karpathy_sleep_tracker_explained_variance.py

import numpy as np 

# Variable names.
# The first variable is a latent sleep-quality factor.
# The remaining variables are observed sleep trackers.
names = [
    'latent',
    'Whoop',
    '8Sleep',
    'Autosleep'
]

# Correlation matrix.
# - The first row/column gives correlations between the latent factor
#   and each sleep tracker. These values are interpreted as factor loadings.
# - The remaining entries are raw correlations between sleep trackers.
correlation_matrix = np.array([
    [1.00, 0.64, 0.60, 0.46],
    [0.64, 1.00, 0.38, 0.14],
    [0.60, 0.38, 1.00, 0.42],
    [0.46, 0.14, 0.42, 1.00]
])

# Iteratively include the top k sleep trackers as predictors of the latent factor
for i in range(1, 3 + 1):
    num_predictors = i

    # Report which sleep trackers are being used
    print(f"Using top {num_predictors} predictor(s): {', '.join(names[1:num_predictors+1])}")

    # Subset the correlation matrix to include:
    # - the latent factor (index 0)
    # - the first k sleep trackers
    num_variables = num_predictors + 1
    corr = correlation_matrix[:num_variables, :num_variables]

    # Invert the correlation matrix to obtain the precision matrix
    # The precision matrix encodes conditional independencies
    precision_matrix = np.linalg.inv(corr)

    # The reciprocal of each diagonal element of the precision matrix
    # gives the conditional variance of that variable given the others
    partial_variances = 1 / np.diag(precision_matrix)

    # For the latent variable (index 0), the conditional variance
    # is the unexplained variance after regressing it on the sleep trackers
    explained_variance = 1 - partial_variances[0]

    # The square root gives the multiple correlation between
    # the latent sleep score and the included sleep trackers
    print(f"Estimated Correlation: {np.sqrt(explained_variance):.2f}")
    print()