Documentation |
label = predict(mdl,Xnew)
[label,score]
= predict(mdl,Xnew)
[label,score,cost]
= predict(mdl,Xnew)
label = predict(mdl,Xnew) returns a vector of predicted class labels for a matrix Xnew, based on mdl, a ClassificationKNN model.
[label,score] = predict(mdl,Xnew) returns a matrix of scores, indicating the likelihood that a label comes from a particular class.
[label,score,cost] = predict(mdl,Xnew) returns a matrix of costs; label is the vector of minimal costs for each row of cost
label |
Predicted class labels for the points in Xnew, a vector with length equal to the number of rows of Xnew. The label is the class with minimal expected cost. See Predicted Class Label. |
score |
Numeric matrix of size N-by-K, where N is the number of observations (rows) in Xnew, and K is the number of classes (in mdl.ClassNames). score(i,j) is the posterior probability that row i of Xnew is of class j. See Posterior Probability. |
cost |
Matrix of expected costs of size N-by-K, where N is the number of observations (rows) in Xnew, and K is the number of classes (in mdl.ClassNames). cost(i,j) is the cost of classifying row i of X as class j. See Expected Cost. |
predict classifies so as to minimize the expected classification cost:
$$\widehat{y}=\underset{y=1,\mathrm{...},K}{\mathrm{arg}\mathrm{min}}{\displaystyle \sum _{k=1}^{K}\widehat{P}\left(k|x\right)C\left(y|k\right)},$$
where
$$\widehat{y}$$ is the predicted classification.
K is the number of classes.
$$\widehat{P}\left(k|x\right)$$ is the posterior probability of class k for observation x.
$$C\left(y|k\right)$$ is the cost of classifying an observation as y when its true class is k.
For a vector (single query point) Xnew and model mdl, let:
K be the number of nearest neighbors used in prediction, mdl.NumNeighbors
nbd(mdl,Xnew) be the K nearest neighbors to Xnew in mdl.X
Y(nbd) be the classifications of the points in nbd(mdl,Xnew), namely mdl.Y(nbd)
W(nbd) be the weights of the points in nbd(mdl,Xnew)
prior be the priors of the classes in mdl.Y
If there is a vector of prior probabilities, then the observation weights W are normalized by class to sum to the priors. This might involve a calculation for the point Xnew, because weights can depend on the distance from Xnew to the points in mdl.X.
The posterior probability p(j|Xnew) is
$$p\left(j|\text{Xnew}\right)=\frac{{\displaystyle \sum _{i\in \text{nbd}}W(i){1}_{Y(X(i)=j)}}}{{\displaystyle \sum _{i\in \text{nbd}}W(i)}}.$$
Here, $${1}_{Y(X(i)=j)}$$ means 1 when mdl.Y(i) = j, and 0 otherwise.
There are two costs associated with KNN 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 run fitcknn. 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 KNN classification: the true misclassification cost per class, and the expected misclassification cost per observation. The third output of predict is the expected misclassification cost per observation.
Suppose you have Nobs observations that you want to classify with a trained classifier mdl. Suppose you have K classes. You place the observations into a matrix Xnew with one observation per row. The command
[label,score,cost] = predict(mdl,Xnew)
returns, among other outputs, a cost matrix of size Nobs-by-K. Each row of the cost matrix contains the expected (average) cost of classifying the observation into each of the K classes. cost(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.