基本概念: 均值，期望，中位数，众数，极差，总体方差，样本方差，协方差

# --*-- coding:utf-8 --*--
import math
import itertools


def Mean(t):
    """均值"""
    return float(sum(t)) / len(t)


def E(x, p):
    """
    离散性随即变量的数学期望(也称为均值): 随机变量X与其概率P乘积的和
    """
    return sum([x[i] * p[i] for i in range(len(x))])


def Median(t):
    """中位数"""
    arr = sorted(t)
    idx = (len(arr) - 1) / 2
    if type(idx) is int:
        return arr[idx]
    if type(idx) is float:
        return Mean(arr[int(math.floor(idx)):int(math.ceil(idx)) + 1])


def Mode(t):
    """众数"""
    if not t:
        return None
    arr = __getfreq(t)
    if arr[0][0] == 1:
        return None
    else:
        for k, g in itertools.groupby(arr, key=lambda x: x[0]):
            return [t[1] for t in g]


def __getfreq(t):
    """获取t中每个值及其出现次数"""
    arr = sorted(t)
    alist = []
    for k, g in itertools.groupby(arr):
        alist.append((len(list(g)), k))
    alist.sort(key=lambda x: x[0], reverse=True)
    return alist


def Var(t, mu=None):
    """方差"""
    if mu is None:
        mu = Mean(t)

    # compute the squared deviations and return their mean.
    dev2 = [(x - mu)**2 for x in t]
    var = Mean(dev2)
    return var


def D(x, p):
    """
    离散性随机变量的方差： ((X与X的期望(均值)的差值)的平方)的期望(均值)
    """
    # 由定义计算_0
    # e = E(x, p)
    # return sum([(x[i] - e) ** 2 * p[i] for i in range(len(x))])

    # 由定义计算_1,构造新的随机变量Y
    # e = E(x, p)
    # y = [(x[i] - e) ** 2 for i in range(len(x))]
    # return E(y, p)

    # 由简化公式计算
    e = E(x, p)
    e_1 = E([x[i] ** 2 for i in range(len(x))], p)
    return e_1 - e ** 2


def SVar(t):
    """样本方差"""
    if not t:
        return None
    mu = Mean(t)
    return sum([(x - mu) ** 2 for x in t]) / (len(t) - 1)


def MeanVar(t):
    """均值和方差"""
    mu = Mean(t)
    var = Var(t, mu)
    return mu, var


def StdVar(t, mu=None):
    """标准差"""
    if mu is None:
        mu = Mean(t)
    import math
    return math.sqrt(Var(t, mu))


def Range(t):
    """极差"""
    if not t:
        return None
    return max(t) - min(t)


def Cov(X, Y):
    """
    协方差
    X与Y的对应离均差(x-mu)的乘积的均值
    功能：
    如果离均差变化方向一致，则正负号相同，乘积为正数
    缺陷：
    X Y 例如是身高和体重， cm * kg 没有意义,所以一般用标准分数来解决
    详见相关系数 ==> def pearson_correlation(X, Y): 标准分数单位为1,均值为0,方差为1
    相关系数的单位为1,相比于协方差的单位更好理解
    """
    mu_x = Mean(X)
    mu_y = Mean(Y)
    # 计算离均差：如果X Y的变化方向一致，那么X，Y的离均差应该有相同的正负号
    # d_x = [x - mu_x for x in X]
    # d_y = [y - mu_y for y in Y]
    # # 离均差的均值 ==> 协方差
    # return Mean([d_x[i] * d_y[i] for i in range(len(X))])

    total = 0.0
    for x, y in zip(X, Y):
        total += (x-mu_x) * (y-mu_y)
    return total / len(X)


def standardsocre(x, mu, sigma):
    """
    标准分数
    x-mu ==> 离差：x与均值的差
    x-mu / sigma 实现归一化
    功能：转换后的标准化变量Z的单位为 1 , 均值为0, 方差为 1
    """
    return (x - mu) / sigma


def pearson_correlation(X, Y):
    """
    相关系数：
    将协方差中的X，Y 转化为标准分数，标准分数的乘积的均值即为相关系数
    (x-mu）/ sigma 实现了 归一化
    相关系数的单位为1,相比于协方差的单位更好理解
    协方差为1，二者完全相关，知道其中的一个值，可以准确预测另外一个值
    协方差为-1,而这完全负相关
    """
    # 计算均值（期望）
    mu_x = Mean(X)
    mu_y = Mean(Y)
    # 计算标准差
    sigma_x = StdVar(X)
    sigma_y = StdVar(Y)
    # 标准分数的均值即为：相关系数
    p = [standardsocre(X[i], mu_x, sigma_x) * standardsocre(Y[i], mu_y, sigma_y) for i in range(len(X))]
    return Mean(p)

    # 采用简易公式计算相关系数
    # sigma_x = stdvar(X)
    # sigma_y = stdvar(Y)
    # return Cov(X, Y) / (sigma_x * sigma_y)

    #相关系数=X，Y的协方差除以X的标准差*Y的标准差
    # xbar, varx = MeanVar(X)
    # ybar, vary = MeanVar(Y)
    # corr = Cov(xs, ys) / math.sqrt(varx * vary)
    # return corr


if __name__ == ‘__main__‘:
    astr = ‘93 62 51 93 75 82 93 62 65 51‘
    alist = [int(e) for e in astr.split()]
    print ‘均值:‘, Mean(alist)
    print ‘中位数:‘, Median(alist)
    print ‘众数:‘, Mode(alist)

    print ‘极差:‘, Range(alist)
    print ‘总体方差:‘, Var(alist)
    print ‘样本方差‘, SVar(alist)
    print ‘标准差:‘, StdVar(alist)

    print ‘验证协方差 Cov(X, X) == Var(X)‘
    print Cov(alist, alist)
    print Var(alist)
    print ‘协方差与方差相等，说明公式编写的函数是正确的‘