Classification error
L = loss(tree,TBL,ResponseVarName)
L = loss(tree,TBL,Y)
L = loss(tree,X,Y)
L = loss(___,Name,Value)
[L,se,NLeaf,bestlevel]
= loss(___)
returns a scalar representing how well L
= loss(tree
,TBL
,ResponseVarName
)tree
classifies the
data in TBL
, when TBL.ResponseVarName
contains the true classifications.
When computing the loss, loss
normalizes the
class probabilities in Y
to the class probabilities used
for training, stored in the Prior
property of
tree
.
returns the loss with additional options specified by one or more
L
= loss(___,Name,Value
)Name,Value
pair arguments, using any of the previous
syntaxes. For example, you can specify the loss function or observation
weights.
loss
returns se
and further
outputs only when the LossFun
name-value pair is the default
'classiferror'
.
tree
— Trained classification treeClassificationTree
model object | CompactClassificationTree
model objectTrained classification tree, specified as a ClassificationTree
or CompactClassificationTree
model
object. That is, tree
is a trained classification
model returned by fitctree
or compact
.
TBL
— Sample dataSample data, specified as a table. Each row of TBL
corresponds
to one observation, and each column corresponds to one predictor variable.
Optionally, TBL
can contain additional columns
for the response variable and observation weights. TBL
must
contain all the predictors used to train tree
.
Multi-column variables and cell arrays other than cell arrays of character
vectors are not allowed.
If TBL
contains the response variable
used to train tree
, then you do not need to specify ResponseVarName
or Y
.
If you train tree
using sample data contained
in a table
, then the input data for this method
must also be in a table.
Data Types: table
X
— Data to classifyResponseVarName
— Response variable nameTBL
Response variable name, specified as the name of a variable
in TBL
. If TBL
contains
the response variable used to train tree
, then
you do not need to specify ResponseVarName
.
If you specify ResponseVarName
, then you must do so as a character vector
or string scalar. For example, if the response variable is stored as
TBL.Response
, then specify it as 'Response'
.
Otherwise, the software treats all columns of TBL
, including
TBL.ResponseVarName
, as predictors.
The response variable must be a categorical, character, or string array, logical or numeric vector, or cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.
Data Types: char
| string
Y
— Class labelsClass labels, specified as a categorical, character, or string array, a logical or numeric
vector, or a cell array of character vectors. Y
must be of the same
type as the classification used to train tree
, and its number of
elements must equal the number of rows of X
.
Data Types: categorical
| char
| string
| logical
| single
| double
| cell
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
'LossFun'
— Loss function'mincost'
(default) | 'binodeviance'
| 'classiferror'
| 'exponential'
| 'hinge'
| 'logit'
| 'quadratic'
| function handleLoss function, specified as the comma-separated pair consisting of
'LossFun'
and a built-in, loss-function name or
function handle.
The following table lists the available loss functions. Specify one using its corresponding character vector or string scalar.
Value | Description |
---|---|
'binodeviance' | Binomial deviance |
'classiferror' | Classification error |
'exponential' | Exponential |
'hinge' | Hinge |
'logit' | Logistic |
'mincost' | Minimal expected misclassification cost (for classification scores that are posterior probabilities) |
'quadratic' | Quadratic |
'mincost'
is appropriate for
classification scores that are posterior probabilities.
Classification trees return posterior probabilities as
classification scores by default (see predict
).
Specify your own function using function handle notation.
Suppose that n
be the number of
observations in X
and
K
be the number of distinct classes
(numel(tree.ClassNames)
). Your
function must have this signature
lossvalue = lossfun
(C,S,W,Cost)
The output argument
lossvalue
is a scalar.
You choose the function name
(lossfun
).
C
is an
n
-by-K
logical matrix with rows indicating which class
the corresponding observation belongs. The column
order corresponds to the class order in
tree.ClassNames
.
Construct C
by setting
C(p,q) = 1
if observation
p
is in class
q
, for each row. Set all other
elements of row p
to
0
.
S
is an
n
-by-K
numeric matrix of classification scores. The
column order corresponds to the class order in
tree.ClassNames
.
S
is a matrix of classification
scores, similar to the output of predict
.
W
is an
n
-by-1 numeric vector of
observation weights. If you pass
W
, the software normalizes them
to sum to 1
.
Cost
is a
K-by-K
numeric matrix of misclassification costs. For
example, Cost = ones(K) -
eye(K)
specifies a cost of
0
for correct classification,
and 1
for
misclassification.
Specify your function using
'LossFun',@
.lossfun
For more details on loss functions, see Classification Loss.
Data Types: char
| string
| function_handle
'Weights'
— Observation weightsones(size(X,1),1)
(default) | name of a variable in TBL
| numeric vector of positive valuesObservation weights, specified as the comma-separated pair consisting
of 'Weights'
and a numeric vector of positive values
or the name of a variable in TBL
.
If you specify Weights
as a numeric vector, then
the size of Weights
must be equal to the number of
rows in X
or TBL
.
If you specify Weights
as the name of a variable
in TBL
, you must do so as a character vector or
string scalar. For example, if the weights are stored as
TBL.W
, then specify it as 'W'
.
Otherwise, the software treats all columns of TBL
,
including TBL.W
, as predictors.
loss
normalizes the weights so that
observation weights in each class sum to the prior probability of that
class. When you supply Weights
, loss
computes weighted classification
loss.
Data Types: single
| double
| char
| string
Name,Value
arguments associated with pruning subtrees:
'Subtrees'
— Pruning level'all'
Pruning level, specified as the comma-separated pair consisting
of 'Subtrees'
and a vector of nonnegative integers
in ascending order or 'all'
.
If you specify a vector, then all elements must be at least 0
and
at most max(tree.PruneList)
. 0
indicates
the full, unpruned tree and max(tree.PruneList)
indicates
the completely pruned tree (i.e., just the root node).
If you specify 'all'
, then loss
operates
on all subtrees (i.e., the entire pruning sequence). This specification
is equivalent to using 0:max(tree.PruneList)
.
loss
prunes tree
to
each level indicated in Subtrees
, and then estimates
the corresponding output arguments. The size of Subtrees
determines
the size of some output arguments.
To invoke Subtrees
, the properties PruneList
and PruneAlpha
of tree
must
be nonempty. In other words, grow tree
by setting 'Prune','on'
,
or by pruning tree
using prune
.
Example: 'Subtrees','all'
Data Types: single
| double
| char
| string
'TreeSize'
— Tree size'se'
(default) | 'min'
Tree size, specified as the comma-separated pair consisting of
'TreeSize'
and one of the following
values:
'se'
— loss
returns the highest pruning level with loss within one standard
deviation of the minimum
(L
+se
, where
L
and se
relate to the
smallest value in Subtrees
).
'min'
— loss
returns the element of Subtrees
with smallest
loss, usually the smallest element of
Subtrees
.
L
— Classification lossClassification
loss, returned as a vector the length of
Subtrees
. The meaning of the error depends on the
values in Weights
and LossFun
.
se
— Standard error of lossStandard error of loss, returned as a vector the length of
Subtrees
.
NLeaf
— Number of leaf nodesNumber of leaves (terminal nodes) in the pruned subtrees, returned as a
vector the length of Subtrees
.
bestlevel
— Best pruning levelBest pruning level as defined in the TreeSize
name-value pair, returned as a scalar whose value depends on
TreeSize
:
TreeSize
= 'se'
—
loss
returns the highest pruning level with
loss within one standard deviation of the minimum
(L
+se
, where
L
and se
relate to the
smallest value in Subtrees
).
TreeSize
= 'min'
—
loss
returns the element of
Subtrees
with smallest loss, usually the
smallest element of Subtrees
.
By default, bestlevel
is the pruning level that gives
loss within one standard deviation of minimal loss.
Compute the resubstituted classification error for the ionosphere
data set.
load ionosphere
tree = fitctree(X,Y);
L = loss(tree,X,Y)
L = 0.0114
Unpruned decision trees tend to overfit. One way to balance model complexity and out-of-sample performance is to prune a tree (or restrict its growth) so that in-sample and out-of-sample performance are satisfactory.
Load Fisher's iris data set. Partition the data into training (50%) and validation (50%) sets.
load fisheriris n = size(meas,1); rng(1) % For reproducibility idxTrn = false(n,1); idxTrn(randsample(n,round(0.5*n))) = true; % Training set logical indices idxVal = idxTrn == false; % Validation set logical indices
Grow a classification tree using the training set.
Mdl = fitctree(meas(idxTrn,:),species(idxTrn));
View the classification tree.
view(Mdl,'Mode','graph');
The classification tree has four pruning levels. Level 0 is the full, unpruned tree (as displayed). Level 3 is just the root node (i.e., no splits).
Examine the training sample classification error for each subtree (or pruning level) excluding the highest level.
m = max(Mdl.PruneList) - 1;
trnLoss = resubLoss(Mdl,'SubTrees',0:m)
trnLoss = 3×1
0.0267
0.0533
0.3067
The full, unpruned tree misclassifies about 2.7% of the training observations.
The tree pruned to level 1 misclassifies about 5.3% of the training observations.
The tree pruned to level 2 (i.e., a stump) misclassifies about 30.6% of the training observations.
Examine the validation sample classification error at each level excluding the highest level.
valLoss = loss(Mdl,meas(idxVal,:),species(idxVal),'SubTrees',0:m)
valLoss = 3×1
0.0369
0.0237
0.3067
The full, unpruned tree misclassifies about 3.7% of the validation observations.
The tree pruned to level 1 misclassifies about 2.4% of the validation observations.
The tree pruned to level 2 (i.e., a stump) misclassifies about 30.7% of the validation observations.
To balance model complexity and out-of-sample performance, consider pruning Mdl
to level 1.
pruneMdl = prune(Mdl,'Level',1); view(pruneMdl,'Mode','graph')
Classification loss functions measure the predictive inaccuracy of classification models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.
Consider the following scenario.
L is the weighted average classification loss.
n is the sample size.
For binary classification:
y_{j} is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class, respectively.
f(X_{j}) is the raw classification score for observation (row) j of the predictor data X.
m_{j} = y_{j}f(X_{j}) is the classification score for classifying observation j into the class corresponding to y_{j}. Positive values of m_{j} indicate correct classification and do not contribute much to the average loss. Negative values of m_{j} indicate incorrect classification and contribute significantly to the average loss.
For algorithms that support multiclass classification (that is, K ≥ 3):
y_{j}^{*}
is a vector of K – 1 zeros, with 1 in the
position corresponding to the true, observed class
y_{j}. For example,
if the true class of the second observation is the third class and
K = 4, then
y^{*}_{2}
= [0 0 1 0]′. The order of the classes corresponds to the order in
the ClassNames
property of the input
model.
f(X_{j})
is the length K vector of class scores for
observation j of the predictor data
X. The order of the scores corresponds to the
order of the classes in the ClassNames
property
of the input model.
m_{j} = y_{j}^{*}′f(X_{j}). Therefore, m_{j} is the scalar classification score that the model predicts for the true, observed class.
The weight for observation j is w_{j}. The software normalizes the observation weights so that they sum to the corresponding prior class probability. The software also normalizes the prior probabilities so they sum to 1. Therefore,
$$\sum _{j=1}^{n}{w}_{j}}=1.$$
Given this scenario, the following table describes the supported loss
functions that you can specify by using the 'LossFun'
name-value pair
argument.
Loss Function | Value of LossFun | Equation |
---|---|---|
Binomial deviance | 'binodeviance' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |
Exponential loss | 'exponential' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |
Classification error | 'classiferror' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$ It is the weighted fraction of misclassified observations where $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function. |
Hinge loss | 'hinge' | $$L={\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$$ |
Logit loss | 'logit' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right)}.$$ |
Minimal cost | 'mincost' | Minimal cost. The software computes the weighted minimal cost using this procedure for observations j = 1,...,n.
The weighted, average, minimum cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ |
Quadratic loss | 'quadratic' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}}.$$ |
This figure compares the loss functions (except 'mincost'
) for one
observation over m. Some functions are normalized to pass through [0,1].
There are two costs associated with classification: the true misclassification cost per class, and the expected misclassification cost per observation.
You can set the true misclassification cost per class in the Cost
name-value
pair when you create the classifier using the fitctree
method. Cost(i,j)
is
the cost of classifying an observation into class j
if
its true class is i
. By default, Cost(i,j)=1
if i~=j
,
and Cost(i,j)=0
if i=j
. In other
words, the cost is 0
for correct classification,
and 1
for incorrect classification.
There are two costs associated with classification: the true misclassification cost per class, and the expected misclassification cost per observation.
Suppose you have Nobs
observations that you
want to classify with a trained classifier. Suppose you have K
classes.
You place the observations into a matrix Xnew
with
one observation per row.
The expected cost matrix CE
has size Nobs
-by-K
.
Each row of CE
contains the expected (average)
cost of classifying the observation into each of the K
classes. CE(n,k)
is
$$\sum _{i=1}^{K}\widehat{P}\left(i|Xnew(n)\right)C\left(k|i\right)},$$
where
K is the number of classes.
$$\widehat{P}\left(i|Xnew(n)\right)$$ is the posterior probability of class i for observation Xnew(n).
$$C\left(k|i\right)$$ is the true misclassification cost of classifying an observation as k when its true class is i.
For trees, the score of a classification of a leaf node is the posterior probability of the classification at that node. The posterior probability of the classification at a node is the number of training sequences that lead to that node with the classification, divided by the number of training sequences that lead to that node.
For example, consider classifying a predictor X
as true
when X
< 0.15
or X
> 0.95
, and X
is
false otherwise.
Generate 100 random points and classify them:
rng(0,'twister') % for reproducibility X = rand(100,1); Y = (abs(X - .55) > .4); tree = fitctree(X,Y); view(tree,'Mode','Graph')
Prune the tree:
tree1 = prune(tree,'Level',1); view(tree1,'Mode','Graph')
The pruned tree correctly classifies observations that are less
than 0.15 as true
. It also correctly classifies
observations from .15 to .94 as false
. However,
it incorrectly classifies observations that are greater than .94 as false
.
Therefore, the score for observations that are greater than .15 should
be about .05/.85=.06 for true
, and about .8/.85=.94
for false
.
Compute the prediction scores for the first 10 rows of X
:
[~,score] = predict(tree1,X(1:10)); [score X(1:10,:)]
ans = 10×3
0.9059 0.0941 0.8147
0.9059 0.0941 0.9058
0 1.0000 0.1270
0.9059 0.0941 0.9134
0.9059 0.0941 0.6324
0 1.0000 0.0975
0.9059 0.0941 0.2785
0.9059 0.0941 0.5469
0.9059 0.0941 0.9575
0.9059 0.0941 0.9649
Indeed, every value of X
(the right-most
column) that is less than 0.15 has associated scores (the left and
center columns) of 0
and 1
,
while the other values of X
have associated scores
of 0.91
and 0.09
. The difference
(score 0.09
instead of the expected .06
)
is due to a statistical fluctuation: there are 8
observations
in X
in the range (.95,1)
instead
of the expected 5
observations.
This function supports tall arrays for out-of-memory data with the limitation:
Only one output is supported.
For more information, see Tall Arrays (MATLAB).
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