classify
Discriminant analysis
Syntax
class = classify(sample,training,group)
class = classify(sample,training,group,'type')
class = classify(sample,training,group,'type',prior)
[class,err] = classify(...)
[class,err,POSTERIOR] = classify(...)
[class,err,POSTERIOR,logp] = classify(...)
[class,err,POSTERIOR,logp,coeff] = classify(...)
Description
class = classify(sample,training,group) classifies
each row of the data in sample into one of the groups in
training. sample and
training must be matrices with the same number of columns.
group is a grouping variable for training. Its
unique values define groups; each element defines the group to which the corresponding
row of training belongs. group can be a
categorical variable, a numeric vector, a character array, a string array, or a cell
array of character vectors. training and group
must have the same number of rows. classify treats
<undefined> values, NaNs, empty character
vectors, empty strings, and <missing> string values in
group as missing data values, and ignores the corresponding rows
of training. The output class indicates the group
to which each row of sample has been assigned, and is of the same
type as group.
class = classify(sample,training,group,' allows
you to specify the type of discriminant function. Specify type')type inside
single quotes. type is one of:
linear— Fits a multivariate normal density to each group, with a pooled estimate of covariance. This is the default.diaglinear— Similar tolinear, but with a diagonal covariance matrix estimate (naive Bayes classifiers).quadratic— Fits multivariate normal densities with covariance estimates stratified by group.diagquadratic— Similar toquadratic, but with a diagonal covariance matrix estimate (naive Bayes classifiers).mahalanobis— Uses Mahalanobis distances with stratified covariance estimates.
class = classify(sample,training,group,' allows
you to specify prior probabilities for the groups. type',prior)prior is
one of:
A numeric vector the same length as the number of unique values in
group(or the number of levels defined forgroup, ifgroupis categorical). Ifgroupis numeric or categorical, the order ofpriormust correspond to the ordered values ingroup. Otherwise, the order ofpriormust correspond to the order of the first occurrence of the values ingroup.A 1-by-1 structure with fields:
prob— A numeric vector.group— Of the same type asgroup, containing unique values indicating the groups to which the elements ofprobcorrespond.
As a structure,
priorcan contain groups that do not appear ingroup. This can be useful iftrainingis a subset a larger training set.classifyignores any groups that appear in the structure but not in thegrouparray.The character vector or string scalar
'empirical', indicating that group prior probabilities should be estimated from the group relative frequencies intraining.
prior defaults to a numeric vector
of equal probabilities, i.e., a uniform distribution. prior is
not used for discrimination by Mahalanobis distance, except for error
rate calculation.
[class,err] = classify(...) also
returns an estimate err of the misclassification
error rate based on the training data. classify returns
the apparent error rate, i.e., the percentage of observations in training that
are misclassified, weighted by the prior probabilities for the groups.
[class,err,POSTERIOR] = classify(...) also
returns a matrix POSTERIOR of estimates of the
posterior probabilities that the jth training group
was the source of the ith sample observation, i.e., Pr(group
j|obs i). POSTERIOR is
not computed for Mahalanobis discrimination.
[class,err,POSTERIOR,logp] = classify(...) also
returns a vector logp containing estimates of the
logarithms of the unconditional predictive probability density of
the sample observations, p(obs i)
= ∑p(obs i|group
j)Pr(group j) over
all groups. logp is not computed for Mahalanobis
discrimination.
[class,err,POSTERIOR,logp,coeff] = classify(...) also
returns a structure array coeff containing coefficients
of the boundary curves between pairs of groups. Each element coeff(I,J)
contains information for comparing group I to group J in
the following fields:
type— Type of discriminant function, from thetypeinput.name1— Name of the first group.name2— Name of the second group.const— Constant term of the boundary equation (K)linear— Linear coefficients of the boundary equation (L)quadratic— Quadratic coefficient matrix of the boundary equation (Q)
For the linear and diaglinear types,
the quadratic field is absent, and a row x from
the sample array is classified into group I rather
than group J if 0 < K+x*L.
For the other types, x is classified into group I if 0
< K+x*L+x*Q*x'.
Examples
Classify Using Discriminant Analysis
For training data, use Fisher's sepal measurements for iris versicolor and virginica:
load fisheriris SL = meas(51:end,1); SW = meas(51:end,2); group = species(51:end); h1 = gscatter(SL,SW,group,'rb','v^',[],'off'); set(h1,'LineWidth',2) legend('Fisher versicolor','Fisher virginica',... 'Location','NW')
Classify a grid of measurements on the same scale:
[X,Y] = meshgrid(linspace(4.5,8),linspace(2,4)); X = X(:); Y = Y(:); [C,err,P,logp,coeff] = classify([X Y],[SL SW],... group,'Quadratic');
Visualize the classification:
hold on; gscatter(X,Y,C,'rb','.',1,'off'); K = coeff(1,2).const; L = coeff(1,2).linear; Q = coeff(1,2).quadratic; % Function to compute K + L*v + v'*Q*v for multiple vectors % v=[x;y]. Accepts x and y as scalars or column vectors. f = @(x,y) K + L(1)*x + L(2)*y + Q(1,1)*x.*x + (Q(1,2)+Q(2,1))*x.*y + Q(2,2)*y.*y; h2 = fimplicit(f,[4.5 8 2 4]); set(h2,'Color','m','LineWidth',2,'DisplayName','Decision Boundary') axis([4.5 8 2 4]) xlabel('Sepal Length') ylabel('Sepal Width') title('{\bf Classification with Fisher Training Data}')
Alternative Functionality
The fitcdiscr function also performs
discriminant analysis. You can train a classifier by using the
fitcdiscr function and predict labels of new data by using the
predict function. The
fitcdiscr supports cross-validation and hyperparameter
optimization and does not require you to fit the classifier every time you make a new
prediction or you change prior probabilities.
References
[1] Krzanowski, W. J. Principles of Multivariate Analysis: A User's Perspective. New York: Oxford University Press, 1988.
[2] Seber, G. A. F. Multivariate Observations. Hoboken, NJ: John Wiley & Sons, Inc., 1984.