在Python中对频率权重进行排序

Question

我的数据是一组n个观察对及其频率，即每对（xi，yi）对应一些ki，观察次数（xi，yi）。理想情况下，我想计算Kendall的tau和Spearman的rho来计算这些对的所有副本的集合，其中包括k1 + k2 + ... + kn对。问题是观察总数k1 + k2 + ... + kn是巨大的，并且这样的数据结构将不适合存储器。

当然，我考虑分配第i对的频率，ki /（k1 + k2 + ... + kn）作为其权重，并计算加权集的等级相关性 - 但我找不到任何工具那。在我遇到的等级相关的加权变量（例如，scipy.stats.weightedtau）中，权重代表等级的重要性而不是对，这与我的原因无关。 Pearson的r似乎完全具有我需要的加权选项，但它不符合我的目的，因为x和y无关是线性相关的。我想知道我是否错过了关于加权数据点的广义相关的一些概念。

到目前为止我唯一的想法是通过一些公因子c缩小k1，k2，...，kn，因此第i对的缩放份数是[ki / c]（这里[。]是舍入运算符，因为我们需要每对的整数副本）。通过选择c使得[k1 / c] + [k2 / c] + ... + [kn / c]对可以适合存储器，然后我们可以计算得到的集合的相关系数tau和rho。然而，ki和kj可以相差很多个数量级，因此对于某些ki，c可能非常大，因此舍入ki / c会导致信息丢失。

UPD：可以在具有指定频率权重的数据集上计算Spearman的rho和p值，如下所示：

def frequency_pearsonr(data, frequencies):
    """
    Calculates Pearson's r between columns (variables), given the
    frequencies of the rows (observations).

    :param data: 2-D array with data
    :param frequencies: 1-D array with frequencies
    :return: 2-D array with pairwise correlations,
        2-D array with pairwise p-values
    """
    df = frequencies.sum() - 2
    Sigma = np.cov(data.T, fweights=frequencies)
    sigma_diag = Sigma.diagonal()
    Sigma_diag_pairwise_products = np.multiply.outer(sigma_diag, sigma_diag)
    # Calculate matrix with pairwise correlations.
    R = Sigma / np.sqrt(Sigma_diag_pairwise_products)
    # Calculate matrix with pairwise t-statistics. Main diagonal should
    # get 1 / 0 = inf.
    with np.errstate(divide='ignore'):
        T = R / np.sqrt((1 - R * R) / df)
    # Calculate matrix with pairwise p-values.
    P = 2 * stats.t.sf(np.abs(T), df)

    return R, P


def frequency_rank(data, frequencies):
    """
    Ranks 1-D data array, given the frequency of each value. Same
    values get same "averaged" ranks. Array with ranks is shaped to
    match the input data array.

    :param data: 1-D array with data
    :param frequencies: 1-D array with frequencies
    :return: 1-D array with ranks
    """
    s = 0
    ranks = np.empty_like(data)
    # Compute rank for each unique value.
    for value in sorted(set(data)):
        index_grid = np.ix_(data == value)
        # Find total frequency of the value.
        frequency = frequencies[index_grid].sum()
        ranks[index_grid] = s + 0.5 * (frequency + 1)
        s += frequency    

    return ranks


def frequency_spearmanrho(data, frequencies):
    """
    Calculates Spearman's rho between columns (variables), given the
    frequencies of the rows (observations).

    :param data: 2-D array with data
    :param frequencies: 1-D array with frequencies
    :return: 2-D array with pairwise correlations,
        2-D array with pairwise p-values
    """
    # Rank the columns.
    ranks = np.empty_like(data)
    for i, data_column in enumerate(data.T):
        ranks[:, i] = frequency_rank(data_column, frequencies)
    # Compute Pearson's r correlation and p-values on the ranks.
    return frequency_pearsonr(ranks, frequencies)


# Columns are variables and rows are observations, whose frequencies
# are specified.
data_col1 = np.array([1, 0, 1, 0, 1])
data_col2 = np.array([.67, .25, .75, .2, .6])
data_col3 = np.array([.1, .3, .8, .3, .2])
data = np.array([data_col1, data_col2, data_col3]).T
frequencies = np.array([2, 4, 1, 3, 2])

# Same data, but with observations (rows) actually repeated instead of
# their frequencies being specified.
expanded_data_col1 = np.array([1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1])
expanded_data_col2 = np.array([.67, .67, .25, .25, .25, .25, .75, .2, .2, .2, .6, .6])
expanded_data_col3 = np.array([.1, .1, .3, .3, .3, .3, .8, .3, .3, .3, .2, .2])
expanded_data = np.array([expanded_data_col1, expanded_data_col2, expanded_data_col3]).T

# Compute Spearman's rho for data in both formats, and compare.
frequency_Rho, frequency_P = frequency_spearmanrho(data, frequencies)
Rho, P = stats.spearmanr(expanded_data)
print(frequency_Rho - Rho)
print(frequency_P - P)

上面的特定示例显示两种方法都产生相同的相关性和相同的p值：

[[  0.00000000e+00   0.00000000e+00   0.00000000e+00]
 [  1.11022302e-16   0.00000000e+00  -5.55111512e-17]
 [  0.00000000e+00  -5.55111512e-17   0.00000000e+00]]
[[  0.00000000e+00  -1.35525272e-19   4.16333634e-17]
 [ -9.21571847e-19   0.00000000e+00  -5.55111512e-17]
 [  4.16333634e-17  -5.55111512e-17   0.00000000e+00]]