# PCA Visualization in Python

Visualize Principle Component Analysis (PCA) of your high-dimensional data in Python with Plotly.

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## New to Plotly?

Plotly is a free and open-source graphing library for Python. We recommend you read our Getting Started guide for the latest installation or upgrade instructions, then move on to our Plotly Fundamentals tutorials or dive straight in to some Basic Charts tutorials.

This page first shows how to visualize higher dimension data using various Plotly figures combined with dimensionality reduction (aka projection). Then, we dive into the specific details of our projection algorithm.

We will use Scikit-learn to load one of the datasets, and apply dimensionality reduction. Scikit-learn is a popular Machine Learning (ML) library that offers various tools for creating and training ML algorithms, feature engineering, data cleaning, and evaluating and testing models. It was designed to be accessible, and to work seamlessly with popular libraries like NumPy and Pandas.

## High-dimensional PCA Analysis with `px.scatter_matrix`

¶

The dimensionality reduction technique we will be using is called the Principal Component Analysis (PCA). It is a powerful technique that arises from linear algebra and probability theory. In essense, it computes a matrix that represents the variation of your data (covariance matrix/eigenvectors), and rank them by their relevance (explained variance/eigenvalues). For a video tutorial, see this segment on PCA from the Coursera ML course.

### Visualize all the original dimensions¶

First, let's plot all the features and see how the `species`

in the Iris dataset are grouped. In a Scatter Plot Matrix (splom), each subplot displays a feature against another, so if we have $N$ features we have a $N \times N$ matrix.

In our example, we are plotting all 4 features from the Iris dataset, thus we can see how `sepal_width`

is compared against `sepal_length`

, then against `petal_width`

, and so forth. Keep in mind how some pairs of features can more easily separate different species.

In this example, we will use Plotly Express, Plotly's high-level API for building figures.

```
import plotly.express as px
df = px.data.iris()
features = ["sepal_width", "sepal_length", "petal_width", "petal_length"]
fig = px.scatter_matrix(
df,
dimensions=features,
color="species"
)
fig.update_traces(diagonal_visible=False)
fig.show()
```

### Visualize all the principal components¶

Now, we apply `PCA`

the same dataset, and retrieve **all** the components. We use the same `px.scatter_matrix`

trace to display our results, but this time our features are the resulting *principal components*, ordered by how much variance they are able to explain.

The importance of explained variance is demonstrated in the example below. The subplot between PC3 and PC4 is clearly unable to separate each class, whereas the subplot between PC1 and PC2 shows a clear separation between each species.

In this example, we will use Plotly Express, Plotly's high-level API for building figures.

```
import plotly.express as px
from sklearn.decomposition import PCA
df = px.data.iris()
features = ["sepal_width", "sepal_length", "petal_width", "petal_length"]
pca = PCA()
components = pca.fit_transform(df[features])
labels = {
str(i): f"PC {i+1} ({var:.1f}%)"
for i, var in enumerate(pca.explained_variance_ratio_ * 100)
}
fig = px.scatter_matrix(
components,
labels=labels,
dimensions=range(4),
color=df["species"]
)
fig.update_traces(diagonal_visible=False)
fig.show()
```

### Visualize a subset of the principal components¶

When you will have too many features to visualize, you might be interested in only visualizing the most relevant components. Those components often capture a majority of the explained variance, which is a good way to tell if those components are sufficient for modelling this dataset.

In the example below, our dataset contains 10 features, but we only select the first 4 components, since they explain over 99% of the total variance.

```
import pandas as pd
import plotly.express as px
from sklearn.decomposition import PCA
from sklearn.datasets import load_boston
boston = load_boston()
df = pd.DataFrame(boston.data, columns=boston.feature_names)
n_components = 4
pca = PCA(n_components=n_components)
components = pca.fit_transform(df)
total_var = pca.explained_variance_ratio_.sum() * 100
labels = {str(i): f"PC {i+1}" for i in range(n_components)}
labels['color'] = 'Median Price'
fig = px.scatter_matrix(
components,
color=boston.target,
dimensions=range(n_components),
labels=labels,
title=f'Total Explained Variance: {total_var:.2f}%',
)
fig.update_traces(diagonal_visible=False)
fig.show()
```

## 2D PCA Scatter Plot¶

In the previous examples, you saw how to visualize high-dimensional PCs. In this example, we show you how to simply visualize the first two principal components of a PCA, by reducing a dataset of 4 dimensions to 2D.

```
import plotly.express as px
from sklearn.decomposition import PCA
df = px.data.iris()
X = df[['sepal_length', 'sepal_width', 'petal_length', 'petal_width']]
pca = PCA(n_components=2)
components = pca.fit_transform(X)
fig = px.scatter(components, x=0, y=1, color=df['species'])
fig.show()
```

## Visualize PCA with `px.scatter_3d`

¶

With `px.scatter_3d`

, you can visualize an additional dimension, which let you capture even more variance.

```
import plotly.express as px
from sklearn.decomposition import PCA
df = px.data.iris()
X = df[['sepal_length', 'sepal_width', 'petal_length', 'petal_width']]
pca = PCA(n_components=3)
components = pca.fit_transform(X)
total_var = pca.explained_variance_ratio_.sum() * 100
fig = px.scatter_3d(
components, x=0, y=1, z=2, color=df['species'],
title=f'Total Explained Variance: {total_var:.2f}%',
labels={'0': 'PC 1', '1': 'PC 2', '2': 'PC 3'}
)
fig.show()
```

## Plotting explained variance¶

Often, you might be interested in seeing how much variance PCA is able to explain as you increase the number of components, in order to decide how many dimensions to ultimately keep or analyze. This example shows you how to quickly plot the cumulative sum of explained variance for a high-dimensional dataset like Diabetes.

With a higher explained variance, you are able to capture more variability in your dataset, which could potentially lead to better performance when training your model. For a more mathematical explanation, see this Q&A thread.

```
import plotly.express as px
import numpy as np
import pandas as pd
from sklearn.decomposition import PCA
from sklearn.datasets import load_diabetes
boston = load_diabetes()
df = pd.DataFrame(boston.data, columns=boston.feature_names)
pca = PCA()
pca.fit(df)
exp_var_cumul = np.cumsum(pca.explained_variance_ratio_)
px.area(
x=range(1, exp_var_cumul.shape[0] + 1),
y=exp_var_cumul,
labels={"x": "# Components", "y": "Explained Variance"}
)
```

## Visualize Loadings¶

It is also possible to visualize loadings using `shapes`

, and use `annotations`

to indicate which feature a certain loading original belong to. Here, we define loadings as:

For more details about the linear algebra behind eigenvectors and loadings, see this Q&A thread.

```
import plotly.express as px
from sklearn.decomposition import PCA
from sklearn import datasets
from sklearn.preprocessing import StandardScaler
df = px.data.iris()
features = ['sepal_length', 'sepal_width', 'petal_length', 'petal_width']
X = df[features]
pca = PCA(n_components=2)
components = pca.fit_transform(X)
loadings = pca.components_.T * np.sqrt(pca.explained_variance_)
fig = px.scatter(components, x=0, y=1, color=df['species'])
for i, feature in enumerate(features):
fig.add_shape(
type='line',
x0=0, y0=0,
x1=loadings[i, 0],
y1=loadings[i, 1]
)
fig.add_annotation(
x=loadings[i, 0],
y=loadings[i, 1],
ax=0, ay=0,
xanchor="center",
yanchor="bottom",
text=feature,
)
fig.show()
```

## References¶

Learn more about `px`

, `px.scatter_3d`

, and `px.scatter_matrix`

here:

- https://plot.ly/python/plotly-express/
- https://plot.ly/python/3d-scatter-plots/
- https://plot.ly/python/splom/

The following resources offer an in-depth overview of PCA and explained variance:

- https://en.wikipedia.org/wiki/Explained_variation
- https://scikit-learn.org/stable/modules/decomposition.html#pca
- https://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues/140579#140579
- https://stats.stackexchange.com/questions/143905/loadings-vs-eigenvectors-in-pca-when-to-use-one-or-another
- https://stats.stackexchange.com/questions/22569/pca-and-proportion-of-variance-explained

### What About Dash?¶

Dash is an open-source framework for building analytical applications, with no Javascript required, and it is tightly integrated with the Plotly graphing library.

Learn about how to install Dash at https://dash.plot.ly/installation.

Everywhere in this page that you see `fig.show()`

, you can display the same figure in a Dash application by passing it to the `figure`

argument of the `Graph`

component from the built-in `dash_core_components`

package like this:

```
import plotly.graph_objects as go # or plotly.express as px
fig = go.Figure() # or any Plotly Express function e.g. px.bar(...)
# fig.add_trace( ... )
# fig.update_layout( ... )
import dash
import dash_core_components as dcc
import dash_html_components as html
app = dash.Dash()
app.layout = html.Div([
dcc.Graph(figure=fig)
])
app.run_server(debug=True, use_reloader=False) # Turn off reloader if inside Jupyter
```