如何使用此答案中的快速 PCA 脚本在 3D 主成分分析可视化中绘图
How to Plot in 3D Principal Component Analysis Visualizations, using the fast PCA script from this answer
我在 Stack Overflow 中找到 this fast script 以使用给定的 numpy 数组执行 PCA。
我不知道如何在 3D 中绘制它，也不知道如何在 3D 中绘制累积解释方差和分量数。这个快速脚本是用协方差方法执行的，而不是奇异值分解，也许这就是我无法获得累积方差的原因？
我尝试用 this 绘图，但它不起作用。
这是代码和我的输出：
from numpy import array, dot, mean, std, empty, argsort
from numpy.linalg import eigh, solve
from numpy.random import randn
from matplotlib.pyplot import subplots, show
def cov(X):
"""
Covariance matrix
note: specifically for mean-centered data
note: numpy's `cov` uses N-1 as normalization
"""
return dot(X.T, X) / X.shape[0]
# N = data.shape[1]
# C = empty((N, N))
# for j in range(N):
# C[j, j] = mean(data[:, j] * data[:, j])
# for k in range(j + 1, N):
# C[j, k] = C[k, j] = mean(data[:, j] * data[:, k])
# return C
def pca(data, pc_count = None):
"""
Principal component analysis using eigenvalues
note: this mean-centers and auto-scales the data (in-place)
"""
data -= mean(data, 0)
data /= std(data, 0)
C = cov(data)
E, V = eigh(C)
key = argsort(E)[::-1][:pc_count]
E, V = E[key], V[:, key]
U = dot(data, V)
print(f'Eigen Values: {E}')
print(f'Eigen Vectors: {V}')
print(f'Key: {key}')
print(f'U: {U}')
print(f'shape: {U.shape}')
return U, E, V
data = dftransformed.transpose() # df tranpose and convert to numpy
trans = pca(data, 3)[0]
fig, (ax1, ax2) = subplots(1, 2)
ax1.scatter(data[:50, 0], data[:50, 1], c = 'r')
ax1.scatter(data[50:, 0], data[50:, 1], c = 'b')
ax2.scatter(trans[:50, 0], trans[:50, 1], c = 'r')
ax2.scatter(trans[50:, 0], trans[50:, 1], c = 'b')
show()
特征值&特征向量我看懂了，但是这个键值我看不懂，用户没有在答案中注释这段代码，谁知道打印出来的每个变量是什么意思？
输出：
Eigen Values: [126.30390621 68.48966957 26.03124927]
Eigen Vectors: [[-0.05998409 0.05852607 -0.03437937]
[ 0.00807487 0.00157143 -0.12352761]
[-0.00341751 0.03819162 0.08697668]
...
[-0.0210582 0.06601974 -0.04013712]
[-0.03558994 0.02953385 0.01885872]
[-0.06728424 -0.04162485 -0.01508154]]
Key: [439 438 437]
U: [[-12.70954048 8.97405411 -2.79812235]
[ -4.90853527 4.36517107 0.54129243]
[ -2.49370123 0.48341147 7.26682759]
[-16.07860635 6.16100749 5.81777637]
[ -1.81893291 6.48443689 -5.8655646 ]
[ 9.03939039 2.64196391 4.22056618]
[-14.71731064 9.19532016 -2.79275543]
[ 1.60998654 8.37866823 0.86207034]
[ -4.4503797 10.12688097 -5.12453656]
[ 12.16293556 2.2594413 -2.11730311]
[-15.76505125 9.48537581 -2.73906772]
[ -2.54289959 9.86768111 -4.84802992]
[ -5.78214902 9.21901651 -8.13594627]
[ -1.35428398 5.85550586 6.30553987]
[ 12.87261987 0.96283606 -3.26982121]
[ 24.57767477 -4.28214631 6.29510659]
[ 4.13941679 3.3688288 3.01194055]
[ -2.98318764 1.32775227 7.62610929]
[ -4.44461549 -1.49258339 1.39080386]
[ -0.10590795 -0.3313904 8.46363066]
[ 6.05960739 1.03091753 5.10875657]
[-21.27737352 -3.44453629 3.25115921]
[ -1.1183025 0.55238687 10.75611405]
[-10.6359291 7.58630341 -0.55088259]
[ 4.52557492 -8.05670864 2.23113833]
[-11.07822559 1.50970501 4.66555889]
[ -6.89542628 -19.24672805 -3.71322812]
[ -0.57831362 -17.84956249 -5.52002876]
[-12.70262277 -14.05542691 -2.72417438]
[ -7.50263129 -15.83723295 -3.2635125 ]
[ -7.52780216 -17.60790567 -2.00134852]
[ -5.34422731 -17.29394266 -2.69261597]
[ 9.40597893 0.21140292 2.05522806]
[ 12.12423431 -2.80281266 7.81182024]
[ 19.51224195 4.7624575 -11.20523383]
[ 22.38102384 0.82486072 -1.64716468]
[ -8.60947699 4.12597477 -6.01885407]
[ 9.56268414 1.18190655 -5.44074124]
[ 14.97675455 3.31666971 -3.30012109]
[ 20.47530869 -1.95896058 -1.91238615]]
shape: (40, 3)
trans = pca(data, 3)[0]是U数据，因为[0]选择返回数据的第一个索引，而pcareturnsU, E, V ax2.scatter(trans[:50, 0], trans[:50, 1], c = 'r') 绘制第 0 列的前 50 行与第 1 列的前 50 行，ax2.scatter(trans[50:, 0], trans[50:, 1], c = 'b') 对从 50 到末尾的行执行相同的操作。这来自 this fast script 中给出的示例数据，但您的数据只有 shape: (40, 3)（例如只有 40 行数据）。- 为了将
trans 绘制为 3d 散点图，将 3 列中的每一列提取到一个单独的变量中并绘制为散点图。
# imports as shown in the linked answer
from numpy import array, dot, mean, std, empty, argsort
from numpy.linalg import eigh, solve
from numpy.random import randn
from matplotlib.pyplot import subplots, show
# other imports
import numpy as np
# test data from linked answer (e.g. this fast script)
np.random.seed(365) # makes data repeatable
data = array([randn(8) for k in range(150)]) # creates array with shape (150, 8)
data[:50, 2:4] += 5 # adds 5 to first 50 rows of columns 2:4
data[50:, 2:5] += 5 # adds 5 to to rows from 50 of columns 2:5
# function call
trans = pca(data, 3)[0] # [0] gets U returned by pca(...)
# extract each column to a separate variable
x = trans[:, 0] # all rows of column 0
y = trans[:, 1] # all rows of column 1
z = trans[:, 2] # all rows of column 2
# plot 3d scatter plot
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(x, y, z)
我在 Stack Overflow 中找到 this fast script 以使用给定的 numpy 数组执行 PCA。
我不知道如何在 3D 中绘制它，也不知道如何在 3D 中绘制累积解释方差和分量数。这个快速脚本是用协方差方法执行的，而不是奇异值分解，也许这就是我无法获得累积方差的原因？
我尝试用 this 绘图，但它不起作用。
这是代码和我的输出：
from numpy import array, dot, mean, std, empty, argsort
from numpy.linalg import eigh, solve
from numpy.random import randn
from matplotlib.pyplot import subplots, show
def cov(X):
"""
Covariance matrix
note: specifically for mean-centered data
note: numpy's `cov` uses N-1 as normalization
"""
return dot(X.T, X) / X.shape[0]
# N = data.shape[1]
# C = empty((N, N))
# for j in range(N):
# C[j, j] = mean(data[:, j] * data[:, j])
# for k in range(j + 1, N):
# C[j, k] = C[k, j] = mean(data[:, j] * data[:, k])
# return C
def pca(data, pc_count = None):
"""
Principal component analysis using eigenvalues
note: this mean-centers and auto-scales the data (in-place)
"""
data -= mean(data, 0)
data /= std(data, 0)
C = cov(data)
E, V = eigh(C)
key = argsort(E)[::-1][:pc_count]
E, V = E[key], V[:, key]
U = dot(data, V)
print(f'Eigen Values: {E}')
print(f'Eigen Vectors: {V}')
print(f'Key: {key}')
print(f'U: {U}')
print(f'shape: {U.shape}')
return U, E, V
data = dftransformed.transpose() # df tranpose and convert to numpy
trans = pca(data, 3)[0]
fig, (ax1, ax2) = subplots(1, 2)
ax1.scatter(data[:50, 0], data[:50, 1], c = 'r')
ax1.scatter(data[50:, 0], data[50:, 1], c = 'b')
ax2.scatter(trans[:50, 0], trans[:50, 1], c = 'r')
ax2.scatter(trans[50:, 0], trans[50:, 1], c = 'b')
show()
特征值&特征向量我看懂了，但是这个键值我看不懂，用户没有在答案中注释这段代码，谁知道打印出来的每个变量是什么意思？
输出：
Eigen Values: [126.30390621 68.48966957 26.03124927]
Eigen Vectors: [[-0.05998409 0.05852607 -0.03437937]
[ 0.00807487 0.00157143 -0.12352761]
[-0.00341751 0.03819162 0.08697668]
...
[-0.0210582 0.06601974 -0.04013712]
[-0.03558994 0.02953385 0.01885872]
[-0.06728424 -0.04162485 -0.01508154]]
Key: [439 438 437]
U: [[-12.70954048 8.97405411 -2.79812235]
[ -4.90853527 4.36517107 0.54129243]
[ -2.49370123 0.48341147 7.26682759]
[-16.07860635 6.16100749 5.81777637]
[ -1.81893291 6.48443689 -5.8655646 ]
[ 9.03939039 2.64196391 4.22056618]
[-14.71731064 9.19532016 -2.79275543]
[ 1.60998654 8.37866823 0.86207034]
[ -4.4503797 10.12688097 -5.12453656]
[ 12.16293556 2.2594413 -2.11730311]
[-15.76505125 9.48537581 -2.73906772]
[ -2.54289959 9.86768111 -4.84802992]
[ -5.78214902 9.21901651 -8.13594627]
[ -1.35428398 5.85550586 6.30553987]
[ 12.87261987 0.96283606 -3.26982121]
[ 24.57767477 -4.28214631 6.29510659]
[ 4.13941679 3.3688288 3.01194055]
[ -2.98318764 1.32775227 7.62610929]
[ -4.44461549 -1.49258339 1.39080386]
[ -0.10590795 -0.3313904 8.46363066]
[ 6.05960739 1.03091753 5.10875657]
[-21.27737352 -3.44453629 3.25115921]
[ -1.1183025 0.55238687 10.75611405]
[-10.6359291 7.58630341 -0.55088259]
[ 4.52557492 -8.05670864 2.23113833]
[-11.07822559 1.50970501 4.66555889]
[ -6.89542628 -19.24672805 -3.71322812]
[ -0.57831362 -17.84956249 -5.52002876]
[-12.70262277 -14.05542691 -2.72417438]
[ -7.50263129 -15.83723295 -3.2635125 ]
[ -7.52780216 -17.60790567 -2.00134852]
[ -5.34422731 -17.29394266 -2.69261597]
[ 9.40597893 0.21140292 2.05522806]
[ 12.12423431 -2.80281266 7.81182024]
[ 19.51224195 4.7624575 -11.20523383]
[ 22.38102384 0.82486072 -1.64716468]
[ -8.60947699 4.12597477 -6.01885407]
[ 9.56268414 1.18190655 -5.44074124]
[ 14.97675455 3.31666971 -3.30012109]
[ 20.47530869 -1.95896058 -1.91238615]]
shape: (40, 3)
trans = pca(data, 3)[0]是U数据，因为[0]选择返回数据的第一个索引，而pcareturnsU, E, Vax2.scatter(trans[:50, 0], trans[:50, 1], c = 'r')绘制第 0 列的前 50 行与第 1 列的前 50 行，ax2.scatter(trans[50:, 0], trans[50:, 1], c = 'b')对从 50 到末尾的行执行相同的操作。这来自 this fast script 中给出的示例数据，但您的数据只有shape: (40, 3)（例如只有 40 行数据）。- 为了将
trans绘制为 3d 散点图，将 3 列中的每一列提取到一个单独的变量中并绘制为散点图。
# imports as shown in the linked answer
from numpy import array, dot, mean, std, empty, argsort
from numpy.linalg import eigh, solve
from numpy.random import randn
from matplotlib.pyplot import subplots, show
# other imports
import numpy as np
# test data from linked answer (e.g. this fast script)
np.random.seed(365) # makes data repeatable
data = array([randn(8) for k in range(150)]) # creates array with shape (150, 8)
data[:50, 2:4] += 5 # adds 5 to first 50 rows of columns 2:4
data[50:, 2:5] += 5 # adds 5 to to rows from 50 of columns 2:5
# function call
trans = pca(data, 3)[0] # [0] gets U returned by pca(...)
# extract each column to a separate variable
x = trans[:, 0] # all rows of column 0
y = trans[:, 1] # all rows of column 1
z = trans[:, 2] # all rows of column 2
# plot 3d scatter plot
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(x, y, z)