wsvm: Subject Weighted Support Vector Machines

Description

wsvm is used to train a subject weighted support vector machine. It can be used to carry out general regression and classification (of nu and epsilon-type), as well as density-estimation. A formula interface is provided.

Usage

# S3 method for formula
wsvm(formula, weight, data = NULL, ..., subset, na.action =
na.omit, scale = TRUE)
# S3 method for default
wsvm(x, y = NULL, weight, scale = TRUE, type = NULL, kernel =
"radial", degree = 3, gamma = if (is.vector(x)) 1 else 1 / ncol(x),
coef0 = 0, cost = 1, nu = 0.5,
class.weights = NULL, cachesize = 100, tolerance = 0.001, epsilon = 0.1,
shrinking = TRUE, cross = 0, probability = FALSE, fitted = TRUE,
..., subset, na.action = na.omit)

Value

An object of class "wsvm" containing the fitted model, including:

SV: The resulting support vectors (possibly scaled).
index: The index of the resulting support vectors in the data matrix. Note that this index refers to the preprocessed data (after the possible effect of na.omit and subset)
coefs: The corresponding coefficients times the training labels.
rho: The negative intercept.
sigma: In case of a probabilistic regression model, the scale parameter of the hypothesized (zero-mean) laplace distribution estimated by maximum likelihood.
probA, probB: numeric vectors of length k(k-1)/2, k number of classes, containing the parameters of the logistic distributions fitted to the decision values of the binary classifiers (1 / (1 + exp(a x + b))).

Arguments

formula

a symbolic description of the model to be fit.

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment which ‘wsvm’ is called from.

x

a data matrix, a vector, or a sparse 'design matrix' (object of class Matrix provided by the Matrix package, or of class matrix.csr provided by the SparseM package, or of class simple_triplet_matrix provided by the slam package). Or a kernel matrix of class kernelMatrix by the kernlab package.

y

a response vector with one label for each row/component of x. Can be either a factor (for classification tasks) or a numeric vector (for regression).

weight

the weight of each subject. It should be in the same length of y.

scale

A logical vector indicating the variables to be scaled. If scale is of length 1, the value is recycled as many times as needed. By default, data are scaled internally (both x and y variables) to zero mean and unit variance. The center and scale values are returned and used for later predictions. If x is a design matrix which contains dummy variables, please make these variable NOT scaled.

type

wsvm can be used as a classification machine, as a regression machine, or for novelty detection. Depending of whether y is a factor or not, the default setting for type is C-classification or eps-regression, respectively, but may be overwritten by setting an explicit value.
Valid options are:

C-classification
nu-classification
one-classification (for novelty detection)
eps-regression
nu-regression

kernel

the kernel used in training and predicting. You might consider changing some of the following parameters, depending on the kernel type.

linear:: $u^{'} v$

polynomial:

$(γ u^{'} v + c o e f 0)^{d e g r e e}$

radial basis:

$e^{(} - γ | u - v |^{2})$

sigmoid:

$t a n h (γ u^{'} v + c o e f 0)$

precomputed:

x is a precomputed kernel matrix that contains NO missing values. scale will not work. Cannot use subset and na.action with this kernel.

degree

parameter needed for kernel of type polynomial (default: 3)

gamma

parameter needed for all kernels except linear (default: 1/(data dimension))

coef0

parameter needed for kernels of type polynomial and sigmoid (default: 0)

cost

cost of constraints violation (default: 1)---it is the ‘C’-constant of the regularization term in the Lagrange formulation.

parameter needed for nu-classification, nu-regression, and one-classification

class.weights

a named vector of weights for the different classes, used for asymmetric class sizes. Not all factor levels have to be supplied (default weight: 1). All components have to be named. Specifying "inverse" will choose the weights inversely proportional to the class distribution.

cachesize

cache memory in MB (default 100)

tolerance

tolerance of termination criterion (default: 0.001)

epsilon

epsilon in the insensitive-loss function (default: 0.1)

shrinking

option whether to use the shrinking-heuristics (default: TRUE)

cross

if a integer value k>0 is specified, a k-fold cross validation on the training data is performed to assess the quality of the model: the accuracy rate for classification and the Mean Squared Error for regression. Note the result is not weighted. For weighted results, use tune_wsvm fucntion.

fitted

logical indicating whether the fitted values should be computed and included in the model or not (default: TRUE)

probability

logical indicating whether the model should allow for probability predictions.

...

additional parameters for the low level fitting function wsvm.default

subset

An index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)

na.action

A function to specify the action to be taken if NAs are found. The default action is na.omit, which leads to rejection of cases with missing values on any required variable. An alternative is na.fail, which causes an error if NA cases are found. (NOTE: If given, this argument must be named.)

Author

David Meyer (based on C/C++-code by Chih-Chung Chang and Chih-Jen Lin)
Modified by Tianchen Xu tx2155@columbia.edu

Details

The original libsvm does not support subject/instance weighted svm. From the 'LIBSVM Tools' https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/#weights_for_data_instances, we are able to use a modified version of libsvm to support subject weights.

For multiclass-classification with k levels, k>2, libsvm uses the ‘one-against-one’-approach, in which k(k-1)/2 binary classifiers are trained; the appropriate class is found by a voting scheme.

libsvm internally uses a sparse data representation, which is also high-level supported by the package SparseM.

If the predictor variables include factors, the formula interface must be used to get a correct model matrix or make x a design matrix.

When using the formula interface and na.action is na.omit, we delete any subjects with missing values on x, y (if exists) or weight in the training and predicting procedure (when fitted = TRUE). When using the x, y interface and na.action is na.omit, we delete any subjects with missing values on x, y (if exists) or weight in the training procedure, and retain the subjects with missing values only on weight in the predicting procedure (when fitted = TRUE).

plot.wsvm allows a simple graphical visualization of classification models.

The probability model for classification fits a logistic distribution using maximum likelihood to the decision values of all binary classifiers, and computes the a-posteriori class probabilities for the multi-class problem using quadratic optimization. The probabilistic regression model assumes (zero-mean) laplace-distributed errors for the predictions, and estimates the scale parameter using maximum likelihood.

For linear kernel, the coefficients of the regression/decision hyperplane can be extracted using the coef method (see examples).

References

Chang, Chih-Chung and Lin, Chih-Jen:
LIBSVM: a library for Support Vector Machines
https://www.csie.ntu.edu.tw/~cjlin/libsvm/
Ming-Wei Chang, Hsuan-Tien Lin, Ming-Hen Tsai, Chia-Hua Ho and Hsiang-Fu Yu
Weights for data instances
https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/#weights_for_data_instances
Exact formulations of models, algorithms, etc. can be found in the document:
Chang, Chih-Chung and Lin, Chih-Jen:
LIBSVM: a library for Support Vector Machines
https://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.ps.gz
More implementation details and speed benchmarks can be found on: Rong-En Fan and Pai-Hsune Chen and Chih-Jen Lin:
Working Set Selection Using the Second Order Information for Training SVM
https://www.csie.ntu.edu.tw/~cjlin/papers/quadworkset.pdf

Examples

Run this code

## check what is loaded
dllpath <- getLoadedDLLs()
getDLLRegisteredRoutines(dllpath$WeightSVM[[2]])

## load dataset
data(iris)

## classification mode
# default with factor response:
model1 <- wsvm(Species ~ ., weight = rep(1,150), data = iris) # same weights
model2 <- wsvm(x = iris[,1:4], y = iris[,5],
              weight = c(rep(0.08, 50),rep(1,100))) # less weights to setosa
# alternatively the traditional interface:
x <- subset(iris, select = -Species)
y <- iris$Species
model3 <- wsvm(x, y, weight = rep(10,150)) # similar to model 1,
                               # but larger weights for all subjects

# These models provide error/warning info
try(wsvm(x, y)) # no weight
try(wsvm(x, y, weight = rep(10,100))) # wrong length
try(wsvm(x, y, weight = c(Inf, rep(1,149)))) # contains inf weight

print(model1)
summary(model1)

# test with train data
pred <- predict(model1, iris[,1:4])
# (same as:)
pred <- fitted(model1)

# Check accuracy:
table(pred, y) # model 1, equal weights

# compute decision values and probabilities:
pred <- predict(model1, x, decision.values = TRUE)
attr(pred, "decision.values")[1:4,]

# visualize (classes by color, SV by crosses):
plot(cmdscale(dist(iris[,-5])),
     col = as.integer(iris[,5]),
     pch = c("o","+")[1:150 %in% model1$index + 1]) # model 1
plot(cmdscale(dist(iris[,-5])),
     col = as.integer(iris[,5]),
     pch = c("o","+")[1:150 %in% model2$index + 1])
  # In model 2, less support vectors are based on setosa


## try regression mode on two dimensions
# create data
x <- seq(0.1, 5, by = 0.05)
y <- log(x) + rnorm(x, sd = 0.2)

# estimate model and predict input values
model1 <- wsvm(x, y, weight = rep(1,99))
model2 <- wsvm(x, y, weight = seq(99,1,length.out = 99)) # decreasing weights
# library(kernlab)
# model3 <- wsvm(kernlab::kernelMatrix(kernlab::rbfdot(sigma = 1), x), y,
#      weight = rep(1,99), kernel = 'precomputed') # try user defined kernel

# visualize
plot(x, y)
lines(x, log(x), col = 2)
points(x, fitted(model1), col = 4)
points(x, fitted(model2), col = 3) # better fit for the first few points
# points(x, fitted(model3), col = 5) # similar to model 1 with user defined kernel

## density-estimation
# create 2-dim. normal with rho=0:
X <- data.frame(a = rnorm(1000), b = rnorm(1000))
attach(X)

# formula interface:
model <- wsvm(~ a + b, gamma = 0.1, weight = c(seq(5000,1,length.out = 500),1:500))

# test:
newdata <- data.frame(a = c(0, 4), b = c(0, 4))

# visualize:
plot(X, col = 1:1000 %in% model$index + 1, xlim = c(-5,5), ylim=c(-5,5))
points(newdata, pch = "+", col = 2, cex = 5)

## class weights:
i2 <- iris
levels(i2$Species)[3] <- "versicolor"
summary(i2$Species)
wts <- 100 / table(i2$Species)
wts
m <- wsvm(Species ~ ., data = i2, class.weights = wts, weight=rep(1,150))

## extract coefficients for linear kernel

# a. regression
x <- 1:100
y <- x + rnorm(100)
m <- wsvm(y ~ x, scale = FALSE, kernel = "linear", weight = rep(1,100))
coef(m)
plot(y ~ x)
abline(m, col = "red")

# b. classification
# transform iris data to binary problem, and scale data
setosa <- as.factor(iris$Species == "setosa")
iris2 = scale(iris[,-5])

# fit binary C-classification model
model1 <- wsvm(setosa ~ Petal.Width + Petal.Length,
          data = iris2, kernel = "linear", weight = rep(1,150))
model2 <- wsvm(setosa ~ Petal.Width + Petal.Length,
               data = iris2, kernel = "linear",
               weight = c(rep(0.08, 50),rep(1,100))) # less weights to setosa

# plot data and separating hyperplane
plot(Petal.Length ~ Petal.Width, data = iris2, col = setosa)
(cf <- coef(model1))
abline(-cf[1]/cf[3], -cf[2]/cf[3], col = "red")
(cf2 <- coef(model2))
abline(-cf2[1]/cf2[3], -cf2[2]/cf2[3], col = "red", lty = 2)

# plot margin and mark support vectors
abline(-(cf[1] + 1)/cf[3], -cf[2]/cf[3], col = "blue")
abline(-(cf[1] - 1)/cf[3], -cf[2]/cf[3], col = "blue")
points(model1$SV, pch = 5, cex = 2)
abline(-(cf2[1] + 1)/cf2[3], -cf2[2]/cf2[3], col = "blue", lty = 2)
abline(-(cf2[1] - 1)/cf2[3], -cf2[2]/cf2[3], col = "blue", lty = 2)
points(model2$SV, pch = 6, cex = 2)

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