独立线性度 最佳直线

找到的散点线性拟合方法都是基于最小二乘法的（numpy.polyfit() ， scipy.optimize())

以下是根据 GB/T 18459-2001中附录 A2 提供的独立线性度拟合方法，求得的最佳拟合直线

import math


def find_line(x0, y0):
    '''
    根据散点求得端基直线k,b,并得到每点对端基直线的偏差dy
    :param x0: x坐标数组
    :param y0: y坐标数组
    :return: 每点的偏差dy
    '''
    # 求得端基直线的k,b
    k = (y0[-1] - y0[0]) / (x0[-1] - x0[0])
    d = y0[-1] - (k * x0[-1])

    # 根据端基直线的k,b,求得每点对端基直线的偏差dy
    dy = [(y0[i] - ((k * x0[i]) + d)) for i in range(len(x0))]

    return dy


def find_poly(x_lst, y_lst):
    '''
    找到凸多边形,可以包含全部偏差点dy
    :param x_lst:x数组
    :param y_lst:偏差点dy数组
    :return:最佳凸多边形各个点
    '''
    g_x = []
    g_y = []

    # 从起始点开始，向右（x增大方向），找到最大斜率点，再从最大斜率点开始，向右继续寻找
    start_index = 0
    x_tmp = x_lst[1:]
    y_tmp = y_lst[1:]
    while len(x_tmp) > 0:
        k_lst = [((j - y_lst[start_index]) / (i - x_lst[start_index])) for i, j in zip(x_tmp, y_tmp)]
        start_index = k_lst.index(max(k_lst)) + 1 + start_index
        g_x.append(x_lst[start_index])
        g_y.append(y_lst[start_index])
        x_tmp = x_lst[start_index + 1:]
        y_tmp = y_lst[start_index + 1:]
    # 从最末点开始，向左（x减小方向），也找到最大斜率点（有人说找最小斜率点，但是结果算出来不对），再从最大斜率点开始，向左继续寻找
    start_index = len(x_lst) - 1
    x_tmp = x_lst[:start_index]
    y_tmp = y_lst[:start_index]
    while len(x_tmp) > 0:
        k_lst = [((j - y_lst[start_index]) / (i - x_lst[start_index])) for i, j in zip(x_tmp, y_tmp)]
        start_index = k_lst.index(max(k_lst))
        g_x.append(x_lst[start_index])
        g_y.append(y_lst[start_index])
        x_tmp = x_lst[:start_index]
        y_tmp = y_lst[:start_index]

    return g_x, g_y


def find_cross(p1, p2, p3):
    '''
    根据一个点p1和一条直线(p2和p3的连线)，
    求得该点对直线的铅垂线（平行于纵轴坐标的直线）和直线的交点
    :param p1: 点
    :param p2: 线上一点
    :param p3: 线上另一点
    :return: 铅垂线与线（p2-p3）的交点
    '''
    k = (p2[1] - p3[1]) / (p2[0] - p3[0])
    b = p2[1] - k * p2[0]
    p4 = [p1[0], (k * p1[0]) + b]
    return p4


def find_perfect_line(a, b, x0, y0):
    '''
        根据凸多边形的每个点，找到凸多边形内最长的一根铅垂线，
    与最长垂线相交的直线l1的斜率就是最佳直线的斜率
    过最长铅垂线的中点作直线l2平行于直线l1，直线l2为最佳直线
    :param a:凸多边形的x轴数组
    :param b:凸多边形的y轴数组
    :param x0: 实际x轴数组
    :param y0: 实际y轴数组
    :return:最佳直线的斜率k, 截距b
    '''
    dic = {}
    point_lst = [[i, j] for i, j in zip(a, b)]
    for i in range(len(a)):
        p1 = (a[i], b[i])  # 凸多边形的每个点坐标
        rp1 = (x0[x0.index(p1[0])], y0[x0.index(p1[0])])

        for j in range(len(a)):
            p2 = (a[j], b[j])  # 凸多边形的每个点坐标
            rp2 = (x0[x0.index(p2[0])], y0[x0.index(p2[0])])
            if j == len(a) - 1:
                p3 = (a[0], b[0])
            else:
                p3 = (a[j + 1], b[j + 1])
            rp3 = (x0[x0.index(p3[0])], y0[x0.index(p3[0])])

            if p1 == p2 or p1 == p3:
                pass
            else:
                # print('====+++===', p1, p2, p3)
                # print('====+++===', rp1, rp2, rp3)
                # 铅垂线长度
                s_length = abs(
                    (p2[1] * (p1[0] - p3[0])) + (p1[1] * (p3[0] - p2[0]) + (p3[1] * (p2[0] - p1[0]))) / (p3[0] - p2[0]))

                s = find_cross(p1, p2, p3)
                if isPointinPolygon(s, point_lst):
                    dic[s_length] = [rp1, rp2, rp3]
                    print('铅垂线长度', s_length, s, dic[s_length])

    # print(dic)
    max_s_length = dic[max(dic)]
    rp1 = max_s_length[0]
    rp2 = max_s_length[1]
    rp3 = max_s_length[2]

    # 点和直线两端点中点连线的斜率
    k = (((rp1[1] + rp2[1]) / 2) - ((rp1[1] + rp3[1]) / 2)) / (((rp1[0] + rp2[0]) / 2) - ((rp1[0] + rp3[0]) / 2))
    b = ((rp1[1] + rp2[1]) / 2) - (k * ((rp1[0] + rp2[0]) / 2))
    return k, b

def isPointinPolygon(point, rangelist):  # [[0,0],[1,1],[0,1],[0,0]] [1,0.8]
    print(point)
    # 判断是否在外包矩形内，如果不在，直接返回false
    lnglist = []
    latlist = []
    for i in range(len(rangelist) - 1):
        lnglist.append(rangelist[i][0])
        latlist.append(rangelist[i][1])
    # print(lnglist, latlist)
    maxlng = max(lnglist)
    minlng = min(lnglist)
    maxlat = max(latlist)
    minlat = min(latlist)
    # print(maxlng, minlng, maxlat, minlat)
    if (point[0] > maxlng or point[0] < minlng or
            point[1] > maxlat or point[1] < minlat):
        return False
    count = 0
    point1 = rangelist[0]
    for i in range(1, len(rangelist)):
        point2 = rangelist[i]
        # 点与多边形顶点重合
        if (point[0] == point1[0] and point[1] == point1[1]) or (point[0] == point2[0] and point[1] == point2[1]):
            print("在顶点上")
            return False
        # 判断线段两端点是否在射线两侧 不在肯定不相交 射线（-∞，lat）（lng,lat）
        if (point1[1] < point[1] and point2[1] >= point[1]) or (point1[1] >= point[1] and point2[1] < point[1]):
            # 求线段与射线交点 再和lat比较
            point12lng = point2[0] - (point2[1] - point[1]) * (point2[0] - point1[0]) / (point2[1] - point1[1])
            # print(point12lng)
            # 点在多边形边上
            if (point12lng == point[0]):
                print("点在多边形边上")
                return True
            if (point12lng < point[0]):
                count += 1
        point1 = point2
    print(count)
    if count % 2 == 0:
        return True
    else:
        return True


if __name__ == '__main__':
    x0 = [1.00, 2.00, 3.00, 4.00, 5.00, 6.00]
    y0 = [2.02, 4.00, 5.98, 7.90, 10.10, 12.05]

    dy = find_line(x0, y0)
    print(dy)
    a0, b0 = find_poly(x0, dy)
    print(a0, b0)
    print(find_perfect_line(a0, b0, x0, y0))