DCM

Determinant based on Confusion Matrix

• Introduction

In the Confusion Matrix, We can know how much certain class is misclassified in the other classes. In this document, I consider normalized confusion matrix as an prior conditional probabilistic within classes and use this to determine answer class in the classification problem

• Problem

  There's N classes C_1, C_2, ..., C_N

  f : trained model(function) with Training Set T

  F : Confusion Matrix with Valid Set V(used while training)

  D : normalized(for each row) Confusion Matrix

  W : normalized weight of Valid Set V

  x : fixed input data

  Let's assume f(x) = [p_1, p_2, ..., p_N]

• Solution

  Before we go on, we can consider D(i,j) as P(argmax[f(x)] = j | y = C_i), and W(i) as P(y = C_i) for prior conditional probability P

  I want to maximize P(y = C_k | f(x) = [p_1, p_2, ..., p_N]) for variable k

  i.e We use argmax_k P(y = C_k | f(x) = [p_1, p_2, ..., p_N]) as a predict label

  Apply bayesian rule on above formula

  P(y = C_k | f(x) = [p_1, p_2, ..., p _N]) = sum_j { P(y = C_k, argmax[f(x)] =j | f(x) = [p_1, p_2, ..., p_N])}
  
  = sum_j { P(f(x) = [p_1, p_2, ..., p_N] | y = C_k, argmax[f(x)] = j) * P(argmax[f(x)] = j | y = C_k) * P(y = C_k)
  
  = sum_j { P(f(x) = [p_1, p_2, ..., p_N] | y = C_k, argmax[f(x)] = j) * D(k,j) * W(k)}
  
  = sum_j { S(k,j;l)*D(k,j)*W(k)}
  

  In the term P(f(x) = [p_1, p_2, ..., p_N] | y = C_k, argmax[f(x)] = j),

  if j == k, the probablity is proportional to p_k, else the probablity is inversely proportional to p_k

  Consider the above property, I set P(f(x) = [p_1, p_2, ..., p_N] | y = C_k, argmax[f(x)] = j) as the following function S(k,j ;l)

  (and you can set any other function. please suggest better form)

  S(k,j ;l) = p_k if k==j 
  
            = l(1-p_k) if k!=j
  

  Search l which performs best score on validation set V in the (N+1)-element discrete space [0, 1/N, 2/N, ..., 1]

• Usage

  from DCM import DCM
  from sklearn.metrics import accuracy_score
  
  weight = [0.3, 0.3, 0.4]
  pred = np.array([[0.1,0.2,0.7],[0.3,0.3,0.4],[0.8,0.1,0.1]])
  true = np.array([2,1,0])
  search_space = [0, 0.33, 0.67, 1]
  
  determinant = DCM(n_classes = 3, weight = weight, predict = pred, true = true, metric = accuracy_score)
  determinant.search(space = search_space)
  
  Test_X = load_test_x() ## some function that load test data
  pred_test =  model(Test_X) ## model is defined on somewhere above
  pred_label = determinant.apply(pred_test)