Tôi đang cố gắng ước tính các tham số của phân phối gamma phù hợp nhất với mẫu dữ liệu của mình. Tôi chỉ muốn sử dụng giá trị trung bình , tiêu chuẩn (và do đó là phương sai ) từ mẫu dữ liệu, không phải giá trị thực tế - vì những giá trị này sẽ không có sẵn trong ứng dụng của tôi.
Theo tài liệu này , các công thức sau đây có thể được áp dụng để ước tính hình dạng và tỷ lệ:
Tôi đã thử điều này cho dữ liệu của mình, tuy nhiên kết quả rất khác so với việc phân phối gamma trên dữ liệu thực tế bằng thư viện lập trình python.
Tôi đính kèm dữ liệu / mã của mình để hiển thị vấn đề trong tay:
import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import gamma
data = [91.81, 10.02, 27.61, 50.48, 3.34, 26.35, 21.0, 79.27, 31.04, 8.85, 109.2, 15.52, 11.03, 41.09, 10.75, 96.43, 109.52, 33.28, 7.66, 65.44, 52.43, 19.25, 10.97, 586.52, 56.91, 157.18, 434.74, 16.07, 334.43, 6.63, 108.41, 4.45, 42.03, 39.75, 300.17, 4.37, 343.19, 32.04, 42.57, 29.53, 276.75, 15.43, 117.67, 75.47, 292.43, 457.91, 5.49, 17.69, 10.31, 58.91, 76.94, 37.39, 64.46, 187.25, 30.0, 9.94, 83.05, 51.11, 17.68, 81.98, 4.41, 33.24, 20.36, 8.8, 846.0, 154.24, 311.09, 120.72, 65.13, 25.52, 50.9, 14.27, 17.74, 529.82, 35.13, 124.68, 13.21, 88.24, 12.12, 254.32, 22.09, 61.7, 88.08, 18.75, 14.34, 931.67, 19.98, 50.86, 7.71, 5.57, 8.81, 14.49, 26.74, 13.21, 8.92, 26.65, 10.09, 7.74, 21.23, 66.35, 31.81, 36.61, 92.29, 26.18, 20.55, 17.18, 35.44, 6.63, 69.0, 8.81, 19.87, 5.46, 29.81, 122.01, 57.83, 33.04, 9.91, 196.0, 34.26, 34.31, 36.55, 7.74, 6.68, 6.83, 18.83, 6.6, 50.78, 95.65, 53.91, 81.62, 57.96, 26.72, 76.25, 5.48, 4.43, 133.04, 33.37, 45.26, 30.51, 9.98, 11.08, 28.95, 71.25, 70.65, 3.34, 12.28, 111.67, 139.86, 23.34, 30.0, 26.38, 33.51, 1112.64, 25.87, 148.59, 552.79, 11.11, 47.8, 7.8, 9.98, 7.69, 85.46, 3.59, 122.71, 32.09, 82.51, 12.14, 12.57, 8.8, 49.61, 95.41, 26.99, 13.29, 4.57, 7.78, 4.4, 6.66, 12.17, 12.18, 1533.01, 22.95, 15.93, 14.82, 2.2, 12.04, 9.94, 17.64, 6.66, 18.64, 83.66, 142.99, 30.76, 67.57, 9.88, 46.44, 19.5, 22.2, 43.1, 653.67, 9.86, 7.69, 7.74, 27.19, 38.64, 12.32, 182.34, 43.13, 3.28, 14.32, 69.78, 32.2, 17.66, 18.67, 4.4, 9.05, 56.94, 33.32, 13.2, 15.07, 12.73, 3.32, 35.44, 14.35, 66.68, 51.28, 6.86, 75.49, 5.54, 21.0, 24.2, 38.1, 13.31, 7.78, 5.76, 51.86, 11.09, 20.71, 36.74, 21.97, 10.36, 32.04, 96.94, 13.93, 51.84, 6.88, 27.58, 100.56, 20.97, 828.16, 6.63, 32.15, 19.92, 253.23, 25.35, 23.35, 17.6, 43.18, 19.36, 13.7, 3.31, 22.99, 26.58, 4.43, 2.22, 55.46, 22.34, 13.24, 86.18, 181.29, 52.15, 5.52, 21.12, 34.24, 49.78, 14.37, 39.73, 78.22, 26.6, 20.19, 26.57, 105.8, 11.08, 46.47, 52.82, 13.46, 8.0, 7.74, 49.73, 4.4, 5.44, 51.7, 28.64, 8.95, 9.15, 4.46, 21.03, 29.92, 19.89, 4.38, 19.94, 7.77, 23.43, 57.07, 86.5, 12.82, 103.85, 39.63, 8.83, 42.32, 17.02, 14.29, 16.75, 24.4, 27.97, 8.83, 8.91, 24.23, 6.58, 30.97, 150.58, 122.73, 17.69, 37.11, 11.05, 298.23, 25.58, 9.91, 38.85, 17.24, 82.17, 42.11, 3.29, 38.63, 27.55, 18.22, 127.16, 57.66, 34.45, 41.26, 45.91, 9.88, 34.48, 484.33, 58.42, 30.09, 6.69, 254.49, 1313.58, 39.89, 3.31, 7.83, 10.98, 13.21, 67.78, 7.77, 117.72, 20.03, 83.23, 31.28, 38.97, 6.63, 6.63, 36.6, 22.12, 154.57, 112.65, 19.88, 674.18, 83.31, 5.54, 8.81, 11.06, 178.33, 30.47, 1180.39, 79.33, 37.74, 86.3, 16.61, 53.94, 52.78, 20.83, 11.15, 26.68, 86.04, 180.26, 99.62, 11.17, 28.74, 56.85, 15.51, 95.37, 44.09, 6.68, 12.14, 6.72, 19.81, 10.05, 34.26, 69.84, 14.35, 17.72, 8.81, 20.86, 37.69, 24.62, 72.11, 8.83, 7.69, 60.79, 20.02, 9.41, 13.24, 29.8, 43.09, 25.34, 174.34, 161.6, 119.34, 30.08, 54.15, 7.74, 249.29, 9.98, 21.87, 38.92, 98.45, 95.07, 7.74, 4.45, 81.98, 12.18, 28.66, 5.58, 59.94, 22.15, 9.98, 18.86, 6.69, 134.97, 13.29, 4.43, 8.88, 5.74, 25.16, 122.39, 3.53, 6.68, 3.4, 17.58, 62.51, 584.3, 46.63, 21.19, 22.14, 5.74, 8.19, 7.74, 7.64, 4.41, 3.32, 130.76, 3.29, 31.04, 3.26, 18.83, 168.31, 7.68, 120.19, 43.95, 747.12, 18.75, 306.24, 29.72, 5.57, 6.65, 53.2, 7.96, 25.34, 25.57, 8.85, 93.59, 92.96, 23.4, 60.0, 6.63, 12.15, 49.98, 39.75, 7.77, 5.73, 18.74, 11.58, 281.32, 13.99, 4.59, 13.35, 25.05, 9.98, 5.58, 91.43, 288.94, 15.43, 7.8, 9.92, 18.69, 6.63, 78.38, 18.86, 63.03, 26.38, 166.41, 27.78, 54.21, 173.32, 11.12, 17.85, 14.43, 31.31, 3.37, 16.63, 5.51, 77.74, 8.89, 17.71, 3.24, 9.28, 22.12, 2.2, 19.41, 12.23, 22.31, 9.36, 18.85, 51.5, 8.3, 23.0, 29.7, 29.81, 4.65, 75.77, 55.52, 144.45, 6.68, 13.26, 72.78, 56.71, 46.35, 6.63, 8.88, 6.61, 41.7, 15.09, 5.51, 18.78, 74.09, 487.0, 27.52, 18.99, 44.18, 41.76, 6.65, 23.62, 175.68, 446.38, 87.13, 165.69, 16.57, 7.88, 16.57, 80.17, 135.75, 3.29, 134.16, 25.58, 45.13, 114.23, 471.15, 97.75, 12.2, 32.01, 62.21, 22.36, 193.55, 210.65, 42.39, 27.57, 106.15, 44.76, 16.6, 134.76, 18.81, 14.76, 7.97, 160.59, 39.21, 60.36, 62.45, 72.18, 91.15, 23.71, 105.04, 70.87, 25.57, 122.09, 60.09, 38.8, 133.87, 4.41, 13.28, 45.63, 45.41, 67.81, 26.68, 97.33, 723.5, 5.51, 164.05, 165.32, 4.45, 57.67, 85.82, 11.56, 12.26, 17.97, 31.04, 76.72, 15.01, 35.88, 32.37, 23.63, 85.57, 9.34, 4.45, 90.25, 73.71, 45.99, 14.24, 176.85, 65.21, 9.92, 15.02, 12.9, 21.4, 59.94, 64.62, 37.53, 147.89, 36.52, 97.67, 16.65, 22.1, 23.38, 76.85, 16.58, 7.72, 17.75, 91.25, 9.91, 18.46, 4.45, 3.29, 73.18, 19.5, 5.58, 18.85, 28.64, 7.8, 43.74, 4.43, 7.99, 132.4, 41.48, 14.45, 8.78, 8.14, 9.95, 2.46, 16.61, 32.71, 17.74, 4.46, 68.25, 34.55, 9.92, 181.31, 37.63, 125.22, 25.37, 24.45, 220.92, 11.09, 35.46, 588.56, 58.21, 22.39, 78.55, 135.13, 280.65, 273.41, 381.07, 60.56, 68.63, 40.17, 27.68, 23.68, 23.15, 28.8, 20.94, 21.92, 159.06, 9.94, 127.52, 32.4, 15.93, 99.09, 48.31, 104.66, 257.4, 117.08, 180.32, 66.55, 95.99, 17.74, 30.14, 270.54, 39.8, 54.77, 16.04, 76.99, 5.43, 8.78, 76.96, 10.39, 18.47, 290.11, 48.35, 289.06, 10.44, 57.75, 47.83, 101.62, 96.3, 71.62, 256.97, 149.45, 22.17, 23.15, 89.25, 36.46, 90.03, 69.14, 28.27, 28.72, 17.44, 43.38, 56.72, 84.96, 25.4, 55.06, 47.68, 92.11, 6.65, 30.94, 15.38, 27.44, 516.55, 5.83, 19.45, 41.53, 110.69, 6.82, 54.09, 13.31, 89.8, 25.57, 110.89, 3.32, 93.76, 33.81, 80.87, 30.9, 58.53, 185.22, 4.38, 58.75, 189.53, 7.19, 7.8, 48.97, 28.8, 48.52, 45.96, 309.44, 29.16, 2.22, 255.91, 78.7, 102.67, 33.32, 43.2, 19.5, 91.59, 139.89, 5.51, 213.96, 10.02, 10.03, 39.87, 8.95, 27.74, 7.78, 65.93, 45.41, 263.21, 33.06, 5.54, 59.77, 2.2, 9.95, 14.38, 44.76, 96.45, 15.91, 133.07, 38.03, 36.43, 7.83, 105.41, 20.5, 25.35, 20.55, 119.59, 24.31, 28.81, 101.0, 67.0, 143.85, 20.55, 83.45, 60.62, 25.19, 6.65, 1745.95, 41.62, 44.96, 65.42, 9.92, 24.23, 73.56, 34.35, 75.72, 18.77, 88.59, 312.55, 56.43, 106.61, 11.44, 22.04, 5.73, 197.92, 25.32, 144.83, 145.36, 4.43, 18.33, 48.72, 33.42, 8.83, 18.85, 32.25, 88.56, 14.95, 147.39, 9.25, 35.24, 141.51, 14.41, 5.49, 42.28, 75.69, 16.96, 6.71, 17.33, 710.34, 68.92, 28.39, 24.98, 33.03, 31.06, 46.24, 36.77, 43.74, 11.48, 22.14, 13.21, 15.8, 21.9, 5.51, 20.66, 22.04, 127.0, 21.03, 36.75, 61.45, 42.12, 238.3, 57.43, 28.61, 31.31, 15.43, 8.88, 54.26, 34.01, 5.79, 8.02, 25.68, 19.67, 29.19, 4.38, 15.05, 5.57, 32.31, 81.68, 29.92, 397.98, 119.2, 5.52, 25.54, 12.78, 17.78, 100.97, 253.58, 8.92, 22.04, 22.03, 86.57, 97.27, 106.29, 33.31, 13.34, 35.57, 40.75, 6.57, 23.32, 6.63, 30.09, 62.39, 35.62, 25.23, 5.49, 77.67, 4.41, 8.77, 12.09, 32.0, 7.75, 25.44, 27.57, 25.51, 81.59, 8.83, 64.15, 48.92, 52.25, 2.2, 13.29, 15.52, 320.64, 22.26, 21.03, 79.27, 6.61, 59.38, 40.19, 43.07, 2.26, 20.97, 8.8, 205.43, 51.82, 8.78, 90.72, 6.63, 14.46, 85.62, 72.53, 29.24, 68.81, 67.6, 1.15, 13.15, 17.71, 20.06, 77.42, 167.72, 5.54, 34.45, 5.51, 54.04, 7.8, 79.91, 4.62, 66.39, 164.13, 78.1, 49.72, 19.92, 28.92, 709.25, 18.19, 875.38, 60.92, 5.55, 71.14, 301.2, 27.74, 34.26, 108.78, 88.28, 75.83, 7.82, 8.78, 44.68, 20.98, 41.9, 8.88, 124.18, 198.8, 180.0, 71.61, 119.27, 59.33, 3.28, 43.88, 14.46, 64.34, 158.59, 41.98, 32.28, 14.43, 48.49, 2.36, 14.38, 25.52, 7.83, 2.2, 292.18, 8.97, 36.18, 7.8, 8.89, 43.26, 25.35, 12.29, 6.88, 34.48, 11.09, 16.57, 35.99, 13.45, 6.6, 162.65, 13.23, 26.91, 55.62, 61.4, 48.47, 89.62, 7.77, 6.65, 11.56, 23.28, 6.66, 7.74, 4.62, 5.8, 24.56, 10.16, 8.91, 14.45, 25.37, 6.61, 75.29, 11.03, 36.75, 38.61, 36.52, 17.75, 61.87, 31.92, 120.9, 144.82, 70.98, 19.98, 80.09, 30.17, 35.48, 2.4, 42.15, 24.29, 111.26, 71.9, 158.23, 49.75, 7.75, 13.28, 10.97, 5.51, 34.37, 56.61, 138.83, 231.4, 20.17, 29.89, 20.27, 7.69, 77.35, 12.26, 1144.41, 9.95, 7.72, 196.64, 499.4, 114.38, 24.43, 94.88, 75.15, 4.48, 8.89, 196.05, 95.15, 99.28, 42.36, 234.32, 4.59, 80.97, 237.69, 89.34, 4.51, 6.68, 148.42, 108.58, 5.48, 132.38, 7.94, 204.74, 11.08, 74.24, 146.22, 79.5, 17.68, 10.51, 550.77, 45.35, 23.28, 47.57, 40.56, 114.76, 29.81, 15.51, 11.0, 26.61, 6.74, 142.82, 12.17]
Một số thông tin về dữ liệu:
Giá trị trung bình: 68,71313036020582, Phương sai: 19112.931263699986, Độ lệch chuẩn: 138.24952536518882, Số lượng phần tử trong dữ liệu đào tạo: 1166
Biểu đồ của dữ liệu:
Sử dụng thư viện python để phù hợp:
x = np.linspace(0,300,1000)
# Gamma
shape, loc, scale = gamma.fit(data, floc=0)
print(shape, loc, scale)
y = gamma.pdf(x, shape, loc, scale)
plt.title('Fitted Gamma')
plt.plot(x, y)
plt.show()
Thông số: 0.7369587045435088 0 93.2387797804
Ước tính nó:
def calculateGammaParams(data):
mean = np.mean(data)
std = np.std(data)
shape = (mean/std)**2
scale = (std**2)/mean
return (shape, 0, scale)
eshape, eloc, escale = calculateGammaParams(data)
print(eshape, eloc, escale)
ey = gamma.pdf(x, eshape, eloc, escale)
plt.title('Estimated Gamma')
plt.plot(x, ey)
plt.show()
Các thông số: 0,247031406055 0 278.155443705
Người ta có thể thấy rõ một sự khác biệt rất lớn.