diff --git a/_posts/ipython-notebooks/2015-06-30-principal_component_analysis.html b/_posts/ipython-notebooks/2015-06-30-principal_component_analysis.html index 0bc5594ffa3e..969aa44549f8 100755 --- a/_posts/ipython-notebooks/2015-06-30-principal_component_analysis.html +++ b/_posts/ipython-notebooks/2015-06-30-principal_component_analysis.html @@ -1,17 +1,17 @@ --- permalink: ipython-notebooks/principal-component-analysis/ description: A step by step tutorial to Principal Component Analysis, a simple yet powerful transformation technique. -title: Principal Component Analysis in 3 Simple Steps +name: Principal Component Analysis in 3 Simple Steps has_thumbnail: false thumbnail: /images/static-image -layout: user-guide name: Principal Component Analysis -language: python +ipynb: ~notebook_demo/264 +layout: user-guide page_type: u-guide +language: python --- {% raw %} -
-
+
@@ -22,8 +22,7 @@
-
-
+
@@ -32,8 +31,7 @@

Principal Component Anal

-
-
+
@@ -42,8 +40,7 @@

Introduction&#

-
-
+
@@ -51,8 +48,7 @@

PCA Vs. LDA

-
-
+
@@ -62,8 +58,7 @@

PCA Vs. LDA

-
-
+
@@ -71,19 +66,17 @@

PCA and Dimensionality Reduction

-
-
+
-

Often, the desired goal is to reduce the dimensions of a $d$-dimensional dataset by projecting it onto a $(k)$-dimensional subspace (where $k\;<\;d$) in order to increase the computational efficiency while retaining most of the information. An important question is "what is the size of $k$ that represents the data 'well'?"

+

Often, the desired goal is to reduce the dimensions of a $d$-dimensional dataset by projecting it onto a $(k)$-dimensional subspace (where $k\;<\;d$) in order to increase the computational efficiency while retaining most of the information. An important question is "what is the size of $k$ that represents the data 'well'?"

Later, we will compute eigenvectors (the principal components) of a dataset and collect them in a projection matrix. Each of those eigenvectors is associated with an eigenvalue which can be interpreted as the "length" or "magnitude" of the corresponding eigenvector. If some eigenvalues have a significantly larger magnitude than others that the reduction of the dataset via PCA onto a smaller dimensional subspace by dropping the "less informative" eigenpairs is reasonable.

-
-
+
@@ -91,8 +84,7 @@

A Summary of the PCA Approach -
+
@@ -107,8 +99,7 @@

A Summary of the PCA Approach -
+
@@ -116,8 +107,7 @@

Preparing the Iris Dataset -
+
@@ -141,8 +131,7 @@

Preparing the Iris Dataset -
+
@@ -150,8 +139,7 @@

Loading the Dataset -
+
@@ -165,15 +153,15 @@

Loading the DatasetIn [1]:

-
import pandas as pd
+
import pandas as pd
 
 df = pd.read_csv(
     filepath_or_buffer='https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data', 
-    header=None, 
+    header=None, 
     sep=',')
 
 df.columns=['sepal_len', 'sepal_wid', 'petal_len', 'petal_wid', 'class']
-df.dropna(how="all", inplace=True) # drops the empty line at file-end
+df.dropna(how="all", inplace=True) # drops the empty line at file-end
 
 df.tail()
 

@@ -187,11 +175,26 @@ 
Loading the Dataset
+
 
Out[1]:

 
 
+
 

-

+

+
 
@@ -206,43 +209,43 @@ 
Loading the DatasetLoading the DatasetIn [2]:
 

     

-
# split data table into data X and class labels y
+
# split data table into data X and class labels y
 
-X = df.ix[:,0:4].values
-y = df.ix[:,4].values
+X = df.iloc[:,0:4].values
+y = df.iloc[:,4].values
 

 
 

@@ -271,8 +274,7 @@ 
Loading the Dataset
-

+

 

 

 

@@ -284,8 +286,7 @@ 
Loading the Dataset
-

+

 

 

 

@@ -293,58 +294,48 @@ 
Exploratory Visualization
-

-
In [3]:

-

-    

-
import plotly.plotly as py
-from plotly.graph_objs import *
-import plotly.tools as tls
-

-
-

-

-

-
 

 

 

 
In [4]:

 

     

-
# plotting histograms
+
import plotly.plotly as py
 
-traces = []
+# plotting histograms
+data = []
 
-legend = {0:False, 1:False, 2:False, 3:True}
+legend = {0:False, 1:False, 2:False, 3:True}
 
-colors = {'Iris-setosa': 'rgb(31, 119, 180)', 
-          'Iris-versicolor': 'rgb(255, 127, 14)', 
-          'Iris-virginica': 'rgb(44, 160, 44)'}
+colors = {'Iris-setosa': '#0D76BF', 
+          'Iris-versicolor': '#00cc96', 
+          'Iris-virginica': '#EF553B'}
 
 for col in range(4):
     for key in colors:
-        traces.append(Histogram(x=X[y==key, col], 
-                        opacity=0.75,
-                        xaxis='x%s' %(col+1),
-                        marker=Marker(color=colors[key]),
-                        name=key,
-                        showlegend=legend[col]))
-
-data = Data(traces)
-
-layout = Layout(barmode='overlay',
-                xaxis=XAxis(domain=[0, 0.25], title='sepal length (cm)'),
-                xaxis2=XAxis(domain=[0.3, 0.5], title='sepal width (cm)'),
-                xaxis3=XAxis(domain=[0.55, 0.75], title='petal length (cm)'),
-                xaxis4=XAxis(domain=[0.8, 1], title='petal width (cm)'),
-                yaxis=YAxis(title='count'),
-                title='Distribution of the different Iris flower features')
-
-fig = Figure(data=data, layout=layout)
-py.iplot(fig)
+        trace = dict(
+            type='histogram',
+            x=list(X[y==key, col]),
+            opacity=0.75,
+            xaxis='x%s' %(col+1),
+            marker=dict(color=colors[key]),
+            name=key,
+            showlegend=legend[col]
+        )
+        data.append(trace)
+
+layout = dict(
+    barmode='overlay',
+    xaxis=dict(domain=[0, 0.25], title='sepal length (cm)'),
+    xaxis2=dict(domain=[0.3, 0.5], title='sepal width (cm)'),
+    xaxis3=dict(domain=[0.55, 0.75], title='petal length (cm)'),
+    xaxis4=dict(domain=[0.8, 1], title='petal width (cm)'),
+    yaxis=dict(title='count'),
+    title='Distribution of the different Iris flower features'
+)
+
+fig = dict(data=data, layout=layout)
+py.iplot(fig, filename='exploratory-vis-histogram')
 

 
 

@@ -356,11 +347,13 @@ 
Exploratory Visualization
+
 
Out[4]:

 
 
+
 

-
+
 

 
 

@@ -369,8 +362,7 @@ 
Exploratory Visualization
-

+

 

 

 

@@ -378,8 +370,7 @@ 
Standardizing
 

 

-

-

+

 

 

 

@@ -390,10 +381,10 @@ 
Standardizing
 

 

-
In [5]:

+
In [6]:

 

     

-
from sklearn.preprocessing import StandardScaler
+
from sklearn.preprocessing import StandardScaler
 X_std = StandardScaler().fit_transform(X)
 

 
@@ -402,8 +393,7 @@ 
Standardizing
 
 

-

-

+

 

 

 

@@ -411,8 +401,7 @@ 
1 - Eig
 

 

 

-

-

+

 

 

 

@@ -421,8 +410,7 @@ 
1 - Eig
 

 

 

-

-

+

 

 

 

@@ -430,8 +418,7 @@ 
Covariance Matrix
-

+

 

 

 

@@ -448,13 +435,13 @@ 
Covariance Matrix
 

-
In [6]:

+
In [7]:

 

     

-
import numpy as np
+
import numpy as np
 mean_vec = np.mean(X_std, axis=0)
 cov_mat = (X_std - mean_vec).T.dot((X_std - mean_vec)) / (X_std.shape[0]-1)
-print('Covariance matrix \n%s' %cov_mat)
+print('Covariance matrix \n%s' %cov_mat)
 

 
 

@@ -466,8 +453,10 @@ 
Covariance Matrix
+
 

 
+
 

 
Covariance matrix 
 [[ 1.00671141 -0.11010327  0.87760486  0.82344326]
@@ -482,8 +471,7 @@ 
Covariance Matrix
-

+

 

 

 

@@ -494,10 +482,10 @@ 
Covariance Matrix
 

-
In [7]:

+
In [8]:

 

     

-
print('NumPy covariance matrix: \n%s' %np.cov(X_std.T))
+
print('NumPy covariance matrix: \n%s' %np.cov(X_std.T))
 

 
 

@@ -509,8 +497,10 @@ 
Covariance Matrix
+
 

 
+
 

 
NumPy covariance matrix: 
 [[ 1.00671141 -0.11010327  0.87760486  0.82344326]
@@ -525,8 +515,7 @@ 
Covariance Matrix
-

+

 

 

 

@@ -537,15 +526,15 @@ 
Covariance Matrix
 

-
In [8]:

+
In [9]:

 

     

-
cov_mat = np.cov(X_std.T)
+
cov_mat = np.cov(X_std.T)
 
 eig_vals, eig_vecs = np.linalg.eig(cov_mat)
 
-print('Eigenvectors \n%s' %eig_vecs)
-print('\nEigenvalues \n%s' %eig_vals)
+print('Eigenvectors \n%s' %eig_vecs)
+print('\nEigenvalues \n%s' %eig_vals)
 

 
 

@@ -557,8 +546,10 @@ 
Covariance Matrix
+
 

 
+
 

 
Eigenvectors 
 [[ 0.52237162 -0.37231836 -0.72101681  0.26199559]
@@ -567,7 +558,7 @@ 
Covariance MatrixCovariance Matrix
-

+

 

 

 

@@ -589,15 +579,15 @@ 
Correlation Matrix
 

-
In [9]:

+
In [10]:

 

     

-
cor_mat1 = np.corrcoef(X_std.T)
+
cor_mat1 = np.corrcoef(X_std.T)
 
 eig_vals, eig_vecs = np.linalg.eig(cor_mat1)
 
-print('Eigenvectors \n%s' %eig_vecs)
-print('\nEigenvalues \n%s' %eig_vals)
+print('Eigenvectors \n%s' %eig_vecs)
+print('\nEigenvalues \n%s' %eig_vals)
 

 
 

@@ -609,8 +599,10 @@ 
Correlation Matrix
+
 

 
+
 

 
Eigenvectors 
 [[ 0.52237162 -0.37231836 -0.72101681  0.26199559]
@@ -619,7 +611,7 @@ 
Correlation MatrixCorrelation Matrix
-

+

 

 

 

@@ -640,15 +631,15 @@ 
Correlation Matrix
 

-
In [10]:

+
In [11]:

 

     

-
cor_mat2 = np.corrcoef(X.T)
+
cor_mat2 = np.corrcoef(X.T)
 
 eig_vals, eig_vecs = np.linalg.eig(cor_mat2)
 
-print('Eigenvectors \n%s' %eig_vecs)
-print('\nEigenvalues \n%s' %eig_vals)
+print('Eigenvectors \n%s' %eig_vecs)
+print('\nEigenvalues \n%s' %eig_vals)
 

 
 

@@ -660,8 +651,10 @@ 
Correlation Matrix
+
 

 
+
 

 
Eigenvectors 
 [[ 0.52237162 -0.37231836 -0.72101681  0.26199559]
@@ -670,7 +663,7 @@ 
Correlation MatrixCorrelation Matrix
-

+

 

 

 

@@ -694,8 +686,7 @@ 
Correlation Matrix
-

+

 

 

 

@@ -706,10 +697,10 @@ 
Singular Vector Decomposition
 

-
In [11]:

+
In [12]:

 

     

-
u,s,v = np.linalg.svd(X_std.T)
+
u,s,v = np.linalg.svd(X_std.T)
 u
 

 
@@ -722,7 +713,9 @@ 
Singular Vector Decomposition
-
Out[11]:

+
+
Out[12]:

+
 
 
 
@@ -739,8 +732,7 @@ 
Singular Vector Decomposition
-

+

 

 

 

@@ -748,8 +740,7 @@ 
2 - Selecting Principal Components
 

 

-

-

+

 

 

 

@@ -760,12 +751,12 @@ 
2 - Selecting Principal Components
 

 

-
In [12]:

+
In [13]:

 

     

-
for ev in eig_vecs:
+
for ev in eig_vecs:
     np.testing.assert_array_almost_equal(1.0, np.linalg.norm(ev))
-print('Everything ok!')
+print('Everything ok!')
 

 
 

@@ -777,8 +768,10 @@ 
2 - Selecting Principal Components
+
 

 
+
 

 
Everything ok!
 

@@ -789,12 +782,11 @@ 
2 - Selecting Principal Components
 
 

-

-

+

 

 

 

-
In order to decide which eigenvector(s) can dropped without losing too much information
+
In order to decide which eigenvector(s) can be dropped without losing too much information
 for the construction of lower-dimensional subspace, we need to inspect the corresponding eigenvalues: The eigenvectors with the lowest eigenvalues bear the least information about the distribution of the data; those are the ones can be dropped.

 In order to do so, the common approach is to rank the eigenvalues from highest to lowest in order choose the top $k$ eigenvectors.

 
@@ -803,10 +795,10 @@ 
2 - Selecting Principal Components
 

 

-
In [13]:

+
In [14]:

 

     

-
# Make a list of (eigenvalue, eigenvector) tuples
+
# Make a list of (eigenvalue, eigenvector) tuples
 eig_pairs = [(np.abs(eig_vals[i]), eig_vecs[:,i]) for i in range(len(eig_vals))]
 
 # Sort the (eigenvalue, eigenvector) tuples from high to low
@@ -814,9 +806,9 @@ 
2 - Selecting Principal Componentseig_pairs.reverse()
 
 # Visually confirm that the list is correctly sorted by decreasing eigenvalues
-print('Eigenvalues in descending order:')
+print('Eigenvalues in descending order:')
 for i in eig_pairs:
-    print(i[0])
+    print(i[0])
 

 
 

@@ -828,14 +820,16 @@ 
2 - Selecting Principal Components
+
 

 
+
 

 
Eigenvalues in descending order:
-2.91081808375
-0.921220930707
-0.147353278305
-0.0206077072356
+2.910818083752054
+0.9212209307072242
+0.14735327830509573
+0.020607707235625678
 

 

 

@@ -844,8 +838,7 @@ 
2 - Selecting Principal Components
 
 

-

-

+

 

 

 

@@ -856,31 +849,48 @@ 
2 - Selecting Principal Components
 

 

-
In [14]:

+
In [16]:

 

     

-
tot = sum(eig_vals)
-var_exp = [(i / tot)*100 for i in sorted(eig_vals, reverse=True)]
+
tot = sum(eig_vals)
+var_exp = [(i / tot)*100 for i in sorted(eig_vals, reverse=True)]
 cum_var_exp = np.cumsum(var_exp)
 
-trace1 = Bar(
-        x=['PC %s' %i for i in range(1,5)],
-        y=var_exp,
-        showlegend=False)
-
-trace2 = Scatter(
-        x=['PC %s' %i for i in range(1,5)], 
-        y=cum_var_exp,
-        name='cumulative explained variance')
-
-data = Data([trace1, trace2])
-
-layout=Layout(
-        yaxis=YAxis(title='Explained variance in percent'),
-        title='Explained variance by different principal components')
-
-fig = Figure(data=data, layout=layout)
-py.iplot(fig)
+trace1 = dict(
+    type='bar',
+    x=['PC %s' %i for i in range(1,5)],
+    y=var_exp,
+    name='Individual'
+)
+
+trace2 = dict(
+    type='scatter',
+    x=['PC %s' %i for i in range(1,5)], 
+    y=cum_var_exp,
+    name='Cumulative'
+)
+
+data = [trace1, trace2]
+
+layout=dict(
+    title='Explained variance by different principal components',
+    yaxis=dict(
+        title='Explained variance in percent'
+    ),
+    annotations=list([
+        dict(
+            x=1.16,
+            y=1.05,
+            xref='paper',
+            yref='paper',
+            text='Explained Variance',
+            showarrow=False,
+        )
+    ])
+)
+
+fig = dict(data=data, layout=layout)
+py.iplot(fig, filename='selecting-principal-components')
 

 
 

@@ -892,11 +902,13 @@ 
2 - Selecting Principal Components
-
Out[14]:

+
+
Out[16]:

+
 
 
 

-
+
 

 
 

@@ -905,8 +917,7 @@ 
2 - Selecting Principal Components
 
 

-

-

+

 

 

 

@@ -915,8 +926,7 @@ 
2 - Selecting Principal Components
 

 

-

-

+

 

 

 

@@ -928,13 +938,13 @@ 
2 - Selecting Principal Components
 

 

-
In [15]:

+
In [17]:

 

     

-
matrix_w = np.hstack((eig_pairs[0][1].reshape(4,1), 
+
matrix_w = np.hstack((eig_pairs[0][1].reshape(4,1), 
                       eig_pairs[1][1].reshape(4,1)))
 
-print('Matrix W:\n', matrix_w)
+print('Matrix W:\n', matrix_w)
 

 
 

@@ -946,13 +956,16 @@ 
2 - Selecting Principal Components
+
 

 
+
 

-
('Matrix W:\n', array([[ 0.52237162, -0.37231836],
-       [-0.26335492, -0.92555649],
-       [ 0.58125401, -0.02109478],
-       [ 0.56561105, -0.06541577]]))
+
Matrix W:
+ [[ 0.52237162 -0.37231836]
+ [-0.26335492 -0.92555649]
+ [ 0.58125401 -0.02109478]
+ [ 0.56561105 -0.06541577]]
 

 

 

@@ -961,8 +974,7 @@ 
2 - Selecting Principal Components
 
 

-

-

+

 

 

 

@@ -970,8 +982,7 @@ 
3 - Projection Onto the New F
 

 

 

-

-

+

 

 

 

@@ -983,10 +994,10 @@ 
3 - Projection Onto the New F
 

 

 

-
In [16]:

+
In [19]:

 

     

-
Y = X_std.dot(matrix_w)
+
Y = X_std.dot(matrix_w)
 

 
 

@@ -996,34 +1007,38 @@ 
3 - Projection Onto the New F
 

 

 

-
In [17]:

+
In [20]:

 

     

-
traces = []
+
data = []
 
-for name in ('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'):
-
-    trace = Scatter(
+for name, col in zip(('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'), colors.values()):
+    trace = dict(
+        type='scatter',
         x=Y[y==name,0],
         y=Y[y==name,1],
         mode='markers',
         name=name,
-        marker=Marker(
+        marker=dict(
+            color=col,
             size=12,
-            line=Line(
+            line=dict(
                 color='rgba(217, 217, 217, 0.14)',
                 width=0.5),
-            opacity=0.8))
-    traces.append(trace)
-
-
-data = Data(traces)
-layout = Layout(showlegend=True,
-                scene=Scene(xaxis=XAxis(title='PC1'),
-                yaxis=YAxis(title='PC2'),))
-
-fig = Figure(data=data, layout=layout)
-py.iplot(fig)
+            opacity=0.8)
+    )
+    data.append(trace)
+
+layout = dict(
+    showlegend=True,
+    scene=dict(
+        xaxis=dict(title='PC1'),
+        yaxis=dict(title='PC2')
+    )
+)
+
+fig = dict(data=data, layout=layout)
+py.iplot(fig, filename='projection-matrix')
 

 
 

@@ -1035,11 +1050,13 @@ 
3 - Projection Onto the New F
 
 
 

-
Out[17]:

+
+
Out[20]:

+
 
 
 

-
+
 

 
 

@@ -1048,8 +1065,7 @@ 
3 - Projection Onto the New F
 

 
 

-

-

+

 

 

 

@@ -1059,8 +1075,7 @@ 
3 - Projection Onto the New F
 

 

 

-

-

+

 

 

 

@@ -1068,8 +1083,7 @@ 
Shortcut - PCA in scikit-learn
-

+

 

 

 

@@ -1080,10 +1094,10 @@ 
Shortcut - PCA in scikit-learn
 

-
In [18]:

+
In [21]:

 

     

-
from sklearn.decomposition import PCA as sklearnPCA
+
from sklearn.decomposition import PCA as sklearnPCA
 sklearn_pca = sklearnPCA(n_components=2)
 Y_sklearn = sklearn_pca.fit_transform(X_std)
 

@@ -1095,32 +1109,35 @@ 
Shortcut - PCA in scikit-learn
 

-
In [19]:

+
In [23]:

 

     

-
traces = []
+
data = []
 
-for name in ('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'):
+for name, col in zip(('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'), colors.values()):
 
-    trace = Scatter(
+    trace = dict(
+        type='scatter',
         x=Y_sklearn[y==name,0],
         y=Y_sklearn[y==name,1],
         mode='markers',
         name=name,
-        marker=Marker(
+        marker=dict(
+            color=col,
             size=12,
-            line=Line(
+            line=dict(
                 color='rgba(217, 217, 217, 0.14)',
                 width=0.5),
-            opacity=0.8))
-    traces.append(trace)
-
-
-data = Data(traces)
-layout = Layout(xaxis=XAxis(title='PC1', showline=False),
-                yaxis=YAxis(title='PC2', showline=False))
-fig = Figure(data=data, layout=layout)
-py.iplot(fig)
+            opacity=0.8)
+    )
+    data.append(trace)
+
+layout = dict(
+        xaxis=dict(title='PC1', showline=False),
+        yaxis=dict(title='PC2', showline=False)
+)
+fig = dict(data=data, layout=layout)
+py.iplot(fig, filename='pca-scikitlearn')
 

 
 

@@ -1132,11 +1149,13 @@ 
Shortcut - PCA in scikit-learn
-
Out[19]:

+
+
Out[23]:

+
 
 
 

-
+
 

 
 

diff --git a/_posts/ipython-notebooks/principal_component_analysis.ipynb b/_posts/ipython-notebooks/principal_component_analysis.ipynb
index 59baad29468b..c1973c04a5c2 100755
--- a/_posts/ipython-notebooks/principal_component_analysis.ipynb
+++ b/_posts/ipython-notebooks/principal_component_analysis.ipynb
@@ -128,7 +128,20 @@
     {
      "data": {
       "text/html": [
-       "
\n",
+       "
\n",
+       "\n",
        "

   
     
\n",
        "  \n",
        "    \n",
@@ -143,43 +156,43 @@
        "  \n",
        "    \n",
        "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
        "    \n",
        "    \n",
        "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
        "    \n",
        "    \n",
        "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
        "    \n",
        "    \n",
        "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
        "    \n",
        "    \n",
        "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
-       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
+       "      \n",
        "    \n",
        "  \n",
        "
145 6.7 3.0 5.2 2.3 Iris-virginica6.73.05.22.3Iris-virginica
146 6.3 2.5 5.0 1.9 Iris-virginica6.32.55.01.9Iris-virginica
147 6.5 3.0 5.2 2.0 Iris-virginica6.53.05.22.0Iris-virginica
148 6.2 3.4 5.4 2.3 Iris-virginica6.23.45.42.3Iris-virginica
149 5.9 3.0 5.1 1.8 Iris-virginica5.93.05.11.8Iris-virginica
\n",
@@ -216,15 +229,13 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "# split data table into data X and class labels y\n",
     "\n",
-    "X = df.ix[:,0:4].values\n",
-    "y = df.ix[:,4].values"
+    "X = df.iloc[:,0:4].values\n",
+    "y = df.iloc[:,4].values"
    ]
   },
   {
@@ -247,19 +258,6 @@
     "To get a feeling for how the 3 different flower classes are distributes along the 4 different features, let us visualize them via histograms."
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": [
-    "import plotly.plotly as py\n",
-    "from plotly.graph_objs import *\n",
-    "import plotly.tools as tls"
-   ]
-  },
   {
    "cell_type": "code",
    "execution_count": 4,
@@ -268,7 +266,7 @@
     {
      "data": {
       "text/html": [
-       ""
+       ""
       ],
       "text/plain": [
        ""
@@ -280,37 +278,42 @@
     }
    ],
    "source": [
-    "# plotting histograms\n",
+    "import plotly.plotly as py\n",
     "\n",
-    "traces = []\n",
+    "# plotting histograms\n",
+    "data = []\n",
     "\n",
     "legend = {0:False, 1:False, 2:False, 3:True}\n",
     "\n",
-    "colors = {'Iris-setosa': 'rgb(31, 119, 180)', \n",
-    "          'Iris-versicolor': 'rgb(255, 127, 14)', \n",
-    "          'Iris-virginica': 'rgb(44, 160, 44)'}\n",
+    "colors = {'Iris-setosa': '#0D76BF', \n",
+    "          'Iris-versicolor': '#00cc96', \n",
+    "          'Iris-virginica': '#EF553B'}\n",
     "\n",
     "for col in range(4):\n",
     "    for key in colors:\n",
-    "        traces.append(Histogram(x=X[y==key, col], \n",
-    "                        opacity=0.75,\n",
-    "                        xaxis='x%s' %(col+1),\n",
-    "                        marker=Marker(color=colors[key]),\n",
-    "                        name=key,\n",
-    "                        showlegend=legend[col]))\n",
-    "\n",
-    "data = Data(traces)\n",
+    "        trace = dict(\n",
+    "            type='histogram',\n",
+    "            x=list(X[y==key, col]),\n",
+    "            opacity=0.75,\n",
+    "            xaxis='x%s' %(col+1),\n",
+    "            marker=dict(color=colors[key]),\n",
+    "            name=key,\n",
+    "            showlegend=legend[col]\n",
+    "        )\n",
+    "        data.append(trace)\n",
     "\n",
-    "layout = Layout(barmode='overlay',\n",
-    "                xaxis=XAxis(domain=[0, 0.25], title='sepal length (cm)'),\n",
-    "                xaxis2=XAxis(domain=[0.3, 0.5], title='sepal width (cm)'),\n",
-    "                xaxis3=XAxis(domain=[0.55, 0.75], title='petal length (cm)'),\n",
-    "                xaxis4=XAxis(domain=[0.8, 1], title='petal width (cm)'),\n",
-    "                yaxis=YAxis(title='count'),\n",
-    "                title='Distribution of the different Iris flower features')\n",
+    "layout = dict(\n",
+    "    barmode='overlay',\n",
+    "    xaxis=dict(domain=[0, 0.25], title='sepal length (cm)'),\n",
+    "    xaxis2=dict(domain=[0.3, 0.5], title='sepal width (cm)'),\n",
+    "    xaxis3=dict(domain=[0.55, 0.75], title='petal length (cm)'),\n",
+    "    xaxis4=dict(domain=[0.8, 1], title='petal width (cm)'),\n",
+    "    yaxis=dict(title='count'),\n",
+    "    title='Distribution of the different Iris flower features'\n",
+    ")\n",
     "\n",
-    "fig = Figure(data=data, layout=layout)\n",
-    "py.iplot(fig)"
+    "fig = dict(data=data, layout=layout)\n",
+    "py.iplot(fig, filename='exploratory-vis-histogram')"
    ]
   },
   {
@@ -329,10 +332,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
-   "metadata": {
-    "collapsed": true
-   },
+   "execution_count": 6,
+   "metadata": {},
    "outputs": [],
    "source": [
     "from sklearn.preprocessing import StandardScaler\n",
@@ -377,7 +378,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
@@ -408,7 +409,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -436,7 +437,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
@@ -450,7 +451,7 @@
       " [ 0.56561105 -0.06541577  0.6338014   0.52354627]]\n",
       "\n",
       "Eigenvalues \n",
-      "[ 2.93035378  0.92740362  0.14834223  0.02074601]\n"
+      "[2.93035378 0.92740362 0.14834223 0.02074601]\n"
      ]
     }
    ],
@@ -474,7 +475,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -488,7 +489,7 @@
       " [ 0.56561105 -0.06541577  0.6338014   0.52354627]]\n",
       "\n",
       "Eigenvalues \n",
-      "[ 2.91081808  0.92122093  0.14735328  0.02060771]\n"
+      "[2.91081808 0.92122093 0.14735328 0.02060771]\n"
      ]
     }
    ],
@@ -510,7 +511,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
@@ -524,7 +525,7 @@
       " [ 0.56561105 -0.06541577  0.6338014   0.52354627]]\n",
       "\n",
       "Eigenvalues \n",
-      "[ 2.91081808  0.92122093  0.14735328  0.02060771]\n"
+      "[2.91081808 0.92122093 0.14735328 0.02060771]\n"
      ]
     }
    ],
@@ -558,7 +559,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
@@ -570,7 +571,7 @@
        "       [-0.56561105, -0.06541577, -0.6338014 ,  0.52354627]])"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -596,7 +597,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
@@ -617,14 +618,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "In order to decide which eigenvector(s) can dropped without losing too much information\n",
+    "In order to decide which eigenvector(s) can be dropped without losing too much information\n",
     "for the construction of lower-dimensional subspace, we need to inspect the corresponding eigenvalues: The eigenvectors with the lowest eigenvalues bear the least information about the distribution of the data; those are the ones can be dropped.  \n",
     "In order to do so, the common approach is to rank the eigenvalues from highest to lowest in order choose the top $k$ eigenvectors."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [
     {
@@ -632,10 +633,10 @@
      "output_type": "stream",
      "text": [
       "Eigenvalues in descending order:\n",
-      "2.91081808375\n",
-      "0.921220930707\n",
-      "0.147353278305\n",
-      "0.0206077072356\n"
+      "2.910818083752054\n",
+      "0.9212209307072242\n",
+      "0.14735327830509573\n",
+      "0.020607707235625678\n"
      ]
     }
    ],
@@ -662,19 +663,19 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       ""
+       ""
       ],
       "text/plain": [
        ""
       ]
      },
-     "execution_count": 14,
+     "execution_count": 16,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -684,24 +685,41 @@
     "var_exp = [(i / tot)*100 for i in sorted(eig_vals, reverse=True)]\n",
     "cum_var_exp = np.cumsum(var_exp)\n",
     "\n",
-    "trace1 = Bar(\n",
-    "        x=['PC %s' %i for i in range(1,5)],\n",
-    "        y=var_exp,\n",
-    "        showlegend=False)\n",
+    "trace1 = dict(\n",
+    "    type='bar',\n",
+    "    x=['PC %s' %i for i in range(1,5)],\n",
+    "    y=var_exp,\n",
+    "    name='Individual'\n",
+    ")\n",
     "\n",
-    "trace2 = Scatter(\n",
-    "        x=['PC %s' %i for i in range(1,5)], \n",
-    "        y=cum_var_exp,\n",
-    "        name='cumulative explained variance')\n",
+    "trace2 = dict(\n",
+    "    type='scatter',\n",
+    "    x=['PC %s' %i for i in range(1,5)], \n",
+    "    y=cum_var_exp,\n",
+    "    name='Cumulative'\n",
+    ")\n",
     "\n",
-    "data = Data([trace1, trace2])\n",
+    "data = [trace1, trace2]\n",
     "\n",
-    "layout=Layout(\n",
-    "        yaxis=YAxis(title='Explained variance in percent'),\n",
-    "        title='Explained variance by different principal components')\n",
+    "layout=dict(\n",
+    "    title='Explained variance by different principal components',\n",
+    "    yaxis=dict(\n",
+    "        title='Explained variance in percent'\n",
+    "    ),\n",
+    "    annotations=list([\n",
+    "        dict(\n",
+    "            x=1.16,\n",
+    "            y=1.05,\n",
+    "            xref='paper',\n",
+    "            yref='paper',\n",
+    "            text='Explained Variance',\n",
+    "            showarrow=False,\n",
+    "        )\n",
+    "    ])\n",
+    ")\n",
     "\n",
-    "fig = Figure(data=data, layout=layout)\n",
-    "py.iplot(fig)"
+    "fig = dict(data=data, layout=layout)\n",
+    "py.iplot(fig, filename='selecting-principal-components')"
    ]
   },
   {
@@ -722,17 +740,18 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "('Matrix W:\\n', array([[ 0.52237162, -0.37231836],\n",
-      "       [-0.26335492, -0.92555649],\n",
-      "       [ 0.58125401, -0.02109478],\n",
-      "       [ 0.56561105, -0.06541577]]))\n"
+      "Matrix W:\n",
+      " [[ 0.52237162 -0.37231836]\n",
+      " [-0.26335492 -0.92555649]\n",
+      " [ 0.58125401 -0.02109478]\n",
+      " [ 0.56561105 -0.06541577]]\n"
      ]
     }
    ],
@@ -760,10 +779,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
-   "metadata": {
-    "collapsed": true
-   },
+   "execution_count": 19,
+   "metadata": {},
    "outputs": [],
    "source": [
     "Y = X_std.dot(matrix_w)"
@@ -771,49 +788,54 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 20,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       ""
+       ""
       ],
       "text/plain": [
        ""
       ]
      },
-     "execution_count": 17,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "traces = []\n",
     "\n",
-    "for name in ('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'):\n",
+    "data = []\n",
     "\n",
-    "    trace = Scatter(\n",
+    "for name, col in zip(('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'), colors.values()):\n",
+    "    trace = dict(\n",
+    "        type='scatter',\n",
     "        x=Y[y==name,0],\n",
     "        y=Y[y==name,1],\n",
     "        mode='markers',\n",
     "        name=name,\n",
-    "        marker=Marker(\n",
+    "        marker=dict(\n",
+    "            color=col,\n",
     "            size=12,\n",
-    "            line=Line(\n",
+    "            line=dict(\n",
     "                color='rgba(217, 217, 217, 0.14)',\n",
     "                width=0.5),\n",
-    "            opacity=0.8))\n",
-    "    traces.append(trace)\n",
-    "\n",
+    "            opacity=0.8)\n",
+    "    )\n",
+    "    data.append(trace)\n",
     "\n",
-    "data = Data(traces)\n",
-    "layout = Layout(showlegend=True,\n",
-    "                scene=Scene(xaxis=XAxis(title='PC1'),\n",
-    "                yaxis=YAxis(title='PC2'),))\n",
+    "layout = dict(\n",
+    "    showlegend=True,\n",
+    "    scene=dict(\n",
+    "        xaxis=dict(title='PC1'),\n",
+    "        yaxis=dict(title='PC2')\n",
+    "    )\n",
+    ")\n",
     "\n",
-    "fig = Figure(data=data, layout=layout)\n",
-    "py.iplot(fig)"
+    "fig = dict(data=data, layout=layout)\n",
+    "py.iplot(fig, filename='projection-matrix')"
    ]
   },
   {
@@ -840,10 +862,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
-   "metadata": {
-    "collapsed": true
-   },
+   "execution_count": 21,
+   "metadata": {},
    "outputs": [],
    "source": [
     "from sklearn.decomposition import PCA as sklearnPCA\n",
@@ -853,52 +873,55 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       ""
+       ""
       ],
       "text/plain": [
        ""
       ]
      },
-     "execution_count": 19,
+     "execution_count": 23,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "traces = []\n",
+    "data = []\n",
     "\n",
-    "for name in ('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'):\n",
+    "for name, col in zip(('Iris-setosa', 'Iris-versicolor', 'Iris-virginica'), colors.values()):\n",
     "\n",
-    "    trace = Scatter(\n",
+    "    trace = dict(\n",
+    "        type='scatter',\n",
     "        x=Y_sklearn[y==name,0],\n",
     "        y=Y_sklearn[y==name,1],\n",
     "        mode='markers',\n",
     "        name=name,\n",
-    "        marker=Marker(\n",
+    "        marker=dict(\n",
+    "            color=col,\n",
     "            size=12,\n",
-    "            line=Line(\n",
+    "            line=dict(\n",
     "                color='rgba(217, 217, 217, 0.14)',\n",
     "                width=0.5),\n",
-    "            opacity=0.8))\n",
-    "    traces.append(trace)\n",
+    "            opacity=0.8)\n",
+    "    )\n",
+    "    data.append(trace)\n",
     "\n",
-    "\n",
-    "data = Data(traces)\n",
-    "layout = Layout(xaxis=XAxis(title='PC1', showline=False),\n",
-    "                yaxis=YAxis(title='PC2', showline=False))\n",
-    "fig = Figure(data=data, layout=layout)\n",
-    "py.iplot(fig)"
+    "layout = dict(\n",
+    "        xaxis=dict(title='PC1', showline=False),\n",
+    "        yaxis=dict(title='PC2', showline=False)\n",
+    ")\n",
+    "fig = dict(data=data, layout=layout)\n",
+    "py.iplot(fig, filename='pca-scikitlearn')"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 24,
    "metadata": {},
    "outputs": [
     {
@@ -929,7 +952,21 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Requirement already up-to-date: publisher in /Users/chelsea/venv/venv2.7/lib/python2.7/site-packages\r\n"
+      "Requirement already up-to-date: publisher in c:\\anaconda\\anaconda3\\lib\\site-packages (0.13)\n"
+     ]
+    },
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "C:\\Anaconda\\Anaconda3\\lib\\site-packages\\IPython\\nbconvert.py:13: ShimWarning:\n",
+      "\n",
+      "The `IPython.nbconvert` package has been deprecated since IPython 4.0. You should import from nbconvert instead.\n",
+      "\n",
+      "C:\\Anaconda\\Anaconda3\\lib\\site-packages\\publisher\\publisher.py:53: UserWarning:\n",
+      "\n",
+      "Did you \"Save\" this notebook before running this command? Remember to save, always save.\n",
+      "\n"
      ]
     }
    ],
@@ -946,36 +983,36 @@
     "    'ipython-notebooks/principal-component-analysis/', \n",
     "    'Principal Component Analysis in 3 Simple Steps', \n",
     "    'A step by step tutorial to Principal Component Analysis, a simple yet powerful transformation technique.',\n",
-    "     name='Principal Component Analysis')"
+    "     name='Principal Component Analysis',\n",
+    "     ipynb='~notebook_demo/264'\n",
+    ")"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": []
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 2",
+   "display_name": "Python 3",
    "language": "python",
-   "name": "python2"
+   "name": "python3"
   },
   "language_info": {
    "codemirror_mode": {
     "name": "ipython",
-    "version": 2
+    "version": 3
    },
    "file_extension": ".py",
    "mimetype": "text/x-python",
    "name": "python",
    "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython2",
-   "version": "2.7.10"
+   "pygments_lexer": "ipython3",
+   "version": "3.6.8"
   }
  },
  "nbformat": 4,