Singular Value Decomposition Method To Determine Distance Distributions in Pulsed Dipolar Electron Spin Resonance: II. Estimating Uncertainty

Abstract

This paper is a continuation of the method introduced by Srivastava and Freed (2017) that is a new method based on truncated singular value decomposition (TSVD) for obtaining physical results from experimental signals without any need for Tikhonov regularization or other similar methods that require a regularization parameter. We show here how to estimate the uncertainty in the SVD-generated solutions. The uncertainty in the solution may be obtained by finding the minimum and maximum values over which the solution remains converged. These are obtained from the optimum range of singular value contributions, where the width of this region depends on the solution point location (e.g., distance) and the signal-to-noise ratio (SNR) of the signal. The uncertainty levels typically found are very small with substantial SNR of the (denoised) signal, emphasizing the reliability of the method. With poorer SNR, the method is still satisfactory but with greater uncertainty, as expected. Pulsed dipolar electron spin resonance spectroscopy experiments are used as an example, but this TSVD approach is general and thus applicable to any similar experimental method wherein singular matrix inversion is needed to obtain the physically relevant result. We show that the Srivastava-Freed TSVD method along with the estimate of uncertainty can be effectively applied to pulsed dipolar electron spin resonance signals with SNR > 30, and even for a weak signal (e.g., SNR ≈ 3) reliable results are obtained by this method, provided the signal is first denoised using wavelet transforms (WavPDS).

Figure 1.

Comparison of distance distributions of a model signal (A) using Tikhonov regularization with different λ values. The model P(r) is a Gaussian distribution centered at 5 nm with standard deviation of 0.3 nm (B). The maximum entropy method (MEM) was used to constrain P(r) ≥ 0, suppressing the regions of P(r) < 0 in the original Tikhonov result. Reprinted with permission from ref . Copyright 2017 ACS.

Figure 2.

Model data: Exact solution using the SVD method: P(r) versus number of singular value contributions (SVCs). Unimodal distance distribution: (A1) Noise-free dipolar signal. Comparison of model distribution with P(r) generated from (B1) fewer SVCs k = 3, σ_k = 8.04, (C1) exact solution obtained for k = 83, σ_k = 4.8 × 10⁻⁶, (D1) more SVCs k = 85, σ_k = 10⁻⁹, and (E1) Picard plot of ${\log }_{10}\left({\sum }_i{\mid {\sum }_{l=1}^M{\left({\Sigma }^{-1}\right)}_{\mathit{ii}}{U_{\mathit{il}}}^T{S}_l\mid }^2\right)$ versus the number of singular values from k = 1 to 200 starting from largest value. It shows the contributions of singular values that lead to stable and unstable distributions. Bimodal distance distribution: (A2) Noise-free dipolar signal. Comparison of model distribution with P(r) generated from (B2) fewer SVCs k = 3, σ_k = 8.04, (C2) exact solution obtained for k = 82, σ_k = 5.1 × 10, (D2) more SVCs k = 84, σ_k = 10⁻⁷, and (E2) Picard plot of ${\log }_{10}\left({\sum }_i{\mid {\sum }_{l=1}^M{\left({\Sigma }^{-1}\right)}_{\mathit{ii}}{U_{\mathit{il}}}^T{S}_l\mid }^2\right)$ versus the number of singular values from k = 1 to 200 starting from largest value. It shows the contributions of singular values that lead to stable and unstable distributions. Reprinted with permission from ref .

Figure 3.

Model data: Orthogonality validation for the noiseless data and data with some noise (SNR ≈ 850). Unimodal distance distribution: (Al) noiseless data, (B1) data with some noise, and (C1) comparison of results of ${\sum }_{l=1}^M{U_{\mathit{il}}}^T{S}_l$ at the ith column vector for noiseless data and data with some noise. Bimodal distance distribution: (A2) noiseless data, (B2) data with some noise, and (C2) comparison of results of ${\sum }_{l=1}^M{U_{\mathit{il}}}^T{S}_l$ at the ith column vector for noiseless data and data with some noise. Signal in red is noise added to the noise-free data (blue) to obtain data with some noise (black). The orthogonality requirement shown in panel C is clearly met for the noiseless data but not for data with some noise. Reprinted with permission from ref .

Figure 4.

Model data with some noise (SNR ≈ 850): Bimodal distance distribution. (A) Noisy model data dipolar signal. (B) Picard plot of the bimodal distribution from the noisy model data at different number of singular value contributions (SVCs) represented by i, the enlarged inset covers SVCs from 55 to 90. (C–J) Comparison of model distribution with the distance distribution generated from (C) k = 3, σ_k = 8.04, (D) k = 5, σ_k = 3.63, (E) k = 8, σ_k = 2.04, (F) k = 40, σ_k = 0.33, (G) k = 67, σ_k = 0.01, (H) k = 69, σ_k = 3 × 10⁻³, (I) k = 71, σ_k = 1 × 10⁻³, and (J) k = 82, σ_k = 10⁻⁵. Reprinted with permission from ref .

Figure 5.

Bimodal model: Reconstruction of distance distribution for noise-free model data and noisy model data (SNR ≈ 850) using the new SVD method. (A) Model dipolar signal. (B) Model dipolar signal with added noise (see added noise in red plot). (C) Singular value cutoff at each distance (nm) for the model dipolar signal. (D) Singular value cutoff at each distance range (nm) for the model dipolar signal with added noise. (E) Distance distribution reconstructed from the model dipolar signal and model dipolar signal with noise using the singular value cutoffs shown in panels C and D, respectively. Note that the added noise is so small that panels A and B still appear identical, but convergence to the virtually identical final results requires the segmentation shown in panel D in the latter case. Reprinted with permission from ref .

Figure 6.

Stepwise process of obtaining uncertainty in the distance distribution from the dipolar signal. (A) Denoised experimental dipolar signal using WavPDS (cf. Figure 7 for noisy data). (B) Distance distribution reconstructed using the SVD method. (C) Modified Picard plot for the 2 nm distance, revealing that the solution for this distance never diverges. Hence, the last data point is selected as SVC cutoff. (D) Modified Picard plot for the 4.3 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (E) P(r) values obtained at the 2 nm distance by different SVCs until the SVC cutoff; if it never diverges as in this case then the last SVC is selected as cutoff. (F) P(r) values obtained at the 4.3 nm distance by different SVCs until the SVC cutoff. (G) Distance distribution with uncertainty in distribution shown in red.

Figure 7.

Case 1: SVD reconstruction with uncertainty analysis of the noisy and denoised dipolar signal for unimodal distribution. (A1) Noisy data. (B1) Distance distribution from noisy data with uncertainty (in red). (C1) Modified Picard plot for the 2 nm distance, revealing that the solution for this distance never diverges. Hence, the last data point is selected as SVC cutoff. (D1) Modified Picard plot for the 4.3 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (E1) P(r) values obtained at the 2 nm distance by different SVCs until the SVC cutoff; if it never diverges, as in this case, then the last SVC is selected as cutoff. (F1) P(r) values obtained at the 4.3 nm distance by different SVCs until the SVC cutoff. (A2) Denoised data using WavPDS. (B2) Distance distribution from denoised data with uncertainty (in red). (C2) Modified Picard plot for the 2 nm distance, revealing that the solution for this distance never diverges. Hence, the last data point is selected as SVC cutoff. (D2) Modified Picard plot for the 4.3 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (E2) P(r) values obtained at the 2 nm distance by different SVCs until the SVC cutoff; if it never diverges, as in this case, then the last SVC is selected as cutoff. (F2) P(r) values obtained at the 4.3 nm distance by different SVCs until the SVC cutoff.

Figure 8.

Case 2: SVD reconstruction with uncertainty analysis of the noisy and denoised dipolar signal for bimodal distribution. (A1) Noisy data. (B1) Distance distribution from noisy data with uncertainty (in red). (C1) Modified Picard plot for the 3.2 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (D1) Modified Picard plot for the 4.3 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (E1) P(r) values obtained at the 3.2 nm distance by different SVCs until the SVC cutoff; if it never diverges, as in this case, then the last SVC is selected as cutoff. (F1) P(r) values obtained at the 4.3 nm distance by different SVCs until the SVC cutoff. (A2) Denoised data using WavPDS. (B2) Distance distribution from denoised data with uncertainty (in red). (C2) Modified Picard plot for the 2 nm distance, revealing that the solution for this distance never diverges. Hence, the last data point is selected as SVC cutoff. (D2) Modified Picard plot for the 4.3 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (E2) P(r) values obtained at the 2 nm distance by different SVCs until the SVC cutoff; if it never diverges, as in this case, then the last SVC is selected as cutoff. (F2) P(r) values obtained at the 4.3 nm distance by different SVCs until the SVC cutoff.

Figure 9.

Case 3: SVD reconstruction with uncertainty analysis of the noisy and denoised dipolar signal at submicromolar concentration. (A) Noisy (red) and denoised (black) data with baseline. (B1) Noisy data. (C1) Distance distribution from noisy data with uncertainty (in red). (D1) Modified Picard plot for the 3 nm distance, revealing that the solution for this distance never diverges. Hence, the last data point is selected as SVC cutoff. (E1) Modified Picard plot for the 3.7 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (F1) P(r) values obtained at the 3 nm distance by different SVCs until the SVC cutoff; if it never diverges, as in this case, then the last SVC is selected as cutoff. (G1) P(r) values obtained at the 3.7 nm distance by different SVCs until the SVC cutoff. (B2) Denoised data using WavPDS. (C2) Distance distribution from denoised data with uncertainty (in red). (D2) Modified Picard plot for the 3 nm distance, revealing that the solution for this distance never diverges. Hence, the last data point is selected as SVC cutoff. (E2) Modified Picard plot for the 3.7 nm distance, revealing the singular value cutoff before which solution at this distance diverges. (F2) P(r) values obtained at the 3 nm distance by different SVCs until the SVC cutoff; if it never diverges, as in this case, then the last SVC is selected as cutoff. (G2) P(r) values obtained at the 3.7 nm distance by different SVCs until the SVC cutoff.