Optimizing a multivariable function parameters using a random method, genetic algorithm and simulated annealing in R

Say that you are implementing a non-linear regression analysis, which is shortly described by wikipedia as:

"In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables."

For the training set, we have the following:

And the function to optimize the parameters is:

Which leads us to the following equality:

In other words, we want to optimize the value of theta in order to minimize the sum of the error among y and predicted.y:

Given theta (each parameter a0,..a3 has a range from 0 to 15):

And the error function:

finally, the goal function:

In other words, the goal function searches for the value of theta that minimizes the error.

COMPUTATIONS BEGIN

This is the scatter plot of the training set:

Here is the implementation in R, you can download the file clicking here

Here is a result plot using the genetic algorithm:

Benjamin

Lieutenant Dan You Got a New Interface..

After some days of thinking, I realize that this blog deserved a little bit more attention, so I decided to change the interface and I'am happy about how it looks now.

Hope to see you again in my personal blog.

keep on programming!

Benjamin

Accuracy versus F score: Machine Learning for the RNA Polymerases

Hello, today I'm going to show you the difference of using two different common performance measures (useful not only for Machine Learning purposes, is useful in every scientific field). Until now, I have found more the accuracy values than F scores in the performance measuring of some methods which ranges from metaheuristics (Genetic Algorithms fitness functions) to promoter recognition programs, diagnose methods and so on.

But, I would really recommend to avoid using the accuracy measure. The reason is shown below with a nice example in R programming language (all the functions used in the simulation are included,  you can download them clicking here).

Case study 1:

Imagine that you are in a Computer Vision project and your task is to "teach" a program to recognize among  electric guitars and acoustic guitars showing the program pictures of different guitars.

Suppose that you've already developed that program and now you want to measure the performance of this Boolean classifier (this is for example, you show the program a picture of a an electric guitar, and the program has to decide whether it will recognize and "classify" it as an electric or as an acoustic guitar).

For the function of this post, lets write down some useful concepts

Consider the following:

TP: a true positive is when the program classifies an electric guitar as an electric guitar, we will use the letter "E" to denote the electric guitar "class"

FP: a false positive is when the program classifies an acoustic guitar as an electric guitar, we will use the letter "A" to denote the acoustic guitar "class"

FN: a false negative is when the program classifies an electric guitar as an acoustic guitar

TN: a true negative is when the program classifies an acoustic guitar as an acoustic guitar

Now that we are ready, we shall begin with the calculations

In R, I have simulated the results of the program. Say, for 1,000 electric guitar pictures and 1,000 acoustic guitar pictures

The program prompt the following results:

        PREDICTED.E PREDICTED.A
TRUE.E         485         515
TRUE.A           9         991

If you notice, from the 1000 electric guitar pictures, only 485 were labeled as electric (TP=485), the rest were labeled as acoustic (FN=515). I feel bad for the hypothetical programmer of this hypothetical example.

On the other hand, from the 1000 acoustic guitars, 991 were labeled as acoustic (TN=991) and only 9 of them were labeled as electric (FP=9). Well not bad!..... or it is?

The accuracy value of this program is = 0.738

And, for computing the F score is necessary to compute the precision and the recall first, where:

precision = 0.9817814 and recall = 0.485

Then, the F score is equal to 0.6492637

Well, the F scores seems to be more "strict", and in fact it is in comparison of the accuracy performance measure. But this example is not very "cool". Lets pass to the case study 2

Case study 2:

Now we have 1,000 electric guitar pictures and 100,000 acoustic guitar pictures, the confusion matrix of the results are:

        PREDICTED.E PREDICTED.A
TRUE.E         493         507
TRUE.A        1017       98983

If you notice, from the 1,000 electric guitar pictures, only 493 were labeled as electric (TP=493), the rest were labeled as acoustic (FN=507)

On the other hand, from the 100,000 acoustic guitars, 98983 were labeled as acoustic (TN=98983) and only 1017 of them were labeled as electric (FP=1017)

Now (cha cha chan!), the performance values are:

Accuracy: 0.9849109
Precision: 0.3264901
Recall: 0.493
F score: 0.3928287

Now you see it?, how come or how is possible that missing almost the 50% of the labels of the electric guitars, the performance of the program in accuracy is almost 0.99?, despite of having a precision and recall not greater than 0.50. Then we have a winner and is the F score measure.

for references visit the following pages:

http://en.wikipedia.org/wiki/Accuracy
http://en.wikipedia.org/wiki/F1_score
http://en.wikipedia.org/wiki/Precision_and_recall

Make your LaTex presentations using the Beamer class

Hello, I have been working with this tutorial. All you need to do is download the source files and take a look at the source code and the PDF file.

Download link here

Hope you find LaTex as useful as I do

Benjamin

Póster presentado en la XIV Escuela de Otoño en Biología Matemática, México

Póster presentado en la XIV Escuela de Otoño en Biología Matemática - 8vo Encuentro de Biología Matemática celebrado en San Luis Potosí, S.L.P, México:

"Predicción de promotores RNA POL-II en Drosophila melanogaster utilizando propiedades de señal, contexto y estructura a partir de secuencias nucleotídicas"

Póster disponible aquí

Brownian motion simulation in R

I have based this post on a very useful piece of code which basically is the core of my own implementation of a Brownian Motion simulation in R. The original reference code <- http://landshape.org/enm/r-code-for-brownian-motion/

According to Wikipedia the mathematical model for Brownian motion (also known as random walks) can also be used to describe many phenomena as well as the random movements of minute particles,
such as stock market fluctuations and the evolution of physical characteristics in the fossil record. The simple form of the mathematical model for Brownian motion has the form:

S_t = eS_t-1

where e is drawn from a probability distribution.

The source code is here

After loading the source code, there are two functions:

The first one, brownian will plot in an R graphics window the resulting simulation in an animated way.

The second function, export.brownian will export each step of the simulation in independent PNG files.

Example of running:

> source("brownian.motion.R")
> brownian(500)

The second function will produce this output
> export.brownian(500)
CODE:

# *******************************
# BROWNIAN MOTION SIMULATION
# December 2012 | Benjamin Tovar
# *******************************
#
#   REFERENCES
#   http://landshape.org/enm/r-code-for-brownian-motion/
#
#   According to Wikipedia the mathematical model for Brownian motion 
#   (also known as random walks) can also be used to describe many 
#   phenomena as well as the random movements of minute particles, 
#   such as stock market fluctuations and the evolution of physical 
#   characteristics in the fossil record. The simple form of the 
#   mathematical model for Brownian motion has the form:
#
#    S_t = eS_t-1
#
#    where e is drawn from a probability distribution.
#
#######################################################################

brownian <- function(n.times){
    x <- y <- x.new <- y.new <- x.new.p <- y.new.p <- vector()
    for(i in 1:n.times){
        # Initialize variables
        x <- rnorm(1)
        y <- rnorm(1)
        # concatenate variables 
        # to increase the vector size
        x.new <- c(x.new,x)
        y.new <- c(y.new,y)
        # sum the vector numbers
        x.new.p <- cumsum(x.new)
        y.new.p <- cumsum(y.new)  
        # plot the model
        plot(x.new.p,y.new.p,type="b",
             main=paste("Brownian motion simulation in R\nTime =",i,sep=" "),
             xlab="x coordinates",ylab="y coordinates",
             col=c(rep("gray",i-1),"red"),
             pch=c(rep(20,i-1),1))    
    }
}

# Test the function
# brownian(500)

# ****************************************
# EXPORT BROWNIAN MOTION SIMULATION IMAGES
# ****************************************

export.brownian <- function(n.times){
    x <- y <- x.new <- y.new <- x.new.p <- y.new.p <- vector()
    for(i in 1:n.times){
        # Initialize variables
        x <- rnorm(1)
        y <- rnorm(1)
        # concatenate variables to increase the
        # vector size
        x.new <- c(x.new,x)
        y.new <- c(y.new,y)
        # sum the vector numbers
        x.new.p <- cumsum(x.new)
        y.new.p <- cumsum(y.new)  
        # plot the model
        png(paste("image",i,"png",sep="."),width=600,height=600)
            plot(x.new.p,y.new.p,type="b",
                 main=paste("Brownian motion simulation in R\nTime =",
                 i,sep=" "),
                 xlab="x coordinates",ylab="y coordinates",
                 col=c(rep("gray",i-1),"red"),
                 pch=c(rep(20,i-1),1))
                 cat("image",i,"DONE",date(),"\n")
        dev.off()
    }
}

# Test the function
# export.brownian(500)

Extracting upstream regions of a RefSeq human gene list in R using Bioconductor

Suppose that you want to do local mapping of upstream regions of a given RefSeq IDs in a particular genome in R using Bioconductor. Download the script here.

In this case, you may take a look at the Bioconductor AnnotationData Packages here: http://www.bioconductor.org/packages/release/data/annotation/

The goal of this post is that for example I have the following RefSeq IDs and want to extract 250 bases upstream of each gene in a single list with another useful information such the entrez.id, the symbol and the gene description.

# RefSeqs IDs:
gene.list.refseq <- c("NM_003588","NM_001145436", "NM_001135188","NM_020760","NM_173362", "NM_198393","NM_022736","NM_025074","NM_033449","NM_015726", "NM_022110","NM_016478","NM_020634","NM_002291","NM_000418", "NM_001862","NM_017752","NM_006591","NM_000124","NM_144610")

# How many bases upstream of each gene:    
bases.upstream <- 250

Before starting, please download the following packages in R:

       source("http://bioconductor.org/biocLite.R")
       biocLite("BSgenome.Hsapiens.UCSC.hg19")                              
       biocLite("Biostrings")
       biocLite("org.Hs.eg.db")

Load the function using:

source("extract.five.utr.sequence.R")


And finally make the computations:

output.sequences <- extract.five.utr.sequence(gene.list.refseq,bases.upstream)

CODE:

# ******************************************************************************
#       FUNCTION: extract.five.utr.sequence | 29/JULY/2012 | BENJAMIN TOVAR
# ******************************************************************************
extract.five.utr.sequence <- function(gene.list.refseq,
                                      number.bases.upstream=1000){
    
    
    # *****************************************************************
    # RUNNING NOTES: Please download this packages from Bioconductor
    # http://www.bioconductor.org/packages/release/data/annotation/
    # ***************************************************************** 
    #   source("http://bioconductor.org/biocLite.R")
    #   biocLite("BSgenome.Hsapiens.UCSC.hg19")                                
    #   biocLite("Biostrings")
    #   biocLite("org.Hs.eg.db")
                                      
    # *****************************************************************
    # RUNNING EXAMPLE
    # *****************************************************************                              
    #
    ## Extract 250 bases upstream of each gene in gene.list.refseq 
    #
    # gene.list.refseq <- c("NM_003588","NM_001145436", "NM_001135188","NM_020760","NM_173362",   
    #                    "NM_198393","NM_022736","NM_025074","NM_033449","NM_015726",   
    #                    "NM_022110","NM_016478","NM_020634","NM_002291","NM_000418",   
    #                    "NM_001862","NM_017752","NM_006591","NM_000124","NM_144610") 
    #
    # bases.upstream <- 250
    #  
    # result <- extract.five.utr.sequence(gene.list.refseq,bases.upstream)                             
                             
    # *****************************************************************
    # LOAD THE LIBRARIES
    # ***************************************************************** 
    cat("Loading libraries",date(),"\n")   
    # human genome DNA sequences
        library(BSgenome.Hsapiens.UCSC.hg19) 
    # human genome wide annotations 
        library(org.Hs.eg.db) 
    # load IDs, symbol and gene descriptions
        refseq.id <- toTable(org.Hs.egREFSEQ)
        symbol <- toTable(org.Hs.egSYMBOL)
        gene.description <- toTable(org.Hs.egGENENAME)
    
    # *****************************************************************
    # LOAD THE UPSTREAM SEQUENCES
    # *****************************************************************

    if(number.bases.upstream <= 1000){
        # load the 1000 bases upstream of all genes in the human genome 
        # in the BSgenome.Hsapiens.UCSC.hg19 package
        upstream <- Hsapiens$upstream1000
    }
    
    if(number.bases.upstream > 1000 && number.bases.upstream <= 2000){
        # load the 1000 bases upstream of all genes in the human genome 
        # in the BSgenome.Hsapiens.UCSC.hg19 package
        upstream <- Hsapiens$upstream2000
    }
    
    if(number.bases.upstream > 2000 && number.bases.upstream <= 5000){
        # load the 1000 bases upstream of all genes in the human genome 
        # in the BSgenome.Hsapiens.UCSC.hg19 package
        upstream <- Hsapiens$upstream5000
    }        
    
    if(number.bases.upstream > 5000){
        cat("ERROR: The number of bases upstream of 5'UTR is too large (max value = 5000)\n")
        return(0);
    } 
    
    if(number.bases.upstream < 1){
        cat("ERROR: The number of bases upstream of 5'UTR cannot be less than 1 nucleotide (min value = 1)\n")
        return(0);
    }                   
        
    # extract the names of each gene
    names.genes.upstream <- names(upstream)

    # extract the refseq of each gene  
    refseq.upstream.id <- vector("character",length(names.genes.upstream))
    for(i in 1:length(names.genes.upstream)){
        refseq.upstream.id[i] <- gsub(", ","_",toString(unlist(strsplit(names.genes.upstream[i],
                                      "\\_"))[c(1,2)],sep=""))
    }    
    names(refseq.upstream.id) <- 1:length(refseq.upstream.id)

    # *****************************************************************
    # CREATE OUTPUT OBJECT
    # *****************************************************************    
    output.params <- c("entrez.id","symbol","gene.description","five.UTR.sequence")
                        
    output.list <- list()
    for(i in 1:length(gene.list.refseq)){
        output.list[[i]] <- list()
        for(j in 1:length(output.params)){
            output.list[[i]][[j]] <- list()
        }
        names(output.list[[i]]) <- output.params
    }
    names(output.list) <- gene.list.refseq

    # *****************************************************************
    # LET THE COMPUTER DECIDE
    # *****************************************************************     
    cat("Extracting data",date(),"\n")
    
    for(i in 1:length(gene.list.refseq)){
        cat(rep(".",1))

        # extract the entrez.id of each gene in geneList 
        output.list[[i]]$entrez.id <- refseq.id[which(refseq.id$accession==gene.list.refseq[i]),]$gene_id

        # extract the symbol of each gene in geneList 
        output.list[[i]]$symbol <- symbol[which(symbol==output.list[[i]]$entrez.id),]$symbol
        
        # extract the gene description of each gene in geneList 
        output.list[[i]]$gene.description <- gene.description[which(gene.description==output.list[[i]]$entrez.id),]$gene_name  
        
        # Return the index of the target genes
        index.target.gene <- sequence <- NA;
        index.target.gene <- as.numeric(names(refseq.upstream.id[which(refseq.upstream.id==gene.list.refseq[i])]))
        
        # NOTE: some genes have repeated RefSeq.id which corresponds to the same gene in other Chromosomes.
        # by using index.target.gene[1] we are selecting only the first RefSeq entry.
        sequence <- upstream[index.target.gene[1]]
        sequence.length <- nchar(sequence)
        start.position <- ((sequence.length-number.bases.upstream)+1)
        sequence <- substring(toString(sequence),1:sequence.length,1:sequence.length)
        sequence <- sequence[start.position:sequence.length]
        output.list[[i]]$five.UTR.sequence <- sequence
    }
    cat("\n")
    cat("Computations DONE",date(),"\n")
    return(output.list)
}

Benjamin.