# svm

##### Support Vector Machines

`svm`

is used to train a 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
svm(formula, data = NULL, ..., subset, na.action =
na.omit, scale = TRUE)
# S3 method for default
svm(x, y = NULL, 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 = 40, tolerance = 0.001, epsilon = 0.1,
shrinking = TRUE, cross = 0, probability = FALSE, fitted = TRUE,
..., subset, na.action = na.omit)
```

##### 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 ‘svm’ is called from.

- x
a data matrix, a vector, or a sparse 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).- 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).- 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. Per 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.- type
`svm`

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:
\((\gamma u'v + coef0)^{degree}\)

- radial basis:
\(e^(-\gamma |u-v|^2)\)

- sigmoid:
\(tanh(\gamma u'v + coef0)\)

- 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.

- nu
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 40)

- 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

- 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

`svm.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

`NA`

s 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.)

##### Details

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.

`plot.svm`

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).

##### Value

An object of class `"svm"`

containing the fitted model, including:

The resulting support vectors (possibly scaled).

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`

)

The corresponding coefficients times the training labels.

The negative intercept.

In case of a probabilistic regression model, the scale parameter of the hypothesized (zero-mean) laplace distribution estimated by maximum likelihood.

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))).

##### Note

Data are scaled internally, usually yielding better results.

Parameters of SVM-models usually *must* be tuned to yield sensible results!

##### References

Chang, Chih-Chung and Lin, Chih-Jen:

*LIBSVM: a library for Support Vector Machines*http://www.csie.ntu.edu.tw/~cjlin/libsvmExact 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*http://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.ps.gzMore 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*http://www.csie.ntu.edu.tw/~cjlin/papers/quadworkset.pdf

##### See Also

`predict.svm`

`plot.svm`

`tune.svm`

`matrix.csr`

(in package SparseM)

##### Examples

```
# NOT RUN {
data(iris)
attach(iris)
## classification mode
# default with factor response:
model <- svm(Species ~ ., data = iris)
# alternatively the traditional interface:
x <- subset(iris, select = -Species)
y <- Species
model <- svm(x, y)
print(model)
summary(model)
# test with train data
pred <- predict(model, x)
# (same as:)
pred <- fitted(model)
# Check accuracy:
table(pred, y)
# compute decision values and probabilities:
pred <- predict(model, 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% model$index + 1])
## try regression mode on two dimensions
# create data
x <- seq(0.1, 5, by = 0.05)
y <- log(x) + rnorm(x, sd = 0.2)
# estimate model and predict input values
m <- svm(x, y)
new <- predict(m, x)
# visualize
plot(x, y)
points(x, log(x), col = 2)
points(x, new, col = 4)
## density-estimation
# create 2-dim. normal with rho=0:
X <- data.frame(a = rnorm(1000), b = rnorm(1000))
attach(X)
# traditional way:
m <- svm(X, gamma = 0.1)
# formula interface:
m <- svm(~., data = X, gamma = 0.1)
# or:
m <- svm(~ a + b, gamma = 0.1)
# test:
newdata <- data.frame(a = c(0, 4), b = c(0, 4))
predict (m, newdata)
# visualize:
plot(X, col = 1:1000 %in% m$index + 1, xlim = c(-5,5), ylim=c(-5,5))
points(newdata, pch = "+", col = 2, cex = 5)
## weights: (example not particularly sensible)
i2 <- iris
levels(i2$Species)[3] <- "versicolor"
summary(i2$Species)
wts <- 100 / table(i2$Species)
wts
m <- svm(Species ~ ., data = i2, class.weights = wts)
## extract coefficients for linear kernel
# a. regression
x <- 1:100
y <- x + rnorm(100)
m <- svm(y ~ x, scale = FALSE, kernel = "linear")
coef(m)
plot(y ~ x)
abline(m, col = "red")
# b. classification
# transform iris data to binary problem, and scale data
setosa <- as.factor(iris$Species == "setosa")
iris2 = scale(iris[,-5])
# fit binary C-classification model
m <- svm(setosa ~ Petal.Width + Petal.Length,
data = iris2, kernel = "linear")
# plot data and separating hyperplane
plot(Petal.Length ~ Petal.Width, data = iris2, col = setosa)
(cf <- coef(m))
abline(-cf[1]/cf[3], -cf[2]/cf[3], col = "red")
# 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(m$SV, pch = 5, cex = 2)
# }
```

*Documentation reproduced from package e1071, version 1.7-3, License: GPL-2 | GPL-3*