Asssing the reliability of a QSAR model’s predictions
Linnan He,Peter C.Jurs*
Department of Chemistry,The Pennsylvania State University,104Chemistry Building,University Park,PA16802,USA
Received11November2004;received in revid form1March2005;accepted1March2005
Available online17May2005
Abstract
Quantitative structure activity relationships(QSAR)are one of the well-developed areas in computational chemistry.In thisfield,many successful predictive models have been developed for various property,activity or toxicity predictions.However,the predictive power of models for new query compounds is often not well characterized.The breadth of applicability of models is often not characterized.In other words,with a given QSAR model and a specific query compound to be predicted,can the model be ud reliably for the desired prediction?In this study,we assd the reliability of QSAR models’prediction on query compounds.Our approach,employing hierarchical clustering, was developed and tested using a test datat containing322organic compounds with fat
head minnow acute aquatic toxicity as the activity of interest.The hypothesis of the approach was that if a query compound is more similar to the compounds ud to generate the QSAR model,it should be predicted more accurately.Thus,the core of the approach is to determine the relationship between the similarity of query compounds to the training t compounds of the QSAR model and the prediction accuracy given by that model.This relationship determination was achieved by comparing the results given by the two major components of the approach:objects clustering and activity prediction.With the resultant information from the two steps,a direct relationship was shown.
#2005Elvier Inc.All rights rerved.
Keywords:QSAR model;Hierarchical clustering;Fathead minnow acute aquatic toxicity
1.Introduction
Similar molecules with just a slight variation in their structures can have quite different biological activities.This kind of relationship between molecular structure and changes in biological activity is the center of focus for thefield of quantitative structure activity relationships (QSAR).In thefield of QSAR,the main objective is to investigate the relationships by building mathematical models that explain the relationship in a statistical way. QSARfirst dates back to the19th century,with A.F.A.Cros’
discovery of the inver relationship between the water solubility of alcohols and the toxicity of alcohols to mammals[1].Today,QSARs are being applied in many disciplines with much emphasis in drug design.Over the years of development,many methods,algorithms and techniques have been discovered and applied in QSAR studies.With the success of their applications,QSAR has becoming one of the well-developed areas in computational chemistry.
Clearly,QSAR has matured,however,there are still many interesting questions raid in thefield.One of them is the ability of a previously developed QSAR model to predict a query compound’s activity.For instance,many successful QSAR models with high predictability have been reported in the literature for the prediction of fathead minnow acute toxicity[2–5].However,the good predictability was only applicable to the group of compounds that were excluded from the original datat prior to model building.Thus,it would be interesting to determine if the good predictability also applies to a new query compound,a compound not included in the original datat.
In general,the question is:given a QSAR model and a query compound for prediction,can the developed QSAR model be reliably ud to provide an accurate and reliable prediction?To address this question,we have developed a new approach using hierarchical clustering combined with the usual QSAR model generation methods.The main task of this new approach involves testing the foll
owing
homework的音标/locate/JMGM *Corresponding author.Tel.:+18148653739;fax:+18148653314.
E-mail address:pcj@psu.edu(P.C.Jurs).
1093-3263/$–e front matter#2005Elvier Inc.All rights rerved.足球下载
doi:10.1016/j.jmgm.2005.03.003
hypothesis:if a query compound is found to be more similar in structure to the compounds ud to generate the QSAR model,and then it should be predicted more reliably and more accurately by that model.If the hypothesis is shown to be true,then we can answer the question and conclude that a QSAR model can reliably predict a query compound ’s activity if the query compound is suf ficiently s
imilar to the compounds ud to generate the QSAR model.Thus,finding a correlation between the similarity of a query compound to the compounds ud to build the QSAR model and the prediction accuracy of the query compound given by that QSAR model was the core of this approach.
In this study,our main objective is to asss QSAR models ’reliability for activity prediction of new com-pounds.The designed approach was tested on 322organic compounds with fathead minnow acute toxicity as the activity of interest.Even though only one particular datat was studied,we believe our approach is quite general,and it can assist in the asssment of the reliability of a QSAR model ’s prediction in general.
psychology2.Experimental methods
Our approach consisted of two major steps,object clustering and activity prediction.The clustering step involved grouping datat compounds into clusters using hierarchical clustering.The main purpo,here,was to form dissimilar clusters of objects,to which the query compounds would be compared to for determination of degree of similarity.We choo hierarchical clustering for this similarity comparison step over other more sophisticated similarity determination methods becau of its simplicity and speed.The results showed that its similarity determina-tion ability was suf ficient f
or this application.In the cond step,a QSAR model was developed with each of the formed clusters to predict the query compounds ’activity of interest.This step gave us an idea about the prediction accuracy of each QSAR model for each query compound ’s activity prediction.Four different trials were designed with this approach.In the trials,the number of query compounds,the number of clusters and the method of clustering varied.The overall approach is illustrated in Fig.1.
L.He,P .C.Jurs /Journal of Molecular Graphics and Modelling 23(2005)503–523
504Fig.1.Study schema.
Table 1
Datat information No.CAS#Name
Exp.1110-82-7Cyclohexane 2.96267-64-1Acetone 0.85378-93-32-Butanone 1.354107-87-92-Pentanone 1.845591-78-62-Hexanone 2.376110-43-02-Heptanone 2.947111-13-72-Octanone 3.458821-55-62-Nonanone 3.979693-54-92-Decanone 4.5106175-49-12-Dodecanone
5.1911563-80-43-Methyl-2-butanone 212108-10-14-Methyl-2-pentanone 2.271375-97-83,3-Dimethyl-2-butanone 3.061496-22-03-Pentanone
1.7515110-12-35-Methyl-2-hexanone
2.8616502-56-75-Nonanone
3.6617108-94-1Cyclohexanone 2.191876-22-2Camphor 3.141967-56-1Methanol 0.052064-17-5Ethanol 0.522171-23-81-Propanol 1.122271-36-31-Butanol 1.592371-41-01-Pentanol 2.2124111-27-31-Hexanol 2.9425111-70-61-Heptanol 3.5126111-87-51-Octanol 3.9827143-08-81-Nonanol
4.4128112-30-11-Decanol 4.8229112-42-51-Undecanol
5.2230112-53-81-Dodecanol 5.273167-63-0Isopropanol 0.783278-92-22-Butanol
1.3133104-76-72-Ethyl-1-hexanol 3.663478-83-12-Methyl-1-propanol 1.693575-65-02-Methyl-2-propanol 1.0636600-36-22,4-Dimethyl-3-pentanol
2.853777-74-73-Methyl-3-pentanol 2.1838108-93-0Cyclohexanol 2.0639111-46-6Diethyleneglycol 0.1540107-21-1Ethyleneglycol 0.0441107-41-5Hexyleneglycol 1.134257-55-61,2-Propyleneglycol 0.13432216-51-5L -Menthol
3.924475-09-2Dichloromethane 2.424567-66-3Chloroform
3.064656-23-5Carbontetrachloride 3.5647107-06-21,2-Dichloroethane 2.94871-55-61,1,1-Trichloroethane 3.44979-00-51,1,2-Trichloroethane 3.215079-34-51,1,2,2-Tetrachloroethane 3.925176-01-7Pentachloroethane
4.435267-72-1Hexachloroethane
confidentiality5.235358-89-9Hexachlorocyclohexane
6.525478-87-51,2-Dichloropropane 2.9355142-28-91,3-Dichloropropane 2.945696-18-41,2,3-Trichloropropane 3.4157110-56-51,4-Dichlorobutane 3.3958628-76-21,5-Dichloropentane 3.7559627-30-53-Chloro-1-propanol 2.0760115-20-82,2,2-Trichloroethanol
英文名男
individualized2.86157-15-81,1,1-Trichloro-2-methyl-2-propanol
3.1262106-94-51-Bromopropane 3.2663
109-65-9
1-Bromobutane
3.57
L.He,P.C.Jurs/Journal of Molecular Graphics and Modelling23(2005)503–523505
Table1(Continued)
No.CAS#Name Exp. 64111-25-11-Bromohexane 4.68 65629-04-91-Bromoheptane 5.09 66111-83-11-Bromoocatne 5.36 67109-64-81,3-Dibromopropane 5.05 68107-10-8n-Propylamine 2.28 69109-73-9n-Butylamine 2.44 7033966-50-6Sec-butylamine 2.42 71110-58-7n-Pentylamine 2.69 72111-26-2n-Hexylamine 3.25 73111-68-2N-Heptylamine 3.72 74111-86-4n-Octylamine 4.4 75112-20-9n-Nonylamine 4.82 762016-57-1n-Decylamine 5.18 777307-55-3n-Undecylamine 2.91 78124-22-1n-Dodecylamine 6.26 792869-34-3n-Tridecylamine 6.45 8013952-84-62-Butanamine 2.42 815
98-74-31,2-Dimethylpropylamine 2.49 825813-64-92,2-Dimethyl-1-propylamine 2.26 8315673-00-43,3-Dimethylbutylamine 2.23 84107-45-9tert-Octylamine 3.72 85693-16-31-Methylheptylamine 4.4 86141-43-52-Aminoethanol 1.47 8778-96-61-Amino-2-propanol 1.47 88109-85-32-Methoxyethylamine 2.16 89109-89-7Diethylamine 1.93 90143-16-8Di-n-hexylamine 5.38 91110-73-62-(Ethylamine)ethanol 1.78 92100-37-8N,N-Diethylethanolamine 1.82 9396-80-02-(Diisopropylamino)-ethanol 2.86 94105-14-65-(Diethylamino)-2-pentanone 2.67 95102-69-2Tripropylamine 3.45 9691-65-6N,N-Diethylcyclohexylamine 3.86 97103-76-41-(2-Hydroxyethyl)piperazine 1.31 98140-31-8N-Aminoethylpiperazine 1.82 9978-90-01,2-Propanediamine 1.81 100107-15-3Ethylenediamine 2.55 10196-29-7Methylethylketoxime 2.01 102127-06-0Acetoneoxime 2.12 103100-64-1Cyclohexanoneoxime 2.74 104761-65-9N,N-Dibutylformamide 3.25 10568-12-2N,N-Dimethylformamide0.84 1061634-04-4Methyltert-butylether 2.12 107142-96-1Di-n-butylether 3.61 10860-29-7Diethylether 1.46 109108-20-3Diisopropylether 2.11 110693-65-2Di-n-pentylether 4.71 111109-87-5Dimethoxymethane 1.04 112123-91-11,4-Dioxane0.93 113110-88-3Trioxane 1.18 114109-99-9Tetrahydrofuran 1.52 115470-82-61,8-Epoxy-p-menthane 3.18 11664-19-7Aceticacid 2.85 117109-52-4n-Pentanoicacid 3.12 118142-62-1n-Hexanoicacid 2.76 119112-05-0n-Nonanoicacid 3.18 120124-04-9Adipicacid 3.18 12175-07-0Acetaldehyde 3.11 122123-72-8Butanal 3.65 123110-62-3Pentanal 3.82 12466-25-1Hexanal 3.6
6 125590-86-33-Methyl-butanal 4.42 126111-30-8Glutaraldehyde 3.94 127107-22-2Glyoxal 2.43 128141-78-6Ethylacetate 2.58Table1(Continued)
河南四级听力频道是多少No.CAS#Name Exp. 12979-20-9Methylacetate 2.27 130109-21-7n-Butyln-butyrate 4.09 131123-86-4n-Butylacetate 3.81 132540-88-5tert-Butylacetate 2.55 133142-92-7n-Hexylacetate 4.56 134123-66-0Ethylhexanoate 4.21 135109-60-4n-Propylacetate 3.23 136111-15-92-Ethoxyethylacetate 3.5 137108-59-8Dimethylmalonate 4.03 13887-91-2Diethyll-(+)-tartrate 2.5 139105-53-3Diethylmalonate 4.01 140123-25-1Diethylsuccinate 3.09 141141-28-6Diethyladipate 4.05 142110-40-7Diethylbacate 4.98 143105-99-7Dibutyladipate 4.85 144818-61-12-Hydroxyethylacrylate 4.38 145140-88-5Ethylacrylate 4.6 146106-63-8Isobutylacrylate 4.79 147999-61-12-Hydroxypropylacrylate 4.59 148868-77-92-Hydroxyethylmethacrylate 2.76 14980-62-6Methylmethacrylate 2.5 15075-05-8Acetonitrile 1.49 151107-12-0Propionitrile 1.56 1522243-27-8n-Octylcyanide 4.3 153764-13-62,5-Dimethyl-2,4-hexadiene 4.46 154513-81-52,3-Dimethyl-1,3-butadiene 4.08 1555194-50-32,4-Hexadiene 3.61 1565989-27-5d-Limonene 5.29 15777-73-6Dicyclopentadiene 3.63 1581647-16-11,9-Decadiene 5.68 15978-79-5Isoprene 2.95 16075-35-41,1-Dichloroethylene 2.84 16179-01-6Trichloroethylene 3.47 162127-18-4Tetrachloroethylene 3.91 163107-19-7Propargylalcohol 4.56 164818-72-41-Octyn-3-ol 5.49 165110-65-62-Butyne-1,4-di
ol 3.21 166764-01-22-Butyn-1-ol 3.84 16795-63-61,2,4-Trimethylbenzene 4.19 1682416-94-62,3,6-Trimethylphenol 4.22 169527-60-62,4,6-Trimethylphenol 4.02 17025167-83-3Tetrachlorophenol 6.13 17198-86-2Acetophenone 2.87 172100-51-6Benzylalcohol 2.37 173623-25-61,4-Bis(chloromethyl)benzene 6.65 174100-44-7Benzylchloride 4.4 175100-46-9Benzylamine 3.02 176100-52-7Benzaldehyde 3.93 177100-10-74-(Dimethylamino)-benzaldehyde 3.51 178122-03-24-Isopropylbenzaldehyde 4.35 179446-52-62-Fluorobenzaldehyde 4.96 180104-88-14-Chlorobenzaldehyde 4.81 181613-45-62,4-Demethoxybenzaldehyde 3.92 1821761-61-12-Hydroxy-5-bromobenzaldehyde 5.19 183635-93-82-Hydroxy-5-chlorobenzaldehyde 5.31 18490-02-82-Hydroxybenzaldehyde 4.73 185121-33-53-Methoxy-4-hyroxybenzaldehyde 3.12 186708-76-92-Hydroxy-4,6-dimethoxybenzaldehye 4.83 187653-37-2Pentafluorobenzaldehyde 5.25 188387-45-12-Chloro-6-fluorobenzaldehyde 4.23 189874-42-02,4-Dichlorobenzaldehyde 4.99 19058-90-22,3,4,6-Tetrachlorophenol 5.35 1914901-51-32,3,4,5-Tetrachlorophenol 5.74 1923481-20-72,3,5,6-Tetrachloroaniline 5.93 193732-26-32,4,6-tri-tert-Butylphenol 6.39
L.He,P.C.Jurs/Journal of Molecular Graphics and Modelling23(2005)503–523 506
Table1(Continued)
No.CAS#Name Exp. 194150-76-54-Methoxyphenol 3.27 195120-07-0N-Phenyldiethanolamine 2.39 196103-83-3N,N-Dimethylbenzylamine 3.55 197150-19-61-Hydroxy-3-methoxybenzene 3.22 198150-78-71,4-Dimethoxybenzene 3.07 1995673-07-42,6-Dimethoxytoluene 3.88 20013608-87-22,3,4-Trichloroacetophenone 5.05 20195-95-42,4,5-Trichlorophenol 5.34 20288-06-22,4,6-Trichlorophenol 4.64 203937-20-22,4-Dichloroacetophenone 4.21 20470-69-94-Aminopropiophenone 3.01 205102-27-2N-Ethyl-m-toluidine 3.44 206100-61-8N-Methylaniline 3.03 207121-69-7N,N-Dimethylaniline 3.27 20891-66-7N,N-Diethylaniline 3.96 20959-50-7P-Chloro-m-cresol 4.4 21024544-04-52,6-Diisopropylaniline 4.1 21110031-82-04-Ethoxybenzaldehyde 3.74 212634-93-52,4,6-Trichloroaniline 4.59 213634-67-32,3,4-Trichloroaniline 4.74 214615-65-62-Chloro-4-methylaniline 3.59 21571-43-2Benzene 3.5 2161746-23-2p-t-Butylstyrene 3.51 217100-42-5Styrene 3.51 2181745-81-92-Allylphenol 3.95 21997-53-0Eugenol 3.84 220108-86-1Bromobenzene 4.45 221608-71-9Pentabromophenol 6.72 222106-37-61,4-Dibromobenzene 5.28 223118-79-61,3,5-Tribromo2-hydroxybenzene 4.71 224106-40-14-Bromoaniline 3.56 225108-88-3Toluene 3.42 226108-90-7Chlorobenzene 3.7 227108-41-81-Chloro-3-methylbenzene 3.84 228106-43-41-Chloro-4-methylbenzene 4.33 22995-50-1o-Dichlorobenzene 4.19 230541-73-1m-Dichlorobenzene 4.27 231106-46-7p-Dichlorobenzene 4.27 23295-49-82-Chlorotoluene 4.23 23395-73-82,4-Dichlorotoluene 4.54 234120-82-11,2,4-Trichlorobenzene 4.8 23587-61-61,2,3-Trich
lorobenzene 4.89 236108-70-31,3,5-Trichlorobenzene 4.74 23795-94-31,2,4,5-Tetrachlorobenzene 5.83 238634-66-21,2,3,4-Tetrachlorobenzene 5.29 23987-86-5Pentachlorophenol 6.04 240771-60-8Pentafluoroaniline 3.69 241371-40-41-Amino-4-fluorobenzene 3.82 242128-37-02,6-Di-tert-butyl-4-methylphenol 5.78 243108-95-2Phenol 3.5 244105-67-92,4-Xylenol 3.86 24595-65-83,4-Xylenol 3.94 24695-75-03,4-Dichlorotoluene 4.74 2471126-79-0n-Butylphenylether 4.42 24839905-57-24-Hexyloxyaniline 4.78 249122-99-62-Phenoxyethanol 2.6 25095-48-7o-Cresol 3.9 251108-39-4m-Cresol 3.29 252106-44-5p-Cresol 3.76 25395-57-8o-Chlorophenol4 254106-48-9p-Chlorophenol 4.46 255120-83-22,4-Dichlorophenol 4.3 25662-53-3Aniline 3.03 25795-51-2o-Chloroaniline 4.35 258106-47-8p-Chloroaniline 3.62Table1(Continued)
No.CAS#Name Exp. 25995-76-13,4-Dichloroaniline 4.32 260554-00-72,4-Dichloroaniline 4.07 261106-49-04-Methylaniline 2.83 26295-47-6o-Xylene 3.81 263108-38-3m-Xylene 3.82 264106-42-3p-Xylene 4.21 2651689-84-51-Cyano-3,5-dibromo-4-hydroxybenzene 4.3 2665922-60-11-Cyano-2-amino-5-chlorobenzene 3.73 2676575-09-32-Cyano-6-methyl-benzonitrile4 268529-19-11-Cyano-2-methylbenzene 3.42 269100-47-0Benzonitrile 2.98 270100-41-4Ethylbenzene 3.59 271141-93-5m-Diethylbenzene 4.51 272123-07-9p-Ethylphenol 4.07 273589-16-24-Ethylaniline 3.22 274104-13-24-Butylaniline 4.17 27516245-79-74-Octylaniline 6.24 27637529-30-94-Decylanili
ne 6.57 27780-46-6p-tert-Amylphenol 4.8 27898-54-4p-tert-Butylphenol 4.47 27989-83-8Thymol 4.67 28098-82-8Cumene 4.23 281538-68-1n-Pentylbenzene 4.94 28225154-52-3Nonylphenol 6.22 283131-11-3Dimethylphthalate 3.7 28484-66-2Diethylphthalate 4.12 28584-69-5Diisobutylphthalate 5.49 28684-74-2Dibutylphthalate 5.33 28784-62-8Diphenylphthalate 6.6 28893-89-0Ethylbenzoate 4.23 2891129-35-7Methyl4-cyanobenzoate 3.54 29094-09-7Ethyl4-aminobenzoate 3.67 2911126-46-1Methyl4-chlorobenzoate 4.15 292368-77-43-(Trifluoromethyl)benzonitrile 3.56 293831-82-34-Phenoxyphenol 4.58 294119-61-9Benzophenone 4.11 295101-84-8Diphenylether 4.63 29614548-46-04-Benzoylpyridine 3.25 297122-39-4Diphenylamine 4.65 298118-55-8Phenylsalicylate 5.27 29997-23-42,20-Methylenebis(4-chloro)phenol 5.94 30091-94-13,30-Dichlorobenzidine 5.09 30192-52-4Biphenyl 4.8 30290-43-7o-Phenylphenol 4.5 303109-06-82-Methylpyridine 2.02 304108-99-63-Methylpyridine 2.81 305108-89-44-Methylpyridine 2.36 306100-70-92-Pyridinecarbonitrile 2.16 307500-22-13-Pyridinecarboxaldehyde 3.81 308104-90-55-Ethyl-2-methylpyridine 3.17 3091122-54-94-Acetylpyridine 2.86 3105683-33-02-Dimethylaminopyridine 2.98 311110-86-1Pyridine 2.9 3122859-67-83-Pyridinepropanol 2.96 3132176-62-7Pentachloropyridine 5.73 314939-23-14-Phenylpyridine 3.98 31591-20-3Naphthalene 4.32 31690-15-31-Naphthalenol 4.54 317135-19-32-Naphthol 4.62 31890-12-01-Methylnaphthalene 4.2 31990-13-11-Chloronaphthalene 4.85 32091-22-5Quinoline 3.45 321260-94-6Acridine 4.89 322253-52-1Phthalazine 3.11
2.1.Compounds
The majority of the data for this study came from the AQUIRE databa [4].Additional data were collected from the literature [5–7].The final datat included 322organic compounds with fathead minnow acute aquatic toxicity,expresd as:Àlog (mmol/L),as the activity of interest.A list of the 322compounds is prented in Table 1.The most toxicologically active compound in this datat was pentabromophenol with a toxicity of 6.72log units,and the least toxicologically active compound was ethylene glycol with a toxicity of 0.04log units.The compounds included in the datat were lected bad on the requirements of the ADAPT software package [8–11].The system handles only neutral organic compounds,but not charged species,and it supports atom types C,H,O,N,P halogens and sulfur compounds.All computations in this study were performed on a DEC 300AXP Model 500workstation with the ADAPT software package.2.2.Objects clustering
In the cluster analysis step,hierarchical cluster analysis was carried out using MATLAB for windows [12]and using a t of descriptors generated by the ADAPT software package.Descriptor generation will be described in detail later in this paper.The next few paragraphs describe the cluster algorithm ud in this study.关雎翻译
有道下载2.2.1.Summary of hierarchical clustering
The aim of the clustering was the recognition of groups of objects bad on their similarity.It involves grouping a collection of objects into clusters (subts),such that objects within each cluster are more cloly related to one another than objects in different clusters.In this study,hierarchical clustering was ud [13].In hierarchical clustering,the objects are clustered into subgroups in a ries of partitions.The basic process of hierarchical clustering starts by assigning each object as a cluster,so that N objects will form N clusters with each cluster containing just one item.Then the similarities between the clusters are determined by the distances (similarities)between the items they contain.The next step is to find the most similar (clost)pair of clusters and merge them into a new cluster,so that now the number of clusters becomes N –1.Then the distances (similarities)between the newly formed cluster and each of the old clusters are computed.The previous two steps are iterated until all items are clustered into a single cluster
containing N objects.Within hierarchical clustering,there are two subdivisions:agglomerative methods and divisive methods [13].The algorithm just described is an agglom-erative method,and it is the method implemented in this study.The divisive method works in the opposite way.It parates N objects into finer groups.Every object is initially in a huge cluster.Then the cluster is divided continuously until the desired number of clusters is formed.In brief,agglomerative method is a bottom
–up clustering method,whereas divisive method us a top –down approach.The graphical reprentation of the hierarchical clustering and agglomerative methods are prented in Figs.2and 3.2.2.1.1.Linkage methods.In hierarchical clustering,the distances (similarities)computation (distance calculation between clusters)can be done in different ways.This cluster fusion step is often called the linkage method,which is what distinguishes single linkage from average linkage and complete linkage clustering [13].In single linkage cluster-ing,the shortest distance from any object in one cluster to any object in the other cluster is considered as the distance between one cluster and another cluster.In average linkage clustering,the distance between two clusters is the average distance from any object in one cluster to any object in the other cluster.Complete linkage clustering considers the longest distance from any object in one cluster to any object in the other cluster.In this study,single linkage and average linkage clustering were ud in different parts of the study.Single linkage is one of the simplest agglomerative hier-archical clustering methods.The distance between clusters is given by the value of the shortest link between the clusters,illustrated in Fig.4.The distance D (x ,y )is computed as D ðx ;y Þ¼Min ðd ði ;j ÞÞ
where object i is in cluster x and object j is in cluster y .
L.He,P .C.Jurs /Journal of Molecular Graphics and Modelling 23(2005)503–523欢笑之歌
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Fig.3.Agglomerative method.
