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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.
Note:

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
.
Multicolumn 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. 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 or character 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.
Y
— Class labelsClass labels, specified as a categorical or character 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: single
 double
 categorical
 char
 logical
 cell
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single 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 commaseparated pair consisting
of 'LossFun'
and a builtin, lossfunction name
or function handle.
The following lists available loss functions. Specify one using its corresponding character vector.
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
byK
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
byK
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
by1
numeric vector of observation weights. If you pass W
,
the software normalizes them to sum to 1
.
Cost
is a KbyK
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.
'Weights'
— Observation weightsones(size(X,1))
(default)  name of a variable in TBL
 numeric vector of positive valuesObservation weights, specified as the commaseparated 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
vectors. 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
Name,Value
arguments associated with pruning
subtrees:
'Subtrees'
— Pruning level'all'
Pruning level, specified as the commaseparated 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
indicates
the completely pruned tree (i.e., just the root node).max(tree.PruneList)
If you specify 'all'
, then CompactClassificationTree.loss
operates
on all subtrees (i.e., the entire pruning sequence). This specification
is equivalent to using 0:max(tree.PruneList)
.
CompactClassificationTree.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'
'TreeSize'
— Tree size'se'
(default)  'min'
Tree size, specified as the commaseparated pair consisting
of 'TreeSize'
and one of the following character
vectors:
'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
namevalue
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 outofsample performance is to prune a tree (or restrict its growth) so that insample and outofsample 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 = 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 = 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 outofsample 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 comparing the same type of loss among many models, lower loss indicates a better predictive model.
Suppose that:
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 to the average loss.
For algorithms that support multiclass classification (that is, K ≥ 3):
y_{j}^{*} is
a vector of K – 1 zeros, and a 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,
$$\sum _{j=1}^{n}{w}_{j}}=1.$$
The supported loss functions are:
Binomial deviance, specified using 'LossFun','binodeviance'
.
Its equation is
$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[2{m}_{j}\right]\right\}}.$$
Exponential loss, specified using 'LossFun','exponential'
.
Its equation is
$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left({m}_{j}\right)}.$$
Classification error, specified using 'LossFun','classiferror'
.
It is the weighted fraction of misclassified observations, with equation
$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$
$${\widehat{y}}_{j}$$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.
Hinge loss, specified using 'LossFun','hinge'
.
Its equation is
$$L={\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1{m}_{j}\right\}.$$
Logit loss, specified using 'LossFun','logit'
.
Its equation is
$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left({m}_{j}\right)\right)}.$$
Minimal cost, specified using 'LossFun','mincost'
.
The software computes the weighted minimal cost using this procedure
for observations j = 1,...,n:
Estimate the 1byK vector of expected classification costs for observation j
$${\gamma}_{j}=f{\left({X}_{j}\right)}^{\prime}C.$$
f(X_{j})
is the column vector of class posterior probabilities for binary and
multiclass classification. C is the cost matrix
the input model stores in the property Cost
.
For observation j, predict the class label corresponding to the minimum, expected classification cost:
$${\widehat{y}}_{j}=\underset{j=1,\mathrm{...},K}{\mathrm{min}}\left({\gamma}_{j}\right).$$
Using C, identify the cost incurred (c_{j}) for making the prediction.
The weighted, average, minimum cost loss is
$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$
Quadratic loss, specified using 'LossFun','quadratic'
.
Its equation is
$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1{m}_{j}\right)}^{2}}.$$
This figure compares some of the loss functions 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
namevalue
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
byK
.
Each row of CE
contains the expected (average)
cost of classifying the observation into each of the K
classes. CE(n,k)
is
$$\sum _{i=1}^{K}\widehat{P}\left(iXnew(n)\right)C\left(ki\right)},$$
where
K is the number of classes.
$$\widehat{P}\left(iXnew(n)\right)$$ is the posterior probability of class i for observation Xnew(n).
$$C\left(ki\right)$$ 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 = 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 rightmost
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 outofmemory data with the limitation:
Only one output is supported.
For more information, see Tall Arrays (MATLAB).
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