tokens: 3 . The amplitudes of the PCs have been artificially normalized to unity for easier comparison. We find in general that at a particular redshift, an individual PC tends to be smooth and positively correlated along k , and at a particular k shows negative and positive correlations across redshift. This is a reflection of the underlying smoothness of the power spectra across k , and the fact that physical processes such as reionization, X-ray heating, and Ly α coupling tend to produce redshift-dependent peaks and troughs in the power spectrum (see, e.g., Figure 3 ). The reason why the PCs lose strength at high k is because our rescaling matrix downweights our data covariance matrix at high k . As we will see in Section 4.1 , the bulk of a 21 cm experiment’s sensitivity to the power spectrum is located at lower k . Figure 1. Top: scree plot showing the eigenvalues of the 30 principal components formed from the training data of . Bottom: the first nine principal components of the power spectrum data at each unique k – z combination. The color scale is artificially normalized to [−1, 1] for easier comparison. 2.3. GP Regression For the purposes of emulation, we are not interested in merely reconstructing our original training set data at their corresponding points in parameter space j , but are interested in constructing a prediction of a new data vector, new , at a new position in our parameter space, new . We can construct a prediction of the data vector at this new point in parameter space by evaluating Equation ( 7 ) with new ; however, we do not know this weight vector a priori. To estimate it at any point in parameter space, we require a predictive function for each element of spanning the entire parameter space. In other words, we need to interpolate over our parameter space. To do this, we adopt a GP model, which is a highly flexible and non-parametric regressor. See Rasmussen Williams ( 2006 ) for a review on GPs for regression. A GP is fully specified by its mean function and covariance kernel. The GP mean function can be thought of as the global trend of the data, while its covariance kernel describes the correlated random Gaussian fluctuations about the mean trend. In practice, because we center our data about zero before constructing the PCs and their weights (Equation ( 2 )), we set our mean function to be identically zero. For the covariance kernel, we employ a standard squared-exponential kernel, which is fully stationary, infinitely differentiable, produces smooth realizations of a correlated random Gaussian field, and is given in multivariate form aswhere and ′ denote two position vectors in our parameter space, is a diagonal matrix containing the characteristic scale length of correlations l across each parameter, and σ A is the characteristic amplitude of the covariance. is a tunable hyperparameter of the kernel function that must be selected a priori. We discuss how we make these choices in Section 2.3.1 . We set σ A = 1, and
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