Dimensionality reduction

The curse of dimensionality

Fitting and overfitting get worse with ‘’curse of dimensionality’’ Bellman 1961

Think about a hypersphere. Its volume is given by

\[ V_D(r) = \frac{2r^D\pi^{D/2}}{D\ \Gamma(D/2)}\]

where \(\Gamma(z)\) is the complete gamma function, \(D\) is the dimension, and \(r\) the radius of the sphere.

If you populated a hypercube of size \(2r\) how much data would be enclosed by the hypersphere

  • as \(D\) increases the fractional volume enclosed by the hypersphere goes to 0!

For example: the SDSS comprises a sample of 357 million sources.

  • each source has 448 measured attributes

  • selecting just 30 (e.g., magnitude, size..) and normalizing the data range \(-1\) to \(1\)

probability of having one of the 357 million sources reside within a unit hypersphere 1 in 1.4\(\times 10^5\).

Principal Component Analysis

Points are correlated along a particular direction which doesn’t align with the initial choice of axes.

  • we should rotate our axes to align with this correlation.

  • rotation preserves the relative ordering of data

Choose rotation to maximize the ability to discriminate between the data points

  • first axis, or principal component, is direction of maximal variance

  • second principal component is orthogonal to the first component and maximizes the residual variance

In the following example a distribution of points drawn from a bivariate Gaussian and centered on the origin of \(x\) and \(y\). PCA defines a rotation such that the new axes (\(x’\) and \(y’\)) are aligned along the directions of maximal variance (the principal components) with zero covariance. This is equivalent to minimizing the square of the perpendicular distances between the points and the principal components.

import numpy as np
from matplotlib import pyplot as plt
from matplotlib.patches import Ellipse
from matplotlib import ticker
np.random.seed(42)
r = 0.9

sigma1 = 0.25
sigma2 = 0.08
rotation = np.pi / 6
s = np.sin(rotation)
c = np.cos(rotation)

X = np.random.normal(0, [sigma1, sigma2], size=(100, 2)).T
R = np.array([[c, -s],
              [s, c]])
X = np.dot(R, X)
#------------------------------------------------------------
# Plot the diagram
fig = plt.figure(figsize=(5, 5), facecolor='w')
ax = plt.axes((0, 0, 1, 1), xticks=[], yticks=[], frameon=False)

# draw axes
ax.annotate(r'$x$', (-r, 0), (r, 0),
            ha='center', va='center',
            arrowprops=dict(arrowstyle='<->', color='k', lw=1))
ax.annotate(r'$y$', (0, -r), (0, r),
            ha='center', va='center',
            arrowprops=dict(arrowstyle='<->', color='k', lw=1))

# draw rotated axes
ax.annotate(r'$x^\prime$', (-r * c, -r * s), (r * c, r * s),
            ha='center', va='center',
            arrowprops=dict(color='k', arrowstyle='<->', lw=1))
ax.annotate(r'$y^\prime$', (r * s, -r * c), (-r * s, r * c),
            ha='center', va='center',
            arrowprops=dict(color='k', arrowstyle='<->', lw=1))

# scatter points
ax.scatter(X[0], X[1], s=25, lw=0, c='k', zorder=2)

# draw lines
vnorm = np.array([s, -c])
for v in (X.T):
    d = np.dot(v, vnorm)
    v1 = v - d * vnorm
    ax.plot([v[0], v1[0]], [v[1], v1[1]], '-k')

# draw ellipses
for sigma in (1, 2, 3):
    ax.add_patch(Ellipse((0, 0), 2 * sigma * sigma1, 2 * sigma * sigma2,
                         rotation * 180. / np.pi,
                         ec='k', fc='gray', alpha=0.2, zorder=1))

ax.set_xlim(-1, 1)
ax.set_ylim(-1, 1)

plt.show()
../_images/astroml_chapter7_Dimensionality_Reduction_6_0.png

Derivation of principal component analyses

Set of data \(X\): \(N\) observations by \(K\) measurements

Center data by subtracting the mean

The covariance is

\[ C_X=\frac{1}{N-1}X^TX,\]

\(N-1\) as the sample covariance matrix.

We want a projection, \(R\), aligned with the directions of maximal variance (\(Y= X R\)) with covariance

\[C_{Y} = R^T X^T X R = R^T C_X R\]

Derive principal component by maximizing its variance (using Lagrange multipliers and constraint)

\[\phi(r_1,\lambda_1) = r_1^TC_X r_1 - \lambda_1(r_1^Tr_1-1)\]

derivative of \(\phi(r_1,\lambda)\) with respect to \(r_1\) set to 0

\[C_Xr_1 - \lambda_1 r_1 = 0\]

\(\lambda_1\) is the root of the equation \(\det(C_X - \lambda_1 {\bf I})=0\) and the largest eigenvalue

\[\lambda_1 = r_1^T C_X r_1\]

Other principal components derived by applying additional constraint that components are uncorrelated (e.g., \(r^T_2 C_X r_1 = 0\)).

Singular value decomposition (SVD)

Common approach is eigenvalue decomposition of the covariance or correlation matrix, or singular value decomposition (SVD) of the data matrix

\[U \Sigma V^T = \frac{1}{\sqrt{N - 1}} X\]

columns of \(U\) are left-singular vectors

columns of \(V\) are the right-singular vectors

The columns of \(U\) and \(V\) form orthonormal bases (\(U^TU = V^TV = I\))

Covariance matrix is

\[\begin{split} C_X = \left[\frac{1}{\sqrt{N - 1}}X\right]^T \left[\frac{1}{\sqrt{N - 1}}X\right]\nonumber\\ = V \Sigma U^T U \Sigma V^T\nonumber\\ = V \Sigma^2 V^T. \end{split}\]

right singular vectors \(V\) are the principal components so principal from the SVD of \(X\) dont need \(C_X\).

Application of PCA

In the following example we have 100 data points in 3 dimensions, \(X\), and \(R\) as the projection matrix. To compute the PCA components (4 in our case) we use PCA from scikit-learn.

from sklearn.decomposition import PCA

X = np.random.normal(size=(100, 3))
R = np.random.random((3, 10))
X = np.dot(X, R)

pca = PCA(n_components=4)
pca.fit(X)
comp = pca.transform(X)

mean = pca.mean_
components = pca.components_
var = pca.explained_variance_

For our astronomy use case, we are using the SDSS spectroscopic dataset. The SDSS spectra come from galaxies at a range of redshifts, and have sections of unreliable or missing data due to sky absorption, cosmic rays, bad detector pixels, or other effects. AstroML provides a set of spectra which have been moved to rest frame, corrected for masking using an iterative PCA reconstruction technique, and resampled to 1000 common wavelength bins.

The spectra can be downloaded using fetch_sdss_corrected_spectra(). In the following example we plot 15 of these spectra:

import numpy as np
from matplotlib import pyplot as plt
from astroML.datasets import sdss_corrected_spectra

#----------------------------------------------------------------------
# Use pre-computed PCA to reconstruct spectra
data = sdss_corrected_spectra.fetch_sdss_corrected_spectra()
spectra_raw = data['spectra']
spectra_corr = sdss_corrected_spectra.reconstruct_spectra(data)
wavelengths = sdss_corrected_spectra.compute_wavelengths(data)

#------------------------------------------------------------
# select random spectra
np.random.seed(5)
nrows = 5
ncols = 3
ind = np.random.randint(spectra_corr.shape[0], size=nrows * ncols)
spec_sample_raw = spectra_raw[ind]
spec_sample_corr = spectra_corr[ind]

We show the SDSS downloaded spectra with black, and our corrected spectra with blue.

fig = plt.figure(figsize=(10, 8))

fig.subplots_adjust(left=0.05, right=0.95, wspace=0.05,
                    bottom=0.1, top=0.95, hspace=0.05)

for i in range(ncols):
    for j in range(nrows):
        ax = fig.add_subplot(nrows, ncols, ncols * j + 1 + i)
        ax.plot(wavelengths, spec_sample_raw[ncols * j + i], '-k', lw=1)
        ax.plot(wavelengths, spec_sample_corr[ncols * j + i], '-k', lw=1, c='blue')               
        ax.set_xlim(3100, 7999)

        ax.yaxis.set_major_formatter(plt.NullFormatter())
        ax.xaxis.set_major_locator(plt.MultipleLocator(1000))
        if j < nrows - 1:
            ax.xaxis.set_major_formatter(plt.NullFormatter())
        else:
            plt.xlabel(r'wavelength $(\AA)$')

        ylim = ax.get_ylim()
        dy = 0.05 * (ylim[1] - ylim[0])
        ax.set_ylim(ylim[0] - dy, ylim[1] + dy)

plt.show()
../_images/astroml_chapter7_Dimensionality_Reduction_15_0.png

PCA Reconstruction of a spectrum

Reconstruction of spectrum, \({x}(k)\), from the eigenvectors, \({e}_i(k)\)

\[ {x}_i(k) = {\mu}(k) + \sum_j^R \theta_{ij} {e}_j(k),\]

Truncating this expansion (i.e., \(r<R\))

\[ {x}_i(k) = {\mu}(k) + \sum_i^{r<R} \theta_i {e}_i(k), \]
  • eigenvectors ordered by their associated eigenvalues

  • eigenvalues reflect variance within each eigenvector (sum of the eigenvalues is total variance of the system).

  • project a each spectrum onto these first few eigenspectra is a compression of the data

This is the sense in which PCA gives for dimensionality reduction.

In the following example, the input spectrum is shown in gray, and the partial reconstruction for progressively more terms is shown in black. The top panel shows only the mean of the set of spectra. By the time 20 PCA components are added, the reconstruction is very close to the input, as indicated by the expected total variance of 94%.

#------------------------------------------------------------
# Compute PCA components

# Eigenvalues can be computed using PCA as in the commented code below:

#from sklearn.decomposition import PCA
#pca = PCA()
#pca.fit(spectra_corr)
#evals = pca.explained_variance_ratio_
#evals_cs = evals.cumsum()

#  because the spectra have been reconstructed from masked values, this
#  is not exactly correct in this case: we'll use the values computed
#  in the file compute_sdss_pca.py
evals = data['evals'] ** 2
evals_cs = evals.cumsum()
evals_cs /= evals_cs[-1]
evecs = data['evecs']
spec_mean = spectra_corr.mean(0)

#------------------------------------------------------------
# Find the coefficients of a particular spectrum
spec = spectra_corr[1]
coeff = np.dot(evecs, spec - spec_mean)

#------------------------------------------------------------
# Plot the sequence of reconstructions
fig = plt.figure(figsize=(5, 5))
fig.subplots_adjust(hspace=0, top=0.95, bottom=0.1, left=0.12, right=0.93)

for i, n in enumerate([0, 4, 8, 20]):
    ax = fig.add_subplot(411 + i)
    ax.plot(wavelengths, spec, '-', c='gray')
    ax.plot(wavelengths, spec_mean + np.dot(coeff[:n], evecs[:n]), '-k')

    if i < 3:
        ax.xaxis.set_major_formatter(plt.NullFormatter())

    ax.set_ylim(-2, 21)
    ax.set_ylabel('flux')

    if n == 0:
        text = "mean"
    elif n == 1:
        text = "mean + 1 component\n"
        text += r"$(\sigma^2_{tot} = %.2f)$" % evals_cs[n - 1]
    else:
        text = "mean + %i components\n" % n
        text += r"$(\sigma^2_{tot} = %.2f)$" % evals_cs[n - 1]

    ax.text(0.02, 0.93, text, ha='left', va='top', transform=ax.transAxes)

fig.axes[-1].set_xlabel(r'${\rm wavelength\ (\AA)}$')
plt.show()
../_images/astroml_chapter7_Dimensionality_Reduction_18_0.png

Eigenvalues

The eigenvalues for the PCA decomposition of the SDSS spectra described in the previous section. The top panel shows the decrease in eigenvalue as a function of the number of eigenvectors, with a break in the distribution at ten eigenvectors. The lower panel shows the cumulative sum of eigenvalues normalized to unity. 94% of the variance in the SDSS spectra can be captured using the first ten eigenvectors.

fig = plt.figure(figsize=(10, 7.5))
fig.subplots_adjust(hspace=0.05, bottom=0.12)

ax = fig.add_subplot(211, xscale='log', yscale='log')
ax.grid()
ax.plot(evals, c='k')
ax.set_ylabel('Normalized Eigenvalues')
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.set_ylim(5E-4, 100)

ax = fig.add_subplot(212, xscale='log')
ax.grid()
ax.semilogx(evals_cs, color='k')
ax.set_xlabel('Eigenvalue Number')
ax.set_ylabel('Cumulative Eigenvalues')
ax.set_ylim(0.65, 1.00)

plt.show()
../_images/astroml_chapter7_Dimensionality_Reduction_20_0.png

PCA with missing data

Observed spectrum, \(x^o\) is the true spectrum, \({x}\) plus a wavelength-dependent weight, \({w}\).

Weight is zero where data are missing and \(1/{\sigma}^2\) for rest

Minimizing the quadratic deviation between \({x}^o\) truncated reconstruction, \(\sum_i \theta_i {e}_i\) and solving for \(\theta_i\) gives

\[ \sum_k \theta_i {w}(k) {e}_i(k) {e}_j(k) = \sum_k {w}(k) {x}^o(k) {e}_j(k), \]

If \(M_{ij} = \sum_k {w}(k) {e}_i(k) {e}_j(k)\) and \(F_i = \sum_k {w}(k) {x}^o(k) {e}_i(k)\) then

\[ \theta_i = \sum_j M_{ij}^{-1} F_{j}, \]
  • \(F_j\) are coefficients derived from gappy data

  • \(M_{ij}^{-1}\) shows how correlated eigenvectors are over the missing regions.

An estimate of the uncertainty on the reconstruction coefficients is given by

\[ %Cov(\theta_i,\theta_j) = \frac{1}{N}M_{ij}^{-1} {\rm Cov}(\theta_i,\theta_j) = M_{ij}^{-1}. \]

Accuracy of this reconstruction will depend on the distribution of the gaps within the data vector.

The principal component vectors defined for the SDSS spectra can be used to interpolate across or reconstruct missing data. Examples of three masked spectral regions are shown comparing the reconstruction of the input spectrum (black line) using the mean and the first ten eigenspectra (blue line) The gray bands represent the masked region of the spectrum.

evecs = data['evecs']
mu = data['mu']
norms = data['norms']
mask = data['mask']

#------------------------------------------------------------
# plot the results
i_plot = ((wavelengths > 5750) & (wavelengths < 6350))
wavelength = wavelengths[i_plot]

specnums = [20, 8, 9]
subplots = [311, 312, 313]

fig = plt.figure(figsize=(10, 12.5))
fig.subplots_adjust(left=0.09, bottom=0.08, hspace=0, right=0.92, top=0.95)

for subplot, i in zip(subplots, specnums):
    ax = fig.add_subplot(subplot)

    # compute eigen-coefficients
    spectra_i_centered = spectra_corr[i] / norms[i] - mu
    coeffs = np.dot(spectra_i_centered, evecs.T)

    # blank out masked regions
    spectra_i = spectra_corr[i]
    mask_i = mask[i]
    spectra_i[mask_i] = np.nan

    # plot the raw masked spectrum
    ax.plot(wavelength, spectra_i[i_plot], '-', color='k',
            label='True spectrum', lw=1.5)

    # plot two levels of reconstruction
    for nev in [10]:
        if nev == 0:
            label = 'mean'
        else:
            label = 'reconstruction\n(nev=%i)' % nev
        spectra_i_recons = norms[i] * (mu + np.dot(coeffs[:nev], evecs[:nev]))
        ax.plot(wavelength, spectra_i_recons[i_plot], label=label, color='grey')

    # plot shaded background in masked region
    ylim = ax.get_ylim()
    mask_shade = ylim[0] + mask[i][i_plot].astype(float) * ylim[1]
    plt.fill(np.concatenate([wavelength[:1], wavelength, wavelength[-1:]]),
             np.concatenate([[ylim[0]], mask_shade, [ylim[0]]]),
             lw=0, fc='k', alpha=0.2)

    ax.set_xlim(wavelength[0], wavelength[-1])
    ax.set_ylim(ylim)
    ax.yaxis.set_major_formatter(ticker.NullFormatter())

    if subplot == 311:
        ax.legend(loc=1)

    ax.set_xlabel('$\lambda\ (\AA)$')
    ax.set_ylabel('normalized flux')

plt.show()
../_images/astroml_chapter7_Dimensionality_Reduction_24_0.png

Comparing PCA, NMF and ICA

Nonnegative Matrix Factorization

Eigenvectors are defined relative to mean data vector. Principal components that can be positive or negative but for many physical systems we know data are a linear sum of positive components (e.g. galaxy spectrum is a linear sum of stellar components

Nonnegative matrix factorization (NMF) applies positivity constraint

from sklearn.decomposition import NMF

X = np.random.random((100, 3))  # 100 points in 3 dims, all positive
nmf = NMF(n_components=3)  # setting n_components is optional
nmf.fit(X)
proj = nmf.transform(X)  # project to 3 dimensions

comp = nmf.components_  # 3 x 10 array of components
err = nmf.reconstruction_err_  # how well 3 components captures data
/Users/bsipocz/.pyenv/versions/3.9.1/lib/python3.9/site-packages/sklearn/decomposition/_nmf.py:312: FutureWarning: The 'init' value, when 'init=None' and n_components is less than n_samples and n_features, will be changed from 'nndsvd' to 'nndsvda' in 1.1 (renaming of 0.26).
  warnings.warn(("The 'init' value, when 'init=None' and "

Independent component analysis

from sklearn.decomposition import FastICA

X = np.random.normal(size=(100, 2))  # 100 pts in 2 dims
R = np.random.random((2, 5))  # mixing matrix
X = np.dot(X, R)  # X is now 2D data in 5D space
ica = FastICA(2)  # fit two components

sources = ica.fit_transform(X)
proj = ica.transform(X)  # 100 x 2 projection of data
comp = ica.components_  # the 2 x 5 matrix of indep. components

Comparing PCA, NMF and ICA

A comparison of the decomposition of SDSS spectra using PCA (left panel), ICA (middle panel) and NMF (right panel). The rank of the component increases from top to bottom. For the ICA and PCA the first component is the mean spectrum (NMF does not require mean subtraction). All of these techniques isolate a common set of spectral features (identifying features associated with the continuum and line emission). The ordering of the spectral components is technique dependent.

from sklearn.decomposition import NMF, FastICA, PCA

data = sdss_corrected_spectra.fetch_sdss_corrected_spectra()
spectra = sdss_corrected_spectra.reconstruct_spectra(data)
wavelengths = sdss_corrected_spectra.compute_wavelengths(data)

#----------------------------------------------------------------------
# Compute PCA, ICA, and NMF components
def compute_PCA_ICA_NMF(n_components=5):
    spec_mean = spectra.mean(0)

    # PCA: use randomized PCA for speed
    pca = PCA(n_components - 1, random_state=0, svd_solver='randomized')
    pca.fit(spectra)
    pca_comp = np.vstack([spec_mean,
                          pca.components_])

    # ICA treats sequential observations as related.  Because of this, we need
    # to fit with the transpose of the spectra
    ica = FastICA(n_components - 1, random_state=0)
    ica.fit(spectra.T)
    ica_comp = np.vstack([spec_mean,
                          ica.transform(spectra.T).T])

    # NMF requires all elements of the input to be greater than zero
    spectra[spectra < 0] = 0
    nmf = NMF(n_components, random_state=0)
    nmf.fit(spectra)
    nmf_comp = nmf.components_

    return pca_comp, ica_comp, nmf_comp

n_components = 5
decompositions = compute_PCA_ICA_NMF(n_components)

#----------------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(10, 8))
fig.subplots_adjust(left=0.05, right=0.95, wspace=0.05,
                    bottom=0.1, top=0.95, hspace=0.05)

titles = ['PCA components', 'ICA components', 'NMF components']

for i, comp in enumerate(decompositions):
    for j in range(n_components):
        ax = fig.add_subplot(n_components, 3, 3 * j + 1 + i)

        ax.yaxis.set_major_formatter(plt.NullFormatter())
        ax.xaxis.set_major_locator(plt.MultipleLocator(1000))
        if j < n_components - 1:
            ax.xaxis.set_major_formatter(plt.NullFormatter())
        else:
            ax.xaxis.set_major_locator(
                plt.FixedLocator(list(range(3000, 7999, 1000))))
            ax.set_xlabel(r'wavelength ${\rm (\AA)}$')

        ax.plot(wavelengths, comp[j], '-k', lw=1)

        # plot zero line
        xlim = [3000, 8000]
        ax.plot(xlim, [0, 0], '-', c='gray', lw=1)

        if j == 0:
            ax.set_title(titles[i])

        if titles[i].startswith('PCA') or titles[i].startswith('ICA'):
            if j == 0:
                label = 'mean'
            else:
                label = 'component %i' % j
        else:
            label = 'component %i' % (j + 1)

        ax.text(0.03, 0.94, label, transform=ax.transAxes,
                ha='left', va='top')

        for l in ax.get_xticklines() + ax.get_yticklines():
            l.set_markersize(2)

        # adjust y limits
        ylim = plt.ylim()
        dy = 0.05 * (ylim[1] - ylim[0])

        ax.set_ylim(ylim[0] - dy, ylim[1] + 4 * dy)
        ax.set_xlim(xlim)


plt.show()
/Users/bsipocz/.pyenv/versions/3.9.1/lib/python3.9/site-packages/sklearn/decomposition/_nmf.py:312: FutureWarning: The 'init' value, when 'init=None' and n_components is less than n_samples and n_features, will be changed from 'nndsvd' to 'nndsvda' in 1.1 (renaming of 0.26).
  warnings.warn(("The 'init' value, when 'init=None' and "
/Users/bsipocz/.pyenv/versions/3.9.1/lib/python3.9/site-packages/sklearn/decomposition/_nmf.py:1090: ConvergenceWarning: Maximum number of iterations 200 reached. Increase it to improve convergence.
  warnings.warn("Maximum number of iterations %d reached. Increase it to"
../_images/astroml_chapter7_Dimensionality_Reduction_32_1.png

Summary of pracical properties of PCA, NMF, and ICA

The following table is a simple summary of the trade-offs along our axes of accuracy, interpretability, simplicity, and speed in dimension reduction methods, expressed in terms of high (H), medium (M), and low (L) categories.

Method

Accuracy

Interpretability

Simplicity

Speed

Principal component analysis

H

H

H

H

Nonnegative matrix factorization

H

H

M

M

Independent component analysis

M

M

L

L