Reviews in Computational Chemistry Volume 18

More documents

Recommendations

Info

Progress in Clustering Methodology 25 of natural clusters and that the seeds chosen are reasonably close to the centers of these clusters. Epter, Krishnamoorthy, and Zaki 113 published one of the few papers addressing these issues for large data sets. Their solution is applicable to distance-based clustering and involves analysis of the histogram of pairwise distances between data items. For small data sets, all pairwise distances can be used, whereas for large data sets, random sampling (up to 10% of the data set) can be used to lessen the quadratic increase in time needed to generate the distances. For the distances calculated, the corresponding histogram is generated and then scanned to find the first spike (a large maximum followed by a large minimum). This point is used as the threshold for intracluster distance. The graph containing distances within this threshold contains connected components used to determine both the number of clusters present in the data set and the initial starting points from within these clusters. Assuming that a reasonable value for k is known, Fayyad, Reima, and Bradley 114,115 showed that one can minimize the problem of poor initial starting points by sampling the data set to derive a better set of starting points. A series of randomly selected subsets, larger than k, are extracted, clustered by kmeans, amalgamated, and then clustered again using each solution from the subsets. The starting points from the subset giving the best clustering of the amalgamated subset are then chosen as the set of refined points for the main clustering, where ‘‘best’’ means the clustering that gives the minimal ‘‘distortion,’’ that is, minimum error across the amalgamated subset. The method aims to avoid selecting outliers, which may occur with other selection methods such as MaxMin. In hierarchical clustering, each level defines a partition of the data set into clusters. However, there is no associated information indicating which level is best in terms of splitting the data set into the ‘‘natural’’ number of clusters present and with each cluster containing the most appropriate compounds. Many methods and criteria have been proposed to try to derive such information from the hierarchy so that the ‘‘best’’ level is selected. Milligan and Cooper 116 published the first comprehensive comparison of hierarchy level selection methods, using psychology data. Thirty methods were tested for their ability to retrieve the correct number of clusters from several small data sets containing from 2 to 5 ‘‘natural’’ clusters. Fifteen years later, Wild and Blankley 117 published a major comparison of hierarchy level selection methods using chemical data sets. As part of that study, Ward clustering with 2D fingerprints was used to evaluate the performance of nine hierarchy level selection methods. The methods chosen were those that would be easy to implement and that did not require parameters. Eight of those methods were ones that Milligan and Cooper had previously examined; the ninth was a more recent method published by Kelley, Gardner, and Sutcliffe. 118 The study by Wild and Blankley concluded that the point biseral, 119 variance ratio criterion, 120 and Kelley methods gave the best overall results, with the Kelley method being more computationally efficient than the others [scaling at less
26 Clustering Methods and Their Uses in Computational Chemistry feature 2 50 40 30 20 10 0 0 Test dataset (100 objects, 5 clusters) 10 20 30 40 50 feature 1 Figure 3 An example data set of 100 objects, represented by 2 features, that fall into 5 natural clusters. than OðN 2 Þ]. A test data set of 100 objects, represented by 2 features and grouped into 5 natural clusters, is shown in Figure 3. The corresponding plot of penalty values (calculated using the Kelley method) against the number of clusters (Figure 4) shows a clear minimum at 5 clusters. penalty value 100 80 60 40 20 0 Kelley plot (100 objects, 5 clusters) number of clusters (0 to 100) Figure 4 Kelley plot of the penalty value against number of clusters for the data set of 100 items in Figure 3, showing the minimum at 5 clusters.
Page 1 and 2:
Reviews in Computational Chemistry
Page 3 and 4: Kenny B. Lipkowitz Department of Ch
Page 5 and 6: vi Preface three-dimensional struct
Page 7 and 8: viii Preface some descriptors and i
Page 9 and 10: Epilogue and Dedication My associat
Page 11 and 12: Contents 1. Clustering Methods and
Page 13 and 14: Contents xv Electron Transfer in Po
Page 15 and 16: Contributors John M. Barnard, Barna
Page 17 and 18: Contributors to Previous Volumes *
Page 19 and 20: Volume 3 (1992) Tamar Schlick, Opti
Page 21 and 22: Volume 7 (1996) Geoffrey M. Downs a
Page 23 and 24: Volume 11 (1997) Mark A. Murcko, Re
Page 25 and 26: T. Daniel Crawford* and Henry F. Sc
Page 27 and 28: Topics Covered in Volumes 1-18 * Ab
Page 29 and 30: Reviews in Computational Chemistry
Page 31 and 32: 2 Clustering Methods and Their Uses
Page 39 and 40: 10 Clustering Methods and Their Use
Page 53: 24 Clustering Methods and Their Use
Page 71 and 72: 42 The Use of Scoring Functions in
Page 79 and 80: Table 1 Reference List for the Most
Page 105 and 106:
76 The Use of Scoring Functions in
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
CHAPTER 3 Potentials and Algorithms
Page 119 and 120:
Vn (1 + cos(nω + γ)) 2 K θ (θ
Page 121 and 122:
are modified by their environment w
Page 123 and 124:
Table 1 Polarizability Parameters f
Page 125 and 126:
The polarizable point dipole models
Page 127 and 128:
two to three orders of magnitude sl
Page 129 and 130:
M i z i + q i d i k i and shell cha
Page 131 and 132:
Shell Models 103 on estimates of sh
Page 133 and 134:
minimization can be replaced by mor
Page 135 and 136:
The energy required to create a cha
Page 137 and 138:
for all i ði:e:; 8 iÞ: @U @qi l
Page 139 and 140:
where we have used q Cl ¼ qNa. The
Page 141 and 142:
Electronegativity Equalization Mode
Page 143 and 144:
Electronegativity Equalization Mode
Page 145 and 146:
of N molecules is taken as a Hartre
Page 147 and 148:
is treated using variable charges.
Page 149 and 150:
water have been developed, includin
Page 151 and 152:
Applications 123 classical and rigi
Page 153 and 154:
developing polarizable models. A va
Page 155 and 156:
Comparison of the Polarization Mode
Page 157 and 158:
negligible errors in such propertie
Page 159 and 160:
ecome significant at field strength
Page 161 and 162:
noteworthy in this regard because t
Page 163 and 164:
References 135 9. P. Cieplak and P.
Page 165 and 166:
References 137 48. E. L. Pollock an
Page 167 and 168:
References 139 Computational Chemis
Page 169 and 170:
References 141 139. J. Hinze and H.
Page 171 and 172:
References 143 178. J. J. P. Stewar
Page 173 and 174:
References 145 216. M. W. Mahoney a
Page 175 and 176:
CHAPTER 4 New Developments in the T
Page 177 and 178:
Introduction 149 applications). For
Page 179 and 180:
FCWD(∆E ) FCWD(0) ∆E = 0 ∆E
Page 181 and 182:
Introduction 153 Equations [6]-[12]
Page 183 and 184:
Paradigm of Free Energy Surfaces 15
Page 185 and 186:
Paradigm of Free Energy Surfaces 15
Page 187 and 188:
In Eqs. [18] and [19], F0i is the e
Page 189 and 190:
where ‘‘þ’’ and ‘‘ ’
Page 191 and 192:
is, however, small for the usual co
Page 193 and 194:
solute-solvent coupling through the
Page 195 and 196:
where Z is the electrode overpotent
Page 197 and 198:
energy surfaces of ET. 33,50-56 It
Page 199 and 200:
This result indicates a fundamental
Page 201 and 202:
βF i (X ) 40 20 0 −20 1 2 βλ 1
Page 203 and 204:
Table 1 Main Features of the Two-Pa
Page 205 and 206:
and Hss ¼ U rep ss 1 2 X j;k ðmj
Page 207 and 208:
λ i /eV 4 3 2 1 0 0 10 20 30 40 Bo
Page 209 and 210:
the width/Stokes shift relation (Eq
Page 211 and 212:
Table 3 Mapping of the Q Model on S
Page 213 and 214:
This situation is of course not sat
Page 215 and 216:
F ± (Y ad )/λ I 0.8 0.4 0 −0.4
Page 217 and 218:
it by choosing the GMH basis set 7
Page 219 and 220:
Anharmonic higher order terms gain
Page 221 and 222:
of the radiation is the perturbatio
Page 223 and 224:
individual vibrational excitations
Page 225 and 226:
m 12 /D 6 5.5 5 4.5 4 3.5 17 18 19
Page 227 and 228:
In Eq. [144], the coordinates Y km
Page 229 and 230:
ε/M −1 cm −1 4000 2000 0 analy
Page 231 and 232:
βσ 2 /10 3 cm −1 14 12 10 8 Opt
Page 233 and 234:
0.2 0.1 0 12 16 20 24 28 32 The app
Page 235 and 236:
References 207 7. M. D. Newton, Adv
Page 237 and 238:
References 209 59. R. Kubo and Y. T
show all

Reviews in Computational Chemistry Volume 18

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?