Principal Component Analysis (PCA)

In this in-class exercise you will be guided through the steps necessary for implementing a PCA on a sequence of human poses. You will work with the poses data, which was used for the exercises in week 2 . The dataset has a shape of $(1403, 100, 25*2)$. This means that there are 1403 pose sequences. Each sequence is a 100-frames time series capturing human poses. Each pose consists of 25 skeletal joints, where each joint is an x and y coordinate ($25*2$). For this exercise, you will use a single pose sequence of 100 frames and apply dimension reduction to the selected sequence.

The following cells loads the libraries, the dataset and provides functions for plotting the poses:

import numpy as np
import matplotlib.pyplot as plt
import warnings
import seaborn as sns
from pca_utils import *

# Suppress the specific warning
warnings.filterwarnings("ignore")

1. Data inspection

The cell below:

1. Loads the data and constructs the data matrix.
2. Reshapes the data into a $100x50$ data-matrix where each row contains a flattened vector of a pose.
3. Selects the first 40 frames from a single pose sequence and plot it.

data = np.load('poses_norm.npy')
print(data.shape)
N,T,D,C = data.shape
reshaped_data = data.reshape(N,T,D*C)
dataset = reshaped_data[19]
print(reshaped_data.shape)

# Define the new desired shape (40, 50)
nr_of_frames=40
new_shape = (nr_of_frames, 50)
# Reshape the array to the new shape
reshaped_data2 = np.empty(new_shape)  # Create an empty array with the new shape
reshaped_data2[:] = dataset[:new_shape[0], :]  
print(dataset.shape)

plot_single_sequence(reshaped_data2,pose_name='Pose',color='blue')

(1403, 100, 25, 2)
(1403, 100, 50)
(100, 50)

2. Covariance matrix

In the following tasks you will construct and inspect the covariance matrix for a given pose sequence.

# Calculate the covariance matrix for the entire dataset
cov_matrix = np.cov(dataset, rowvar=False)
# Plotting
sns.heatmap(cov_matrix, cmap='coolwarm')

Currently, the dataset is organized by frames, with each frame having alternating x and y coordinates in the order: $[x_1, y_1, x_2, y_2, \dots, x_{25}, y_{25}]$. The cell below rearranges the data for illustrative purposes, grouping all x-coordinates first, followed by all y-coordinates: $[x_1, x_2, \dots, x_{25}, y_1, y_2, \dots, y_{25}]$.

# Get the number of rows and columns in the dataset
num_rows, num_columns = dataset.shape

# Separate even and odd columns
even_indexes = np.arange(0, num_columns, 2)  # Even indexes (0, 2, 4, ...)
odd_indexes = np.arange(1, num_columns, 2)   # Odd indexes (1, 3, 5, ...)

# Rearrange the dataset
rearranged_dataset = dataset[:, np.concatenate((even_indexes, odd_indexes))]

cov_matrix_rearranged = np.cov(rearranged_dataset, rowvar=False)
sns.heatmap(cov_matrix_rearranged, cmap='coolwarm')

# write your solution here

3. Eigenvalues and eigenvectors

The following steps involve implementing the eigen decomposition of the covariance matrix.

eigenvalues, eigenvectors = np.linalg.eigh(cov_matrix)

# write your solution here

# Write your solution here

- Are all eigenvalues greater than or equal to 0: True
- Orthogonal eigenvectors: True
- Sum of eigenvalues: 0.56
- Total variance: 0.56

# Write your solution here

cumulative_variance_ratio = np.cumsum(sorted_eigenvalues) / np.sum(sorted_eigenvalues)

# Write your solution here

9 Components keep 95% of the variance

PCA - In-class week 2

4. Mixing parameters

The following section describes how much each variable (x or y coordinates of a shape) contributes to the selected principal components:

# Write your solution here


print(mixing_params.shape)

(50, 9)

num_variables = dataset.shape[1]
bar_width = 0.4  # Width of the bars

# Plotting only the first two principal components
plot_pca_loadings(
    mixing_params=mixing_params,
    dataset=dataset,
    bar_width=0.4,
    colors=['skyblue', 'salmon'],  
    component_indices=[1, 2],      
    show_individual=True,
    show_combined=True,
    save_plots=False               
)

plot_pose_with_numbers(dataset[12])

# Write your solution here

5. Generative process - Projecting to latent space and back

The pose data can be mapped to the latent space spanned by the principal components (eigenvectors) by: $$ \Phi^\top(x-\mu) $$ where

and $\Phi_i$ are the eigenvectors.

The following steps will implement this process.

# Calculate the mean vector
mean_vector = np.mean(dataset, axis=0)

# Subtract the mean from each data point
centered_data = dataset - mean_vector

# Write your solution here


plot_pca_pairwise_scatter(
    dataset=dataset,
    phi=phi,
    num_components=9,                
    figsize=(12, 12),
    marker='.',
    xlim=(-1, 1),
    ylim=(-1, 1)                         
)

print(projected_data.T.shape)

(100, 9)

# Write your solution here


print(reconstructed_data.shape)

(100, 50)

# Write your solution here


# Define the new shape you want (40, 50)
nr_of_frames=40

reshaped_original = dataset[:nr_of_frames, :].reshape(nr_of_frames, -1)
reshaped_reconstructed = reconstructed_data[:nr_of_frames, :].reshape(nr_of_frames, -1)
plot_single_sequence(reshaped_original, pose_name='Original', color='blue')

# Plot reconstructed sequence
plot_single_sequence(reshaped_reconstructed, pose_name='Reconstructed', color='red')

# function for PCA
pca_reconstruction(dataset=dataset, k=9, variance_retained=0.95, num_frames=40)

Number of Components (k): 9
Variance Explained by 9 components: 95.77%