在创建直方图时考虑错误

Question

我有一组N观察(x[i], y[i]), i=0..N在2D空间中作为点分布.每个点都有坐标(e_x[i], e_y[i], i=0..N)中的相关错误以及附加到它的权重(w[i], i=0..N).

我想这些生成的二维直方图N百分点,占不仅为权重,但也为错误,这将导致各点进行传播可能很多箱中如果误差值足够大(假设一个标准高斯分布对于错误,尽管可能会考虑其他分布).

我看到numpy.histogram2d有一个weights参数,所以这是照顾.问题是如何解释每个N观察点的错误.

有没有让我这样做的功能？我愿意在任何事情numpy和scipy.

Answer 1

根据 user1415946 的评论，您可以假设每个点代表一个双变量正态分布，其协方差矩阵由 给出[[e_x[i]**2,0][0,e_y[i]**2]]。但是，生成的分布不是正态分布 - 运行示例后，您会看到直方图根本不像高斯分布，而是一组分布。

要从这组分布中创建直方图，我看到的一种方法是使用numpy.random.multivariate_normal从每个点生成随机样本。请参阅下面的示例代码以及一些人工数据。

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt


# This is a function I like to use for plotting histograms
def plotHistogram3d(hist, xedges, yedges):
    fig = plt.figure()
    ax = fig.add_subplot(111, projection='3d')
    hist = hist.transpose()
    # Transposing is done so that bar3d x and y match hist shape correctly
    dx = np.mean(np.diff(xedges))
    dy = np.mean(np.diff(yedges))

    # Computing the number of elements
    elements = (len(xedges) - 1) * (len(yedges) - 1)
    # Generating mesh grids.
    xpos, ypos = np.meshgrid(xedges[:-1]+dx/2.0, yedges[:-1]+dy/2.0)

    # Vectorizing matrices
    xpos = xpos.flatten()
    ypos = ypos.flatten()
    zpos = np.zeros(elements)
    dx = dx * np.ones_like(zpos) * 0.5  # 0.5 factor to give room between bars.
# Use 1.0 if you want all bars 'glued' to each other
    dy = dy * np.ones_like(zpos) * 0.5
    dz = hist.flatten()

    ax.bar3d(xpos, ypos, zpos, dx, dy, dz, color='b')
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    ax.set_zlabel('Count')
    return

"""
INPUT DATA
"""
#                 x  y ex ey  w
data = np.array([[1, 2, 1, 1, 1],
                 [3, 0, 1, 1, 2],
                 [0, 1, 2, 1, 5],
                 [7, 7, 1, 3, 1]])

"""
Generate samples
"""
# Sample size (100 samples will be generated for each data point)
SAMPLE_SIZE = 100
# I want to fill in a table with columns [x, y, w]. Each data point generates SAMPLE_SIZE
# samples, so we have SAMPLE_SIZE * (number of data points) generated points
points = np.zeros((SAMPLE_SIZE * data.shape[0], 3))  # Initializing this matrix

for i, element in enumerate(data):  # For each row in the data set
    meanVector = element[:2]
    covarianceMatrix = np.diag(element[2:4]**2)  # Diagonal matrix with elements equal to error^2
    # For columns 0 and 1, add generated x and y samples
    points[SAMPLE_SIZE*i:SAMPLE_SIZE*(i+1), :2] = \
        np.random.multivariate_normal(meanVector, covarianceMatrix, SAMPLE_SIZE)
    # For column 2, simply copy original weight
    points[SAMPLE_SIZE*i:SAMPLE_SIZE*(i+1), 2] = element[4]  # weights

hist, xedges, yedges = np.histogram2d(points[:, 0], points[:, 1], weights=points[:, 2])
plotHistogram3d(hist, xedges, yedges)
plt.show()

结果绘制如下：