We introduce a new estimator for the vector of coefficients β in the linear model y = Xβ + z, where X has dimensions n × p with p possibly larger than n. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to minb∈ℝp12‖y−Xb‖ℓ22+λ1false|bfalse|false(1false)+λ2false|bfalse|false(2false)+⋯+λpfalse|bfalse|false(pfalse),where λ1 ≥ λ2 ≥ … ≥ λp ≥ 0 and false|bfalse|false(1false)≥false|bfalse|false(2false)≥⋯≥false|bfalse|false(pfalse) are the decreasing absolute values of the entries of b. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical ℓ1 procedures such as the Lasso. Here, the regularizer is a sorted ℓ1 norm, which penalizes the regression coefficients according to their rank: the higher the rank—that is, stronger the signal—the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] procedure (BH) which compares more significant p-values with more stringent thresholds. One notable choice of the sequence {λi} is given by the BH critical values λBHfalse(ifalse)=zfalse(1−i⋅q/2pfalse), where q ∈ (0, 1) and z(α) is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with λBH provably controls FDR at level q. Moreover, it also appears to have appreciable inferential properties under more general designs X while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.
We present results of a joint computational and experimental study for a series of annulated metalloporphyrins in order to establish structure-property relationships. Specifically, we have examined the effects of substitution by meso-tetraphenylation, tetrabenzo and tetranaphtho annulation, and effects of changing the central metal from zinc (Zn) to palladium (Pd). Utilizing absorption and emission spectroscopy and laser flash photolysis techniques, the photophysical properties of these porphyrins have been determined. Upon the addition of benzo or naphtho groups, we observed an overall red shift in the ground state absorption spectra of both the B-bands and the Q-bands with increased conjugation and an increase in the Q-band to B-band intensity ratios. Time-dependent density functional theory calculations were performed on both series of porphyrins to identify the effects of phenyl, benzo, and naphtho substituents on the spectra. The benzo and naphtho adducts provide a larger contribution (typically 40-90%) to the observed red shifts due to increased π-conjugation, while there is a smaller contribution (typically 0-25%) from distortion of the porphyrin. Similarly, a red shift for the T 1 -T n absorption spectrum and an overall general broadening in the spectrum were found with increased conjugation. An increase in the triplet molar extinction coefficient through the near-infrared region with annulation was also found. Varying the metal has an effect on the overall absorption spectra; i.e., the ground state spectra of the Zn porphyrins are red-shifted relative to the Pd porphyrins. For the triplet excited state spectra there were small effects in the spectra by changing the metal with a significant contribution to the kinetic properties by the heavy atom effect of the Pd.
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