Determining the effect of asymmetric data on the variogram. I. Underlying asymmetry

Matheron's usual variogram estimator can result in unreliable variograms when data are strongly asymmetric or skewed. Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. This paper examines the effects of underlying asymmetry on the variogram and on the accuracy of prediction, and the second one examines the effects arising from outliers. Standard geostatistical texts suggest ways of dealing with underlying asymmetry; however, this is based on informed intuition rather than detailed investigation. To determine whether the methods generally used to deal with underlying asymmetry are appropriate, the effects of different coefficients of skewness on the shape of the experimental variogram and on the model parameters were investigated. Simulated annealing was used to create normally distributed random fields of different size from variograms with different nugget:sill ratios. These data were then modified to give different degrees of asymmetry and the experimental variogram was computed in each case. The effects of standard data transformations on the form of the variogram were also investigated. Cross-validation was used to assess quantitatively the performance of the different variogram models for kriging. The results showed that the shape of the variogram was affected by the degree of asymmetry, and that the effect increased as the size of data set decreased. Transformations of the data were more effective in reducing the skewness coefficient in the larger sets of data. Cross-validation confirmed that variogram models from transformed data were more suitable for kriging than were those from the raw asymmetric data. The results of this study have implications for the 'standard best practice' in dealing with asymmetry in data for geostatistical analyses. (C) 2007 Elsevier Ltd. All rights reserved.

Determining the effect of asymmetric data on the variogram. II. Outliers

Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. The first paper of this series examined the effects of the former on the variogram and this paper examines the effects of asymmetry arising from outliers. Simulated annealing was used to create normally distributed random fields of different size that are realizations of known processes described by variograms with different nugget:sill ratios. These primary data sets were then contaminated with randomly located and spatially aggregated outliers from a secondary process to produce different degrees of asymmetry. Experimental variograms were computed from these data by Matheron's estimator and by three robust estimators. The effects of standard data transformations on the coefficient of skewness and on the variogram were also investigated. Cross-validation was used to assess the performance of models fitted to experimental variograms computed from a range of data contaminated by outliers for kriging. The results showed that where skewness was caused by outliers the variograms retained their general shape, but showed an increase in the nugget and sill variances and nugget:sill ratios. This effect was only slightly more for the smallest data set than for the two larger data sets and there was little difference between the results for the latter. Overall, the effect of size of data set was small for all analyses. The nugget:sill ratio showed a consistent decrease after transformation to both square roots and logarithms; the decrease was generally larger for the latter, however. Aggregated outliers had different effects on the variogram shape from those that were randomly located, and this also depended on whether they were aggregated near to the edge or the centre of the field. The results of cross-validation showed that the robust estimators and the removal of outliers were the most effective ways of dealing with outliers for variogram estimation and kriging. (C) 2007 Elsevier Ltd. All rights reserved.

The synthesis of chiral N-heterocyclic carbene–borane and –diorganoborane complexes and their use in the asymmetric reduction of ketones

Chiral N-heterocyclic carbene–borane complexes have been synthesised, and have been shown to reduce ketones with Lewis acid promotion. Chiral N-heterocyclic carbene–borane and –diorganoborane complexes can reduce ketones with enantioselectivities up to 75% and 85% ee, respectively.

Total synthesis of (+/-) maculalactone A, maculalactone B and maculalactone C and the determination of the absolute configuration of natural (+) maculalactone A by asymmetric synthesis

Maculalactones A, B and C from the marine cyanobacterium Kyrtuthrix maculans are amongst the only compounds based on the tribenzylbutyrolactone skeleton known in nature and (+) maculalactone A from the natural source possesses significant biological activity against various marine herbivores and marine settlers. We now report a concise synthesis of racemic maculalactone A in five steps from inexpensive starting materials. Maculalactones B and C were synthesized by a minor modification to this procedure, and the synthetic design also permitted an asymmetric synthesis of maculalactone A to be achieved in around 85% ee. The (+) and (-) enantiomers of maculalactone A were assigned, respectively, to the S and R configurations on the basis of the chiral selectivity expected for catecholborane reduction of an unsymmetrical ketone in the presence of Corey's oxazoborolidine catalyst. Surprisingly, it appeared that natural (+) maculalactone A was biosynthesized in K. maculans in a partially racemic form, comprising ca. 90-95% of the (S) enantiomer and 5-10% of its (R) enantiomer. Coincidentally therefore, the percentage enantiomeric excess of the product obtained from asymmetric synthesis almost exactly matched that found in nature. (C) 2004 Elsevier Ltd. All rights reserved.

Asymmetric synthesis of (1R,2S,3R)-3-methylcispentacin and (1S,2S,3R)-3-methyltranspentacin by kinetic resolution of tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate

Conjugate addition of lithium dibenzylamide to tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate occurs with high levels of stereocontrol, with preferential addition of lithium dibenzylamide to the face of the cyclic alpha,beta-unsaturated acceptor anti- to the 3-methyl substituent. High levels of enantiorecognition are observed between tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate and an excess of lithium (+/-)-N-benzyl-N-alpha-methylbenzylamide (10 eq.) (E > 140) in their mutual kinetic resolution, while the kinetic resolution of tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate with lithium (S)-N-benzyl-N-alpha-methylbenzylamide proceeds to give, at 51% conversion, tert-butyl (1R, 2S, 3R,alphaS)-3-methyl-2-N-benzyl-N-alpha-methylbenzylaminocyclopentane-1-c arboxylate consistent with E > 130, and in 39% yield and 99 +/- 0.5% de after purification. Subsequent deprotection by hydrogenolysis and ester hydrolysis gives (1R, 2S, 3R)-3-methylcispentacin in > 98% de and 98 +/- 1% ee. Selective epimerisation of tert-butyl (1R, 2S, 3R, alphaS)-3-methyl-2-N- benzyl-N-alpha-methylbenzylaminocyclopentane-1-carboxylate by treatment with (KOBu)-Bu-t in (BuOH)-Bu-t gives tert-butyl (1S, 2S, 3R, alphaS)-3-methyl-2-N-benzyl-N-alpha-methylbenzylaminocyclopentane-1-carb oxylate in quantitative yield and in > 98% de, with subsequent deprotection by hydrogenolysis and ester hydrolysis giving (1S, 2S, 3R)-3-methyltranspentacin hydrochloride in > 98% de and 97 +/- 1% ee.

Stereodivergent approach to the asymmetric synthesis of bacillariolides: A formal synthesis of ent-bacillariolide II

Asymmetric synthesis of densely functionalized bicyclic frameworks for entry into bacillariolides I/III and ent-bacillariolide II is reported. The key features are ring-closing metathesis of a pair of diastereomerically related dienes obtained through a stereodivergent route from a R-(+)-glyceraldehyde derivative, transformation of a nonstereoselective cyclopentene ester enolate alkylation process to a completely stereoselective one through alkylation of a bulky ester enolate with a bulky electrophile, and a remote silyloxymethyl group directed epoxidation.

Asymmetric synthesis of arylalanines via asymmetric aza-darzens (ADZ) reaction

(R)-3-Arylalanines may be prepared in high enantiomeric purity from N-dpp imines by a four-step reaction sequence involving asymmetric aza-Darzens reaction, dephosphinylation, hydrogenolysis and hydrolysis. The amino acids thus obtained were of >95% enantiomeric purity.

Convenient route to enantiopure aryl cyclopentanes via Diels-Alder reaction of asymmetric dienes. Total synthesis of (+)-herbertene and (+)-cuparene

A general route for the synthesis of highly substituted aryl cyclopentanes has been developed involving Diels-Alder reaction of asymmetric dienes prepared from (+)-camphoric acid followed by aromatization of the resulting cyclohexene derivatives. Employing this protocol enantiospecific synthesis of (+)-herbertene and (+)-cuparene has been accomplished. (C) 2003 Elsevier Science Ltd. All rights reserved.

Bidentates versus monodentates in asymmetric hydrogenation catalysis: synergic effects on rate and allosteric effects on enantioselectivity

C-1-Symmetric phosphino/phosphonite ligands are prepared by the reactions of Ph2P(CH2)(2)P(NMe2)(2) with (S)-1,11'-bi-2-naphthol (to give L-A) or (S)-10,10'-bi-9-phenanthrol (to give L-B). Racemic 10,10'-bi-9-phenanthrol is synthesized in three steps from phenanthrene in 44% overall yield. The complexes [PdCl2(L-A,L-B)] (1a,b), [PtCl2(L-A,L-B)] (2a,b), [Rh(cod)(L-A,L-B)]BF4 (3a,b) and [Rh(L-A,L-B)(2)]BF4 (4a,b) are reported and the crystal structure of la has been determined. A P-31 NMR study shows that M, a 1:1 mixture of the monodentates, PMePh2 and methyl monophosphonite L-1a (based on (S)-1,11'-bi-2-naphthol), reacts with 1 equiv of [Rh(cod)(2)]BF4 to give the heteroligand complex [Rh(cod)(PMePh2)(L-1a)]BF4 (5) and homoligand complexes [Rh(cod)(PMePh2)(2)]BF4 (6) and [Rh(cod)(L-1a)(2)]BF4 (7) in the ratio 2:1:1. The same mixture of 5-7 is obtained upon mixing the isolated homoligand complexes 6 and 7 although the equilibrium is only established rapidly in the presence of an excess of PMePh2. The predominant species 5 is a monodentate ligand complex analogue of the chelate 3a. When the mixture of 5-7 is exposed to 5 atm H-2 for 1 h (the conditions used for catalyst preactivation in the asymmetric hydrogenation studies), the products are identified as the solvento species [Rh(PMePh2)(L-1a)(S)(2)]BF4 (5'), [Rh(S)(2)(PMePh2)(2)]BF4 (6') and [Rh(S)(2)(L-1a)(2)]BF4 (7') and are formed in the same 2:1:1 ratio. The reaction of M with 0.5 equiv of [Rh(cod)(2)]BF4 gives exclusively the heteroligand complex cis-[Rh(PMePh2)(2)(L-1a)(2)]BF4 (8), an analogue of 4a. The asymmetric hydrogenation of dehydroamino acid derivatives catalyzed by 3a,b is reported, and the enantioselectivities are compared with those obtained with (a) chelate catalysts derived from analogous diphosphonite ligands L-2a and L-2b, (b) catalysts based on methyl monophosphonites L-1a and L-1b, and (c) catalysts derived from mixture M. For the cinnamate and acrylate substrates studied, the catalysts derived from the phosphino/phosphonite bidentates L-A,L-B generally give superior enantioselectivities to the analogous diphosphonites L-2a and L-2b; these results are rationalized in terms of delta/lambda-chelate conformations and allosteric effects of the substrates. The rate of hydrogenation of acrylate substrate A with heterochelate 3a is significantly faster than with the homochelate analogues [Rh(L-2a)(cod)]BF4 and [Rh(dppe)(cod)]BF4. A synergic effect on the rate is also observed with the monodentate analogues: the rate of hydrogenation with the mixture containing predominantly heteroligand complex 5 is faster than with the monophosphine complex 6 or monophosphonite complex 7. Thus the hydrogenation catalysis carried out with M and [Rh(cod)(2)]BF4 is controlled by the dominant and most efficient heteroligand complex 5. In this study, the heterodiphos chelate 3a is shown to be more efficient and gives the opposite sense of optical induction t the heteromonophos analogue

Application of the asymmetric hetero Diels-Alder reaction for synthesising carbohydrate derivatives and glycosidase inhibitors

This review provides a discussion of recent developments in the asymmetric hetero Diels-Alder reaction (AHDAR), with particular emphasis on the synthesis of carbohydrates, their derivatives, and inhibitors of carbohydrate processing enzymes.

Asymmetric aziridine synthesis by aza-Darzens reaction of N-diphenylphosphinylimines with chiral enolates. Part 2: inversion of diastereoselectivity

The aza-Darzens ('ADZ') reactions of N-diphenylphosphinyl ('N-Dpp') imines with chiral enolates derived from N-bromoacetyl 2S-2,10-camphorsultam proceed in generally good yield to give N-diphenylphosphinyl aziridinoyl sultams. However, the stereoselectivity of the reaction is dependent upon the structure of the imine substituent: when the chiral enolate was reacted with arylimines substituted in the ortho-position, mixtures of cis- and trans-2'R,3'R-aziridines were obtained, often with a complete selectivity in favour of the trans-isomer. (c) 2006 Elsevier Ltd. All rights reserved.

Asymmetric aziridine synthesis by aza-Darzens reaction of N-diphenylphosphinylimines with chiral enolates. Part 1: formation of cis-aziridines

Theaza-Darzens ('ADZ') reactions off-diphenylphosphinyl (W-Dpp') imines with chiral enolates derived from oxazolidinones and camphorsultam have been Studied. Whilst oxazolidinone enolates reacted poorly in terms of aziridination, the use of the chiral enolate derived from both antipodes of N-bromoacetyl 2, 10-camphorsultam, 2R-(5) and 2S-(5), with N-diphehenylphosphinyl aryl and tert-butylimines proceeded in generally good yield to give, respectively, (2'R,3'R)- or (2'S,3'S)-cis-N-diphenylphosphinyl aziridinoyl sultams of high de. (c) 2006 Elsevier Ltd. All rights reserved.

Asymmetric ammonium ylid rearrangements: the effect of nitrogen asymmetry

[2,3]-Sigmatropic rearrangements of allylic ammonium ylids derived from glycinoylcamphorsultams are highly selective in terms of relative and absolute stereocontrol only when acyclic alkenes are present. When chiral esters of ylids derived from N-methyltetrahydro-pyridine ('NMTP') undergo rearrangement, the reactions show exclusive cis-stereoselectivity but the products are obtained with virtually no absolute stereocontrol. These observations support the notion that sigmatropic rearrangements of N-chiral ammonium ylids are controlled by nitrogen stereogenicity. (c) 2006 Elsevier Ltd. All rights reserved.