Getting Started with conditional airport (iv) prediction algorithm for conditional random Airport

The predictive problem of CRF is that given model parameters and input sequence (observation sequence) x, the output sequence (marker sequence) with the most conditional probability is $y ^*$, that is, the observed sequence is labeled. Conditional random field prediction algorithm with HMM or Viterbi algorithm, according to the CRF model can be:

\begin{aligned}
y^* &= \arg \max_yp_w (y|x) \ \
&= \arg \max_y\frac{\exp \left \{w \cdot F (y,x) \right\}}{z_w (x)} \ \
&= \arg \max_y \exp \left \{w \cdot F (y,x) \right\} \ \
&= \arg \max_y \ w \cdot F (y,x)
\end{aligned}

Therefore, the conditional random field prediction problem becomes the best path problem to seek the most non-normalized probability.

\[\arg \max_y \ w \cdot F (y,x) \]

Note that it is only necessary to calculate the non-normalized probabilities, without having to calculate probabilities, which can greatly improve the efficiency. To solve the optimal path, the optimization target is written in the following form:

\[\max_y \sum_{i=1}^n w \cdot f_i (y_{i-1},y_i,x) \]

which

\[f_i (y_{i-1},y_i,x) = \left \{f_1 (y_{i-1},y_i,x), f_2 (y_{i-1},y_i,x),..., f_k (y_{i-1},y_i,x) \right \}^T\]

is a local feature vector.

The Viterbi algorithm is described below. The non-normalized probabilities of j=1,2,... and M are calculated first for each marker of position 1:

\[\delta_1 (j) = W \cdot f_1 (y_0 = start,y_1 = j,x) \]

Generally, by the recursive formula, the individual markers of position I are calculated $l =1,2,... The maximum value of the non-normalized probability of the m$, and the path to the maximum value of the denormalized probability is recorded:

\begin{aligned}
\delta_i (l) &= \max_{1 \le j \le m} \left \{\delta_i (l-1) + w \cdot f_i (y_{i-1} = j,y_i = L,x) \right\}, &\ l= 1 , 2,..., m\\
\psi_i (L) &=\arg\max_{1 \le J \le m} \left \{\delta_{i-1} (L) + w \cdot f_i (y_{i-1} = j,y_i = L,x) \right\}, & \ L =,..., m
\end{aligned}

Until I = n is terminated. At this point the maximum value of the non-normalized probability is

\[\max_y (w \cdot F (y,x)) = \max_{1 \le j \le m} \delta_n (j) \]

And the end of the optimal path

\[y_n^* = \arg \max_{1 \le j \le m} \delta _n (j) \]

From the end of the optimal path, the optimal value of each moment is constantly found:

\[y_i^* = \psi_{i+1} (y^*_{i+1}), \ i = n-1,n-2,..., 1\]

The above is an optimal path, the optimal path is obtained:

\[y^* = (y_1^*,y_2^*,..., y_n^*) ^t\]

This is the Viterbi algorithm that is predicted by the conditional with the airport.

Getting Started with conditional airport (iv) prediction algorithm for conditional random Airport