from scipy.spatial import distance
from scipy.spatial import minkowski_distance
from scipy.spatial.distance import euclidean
import seaborn as sns
def Frechet_distance(exp_data,num_data):
    """
    cal fs by dynamic programming
    :param exp_data: array_like, (M,N) shape represents (data points, dimensions)
    :param num_data: array_like, (M,N) shape represents (data points, dimensions)
    # e.g. P = [[2,1] , [3,1], [4,2], [5,1]]
    # Q = [[2,0] , [3,0], [4,0]]
    :return:
    """
    P=exp_data
    Q=num_data
    p_length = len(P)
    q_length = len(Q)
    distance_matrix = np.ones((p_length, q_length)) * -1

    # fill the first value with the distance between
    # the first two points in P and Q
    distance_matrix[0, 0] = euclidean(P[0], Q[0])

    # load the first column and first row with distances (memorize)
    for i in range(1, p_length):
        distance_matrix[i, 0] = max(distance_matrix[i - 1, 0], euclidean(P[i], Q[0]))
    for j in range(1, q_length):
        distance_matrix[0, j] = max(distance_matrix[0, j - 1], euclidean(P[0], Q[j]))

    for i in range(1, p_length):
        for j in range(1, q_length):
            distance_matrix[i, j] = max(
                min(distance_matrix[i - 1, j], distance_matrix[i, j - 1], distance_matrix[i - 1, j - 1]),
                euclidean(P[i], Q[j]))
    # distance_matrix[p_length - 1, q_length - 1]
    sns.heatmap(distance_matrix, annot=True)
    return distance_matrix[p_length-1,q_length-1] # 最后一步即为弗雷彻距离

参考资料

[1] Thomas Eiter and Heikki Mannila. Computing discrete Frechet distance. Technical report, 1994. http://www.kr.tuwien.ac.at/staff/eiter/et-archive/cdtr9464.pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.937&rep=rep1&type=pdf
[2] similarity_measures python package
[3] https://en.wikipedia.org/wiki/Fr%C3%A9chet_distance
[4] 通俗的数学描述及动态规划求解