Publisher Correction: A hidden Markov model for lymphatic tumor progression in the head and neck

Correction to: Scientific Reports 10.1038/s41598-021-91544-1, published online 10 June 2021

The original version of this Article contained repeated errors in the Equations.

Equations 1, 4, 15, 16, 21 and 26 were incorrectly aligned.

Equation 2 and 14 contained an incorrect separator (|).

Equation 3 contained a duplication of terms.

The $pa(v)$ in Equations 3, 4 and 5 was incorrectly given in italics.

The product ($\prod$), sum ($\sum$) and fraction operators in Equations 6, 18, 19, 20, 22, 23, 24, 27 and 31 were incorrectly given in a lower-height format.

The numbering indicating Equation 15 was omitted.

The original Equations 1, 2, 3, 4, 5, 6, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29 and 31 are included below.

$$\begin{array}{cc}\hfill P_{\mathit{BN}}\left(Z_v^k=z_v^k|X_v=x_v\right)& =\left(z_v^k+{\left(-1\right)}^{z_v^k}·s_P^k\right)\left(1-x_v\right)\hfill \\ \hfill & +\left((1-z_v^k)+{\left(-1\right)}^{1-z_v^k}·s_N^k\right)x_v\hfill \\ \hfill \end{array}$$ 1

$$\begin{array}{c}{P}_{\mathit{BN}}\left(Z_v^k=0|X_v=1\right)=1-s_N^k\end{array}$$ 2

$$\begin{array}{c}{P}_{\mathit{BN}}\left(X_v=x_v|X_{pa\left(v\right)}=x_{pa\left(v\right)},b_v,t_{pa\left(v\right)v}\right)=x_v+{\left(-1\right)}^{x_v}\left(1-b_v\right){\left(1-t_{pa\left(v\right)v}\right)}^{x_{pa\left(v\right)}}=x_v\\ +{(-1)}^{x_v}\left(1-b_v\right){\left(1-t_{pa\left(v\right)v}\right)}^{x_{pa\left(v\right)}}\end{array}$$ 3

$$\begin{array}{cc}& P_{\mathit{BN}}\left(X_v=0|X_{pa\left(v\right)}=0\right)=1-b_v\hfill \\ \hfill & P_{\mathit{BN}}\left(X_v=1|X_{pa\left(v\right)}=0\right)=b_v\hfill \\ \hfill & P_{\mathit{BN}}\left(X_v=0|X_{pa\left(v\right)}=1\right)=\left(1-b_v\right)\left(1-t_{pa\left(v\right)v}\right)\hfill \\ \hfill & \begin{array}{c}{P}_{\mathit{BN}}\left(X_v=1|X_{pa\left(v\right)}=1\right)=1-\left(1-b_v\right)\left(1-t_{pa\left(v\right)v}\right)\end{array}\hfill \\ \hfill \end{array}$$ 4

$$\begin{array}{c}\hfill \begin{array}{c}{P}_{\mathit{BN}}\left(X_v=x_v|\left\{X_{pa\left(v\right)}=x_{pa\left(v\right)}\right\},\left\{t_{pa\left(v\right)v}\right\},b_v\right)=x_v+{\left(-1\right)}^{x_v}\left(1-b_v\right)\prod _{p\in pa\left(v\right)}{\left(1-t_{\mathit{pv}}\right)}^{x_p}\end{array}\end{array}$$ 5

$$\begin{array}{c}{P}_{\mathit{BN}}\left(\mathcal{Z}|\theta \right)=\prod _{n=1}^N\sum _{\mathit{x}\in {\left\{0,1\right\}}^V}\prod _{v=1}^V\prod _{k\in \mathcal{O}}{P}_{\mathit{BN}}\left(z_{\mathit{nv}}^k|x_v\right)P_{\mathit{BN}}\left(x_v|\left\{x_{\text{pa}\left(v\right)}\right\},\left\{t_{\text{pa}\left(v\right)v}\right\},b_v\right)\end{array}$$ 6

$$\begin{array}{cc}& P_{\mathit{HMM}}\left(\mathit{x}\left[t+1\right]|\mathit{x}\left[t\right]\right)\hfill \\ \hfill & =\prod _{v\in V}Q\left(x_v\left[t+1\right];x_v\left[t\right]\right){\left(P_{\mathit{BN}}\left(x_v\left[t+1\right]|\left\{x_{\text{pa}\left(v\right)}\left[t\right]\right\},\left\{{\tilde{t}}_{\text{pa}\left(v\right)v}\right\},{\tilde{b}}_v\right)\right)}^{1-x_v\left[t\right]}\hfill \\ \hfill \end{array}$$ 14

$$\begin{array}{cc}& Q\left(X_v\left[t+1\right]=0|X_v\left[t\right]=0\right)=1\hfill \\ \hfill & Q\left(X_v\left[t+1\right]=0|X_v\left[t\right]=1\right)=0\hfill \\ \hfill & Q\left(X_v\left[t+1\right]=1|X_v\left[t\right]=0\right)=1\hfill \\ \hfill & \begin{array}{c}Q\left(X_v\left[t+1\right]=1|X_v\left[t\right]=1\right)=1\end{array}\hfill \\ \hfill \end{array}$$ 15

$$\begin{array}{cc}& P_{\mathit{HMM}}\left(\mathit{X}\left[t+1\right]={\mathit{\xi }}_7\mathbf{|}\mathit{X}\left[t\right]={\mathit{\xi }}_5\right)\hfill \\ \hfill & =Q\left(X_1\left[t+1\right]=0|X_1\left[t\right]=0\right)P_{\mathit{BN}}{\left(X_1\left[t+1\right]=0|{\tilde{b}}_1\right)}^1\hfill \\ \hfill & ·Q\left(X_2\left[t+1\right]=1|X_2\left[t\right]=1\right)P_{\mathit{BN}}{\left(X_2\left[t+1\right]=1|X_1\left[t\right]=0,{\tilde{t}}_{12},{\tilde{b}}_2\right)}^0\hfill \\ \hfill & ·Q\left(X_3\left[t+1\right]=1|X_3\left[t\right]=0\right)P_{\mathit{BN}}{\left(X_3\left[t+1\right]=1|X_2\left[t\right]=1,{\tilde{t}}_{23},{\tilde{b}}_3\right)}^1\hfill \\ \hfill & ·Q\left(X_4\left[t+1\right]=0|X_4\left[t\right]=0\right)P_{\mathit{BN}}{\left(X_4\left[t+1\right]=0|X_3\left[t\right]=0,{\tilde{t}}_{34},{\tilde{b}}_4\right)}^1\hfill \\ \hfill & =\left(1-{\tilde{b}}_1\right)·1·\left({\tilde{b}}_3+{\tilde{t}}_{23}-{\tilde{b}}_3{\tilde{t}}_{23}\right)·\left(1-{\tilde{b}}_4\right)\hfill \\ \hfill \end{array}$$ 16

$$\begin{array}{c}P\left(\mathit{z}={\mathit{\zeta }}_j\right)=\sum _{t\in \mathbb{T}}p\left(t\right)·P\left(\mathit{z}={\mathit{\zeta }}_j,t\right)={\left[\sum _{t\in \mathbb{T}}p\left(t\right)·{\mathit{\pi }}^{\top }·{\left(\mathit{A}\right)}^t·\mathit{B}\right]}_j\end{array}$$ 18

$$\begin{array}{c}P\left(\mathcal{Z}|\theta \right)=\prod _{i=1}^{V·\left|\mathcal{O}\right|}P{\left({\mathit{\zeta }}_i|\theta \right)}^{f_i}\end{array}$$ 19

$$\begin{array}{c}P\left(\theta |\mathcal{Z}\right)=\frac{P\left(\mathcal{Z}|\theta \right)P\left(\theta \right)}{\int P\left(\mathcal{Z}|{\theta }^{\prime }\right)P\left({\theta }^{\prime }\right)d{\theta }^{\prime }}\end{array}$$ 20

$$P\left(\theta \right)=\left\{\begin{array}{cc}1& \text{if}\theta \in {\mathcal{S}}^{V\left(V-1\right)}\\ 0& \text{otherwise}\end{array}\right)$$ 21

$$\begin{array}{c}logP\left(\mathcal{Z}|\theta \right)=\sum _{T=1}^{4}log\left[\sum _{t\in \mathbb{T}}{p}_T\left(t\right)·{\mathit{\pi }}^{\top }·{\left(\mathit{A}\right)}^t·\mathit{B}\right]·{\mathit{f}}_T\end{array}$$ 22

$$\begin{array}{c}R\left(X_v=1|\mathit{z},\theta \right)=\frac{P\left(\mathit{Z}=\mathit{z}|X_v=1,\theta \right)P\left(X_v=1|\theta \right)}{P\left(\mathit{Z}=\mathit{z}|\theta \right)}=\frac{{\sum }_{\left\{i:{\xi }_{\mathit{iv}}=1\right\}}P\left(\mathit{Z}=\mathit{z}|{\mathit{\xi }}_i,\theta \right)P\left({\mathit{\xi }}_i|\theta \right)}{P\left(\mathit{Z}=\mathit{z}\mathbf{|}\theta \right)}\end{array}$$ 23

$$\begin{array}{c}{\mathbb{E}}_{\mathit{\theta }}\left[R\left(X_v=1|\mathit{z}\right)\right]=\frac{1}{L}\sum _{k=1}^{L}R\left(X_v=1|\mathit{z},{\theta }_k\right)\end{array}$$ 24

$$\begin{array}{c}\text{match}\left(\mathit{d},\mathit{z}\right):=\left\{\begin{array}{cc}\text{true}& \text{if}{d}_v^{\mathcal{O}}=z_v^{\mathcal{O}}\vee d_v^{\mathcal{O}}=\mathrm{\varnothing };\forall v,O\\ \text{false}& \text{else}\end{array}\right)\end{array}$$ 26

$$\begin{array}{c}R\left(X_v=1|\mathit{d},\theta \right)=\frac{P\left(\mathit{d}|X_v=1,\theta \right)P\left(X_v=1|\theta \right)}{P\left(\mathit{d}|\theta \right)}=\frac{{\sum }_{\left\{i:{\xi }_{\mathit{iv}}=1\right\}}P\left(\mathit{d}|{\mathit{\xi }}_i,\theta \right)P\left({\mathit{\xi }}_i|\theta \right)}{P\left(\mathit{d}|\theta \right)}\end{array}$$ 27

$$\begin{array}{c}P\left(\mathit{d}|\theta \right)=\sum _{\left\{j:\text{match}\left(\mathit{d},{\mathit{\zeta }}_j\right)\right\}}{\left[\sum _{t\in \mathbb{T}}{p}_T\left(t\right)·\mathit{\pi }·{\left(\mathit{A}\right)}^t·\mathit{B}\right]}_j\end{array}$$ 29

$$\begin{array}{c}P\left(\mathit{d}|\theta \right)=\sum _{t\in \mathbb{T}}{p}_T\left(t\right)·\mathit{\pi }·{\left(\mathit{A}\right)}^t·\mathit{B}·{\mathit{c}}^{\mathit{d}}\end{array}$$ 31

The original Article has been corrected.