loss
Classification error
Syntax
Description
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.
Input Arguments
tree
— Trained classification tree
ClassificationTree
model object | CompactClassificationTree
model object
Trained 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 data
table
Sample 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 classify
numeric matrix
ResponseVarName
— Response variable name
name of a variable in TBL
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 labels
categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors
Class 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
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
LossFun
— Loss function
'mincost'
(default) | 'binodeviance'
| 'classifcost'
| 'classiferror'
| 'exponential'
| 'hinge'
| 'logit'
| 'quadratic'
| function handle
Loss 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 'classifcost'
Observed misclassification cost 'classiferror'
Misclassified rate in decimal 'exponential'
Exponential loss 'hinge'
Hinge loss 'logit'
Logistic loss 'mincost'
Minimal expected misclassification cost (for classification scores that are posterior probabilities) 'quadratic'
Quadratic loss 'mincost'
is appropriate for classification scores that are posterior probabilities. Classification trees return posterior probabilities as classification scores by default (seepredict
).Specify your own function using function handle notation.
Suppose that
n
be the number of observations inX
andK
be the number of distinct classes (numel(tree.ClassNames)
). Your function must have this signaturewhere:lossvalue =
lossfun
(C,S,W,Cost)The output argument
lossvalue
is a scalar.You choose the function name (
lossfun
).C
is ann
-by-K
logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order intree.ClassNames
.Construct
C
by settingC(p,q) = 1
if observationp
is in classq
, for each row. Set all other elements of rowp
to0
.S
is ann
-by-K
numeric matrix of classification scores. The column order corresponds to the class order intree.ClassNames
.S
is a matrix of classification scores, similar to the output ofpredict
.W
is ann
-by-1 numeric vector of observation weights. If you passW
, the software normalizes them to sum to1
.Cost
is a K-by-K
numeric matrix of misclassification costs. For example,Cost = ones(K) - eye(K)
specifies a cost of0
for correct classification, and1
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 weights
ones(size(X,1),1)
(default) | name of a variable in TBL
| numeric vector of positive values
Observation 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
0 (default) | vector of nonnegative integers | '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
, whereL
andse
relate to the smallest value inSubtrees
).'min'
—loss
returns the element ofSubtrees
with smallest loss, usually the smallest element ofSubtrees
.
Output Arguments
L
— Classification loss
vector of scalar values
Classification
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 loss
vector of scalar values
Standard error of loss, returned as a vector the length of
Subtrees
.
NLeaf
— Number of leaf nodes
vector of integer values
Number of leaves (terminal nodes) in the pruned subtrees, returned as a
vector the length of Subtrees
.
bestlevel
— Best pruning level
scalar value
Best 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
, whereL
andse
relate to the smallest value inSubtrees
).TreeSize
='min'
—loss
returns the element ofSubtrees
with smallest loss, usually the smallest element ofSubtrees
.
By default, bestlevel
is the pruning level that gives
loss within one standard deviation of minimal loss.
Examples
Compute the In-sample Classification Error
Compute the resubstituted classification error for the ionosphere
data set.
load ionosphere
tree = fitctree(X,Y);
L = loss(tree,X,Y)
L = 0.0114
Examine the Classification Error for Each Subtree
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')
More About
Classification Loss
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 (or the first or second class in the
ClassNames
property), respectively.f(X_{j}) is the positive-class 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 theClassNames
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 stored in the
Prior
property. 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 argument.
Loss Function | Value of LossFun | Equation |
---|---|---|
Binomial deviance | 'binodeviance' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |
Observed misclassification cost | 'classifcost' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}{c}_{{y}_{j}{\widehat{y}}_{j}},$$ where $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the maximal score, and $${c}_{{y}_{j}{\widehat{y}}_{j}}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_{j}$$ when its true class is y_{j}. |
Misclassified rate in decimal | 'classiferror' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\},$$ where I{·} is the indicator function. |
Cross-entropy loss | 'crossentropy' |
The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^{n}\frac{{\tilde{w}}_{j}\mathrm{log}({m}_{j})}{Kn}},$$ where the weights $${\tilde{w}}_{j}$$ are normalized to sum to n instead of 1. |
Exponential loss | 'exponential' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |
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 expected misclassification cost | 'mincost' |
The software computes the weighted minimal expected classification cost using this procedure for observations j = 1,...,n.
The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ |
Quadratic loss | 'quadratic' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}}.$$ |
If you use the default cost matrix (whose element value is 0 for correct classification
and 1 for incorrect classification), then the loss values for
'classifcost'
, 'classiferror'
, and
'mincost'
are identical. For a model with a nondefault cost matrix,
the 'classifcost'
loss is equivalent to the 'mincost'
loss most of the time. These losses can be different if prediction into the class with
maximal posterior probability is different from prediction into the class with minimal
expected cost. Note that 'mincost'
is appropriate only if classification
scores are posterior probabilities.
This figure compares the loss functions (except 'classifcost'
,
'crossentropy'
, and 'mincost'
) over the score
m for one observation. Some functions are normalized to pass through
the point (0,1).
True Misclassification Cost
The true misclassification cost is the cost of classifying an observation into an incorrect class.
You can set the true misclassification cost per class by using the 'Cost'
name-value argument when you create the classifier. Cost(i,j)
is the cost
of classifying an observation into class j
when 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.
Expected Misclassification Cost
The expected misclassification cost per observation is an averaged cost of classifying the observation into each class.
Suppose you have Nobs
observations that you want to classify with a trained
classifier, and you have K
classes. You place the observations
into a matrix X
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
$$\sum _{i=1}^{K}\widehat{P}\left(i|X(n)\right)C\left(k|i\right)},$$
where:
K is the number of classes.
$$\widehat{P}\left(i|X(n)\right)$$ is the posterior probability of class i for observation X(n).
$$C\left(k|i\right)$$ is the true misclassification cost of classifying an observation as k when its true class is i.
Score (tree)
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.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
Usage notes and limitations:
Only one output is supported.
You can use models trained on either in-memory or tall data with this function.
For more information, see Tall Arrays.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
loss
executes on a GPU in these cases only:The input argument
X
is agpuArray
.The input argument
tbl
containsgpuArray
predictor variables.The input argument
mdl
was fitted with GPU array input arguments.
If the classification tree model was trained with surrogate splits, these limitations apply:
You cannot specify the input argument
X
as agpuArray
.You cannot specify the input argument
tbl
as a table containinggpuArray
elements.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
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