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.
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 positiveclass 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 stored in the
Prior
property. Therefore,
Given this scenario, the following table describes the supported loss functions that you can specify by using the LossFun
namevalue 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 userspecified 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. 
Crossentropy loss  'crossentropy'  'crossentropy' is appropriate only for neural network models.
The weighted crossentropy loss is
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'  'mincost' is appropriate only if classification scores are posterior
probabilities.
The software computes the weighted minimal
expected classification cost using this procedure for observations
j = 1,...,n.
Estimate the expected misclassification cost of
classifying the observation
X_{j} into
the class k:
f(X_{j})
is the column vector of class posterior probabilities for
the observation
X_{j}.
C is the cost matrix stored in the
Cost property of the model. For observation j, predict the class
label corresponding to the minimal expected
misclassification cost:
Using C, identify the cost incurred
(c_{j}) for
making the prediction.
The weighted average of the minimal expected
misclassification cost loss is

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).