pca
April 23, 2025
0.1 Use Iris dataset and perform PCA on the dataset. Examine the first two
principal components of X.
0.2 Create a scatter plot with each of the 150 rows of X projected onto the
first two principal components. Horizontal axis should be first principal
component and vertical axis should be second principal component.
[ ]: import pandas as pd
df = pd.read_csv("iris.csv")
# Prepare X and Y
X = df.values[:,:-1]
Y = df.values[:,-1]
df.head()
[ ]: sepal length sepal width petal length petal width species
0 5.1 3.5 1.4 0.2 1
1 4.9 3.0 1.4 0.2 1
2 4.7 3.2 1.3 0.2 1
3 4.6 3.1 1.5 0.2 1
4 5.0 3.6 1.4 0.2 1
[ ]: df
[ ]: sepal length sepal width petal length petal width species
0 5.1 3.5 1.4 0.2 1
1 4.9 3.0 1.4 0.2 1
2 4.7 3.2 1.3 0.2 1
3 4.6 3.1 1.5 0.2 1
4 5.0 3.6 1.4 0.2 1
.. … … … … …
145 6.7 3.0 5.2 2.3 3
146 6.3 2.5 5.0 1.9 3
147 6.5 3.0 5.2 2.0 3
148 6.2 3.4 5.4 2.3 3
149 5.9 3.0 5.1 1.8 3
1
[150 rows x 5 columns]
[ ]: X_standard = X - X.mean()
X_standard
[ ]: array([[ 1.6355, 0.0355, -2.0645, -3.2645],
[ 1.4355, -0.4645, -2.0645, -3.2645],
[ 1.2355, -0.2645, -2.1645, -3.2645],
[ 1.1355, -0.3645, -1.9645, -3.2645],
[ 1.5355, 0.1355, -2.0645, -3.2645],
[ 1.9355, 0.4355, -1.7645, -3.0645],
[ 1.1355, -0.0645, -2.0645, -3.1645],
[ 1.5355, -0.0645, -1.9645, -3.2645],
[ 0.9355, -0.5645, -2.0645, -3.2645],
[ 1.4355, -0.3645, -1.9645, -3.3645],
[ 1.9355, 0.2355, -1.9645, -3.2645],
[ 1.3355, -0.0645, -1.8645, -3.2645],
[ 1.3355, -0.4645, -2.0645, -3.3645],
[ 0.8355, -0.4645, -2.3645, -3.3645],
[ 2.3355, 0.5355, -2.2645, -3.2645],
[ 2.2355, 0.9355, -1.9645, -3.0645],
[ 1.9355, 0.4355, -2.1645, -3.0645],
[ 1.6355, 0.0355, -2.0645, -3.1645],
[ 2.2355, 0.3355, -1.7645, -3.1645],
[ 1.6355, 0.3355, -1.9645, -3.1645],
[ 1.9355, -0.0645, -1.7645, -3.2645],
[ 1.6355, 0.2355, -1.9645, -3.0645],
[ 1.1355, 0.1355, -2.4645, -3.2645],
[ 1.6355, -0.1645, -1.7645, -2.9645],
[ 1.3355, -0.0645, -1.5645, -3.2645],
[ 1.5355, -0.4645, -1.8645, -3.2645],
[ 1.5355, -0.0645, -1.8645, -3.0645],
[ 1.7355, 0.0355, -1.9645, -3.2645],
[ 1.7355, -0.0645, -2.0645, -3.2645],
[ 1.2355, -0.2645, -1.8645, -3.2645],
[ 1.3355, -0.3645, -1.8645, -3.2645],
[ 1.9355, -0.0645, -1.9645, -3.0645],
[ 1.7355, 0.6355, -1.9645, -3.3645],
[ 2.0355, 0.7355, -2.0645, -3.2645],
[ 1.4355, -0.3645, -1.9645, -3.2645],
[ 1.5355, -0.2645, -2.2645, -3.2645],
[ 2.0355, 0.0355, -2.1645, -3.2645],
[ 1.4355, 0.1355, -2.0645, -3.3645],
[ 0.9355, -0.4645, -2.1645, -3.2645],
[ 1.6355, -0.0645, -1.9645, -3.2645],
[ 1.5355, 0.0355, -2.1645, -3.1645],
[ 1.0355, -1.1645, -2.1645, -3.1645],
2
[ 0.9355, -0.2645, -2.1645, -3.2645],
[ 1.5355, 0.0355, -1.8645, -2.8645],
[ 1.6355, 0.3355, -1.5645, -3.0645],
[ 1.3355, -0.4645, -2.0645, -3.1645],
[ 1.6355, 0.3355, -1.8645, -3.2645],
[ 1.1355, -0.2645, -2.0645, -3.2645],
[ 1.8355, 0.2355, -1.9645, -3.2645],
[ 1.5355, -0.1645, -2.0645, -3.2645],
[ 3.5355, -0.2645, 1.2355, -2.0645],
[ 2.9355, -0.2645, 1.0355, -1.9645],
[ 3.4355, -0.3645, 1.4355, -1.9645],
[ 2.0355, -1.1645, 0.5355, -2.1645],
[ 3.0355, -0.6645, 1.1355, -1.9645],
[ 2.2355, -0.6645, 1.0355, -2.1645],
[ 2.8355, -0.1645, 1.2355, -1.8645],
[ 1.4355, -1.0645, -0.1645, -2.4645],
[ 3.1355, -0.5645, 1.1355, -2.1645],
[ 1.7355, -0.7645, 0.4355, -2.0645],
[ 1.5355, -1.4645, 0.0355, -2.4645],
[ 2.4355, -0.4645, 0.7355, -1.9645],
[ 2.5355, -1.2645, 0.5355, -2.4645],
[ 2.6355, -0.5645, 1.2355, -2.0645],
[ 2.1355, -0.5645, 0.1355, -2.1645],
[ 3.2355, -0.3645, 0.9355, -2.0645],
[ 2.1355, -0.4645, 1.0355, -1.9645],
[ 2.3355, -0.7645, 0.6355, -2.4645],
[ 2.7355, -1.2645, 1.0355, -1.9645],
[ 2.1355, -0.9645, 0.4355, -2.3645],
[ 2.4355, -0.2645, 1.3355, -1.6645],
[ 2.6355, -0.6645, 0.5355, -2.1645],
[ 2.8355, -0.9645, 1.4355, -1.9645],
[ 2.6355, -0.6645, 1.2355, -2.2645],
[ 2.9355, -0.5645, 0.8355, -2.1645],
[ 3.1355, -0.4645, 0.9355, -2.0645],
[ 3.3355, -0.6645, 1.3355, -2.0645],
[ 3.2355, -0.4645, 1.5355, -1.7645],
[ 2.5355, -0.5645, 1.0355, -1.9645],
[ 2.2355, -0.8645, 0.0355, -2.4645],
[ 2.0355, -1.0645, 0.3355, -2.3645],
[ 2.0355, -1.0645, 0.2355, -2.4645],
[ 2.3355, -0.7645, 0.4355, -2.2645],
[ 2.5355, -0.7645, 1.6355, -1.8645],
[ 1.9355, -0.4645, 1.0355, -1.9645],
[ 2.5355, -0.0645, 1.0355, -1.8645],
[ 3.2355, -0.3645, 1.2355, -1.9645],
[ 2.8355, -1.1645, 0.9355, -2.1645],
[ 2.1355, -0.4645, 0.6355, -2.1645],
3
[ 2.0355, -0.9645, 0.5355, -2.1645],
[ 2.0355, -0.8645, 0.9355, -2.2645],
[ 2.6355, -0.4645, 1.1355, -2.0645],
[ 2.3355, -0.8645, 0.5355, -2.2645],
[ 1.5355, -1.1645, -0.1645, -2.4645],
[ 2.1355, -0.7645, 0.7355, -2.1645],
[ 2.2355, -0.4645, 0.7355, -2.2645],
[ 2.2355, -0.5645, 0.7355, -2.1645],
[ 2.7355, -0.5645, 0.8355, -2.1645],
[ 1.6355, -0.9645, -0.4645, -2.3645],
[ 2.2355, -0.6645, 0.6355, -2.1645],
[ 2.8355, -0.1645, 2.5355, -0.9645],
[ 2.3355, -0.7645, 1.6355, -1.5645],
[ 3.6355, -0.4645, 2.4355, -1.3645],
[ 2.8355, -0.5645, 2.1355, -1.6645],
[ 3.0355, -0.4645, 2.3355, -1.2645],
[ 4.1355, -0.4645, 3.1355, -1.3645],
[ 1.4355, -0.9645, 1.0355, -1.7645],
[ 3.8355, -0.5645, 2.8355, -1.6645],
[ 3.2355, -0.9645, 2.3355, -1.6645],
[ 3.7355, 0.1355, 2.6355, -0.9645],
[ 3.0355, -0.2645, 1.6355, -1.4645],
[ 2.9355, -0.7645, 1.8355, -1.5645],
[ 3.3355, -0.4645, 2.0355, -1.3645],
[ 2.2355, -0.9645, 1.5355, -1.4645],
[ 2.3355, -0.6645, 1.6355, -1.0645],
[ 2.9355, -0.2645, 1.8355, -1.1645],
[ 3.0355, -0.4645, 2.0355, -1.6645],
[ 4.2355, 0.3355, 3.2355, -1.2645],
[ 4.2355, -0.8645, 3.4355, -1.1645],
[ 2.5355, -1.2645, 1.5355, -1.9645],
[ 3.4355, -0.2645, 2.2355, -1.1645],
[ 2.1355, -0.6645, 1.4355, -1.4645],
[ 4.2355, -0.6645, 3.2355, -1.4645],
[ 2.8355, -0.7645, 1.4355, -1.6645],
[ 3.2355, -0.1645, 2.2355, -1.3645],
[ 3.7355, -0.2645, 2.5355, -1.6645],
[ 2.7355, -0.6645, 1.3355, -1.6645],
[ 2.6355, -0.4645, 1.4355, -1.6645],
[ 2.9355, -0.6645, 2.1355, -1.3645],
[ 3.7355, -0.4645, 2.3355, -1.8645],
[ 3.9355, -0.6645, 2.6355, -1.5645],
[ 4.4355, 0.3355, 2.9355, -1.4645],
[ 2.9355, -0.6645, 2.1355, -1.2645],
[ 2.8355, -0.6645, 1.6355, -1.9645],
[ 2.6355, -0.8645, 2.1355, -2.0645],
[ 4.2355, -0.4645, 2.6355, -1.1645],
4
[ 2.8355, -0.0645, 2.1355, -1.0645],
[ 2.9355, -0.3645, 2.0355, -1.6645],
[ 2.5355, -0.4645, 1.3355, -1.6645],
[ 3.4355, -0.3645, 1.9355, -1.3645],
[ 3.2355, -0.3645, 2.1355, -1.0645],
[ 3.4355, -0.3645, 1.6355, -1.1645],
[ 2.3355, -0.7645, 1.6355, -1.5645],
[ 3.3355, -0.2645, 2.4355, -1.1645],
[ 3.2355, -0.1645, 2.2355, -0.9645],
[ 3.2355, -0.4645, 1.7355, -1.1645],
[ 2.8355, -0.9645, 1.5355, -1.5645],
[ 3.0355, -0.4645, 1.7355, -1.4645],
[ 2.7355, -0.0645, 1.9355, -1.1645],
[ 2.4355, -0.4645, 1.6355, -1.6645]])
[ ]: type(X_standard)
[ ]: numpy.ndarray
[ ]: Y_standard = Y
Y_standard
[ ]: array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2.,
2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2.,
2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2.,
2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 3., 3.,
3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3.,
3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3.,
3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3., 3.])
[ ]: type(Y_standard)
[ ]: numpy.ndarray
[ ]: cov = np.cov(X.T) # .T b/c numpy wants varibles along rows rather than down␣
↪columns?
print("covariance matrix =\n",cov)
covariance matrix =
[[ 0.68569351 -0.042434 1.27431544 0.51627069]
[-0.042434 0.18997942 -0.32965638 -0.12163937]
[ 1.27431544 -0.32965638 3.11627785 1.2956094 ]
[ 0.51627069 -0.12163937 1.2956094 0.58100626]]
5
[ ]: from numpy import linalg as LA
lambdas, vs = LA.eig(cov)
[ ]: lambdas
[ ]: array([4.22824171, 0.24267075, 0.0782095 , 0.02383509])
[ ]: vs
[ ]: array([[ 0.36138659, -0.65658877, -0.58202985, 0.31548719],
[-0.08452251, -0.73016143, 0.59791083, -0.3197231 ],
[ 0.85667061, 0.17337266, 0.07623608, -0.47983899],
[ 0.3582892 , 0.07548102, 0.54583143, 0.75365743]])
[ ]: lambdas
[ ]: array([4.22824171, 0.24267075, 0.0782095 , 0.02383509])
[ ]: #sort the eigenvalues in descending order
sorted_index = np.argsort(lambdas)[::-1]
[ ]: sorted_index
[ ]: array([0, 1, 2, 3], dtype=int64)
[ ]: sorted_eigenvalue = lambdas[sorted_index]
[ ]: sorted_eigenvalue
[ ]: array([4.22824171, 0.24267075, 0.0782095 , 0.02383509])
[ ]: sorted_eigenvectors = vs[:,sorted_index]
[ ]: sorted_eigenvectors
[ ]: array([[ 0.36138659, -0.65658877, -0.58202985, 0.31548719],
[-0.08452251, -0.73016143, 0.59791083, -0.3197231 ],
[ 0.85667061, 0.17337266, 0.07623608, -0.47983899],
[ 0.3582892 , 0.07548102, 0.54583143, 0.75365743]])
[ ]: # select the first n eigenvectors, n is desired dimension
# of our final reduced data.
n_components = 2 #you can select any number of components.
eigenvector_subset = sorted_eigenvectors[:,0:n_components]
[ ]: eigenvector_subset
6
[ ]: array([[ 0.36138659, -0.65658877],
[-0.08452251, -0.73016143],
[ 0.85667061, 0.17337266],
[ 0.3582892 , 0.07548102]])
[ ]: #Transform the data
X_reduced = np.dot(eigenvector_subset.transpose(),X_standard.transpose()).
↪transpose()
[ ]: print("Information/Variance in PC1",(sorted_eigenvalue[0]/
↪(sorted_eigenvalue[0]+sorted_eigenvalue[1]))*100)
Information/Variance in PC1 94.57223216899484
[ ]: X_reduced
[ ]: array([[ 2.81823951, -5.64634982],
[ 2.78822345, -5.14995135],
[ 2.61337456, -5.18200315],
[ 2.75702228, -5.0086536 ],
[ 2.7736486 , -5.65370709],
[ 3.2215055 , -6.06828303],
[ 2.68182738, -5.23749119],
[ 2.87622016, -5.49033754],
[ 2.6159824 , -4.74864082],
[ 2.82960933, -5.21317833],
[ 2.99541804, -5.97202148],
[ 2.8896099 , -5.34168252],
[ 2.71625587, -5.09184058],
[ 2.27856139, -4.81555799],
[ 2.85761474, -6.50571721],
[ 3.1163261 , -6.66501491],
[ 2.87883726, -6.13763209],
[ 2.85406843, -5.63880172],
[ 3.30254481, -6.19979162],
[ 2.91437873, -5.84051289],
[ 3.19210892, -5.71829851],
[ 2.9586599 , -5.75994864],
[ 2.28642572, -5.46042065],
[ 3.19963195, -5.42566143],
[ 3.14661108, -5.28967072],
[ 2.99569623, -5.1809357 ],
[ 3.03354506, -5.45790407],
[ 2.94004523, -5.69467143],
[ 2.86283042, -5.63899256],
[ 2.87037575, -5.12999135],
[ 2.91496666, -5.12263409],
7
[ 3.09243264, -5.73787684],
[ 2.8535028 , -6.1403164 ],
[ 2.90362838, -6.42009834],
[ 2.86543825, -5.20563023],
[ 2.63612348, -5.39631705],
[ 2.87712708, -5.9263226 ],
[ 2.70168102, -5.59559631],
[ 2.52186309, -4.83899423],
[ 2.91235882, -5.55599641],
[ 2.73226271, -5.59048011],
[ 2.65299643, -4.385992 ],
[ 2.50495859, -4.98502652],
[ 3.09675065, -5.51582401],
[ 3.29287589, -5.76361572],
[ 2.78791371, -5.07674437],
[ 2.96421687, -5.83072372],
[ 2.66290296, -5.09900701],
[ 2.95927938, -5.9063626 ],
[ 2.79900535, -5.43465866],
[ 6.78719082, -6.01211305],
[ 6.43485366, -5.64528622],
[ 6.96666745, -5.83121539],
[ 5.68568285, -4.49899357],
[ 6.59046839, -5.40154325],
[ 6.14403422, -4.90870571],
[ 6.5974258 , -5.61042085],
[ 4.75324246, -4.32206162],
[ 6.54649696, -5.55531448],
[ 5.49361973, -4.60387067],
[ 4.99452425, -4.06098139],
[ 6.01406369, -5.22297134],
[ 5.76734164, -4.77691611],
[ 6.48729964, -5.20213472],
[ 5.32843976, -5.07209837],
[ 6.43022591, -5.79413207],
[ 6.16264889, -4.97398291],
[ 5.73847013, -4.99334181],
[ 6.44709886, -4.78380703],
[ 5.54759211, -4.7431182 ],
[ 6.61864831, -5.24233572],
[ 5.86025355, -5.25802755],
[ 6.80054901, -4.99916527],
[ 6.42409406, -5.14421478],
[ 6.21721846, -5.47600852],
[ 6.40253951, -5.65545705],
[ 6.83438957, -5.57139345],
[ 7.06016729, -5.59444802],
8
[ 6.31565578, -5.16360228],
[ 5.19678135, -4.95869039],
[ 5.43423864, -4.62178045],
[ 5.31274266, -4.64666581],
[ 5.63879384, -5.01292014],
[ 6.88239157, -4.90599829],
[ 6.09037158, -4.84266516],
[ 6.30922345, -5.52113489],
[ 6.72305602, -5.73457217],
[ 6.31746037, -4.95491552],
[ 5.74832281, -5.05842818],
[ 5.66877835, -4.64502585],
[ 5.96716542, -4.65624103],
[ 6.39318033, -5.29248813],
[ 5.73291316, -4.92256673],
[ 4.79783337, -4.31470435],
[ 5.85934663, -4.82204248],
[ 5.83429961, -5.11429789],
[ 5.87858078, -5.03373365],
[ 6.14494114, -5.34469077],
[ 4.59589527, -4.57085921],
[ 5.80136597, -4.97805477],
[ 8.03355786, -5.31710347],
[ 6.91760101, -4.75203623],
[ 8.11904115, -5.67085573],
[ 7.47389619, -5.14722467],
[ 7.85237105, -5.28669163],
[ 8.89940387, -5.87778925],
[ 6.02359738, -4.13419385],
[ 8.4349522 , -5.68245258],
[ 7.82359395, -5.08312107],
[ 8.4191161 , -6.10974453],
[ 7.16413929, -5.56918098],
[ 7.30576709, -5.11131496],
[ 7.66795693, -5.54322816],
[ 6.84852871, -4.55013423],
[ 7.08829336, -4.78731186],
[ 7.40682151, -5.44620327],
[ 7.45205419, -5.36889584],
[ 8.9894205 , -6.50269191],
[ 9.29801055, -5.58427555],
[ 6.80315685, -4.56580294],
[ 7.93018305, -5.70514859],
[ 6.70136624, -4.72086105],
[ 9.00228517, -5.78762668],
[ 6.89113126, -5.12255325],
[ 7.77779564, -5.66194318],
9
[ 8.11645561, -5.88785393],
[ 6.76087329, -5.14724778],
[ 6.79349719, -5.21028393],
[ 7.62597386, -5.1172231 ],
[ 7.89036815, -5.79159238],
[ 8.34403791, -5.70222174],
[ 8.73303879, -6.70111766],
[ 7.66180278, -5.109675 ],
[ 6.94652637, -5.18353917],
[ 7.28365994, -4.8270509 ],
[ 8.57886506, -6.01503825],
[ 7.64660845, -5.46701678],
[ 7.40746328, -5.3762531 ],
[ 6.67169147, -5.16196231],
[ 7.60997628, -5.69924045],
[ 7.81651984, -5.51060386],
[ 7.42463293, -5.73615604],
[ 6.91760101, -4.75203623],
[ 8.06537851, -5.60481518],
[ 7.92111132, -5.63175077],
[ 7.44647493, -5.51448488],
[ 7.02953175, -4.95163559],
[ 7.26671085, -5.40581143],
[ 7.40330675, -5.44358054],
[ 6.89255399, -5.04429164]])
[ ]: import matplotlib.pyplot as plt
plt.xlabel('Principal Component - 1',fontsize=20)
plt.ylabel('Principal Component - 2',fontsize=20)
plt.scatter(X_reduced[:,0:1],X_reduced[:,-1],c=Y)
plt.show()
10
[ ]:
11
