創建直方圖時記錄錯誤

Question

我有一組N觀測值在一個二維空間中以(x[i], y[i]), i=0..N分佈。每個點在兩個座標（e_x[i], e_y[i], i=0..N）中都有相關的錯誤，並且還有一個與其相關的權重（w[i], i=0..N）。

我想生成這些N點的2D直方圖，佔比不僅爲權重，但也爲錯誤，這將導致每個點是傳播可能在許多垃圾箱如果誤差值大足夠的（假設標準Gaussian distribution爲錯誤，儘管也許可以考慮其他分佈）。

我看到numpy.histogram2d有一個weights參數，所以這是照顧。問題是如何解釋每個觀察點的錯誤。

有沒有讓我這樣做的功能？我願意接受numpy和scipy的任何內容。

Answer 1

建立在user1415946的評論，你可以假設每個點代表bi-variate normal distribution與[[e_x[i]**2,0][0,e_y[i]**2]]給出的協方差矩陣。但是，由此產生的分佈不是一個正態分佈 - 在運行該示例之後，您會看到直方圖根本不像高斯分佈，而是其中的一組分佈。

要從這組分佈中創建直方圖，我看到的一種方法是使用numpy.random.multivariate_normal從每個點中生成隨機樣本。使用一些人造數據查看下面的示例代碼。

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() 

結果下面的曲線：