Analysing microarray data in BioConductor From: http://bioinformatics.knowledgeblog.org/2011/06/20/analysing-microarray-data-in-bioconductor/ Introduction Microarray analysis is a complex procedure, and using R and the associated BioConductor packages can be a daunting task. In this tutorial we are going to show you how to install R and BioConductor, download a dataset from GEO , normalise it, do some quality control checks and finally analyse it to find differentially expressed genes. The following instructions assume that you are working with a fresh install of an Ubuntu based Linux distribution, and includes the installation of all the pre-requisites for the analysis. Some familiarity with Linux is ideal and the instructions were developed on Ubuntu 11.04, R 2.12.1. For a full code listing for this tutorial and figures resulting from it see the second part of the article . Installation Open up a terminal (Applications-Accessories-Terminal from the the toolbar) Install R and BioConductor dependencies, and then start R: ? 1 2 $ sudo apt-get install r-base-core libxml2-dev libcurl4-openssl-dev curl $ R Now at the R prompt: ? 1 2 3 4 5 6 # download the BioC installation routines source("http://bioconductor.org/biocLite.R") # install the core packages biocLite() # install the GEO libraries biocLite("GEOquery") If you are running as an unprivileged user you will be prompted to install the packages into a personal library. We’re also install the GEOquery package at this point. Installing the core BioConductor packages will take some time. GEOquery is an interface to the Gene Expression Omnibus at the NCBI, which holds transcriptomics data in a standardised format. Getting the data For this tutorial we are going to be working with dataset GSE20986 , generated by Dr Andrew Browning. HUVECs are cells often used in research, derived from human umbilical vein, used for investigations into the pathology and physiology of endothelial cells. In this experiment iris, retina and choroidal microvascular endothelial cells were isolated from donor eyes, and compared to HUVECs in order to ascertain if the HUVEC line is a suitable surrogate for studying ocular disorders. The GEO page also includes information on the design of the experiment. We can see that there are triplicate measurements for each set of samples. This splits the samples into four groups “iris” “retina” “HUVEC” and “choroidal”. The GEO Platform is GPL570 – the GEO short code for the Affymetrix Human Genome U133 Plus 2.0 Array , a commonly used microarray chip for human transcriptome studies. If we investigate the GSM links such as GSM524662 we get more detailed information about the experimental conditions used on each chip. They should be the same for each chip, as they are derived from the same experiment. What is captured is information about how the cell lines were extracted, grown, processed for RNA extraction etc. They also include details on how the RNA was processed and hybridised to the microarray chip, and what machine was used to scan the chips. This data is all captured to ensure compliance to MIAME standards. For each chip, the data table shows the probeset alongside a normalised expression value for the probeset. The table header description let you know how the data was normalised. In this case you can see the GC-RMA algorithm was used. For more information about GC-RMA you can read this paper . First of all we need to acquire the raw data directly from GEO. We can do this by loading the GEOquery library, and pointing it to the experiment we want to analyse. This download is approximately 53MB. ? 1 2 library(GEOquery) getGEOSuppFiles("GSE20986") If you open a file browser you will see this creates a GSE20986 directory under the directory you launched R in: ? 1 2 $ ls GSE20986/ filelist.txt GSE20986_RAW.tar The GSE20986_RAW.tar file is a compressed archive of the CEL files (the Affymetrix native file format). Before we can work on them we need to uncompress them. We can achieve this at the R prompt. This uncompresses the tar file (which contains compressed CEL files). The CEL files are subsequently decompressed themselves to the directory in which we are working with R. ? 1 2 3 4 untar("GSE20986/GSE20986_RAW.tar", exdir="data") cels - list.files("data/", pattern = " ") sapply(paste("data", cels, sep="/"), gunzip) cels Describing the experiment We are now ready to start analysis, however in order to do this we need to capture the experimental information. This is just a text file which describes the chip names, and the source of the biological samples hybridised to them. To load the data into R using simpleaffy we need to create a file to represent this information. Open a new terminal window and type: ? 1 $ ls data/*.CEL data/phenodata.txt Open this file in a text editor. Currently this is a single column listing the files just a list of the CEL files. The final file needs 3 tab-delimited columns named ‘Name’ ‘FileName’ and ‘Target’. The FileName and Name columns will be identical in this case, although the Name column can be used to provide a more human readable sample name if desired. The Target column is information about the samples taken from the GEO page. Label the samples appropriately either ‘iris’ ‘retina’ ‘choroid’ or ‘huvec’. This defines our replicate groups for further analysis when we compare the cell line with the tissue samples. The final result should look like this (ensure that there are tabs between the columns and not spaces!). You can produce this file in any spreadsheet program providing you save it as ‘tab-delimited text’. ? 1 2 3 4 5 6 7 8 9 10 11 12 13 Name FileName Target GSM524662.CEL GSM524662.CEL iris GSM524663.CEL GSM524663.CEL retina GSM524664.CEL GSM524664.CEL retina GSM524665.CEL GSM524665.CEL iris GSM524666.CEL GSM524666.CEL retina GSM524667.CEL GSM524667.CEL iris GSM524668.CEL GSM524668.CEL choroid GSM524669.CEL GSM524669.CEL choroid GSM524670.CEL GSM524670.CEL choroid GSM524671.CEL GSM524671.CEL huvec GSM524672.CEL GSM524672.CEL huvec GSM524673.CEL GSM524673.CEL huvec Loading and normalising the data The simpleaffy package provides routines for handling CEL files including normalisation and loading data with sample information. We are going to load the data into an R object called ‘celfiles’ ? 1 2 library(simpleaffy) celfiles - read.affy(covdesc="phenodata.txt", path="data") You can confirm this has worked, and download the chip annotations at the same time (if this is your first run) by typing: ? 1 2 3 4 5 6 7 8 celfiles AffyBatch object size of arrays=1164x1164 features (12 kb) cdf=HG-U133_Plus_2 (54675 affyids) number of samples=12 number of genes=54675 annotation=hgu133plus2 notes= Now we can normalise the data. As with the data deposited in GEO we are going to use the GC-RMA algorithm. If this is the first time you have run it, it will download additional files during the process. ? 1 2 3 4 5 6 7 celfiles.gcrma - gcrma(celfiles) Adjusting for optical effect............Done. Computing affinitiesLoading required package: AnnotationDbi .Done. Adjusting for non-specific binding............Done. Normalizing Calculating Expression If you want to see the data associated with the normalised object you can – you will notice this is no longer an AffyBatch object, but an ExpressionSet object, which holds normalised values. ? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 celfiles.gcrma ExpressionSet (storageMode: lockedEnvironment) assayData: 54675 features, 12 samples element names: exprs protocolData sampleNames: GSM524662.CEL GSM524663.CEL ... GSM524673.CEL (12 total) varLabels: ScanDate varMetadata: labelDescription phenoData sampleNames: GSM524662.CEL GSM524663.CEL ... GSM524673.CEL (12 total) varLabels: sample FileName Target varMetadata: labelDescription featureData: none experimentData: use 'experimentData(object)' Annotation: hgu133plus2 Quality control checks Before we analyse the data it is imperative that we do some quality control checks to make sure there are no issues with the dataset. The first thing we can do is check the effects of the GC-RMA normalisation, by plotting a boxplot of probe intensities before and after normalisation. ? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 # load colour libraries library(RColorBrewer) # set colour palette cols - brewer.pal(8, "Set1") # plot a boxplot of unnormalised intensity values boxplot(celfiles, col=cols) # plot a boxplot of normalised intensity values, affyPLM is required to interrogate celfiles.gcrma library(affyPLM) boxplot(celfiles.gcrma, col=cols) # the boxplots are somewhat skewed by the normalisation algorithm # and it is often more informative to look at density plots # Plot a density vs log intensity histogram for the unnormalised data hist(celfiles, col=cols) # Plot a density vs log intensity histogram for the normalised data hist(celfiles.gcrma, col=cols) From these plots we can conclude that there are no major deviations amongst the 12 chips, and normalisation has brought the intensities from all of the chips into distributions with similar characteristics. To take a closer look at the situation on a per-chip level we can use affyPLM . affyPLM allows us to visualise statistical characteristics of the CEL files. ? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 # Perform probe-level metric calculations on the CEL files: celfiles.qc - fitPLM(celfiles) # Create an image of GSM24662.CEL: image(celfiles.qc, which=1, add.legend=TRUE) # Create an image of GSM524665.CEL # There is a spatial artifact present image(celfiles.qc, which=4, add.legend=TRUE) # affyPLM also provides more informative boxplots # RLE (Relative Log Expression) plots should have # values close to zero. GSM524665.CEL is an outlier RLE(celfiles.qc, main="RLE") # We can also use NUSE (Normalised Unscaled Standard Errors). # The median standard error should be 1 for most genes. # GSM524665.CEL appears to be an outlier on this plot too NUSE(celfiles.qc, main="NUSE") We can also look at the relationships between the samples using heirarchical clustering: ? 1 2 3 4 eset - exprs(celfiles.gcrma) distance - dist(t(eset),method="maximum") clusters - hclust(distance) plot(clusters) This suggests our HUVEC samples are very much a separate group compared to the eye tissues, which show some evidence of clustering by tissue type. GSM524665.CEL does not appear to be a particular outlier in this view. Filtering data Now we have looked at the data, we can go on to analyse it. The first stage of analysis is to filter out uninformative data such as control probesets and other internal controls as well as removing genes with low variance, that would be unlikely to pass statistical tests for differential expression, or are expressed uniformly close to background detection levels. The modifiers to nsFilter below tell nsFilter not to remove probesets without Entrez Gene identifiers, or have duplicated Entrez Gene identifiers. ? 1 2 3 4 5 6 7 8 celfiles.filtered - nsFilter(celfiles.gcrma, require.entrez=FALSE, remove.dupEntrez=FALSE) # What got removed and why? celfiles.filtered$filter.log $numLowVar 27307 $feature.exclude 62 From this we conclude 27,307 probesets have been removed for reasons of low variance, and 62 control probesets have been removed as well. Finding differentially expressed probesets Now we have a filtered dataset, we can send the information to limma for differential gene expression analysis. First of all we need to extract information about the samples: ? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 samples - celfiles.gcrma$Target # check the results of this samples "iris" "retina" "retina" "iris" "retina" "iris" "choroid" "choroid" "choroid" "huvec" "huvec" "huvec" # convert into factors samples - as.factor(samples) # check factors have been assigned samples iris retina retina iris retina iris choroid choroid choroid huvec huvec huvec Levels: choroid huvec iris retina # set up the experimental design design - model.matrix(~0 + samples) colnames(design) - c("choroid", "huvec", "iris", "retina") # inspect the experiment design design choroid huvec iris retina 1 0 0 1 0 2 0 0 0 1 3 0 0 0 1 4 0 0 1 0 5 0 0 0 1 6 0 0 1 0 7 1 0 0 0 8 1 0 0 0 9 1 0 0 0 10 0 1 0 0 11 0 1 0 0 12 0 1 0 0 attr(,"assign") 1 1 1 1 attr(,"contrasts") attr(,"contrasts")$samples "contr.treatment" At this point we have normalised filtered data, and a description of the data and the samples and experimental design. This can be fed into limma for analysis. ? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 library(limma) # fit the linear model to the filtered expression set fit - lmFit(exprs(celfiles.filtered$eset), design) # set up a contrast matrix to compare tissues v cell line contrast.matrix - makeContrasts(huvec_choroid = huvec - choroid, huvec_retina = huvec - retina, huvec_iris - huvec - iris, levels=design) # check the contrast matrix contrast.matrix Contrasts Levels huvec_choroid huvec_retina huvec_iris choroid -1 0 0 huvec 1 1 1 iris 0 0 -1 retina 0 -1 0 # Now the contrast matrix is combined with the per-probeset linear model fit. huvec_fits - contrasts.fit(fit, contrast.matrix) huvec_ebFit - eBayes(huvec_fits) # return the top 10 results for any given contrast # coef=1 is huvec_choroid, coef=2 is huvec_retina topTable(huvec_ebFit, number=10, coef=1) ID logFC AveExpr t P.Value adj.P.Val 6147 204779_s_at 7.367947 4.171874 72.77016 3.290669e-15 8.985500e-11 7292 207016_s_at 6.934796 4.027229 57.37259 3.712060e-14 5.068076e-10 8741 209631_s_at 5.193313 4.003541 51.22423 1.177337e-13 1.071612e-09 26309 242809_at 6.433514 4.167462 48.52518 2.042404e-13 1.394247e-09 6828 205893_at 4.480463 3.544350 40.59376 1.253534e-12 6.845801e-09 20232 227377_at 3.670688 3.209217 36.03427 4.200854e-12 1.911809e-08 6222 204882_at -5.351976 6.512018 -34.70239 6.154318e-12 2.400711e-08 27051 38149_at -5.051906 6.482418 -31.44098 1.672263e-11 5.282592e-08 6663 205576_at 6.586372 4.139236 31.31584 1.741131e-11 5.282592e-08 6589 205453_at 3.623706 3.210306 30.72793 2.109110e-11 5.759136e-08 B 6147 20.25563 7292 19.44681 8741 18.96282 26309 18.70767 6828 17.75331 20232 17.01960 6222 16.77196 27051 16.08854 6663 16.05990 6589 15.92277 To impose a fold change cut off, and see how many genes are returned you can use the lfc modifier for topTable, here we show the results for fold changes of 5,4,3 and 2 in terms of the number of probesets. ? 1 2 3 4 5 6 7 8 9 10 nrow(topTable(huvec_ebFit, coef=1, number=10000, lfc=5)) 88 nrow(topTable(huvec_ebFit, coef=1, number=10000, lfc=4)) 194 nrow(topTable(huvec_ebFit, coef=1, number=10000, lfc=3)) 386 nrow(topTable(huvec_ebFit, coef=1, number=10000, lfc=2)) 1016 # Get a list for probesets with a four fold change or more probeset.list - topTable(huvec_ebFit, coef=1, number=10000, lfc=4) Annotating the results with associated gene symbols In order to annotate the probesets into gene symbols we need to install and load the associated database package and the annotate package, then we can extract the probeset ID’s from the topTable results, and match the symbols ? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 biocLite("hgu133plus2.db") library(hgu133plus2.db) library(annotate) gene.symbols - getSYMBOL(probeset.list$ID, "hgu133plus2") results - cbind(probeset.list, gene.symbols) head(results) ID logFC AveExpr t P.Value adj.P.Val 6147 204779_s_at 7.367947 4.171874 72.77016 3.290669e-15 8.985500e-11 7292 207016_s_at 6.934796 4.027229 57.37259 3.712060e-14 5.068076e-10 8741 209631_s_at 5.193313 4.003541 51.22423 1.177337e-13 1.071612e-09 26309 242809_at 6.433514 4.167462 48.52518 2.042404e-13 1.394247e-09 6828 205893_at 4.480463 3.544350 40.59376 1.253534e-12 6.845801e-09 6222 204882_at -5.351976 6.512018 -34.70239 6.154318e-12 2.400711e-08 B gene.symbols 6147 20.25563 HOXB7 7292 19.44681 ALDH1A2 8741 18.96282 GPR37 26309 18.70767 IL1RL1 6828 17.75331 NLGN1 6222 16.77196 ARHGAP25 write.table(results, "results.txt", sep="\t", quote=FALSE) There is a follow-on article from Simon Cockell on analyising the biological significance of results and another article by Colin Gillespie on visualising data from expression analysis as volcano plots.
Non-stochastic differential equations are models of dynamical systems where the state evolves continuously in time. If they are autonomous, then the state's future values depend only on the present state; if they are non-autonomous, it is allowed to depend on an exogeneous driving term as well. (This may not be the standard way of putting it, but I think it's both correct and more illuminating than the more analytical viewpoints, and anyway is the line taken by V. I. Arnol'd in his book on differential equations.) Stochastic differential equations (SDEs) are, conceptually, ones where the the exogeneous driving term is a stochatic process . --- While differential equation, unmodified, covers both ordinary differential equations, containing only time derivatives, and partial differential equations, containing both time and space derivatives, stochastic differential equation, unmodified, refers only to the ordinary case. Stochastic partial differential equations are just what you'd think. The solution of an SDE is, itself, a stochastic process. Heuristically, the easiest way to think of how this works is via Euler's method for solving differential equations, which is itself about the simplest possible numerical approximation scheme for an ODE. (This line of thought was apparently introduced by Bernstein in the 1920s.) To solve dx/dt = f(x), with initial condition x(0) = y, Euler's method instructs us to pick a small increment of time h, and then say that x(t+h) = x(t) + hf(x), using straight-line interpolation between the points 0, h, 2h, 3h,... Under suitable conditions on vector field f, as h shrinks, the function we obtain in this way actually converges on the correct solution. To accomodate a stochastic term on the right-hand side, say dx/dt = f(x) + E(t), where E(t) is random noise, we approximate x(t+h) - x(t) by hf(x) + E(t+h) - E(t). Then, once again, we let the time-increment shrink to zero. Doing this with full generality requires a theory of the integrals of stochastic processes, which is made especially difficult by the fact that many of the stochastic forces one would most naturally want to use, such as white noise, are ones which don't fit very naturally into differential equations! The necessary theory of stochastic integrals was developed in the 1940s by M. Loeve, K. Ito, and R. Stratonovich (all building on earlier work by, among others, N. Wiener ); the theory of SDEs more strictly by Ito and Stratonovich, in slightly different forms. Most of what one encounters, in applications, as the theory of SDEs assumes that the driving noise is in fact white, i.e., Gaussian and uncorrelated over time. On the one hand, this is less of a restriction than it might seem, because many other natural sorts of noise process can be represented as stochastic integrals of white noise. On the other hand, the same mathematical structure can be used directly to define stochastic integrals and stochastic DEs driven by a far broader class of stochastic processes; on this topic Kallenberg is a very good introduction. Recommended, more introductory: Geoffrey Grimmett and David Stirzaker, Probability and Random Processes Josef Honerkamp, Stochastic Dynamical Systems Joel Keizer, Statistical Thermodynamics of Nonequilibrium Processes Andrzej Lasota and Michael C. Mackey, Chaos, Fractals and Noise: Stochastic Aspects of Dynamics Robert S. Liptser , Lectures on Stochastic Processes Bernt Oksendal, Stochastic Differential Equations Recommended, more advanced: I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes Olav Kallenberg, Foundations of Modern Probability Robert S. Liptser and Albert N. Shiryaev, Statistics of Random Processes Michel Loeve, Probability Theory L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales Modesty forbids me to recommend: CRS with A. Kontorovich, Almost None of the Theory of Stochastic Processes To read: Lakhdar Aggoun and Robert Elliott, Measure Theory and Filtering: Introduction with Applications David Applebaum, Lvy Processes and Stochastic Calculus Yuri Bakhtin and Jonathan C. Mattingly, Stationary Solutions of Stochastic Differential Equation with Memory and Stochastic Partial Differential Equations, math.PR/0509166 Viorel Barbu, Philippe Blanchard, Giuseppe Da Prato, Michael Rckner, Self-organized criticality via stochastic partial differential equations, arxiv:0811.2093 Nicolas Bouleau and Dominique Lpngle, Numerical Methods for Stochastic Process A. A. Budini and M.O. Caceres, Functional characterization of generalized Langevin equations, cond-mat/0402311 Emmanuelle Clment, Arturo Kohatsu-Higa, Damien Lamberton, A duality approach for the weak approximation of stochastic differential equations, math.PR/0610178 = Annals of Applied Probability 16 (2006): 1124--1154 Jacky Cresson and Sbastien Darses, Stochastic embedding of dynamical systems, math.PR/0509713 A. M. Davie, Uniqueness of solutions of stochastic differential equations, arxiv:0709.4147 Hartin Hairer, Exponential Mixing Properties of Stochastic PDEs Through Asymptotic Coupling, math.PR/0109115 David Hochberg, Carmen Molina-Paris, Juan Prez-Mercader and Matt Visser, Effective Action for Stochastic Partial Differential Equations, cond-mat/9904215 Helge Holden, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach Yoshifusa Ito and Izumi Kubo, Calculus on Gaussian and Poisson White Noises, Nagoya Mathematical Journal 111 (1988): 41--84 Gopinath Kallianpur and Jie Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces Karatzas and Shreve, Brownian Motion and Stochastic Calculus Peter Kotelenez, Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations Venkatarama Krishnan, Nonlinear Filtering and Smoothing: An Introduction to Martingales, Stochastic Integrals and Estimation H. Kunita, Stochastic Flows and Stochastic Differential Equations S. V. Lototsky and B. L. Rozovskii Wiener Chaos Solutions of Linear Stochastic Evolution Equations, math.PR/0504558 Stochastic Differential Equations: A Wiener Chaos Approach, math.PR/0504559 Jonathan C. Mattingly, Andrew M. Stuart, M.V. Tretyakov, Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations, arxiv:0908.4450 Anatolii V. Mokshin, Renat M. Yulmetyev, and Peter Hnggi, Simple Measure of Memory for Dynamical Processes Described by a Generalized Langevin Equation, Physical Review Letters 95 (2005): 200601 Esteban Moro and Henri Schurz, Non-negativity preserving numerical algorithms for stochastic differential equations, math.NA/0509724 Cyril Odasso, Exponential mixing for stochastic PDEs: the non-additive case, Probability Theory and Related Fields 140 (2008): 41--82 Fabien Panloup, Recursive computation of the invariant measure of a stochastic differential equation driven by a L\'{e}vy process, math.PR/0509712 S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lvy Noise: An evolution Equation approach Philip Protter, Stochastic Integration and Differential Equations A. J. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, math.DS/0701623 Ramon van Handel, Almost Global Stochastic Stability, math.PR/0411311 Wei Wang and Jinqiao Duan, Invariant manifold reduction and bifurcation for stochastic partial differential equations, math.DS/0607050 http://www.cscs.umich.edu/~crshalizi/notebooks/stoch-diff-eqs.html
仅限学术研究使用，严禁商业用途，作者和出版社如有异议，我立即删除附件。如果觉得本书比较好，请购买正版图书。也欢迎各位博友讨论本书内容。欢迎学术交流。 Introduction In recent years, functional methods have become central to the study of theoretical and applied mathematical problems. An advantage of such an approach is its generality and its potential unifying effect of particular results and techniques. Functional analysis emerged as an independent discipline in the first half of the 20th century, primarily as a result of contributions of S. Banach, D. Hilbert, and F. Riesz. Significant advances have been made in different fields, such as spectral theory, linear semigroup theory (developed by E. Hille, R.S. Phillips, and K. Yosida), the variational theory of linear boundary value problems, etc. At the same time, the study of nonlinear physical models led to the development of nonlinear functional analysis. Today, this includes various independent subfields, such as convex analysis (where H. Brezis, J.J. Moreau, and R.T. Rockafellar have been major contributors), the Leray-Schauder topological degree theory, the theory of accretive and monotone operators (founded by G. Minty, F. Browder, and H. Brezis), and the nonlinear semigroup theory (developed by Y. Komura, T. Kato, H. Brezis, M.G. Grandall, A. Pazy, etc.). As a consequence, there has been significant progress in the study of nonlinear evolution equations associated with monotone or accretive operators (see, e.g., the monographs by H. Brezis , and V. Barbu ). The most important applications of this theory are concerned with boundary value problems for partial differential systems and functional differential equations, including Volterra integral equations. The use of functional methods leads, in some concrete cases, to better results as compared to the ones obtained by classical techniques. In this context, it is essential to choose an appropriate functional framework. As a byproduct of this approach, we will sometimes arrive at mathematical models that are more general than the classical ones, and better describe concrete physical phenomena; in particular, we shall reach a concordance between the physical sense and the mathematical sense for the solution of a concrete problem. The purpose of this monograph is to emphasize the importance of functional methods in the study of a broad range of boundary value problems, as well as that of various classes of abstract differential equations. Functional Methods in Differential Equations / Veli-Matti Hokkanen Gheorghe Morosanu / A CRC Press Company
Differential Evolution (DE) for Continuous Function Optimization (an algorithm by Kenneth Price and Rainer Storn) 发布了很多关于差分进化的资源和研究历程。 http://www.icsi.berkeley.edu/~storn/code.html
标签: Differential
