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SVM classification example with performance measures using R caret

The caret package (short for Classification And REgression Training)

The caret package (short for Classification And REgression Training)

This example is a followup of hyperparameter tuning using the e1071 package in R. This time we’re using the SVM implementation from the R caret package, a binary class classification problem and some extended features that come in handy for many classification problems. For an easy start with caret take a look at one of the many presentations and intros to caret (like this one by Max Kuhn, maintainer of caret).

Desirable features

  • Hyperparameter tuning using grid search
  • Multicore processing for speedup
  • Weighting of samples to compensate for unequal class sizes (not possible with all classifiers, but possible with SVM)
  • Classifier outputting not only predicted classes but prediction probabilities. These can be used for extended performance measures (e.g. ROC curves)

Dataset

We use 2 out of the 3 classes from the standard R iris dataset (the versicolor and virginica classes). We further restrict ourselves to use only the first two features (Sepal.Length and  Sepal.Width) to make well performing classification a bit more difficult. If we would use all features classification would be perfect, therefore performance measures would be rather boring.

> head(iris[order(iris$Species, decreasing = T),])

    Sepal.Length Sepal.Width Petal.Length Petal.Width   Species
101          6.3         3.3          6.0         2.5 virginica
102          5.8         2.7          5.1         1.9 virginica
103          7.1         3.0          5.9         2.1 virginica
104          6.3         2.9          5.6         1.8 virginica
105          6.5         3.0          5.8         2.2 virginica
106          7.6         3.0          6.6         2.1 virginica

Concepts

We first split our data into a training and a test partition. Training data will be used for cross validation (CV) hyperparameter tuning and determining+building a model from the best hyperparameters. Test data will be used exclusively to do performance measures. Second, we create weights for our data samples so that each sample is weighted according to it’s corresponding class size (compensates for smaller and bigger class sizes). Samples of bigger classes will be assigned smaller weights and vice versa. As in our problem classes are exactly of equal size, weights will be equal for all samples too.

Next we search for best hyperparameters and create the according model using caret train. For preprocessing we let train auto scale and center our training data. There exist many other, well known and well working approaches for preprocessing, even implemented in caret (e.g. using PCA or IDA) – but we leave them out for the moment. For the parameter grid to be searched we use exponential parameter series. This way the level of detail gets adjusted along with the parameter size (be careful where to arrange for more and less details) and huge parameter spaces can be covered (with coarse detail) with the first run already. For cross validation we use 10 partitions (performance measures internally of caret will be computed as average partition performance). We allow parallel processing – and request train to create the classification model so that it computes prediction probabilities along with predicted classes.

After having found our model parameters and having created our model we let it predict our test data classes. We do that a) with predicting classes to obtain a confusion matrix and b) with prediction probabilities to create a ROC curve. For the ROC curve we have to define one of our classes as the positive class (we choose versicolor here) and the other (virginica) as the negative class. Finally, we extract the ROC curve points of equal error rate (EER), maximum accuracy, maximum kappa and minimum squared error rate (MSER). They all together give you an idea of how well your model is performing on new data.

Script code

# Example of SVM hyperparameter tuning and performance measures for a binary class problem with the caret package
# Rainhard Findling
# 2014 / 07
# 
library(caret)
library(plyr)
library(ROCR)

# optional - enable multicore processing
library(doMC)
registerDoMC(cores=4)

data(iris)
# create binary problem: only use two classes
x <- iris[,1:2][iris$Species %in% c('versicolor', 'virginica'),]
y <- factor(iris[,5][iris$Species %in% c('versicolor', 'virginica')])

# we see that data will probably not be perfectly seperable using linear separation
featurePlot(x = x, y = y)

# split into train and test data. train = cv parameter grid search and model creation, test = performance analysis
set.seed(123456)
indexes_y_test <- createDataPartition(y = 1:length(y), times = 1, p = 0.3)[[1]]

# creation of weights - also fast for very big datasets
weights <- as.numeric(y[-indexes_y_test])
for(val in unique(weights)) {weights[weights==val]=1/sum(weights==val)*length(weights)/2} # normalized to sum to length(samples)

model <- train(method = 'svmLinear', 
           x = x[-indexes_y_test,], 
           y = y[-indexes_y_test], 
           weights = weights,
           maximize = T,
           tuneGrid = expand.grid(.C=3^(-15:15)),   
           preProcess = c('center', 'scale'),
           trControl = trainControl(method = 'cv', # cross validation
                                    number = 10,   # nr of cv sets
#                                     repeats = 5, # use with method=repeatcv
                                    returnResamp = 'none', # return accuracy per cv partition and parameter setting
                                    classProbs = T, # return prediction probabilities along with predicted classes
#                                     savePredictions=T, # returns all predictions (for all cv paritions) for each tuneGrid parameter set 
                                    returnData = F, # disable return of training data e.g. for big data sets
                                    allowParallel = T
            )
)
# we see some accuracy around 0.7-0.8
model
head(with(model, results[order(results$Kappa, decreasing=T),]))
# confusion matrix: model predicting classes of test data
table(predict.train(object = model, newdata = x[indexes_y_test,], type='raw'), y[indexes_y_test])
# prediction probabilities of test data classes
probs <- predict.train(object = model, newdata = x[indexes_y_test,], type='prob')[,1]
isPositiveClass <- y[indexes_y_test] == 'versicolor' # for a ROC curve there is a positive class (true match rate...) - defining that class here
pred <- prediction(probs, isPositiveClass)
perf <- performance(pred, 'tpr', 'fpr')
# plot: either
plot(perf, lwd=2, col=3)
# or
with(attributes(perf), plot(x=x.values[[1]], y=y.values[[1]], type='l')) 

# some metrics: AUC (area under curve), EER (equal error rate), MSER (minimum squared error rate), Cohen's Kappa etc.
AUC <- attributes(performance(pred, 'auc'))$y.value[[1]] # area under curve
df <- with(attributes(pred), data.frame(cutoffs=cutoffs[[1]], tp=tp[[1]], fn=fn[[1]], tn=tn[[1]], fp=fp[[1]], TMR=tp[[1]]/(tp[[1]]+fn[[1]]), TNMR=tn[[1]]/(tn[[1]]+fp[[1]])))
df$MSER <- with(df, sqrt((1-TMR)**2+(1-TNMR)**2))
MSER <- with(df, sqrt(TMR**2+TNMR**2)) # sqrt of minimum squared error rate = eucl. distance to point TMR=TNMR=1
i_eer <- with(df, which.min(abs(TMR-TNMR)))
EER <- with(df[i_eer,], mean(c(TMR,TNMR))) # equal error rate: mean would not be required when using ROCR as it's always exact
df$acc <- with(df, (tp + tn) / length(isPositiveClass)) # observed accuracy
df$acc_expected <- with(df, sum(isPositiveClass) * (tp+fp) / length(isPositiveClass) + sum(!isPositiveClass) * (tn+fn) / length(isPositiveClass)) / length(isPositiveClass) # expected accuracy
df$kappa <- with(df, (acc - acc_expected) / (1 - acc_expected)) # cohen's kappa
# graphical representation 
matplot(df[,6:11], lty=1:6, lwd=2, type='l'); legend('bottomright', legend=names(df[,6:11]), lty=1:6, lwd=2, col=1:6)

# characteristics for settings with best EER, MSER etc.
df[i_eer,] # curve point for equal error rate
df[order(df$acc, decreasing=T)[[1]],] # curve point for max accuracy
df[order(df$kappa, decreasing=T)[[1]],] # curve point for max kappa
df[order(df$MSER, decreasing=F)[[1]],] # curve point for minimum squared error rate

Results

With a test run the confusion matrix for the test data set was

             versicolor virginica
  versicolor         14         6
  virginica           2        10

which points out that 2 out of 16 versicolor samples were incorrectly classified as virginica and 6 out of 16 virginica samples were incorrectly classified as versicolor. Based on prediction probabilities we generated a ROC curve. The true match rate (TMR) indicates the rate of how much test data set samples were correctly identified as being of the positive class (in our case: versicolor). The false match rate (FMR, counterpart to true non match rate TNMR) indicates the rate of how much test data set samples were incorrectly identified as being of the positive class.rocAlthough ROC curves indicate classification performance well it’s hard to compare classifiers based directly on ROC curves. Here, different other metrics come in handy, such as the area under curve (AUC), equal error rate (EER), maximum accuracy, maximum kappa and minimum squared error rate (MSER = square of minimum euclidean distance between ROC curve and top left corner for TMR = TNMR = 1). In our case the AUC is about 0.8672, the other metrics (and their corresponding ROC curve points) are about:

             cutoffs   TMR TNMR accuracy accuracy_expected kappa  MSER
EER            0.550 0.750 0.75     0.75               0.5   0.5 0.354
Max acc.       0.515 0.875 0.75   0.8125               0.5 0.625 0.280
Max kappa      0.515 0.875 0.75   0.8125               0.5 0.625 0.280
Min MSER       0.515 0.875 0.75   0.8125               0.5 0.625 0.280

mctune: multicore hyperparameter tuning in R on the example of SVM car detection

April 20, 2014 1 comment

mctune

In Machine Learning (ML) tasks finding good hyperparameters for machine learning models is critical (hyperparameter optimization). In R there exist some packages containing routines doing that for you using grid search (constructing and testing all possible parameters as a grid, e.g. in David Meyer’s e1071 package).

Besides the very good routines already contained in those packages a while ago I liked to have multicore hyperparameter tuning and needed true positive rate (true match rate) and true negative rate (true non match rate) alongside overall error rate. Therefore I modified the tune method of the e1071 package to a) use multicore processing and b) return confusion matrizes showing average true positive / true negative rates per sample class and parameter setting.

The script inherits it’s GPL-2 licence from it’s original version. I modified tune specifically for my own needs when tuning SVM hyperparameters, therefore did not validate it for correctness or how it will work with other tune-able functions (e.g. KNN, random forest).

# modified version of "tune" from package "e1071"
# licence: GPL-2
# details: http://cran.r-project.org/web/packages/e1071/index.html
mctune <- function(method, train.x, train.y = NULL, data = list(),
                 validation.x = NULL, validation.y = NULL,
                 ranges = NULL, predict.func = predict,
                 tunecontrol = tune.control(),
                 mc.control=NULL,
                 confusionmatrizes=F,
                 ...
) {
  call <- match.call()

  require('plyr')
  require('parallel')

  ## internal helper functions
  resp <- function(formula, data) {
    model.response(model.frame(formula, data))
  }

  classAgreement2 <- function (tab) {
    n <- sum(tab)
    # correct classification rate
    if (!is.null(dimnames(tab))) {
      lev <- intersect(colnames(tab), rownames(tab))
      d <- diag(tab[lev, lev])
      p0 <- sum(d) / n
    } else {
      m <- min(dim(tab))
      d <- diag(tab[1:m, 1:m])
      p0 <- sum(d) / n
    }
    # confusion matrizes
    if(!confusionmatrizes) {
      list(p0=p0)
    } else if(is.null(dimnames(tab))) {
      stop('tables without dimension names are not allowed when generating confusionmatrizes.')
    }
    else {
      # generate confusion matrix for each class
      classnames <- unique(unlist(dimnames(tab)))
      truepositives <-  unlist(lapply(classnames, function(positiveclassname) { sum(d[positiveclassname])}))
      falsepositives <- unlist(lapply(classnames, function(positiveclassname) { sum(tab[positiveclassname,])-tab[positiveclassname,positiveclassname]}))
      falsenegatives <- unlist(lapply(classnames, function(positiveclassname) { sum(tab[,positiveclassname])-tab[positiveclassname,positiveclassname]}))
      truenegatives <- mapply(FUN=function(tp,fn,fp){ sum(tab)-tp-fn-fp }, truepositives, falsenegatives, falsepositives)
      confusions <- data.frame(classnames, truepositives, truenegatives, falsepositives, falsenegatives, row.names=NULL)
      colnames(confusions) <- c('class', 'tp', 'tn', 'fp', 'fn')
      list(p0=p0, confusions=confusions)
    }
  }

  ## parameter handling
  if (tunecontrol$sampling == "cross")
    validation.x <- validation.y <- NULL
  useFormula <- is.null(train.y)
  if (useFormula && (is.null(data) || length(data) == 0))
    data <- model.frame(train.x)
  if (is.vector(train.x)) train.x <- t(t(train.x))
  if (is.data.frame(train.y))
    train.y <- as.matrix(train.y)

  ## prepare training indices
  if (!is.null(validation.x)) tunecontrol$fix <- 1
  n <- (nrow(if (useFormula) data
            else train.x))
  perm.ind <- sample(n)
  if (tunecontrol$sampling == "cross") {
    if (tunecontrol$cross > n)
      stop(sQuote("cross"), " must not exceed sampling size!")
    if (tunecontrol$cross == 1)
      stop(sQuote("cross"), " must be greater than 1!")
  }
  train.ind <- (if (tunecontrol$sampling == "cross")
                  tapply(1:n, cut(1:n, breaks = tunecontrol$cross), function(x) perm.ind[-x])
                else if (tunecontrol$sampling == "fix")
                  list(perm.ind[1:trunc(n * tunecontrol$fix)])
                else
                ## bootstrap
                  lapply(1:tunecontrol$nboot,
                         function(x) sample(n, n * tunecontrol$boot.size, replace = TRUE))
  )

  ## find best model
  parameters <- (if(is.null(ranges))
                  data.frame(dummyparameter = 0)
                else
                  expand.grid(ranges))
  p <- nrow(parameters)
  if (!is.logical(tunecontrol$random)) {
    if (tunecontrol$random < 1)
      stop("random must be a strictly positive integer")
    if (tunecontrol$random > p) tunecontrol$random <- p
    parameters <- parameters[sample(1:p, tunecontrol$random),]
  }

  ## - loop over all models
  # concatenate arbitrary mc-arguments with explicit X and FUN arguments
  train_results<-do.call(what="mclapply", args=c(mc.control, list(X=1:p, ..., FUN=function(para.set) {
    sampling.errors <- c()
    sampling.confusions <- c()

    ## - loop over all training samples
    for (sample in 1:length(train.ind)) {
      repeat.errors <- c()
      repeat.confusions <- c()

      ## - repeat training `nrepeat' times
      for (reps in 1:tunecontrol$nrepeat) {

        ## train one model
        pars <- if (is.null(ranges))
          NULL
        else
          lapply(parameters[para.set,,drop = FALSE], unlist)

        model <- if (useFormula)
          do.call(method, c(list(train.x,
                                 data = data,
                                 subset = train.ind[[sample]]),
                            pars, list(...)
          )
          )
        else
          do.call(method, c(list(train.x[train.ind[[sample]],],
                                 y = train.y[train.ind[[sample]]]),
                            pars, list(...)
          )
          )

        ## predict validation set
        pred <- predict.func(model,
                             if (!is.null(validation.x))
                               validation.x
                             else if (useFormula)
                               data[-train.ind[[sample]],,drop = FALSE]
                             else if (inherits(train.x, "matrix.csr"))
                               train.x[-train.ind[[sample]],]
                             else
                               train.x[-train.ind[[sample]],,drop = FALSE]
        )

        ## compute performance measure
        true.y <- if (!is.null(validation.y))
          validation.y
        else if (useFormula) {
          if (!is.null(validation.x))
            resp(train.x, validation.x)
          else
            resp(train.x, data[-train.ind[[sample]],])
        } else
          train.y[-train.ind[[sample]]]

        if (is.null(true.y)) true.y <- rep(TRUE, length(pred))

        if (!is.null(tunecontrol$error.fun))
          repeat.errors[reps] <- tunecontrol$error.fun(true.y, pred)
        else if ((is.logical(true.y) || is.factor(true.y)) && (is.logical(pred) || is.factor(pred) || is.character(pred))) { ## classification error
          l <- classAgreement2(table(pred, true.y))
          repeat.errors[reps] <- (1 - l$p0) # wrong classification rate
          if(confusionmatrizes) {
            repeat.confusions[[reps]] <- l$confusions
          }
        } else if (is.numeric(true.y) && is.numeric(pred)) ## mean squared error
          repeat.errors[reps] <- crossprod(pred - true.y) / length(pred)
        else
          stop("Dependent variable has wrong type!")
      }
      sampling.errors[sample] <- tunecontrol$repeat.aggregate(repeat.errors)
      # TODO potentially implement separate aggregation of tp tn fp fn values. currently those are taken with correlate to the least error.
      if(confusionmatrizes) {
        sampling.confusions[[sample]] <- repeat.confusions[repeat.errors == sampling.errors[sample]][[1]]
      }
    }
    # TODO potentially implement separate aggregation of tp tn fp fn values. currently uses the same as for error / variance aggregation
    if(!confusionmatrizes) {
      list(
        model.error=tunecontrol$sampling.aggregate(sampling.errors),
        model.variance=tunecontrol$sampling.dispersion(sampling.errors))
    } else {
      # create one confusion data frame
      confusions <- ldply(sampling.confusions, data.frame)
      # calculate aggregate / disperse values per class
      confusions <- ldply(lapply(X=split(confusions, confusions$class), FUN=function(classdf) {
        class=unique(classdf$class)
        # only take numeric values
        classdf[,c('tp','tn','fp','fn')]
        # calculate aggregate / disperse values for this class
        aggregated <- apply(X=classdf[,c('tp','tn','fp','fn')], MAR=2, FUN=tunecontrol$sampling.aggregate)
        dispersions <- apply(X=classdf[,c('tp','tn','fp','fn')], MAR=2, FUN=tunecontrol$sampling.dispersion)
        # make 1 row dataframe out of it (combine rows later with outer ldply)
        t(data.frame(c(value=aggregated,dispersion=dispersions)))
      }), data.frame)
      colnames(confusions) <- c('class', 'tp.value', 'tn.value', 'fp.value', 'fn.value', 'tp.dispersion', 'tn.dispersion', 'fp.dispersion', 'fn.dispersion')
      # calculate mean confusion matrix values (mean of all classes) for best model
      confusions.mean <- data.frame(t(apply(X=confusions[,c('tp.value','tn.value','fp.value','fn.value','tp.dispersion','tn.dispersion','fp.dispersion','fn.dispersion')], MAR=2, FUN=mean)))
      colnames(confusions.mean) <- c('tp.value', 'tn.value', 'fp.value', 'fn.value', 'tp.dispersion', 'tn.dispersion', 'fp.dispersion', 'fn.dispersion')
      list(
        model.error=tunecontrol$sampling.aggregate(sampling.errors),
        model.variance=tunecontrol$sampling.dispersion(sampling.errors),
        model.confusions=confusions,
        model.confusions.mean=confusions.mean
      )
    }
  })))
#   print('mctune: mclapply done.')
#   print(train_results)
  model.errors <- unlist(lapply(train_results,function(x)x$model.error))
  model.variances <- unlist(lapply(train_results,function(x)x$model.variance))
  if(confusionmatrizes){
    model.confusions <- lapply(train_results,function(x)x$model.confusions)
    model.confusions.mean <- ldply(lapply(train_results,function(x)x$model.confusions.mean), data.frame)
  }

  ## return results
  best <- which.min(model.errors)
  pars <- if (is.null(ranges))
    NULL
  else
    lapply(parameters[best,,drop = FALSE], unlist)
  structure(list(best.parameters  = parameters[best,,drop = FALSE],
                 best.performance = model.errors[best],
                 method           = if (!is.character(method))
                   deparse(substitute(method)) else method,
                 nparcomb         = nrow(parameters),
                 train.ind        = train.ind,
                 sampling         = switch(tunecontrol$sampling,
                                           fix = "fixed training/validation set",
                                           bootstrap = "bootstrapping",
                                           cross = if (tunecontrol$cross == n) "leave-one-out" else
                                             paste(tunecontrol$cross,"-fold cross validation", sep="")
                 ),
                 performances     = if (tunecontrol$performances) cbind(parameters, error = model.errors, dispersion = model.variances),
                 confusionmatrizes = if (confusionmatrizes) model.confusions,
                 confusionmatrizes.mean = if(confusionmatrizes) model.confusions.mean,
                 best.confusionmatrizes = if(confusionmatrizes) model.confusions[[best]],
                 best.confusionmatrizes.mean = if(confusionmatrizes) model.confusions.mean[best,],
                 best.model       = if (tunecontrol$best.model) {
                   modeltmp <- if (useFormula)
                     do.call(method, c(list(train.x, data = data),
                                       pars, list(...)))
                   else
                     do.call(method, c(list(x = train.x,
                                            y = train.y),
                                       pars, list(...)))
                   call[[1]] <- as.symbol("best.tune")
                   modeltmp$call <- call
                   modeltmp
                 }
  ),
            class = "tune"
  )
}

Additional parameters

mc.control

A list of parameters that go to mclapply from the parallel package. Example: mc.control=list(mc.cores=3, mc.preschedule=F)

confusionmatrizes

Binary flag indicating if confusion matrizes should be generated or not.

Additional return values

tune$confusionmatrizes

List of confusion matrizes. The matrizes are sorted the same way as tune$performances is. Each matrix is a data frame listing the sample classes (class), it’s corresponding absolute average values (value) and standard deviation (dispersion) for true positives (TP), true negatives (TN), false positives (FP) and false negatives (FN). An pre-drawn example of such a confusionmatrix from our car detection below could look like this (where “yes” means “car” and “no” means “no car”):

class tp.value tn.value fp.value fn.value tp.dispersion tn.dispersion fp.dispersion fn.dispersion
   no     49.5     53.4      1.6      0.5      5.212165      5.621388     0.8432740     0.5270463
  yes     53.4     49.5      0.5      1.6      5.621388      5.212165     0.5270463     0.8432740

In above example for the “yes” class the tp.value=53.4 means that on average 53.4 samples were correctly classified as “yes” using the corresponding parameter setting.

Car detection

As ML example we do a car detection. The task is to decide if a given image contains or not contains a car (therefore leaving out searching mechanisms like sliding window). As data source we use the UIUC Image Database for Car Detection. In order to reduce the amount of features (pixels) and “blur” images (which discards unnecessary details and likely even increases recognition probabilities) I resized the images from original 100×40 to 50×20 pixels. For a human it’s still easy to decide if an image shows a car or not at this scale:

No car 1

No car 1

No car 2

No car 2

No car 3

No car 3

Car 1

Car 1

Car 2

Car 2

Car 3

Car 3

 

 

As ML model we use a support vector machine with radial (Gaussian) kernel, which leaves us with the cost and gamma parameter to be tuned – which we do using a grid search and 10 fold cross validation.

# load data
print('loading data...')
library('png')
car_neg <- lapply(dir(pattern="neg.*png"),readPNG)
car_pos <- lapply(dir(pattern="pos.*png"),readPNG)
data <- c(car_neg,car_pos)
labels <- factor(c(rep('no',length(car_neg)), rep('yes', length(car_pos))))
# put data to dataframe
data <- t(array(unlist(data),dim=c(length(data[[1]]),length(data))))
data<-data.frame(data)
# look at a car
# image(array(as.numeric(data[600,]),dim=c(20,50)))
# look at correlation between first pixels
# plot(data.frame(data,labels)[,1:8],pch='.') 

library('e1071')
source('mctune.R')
t<-mctune(confusionmatrizes=T,
          mc.control=list(mc.cores=3, mc.preschedule=F),
          method=svm,
          ranges=list(type='C',
                      kernel='radial',
                      gamma=3^(-10:-4),
                      cost=3^(-8:8)),
          train.x=data,
          train.y=labels,
          validation.x=NULL, #validation.x and .y are only used with tune.control sampling='boot' and 'fixed'.
          validation.y=NULL,
          tunecontrol=tune.control(sampling='cross',
                                   cross=10,
                                   performances=T,
                                   nrepeat=1,
                                   best.model=T))
# extract FMR FNMR from our positive class
p <- lapply(X=t$confusionmatrizes, FUN=function(x){
  p <- x[x$class=='yes',]
  p[-1]
})
p <- ldply(p)
t$performances <- data.frame(t$performances, p)
t$performances$FMR <- t$performances$fp.value / (t$performances$tn.value + t$performances$fp.value)
t$performances$FNMR <- t$performances$fn.value / (t$performances$tp.value + t$performances$fn.value)
# print list of errors
t$performances[with(t$performances, order(error)),]

# different plots of parameters and errors
library('scatterplot3d')
scatterplot3d(t$performances$cost, t$performances$gamma, t$performances$error,log='xy', type='h', pch=1, color='red')
plot(t$performances[,3:5],log='xy',pch=4)
# paramters and errors: from best to worst
plot(t$performances[with(t$performances, order(error)),]$cost,log='y', col='blue')
points(t$performances[with(t$performances, order(error)),]$gamma, col='green')
points(t$performances[with(t$performances, order(error)),]$error, col='black')
# points(t$performances[with(t$performances, order(error)),]$FMR, col='red')
# points(t$performances[with(t$performances, order(error)),]$FNMR, col='orange')

Using this setup with the first search we obtain an error rate of about 0.02 for gamma 10^-3 and cost >= 3 (this indicates the search area for the next step). The corresponding false match rate (FMR) and false non match rate (FNMR) are in the range of 0.01 and 0.029.