Nowadays many people have no time, so, it is for such busy person.
The author entrust complement of the logic to reader.
Graphical User Interface: GUI
Or
Or
Then reader will understand what this package is.
Confidence Level | No. of Hits | No. of False alarms |
---|---|---|
5 = definitely present | \(H_{5}\) | \(F_{5}\) |
4 = probably present | \(H_{4}\) | \(F_{4}\) |
3 = equivocal | \(H_{3}\) | \(F_{3}\) |
2 = probably absent | \(H_{2}\) | \(F_{2}\) |
1 = questionable | \(H_{1}\) | \(F_{1}\) |
Note that \(H_{c},F_c \in \mathbb{N} \cup\{0\}\) for \(c=1,2,...,5\).
\[\begin{eqnarray*} H_{c } & \sim &\text{Binomial} ( p_{c}, N_{L} ), \text{ for $c=1,2,...,C$.}\\ F_{c } & \sim &\text{Poisson}( (\lambda _{c} -\lambda _{c+1} )\times N_{I} ), \text{ for $c=1,2,...,C-1$.}\\ \lambda _{c}& =& - \log \Phi ( z_{c } ),\text{ for $c=1,2,...,C$.}\\ p_{c} &=&\Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}), \text{ for $c=1,2,...,C-1$.}\\ p_C & =& 1-\Phi (\frac{z_{C}-\mu}{\sigma}),\\ F_{C} & \sim & \text{Poisson}( (\lambda _{C} - 0)N_I),\\ dz_c=z_{c+1}-z_{c} &\sim& \text{Uniform}(0,\infty), \text{ for $c=1,2,...,C-1$.}\\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).
Or equivalently, if \(C=5\)
\[\begin{eqnarray*} H_{1 } & \sim &\text{Binomial} ( p_{1}, N_{L} ) \\ H_{2 } & \sim &\text{Binomial} ( p_{2}, N_{L} ) \\ H_{3 } & \sim &\text{Binomial} ( p_{3}, N_{L} ) \\ H_{4 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ H_{5 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ F_{1 } & \sim &\text{Poisson}( (\lambda _{1} -\lambda _{2} )\times N_{I} ), \\ F_{2 } & \sim &\text{Poisson}( (\lambda _{2} -\lambda _{3} )\times N_{I} ), \\ F_{3 } & \sim &\text{Poisson}( (\lambda _{3} -\lambda _{4} )\times N_{I} ), \\ F_{4 } & \sim &\text{Poisson}( (\lambda _{4} -\lambda _{5} )\times N_{I} ), \\ F_{5 } & \sim &\text{Poisson}( (\lambda _{5} -0 )\times N_{I} ), \text{Be careful !!:'-D}\\ \lambda _{1}& =& - \log \Phi ( z_{1 } ),\\ \lambda _{2}& =& - \log \Phi ( z_{2 } ),\\ \lambda _{3}& =& - \log \Phi ( z_{3 } ),\\ \lambda _{4}& =& - \log \Phi ( z_{4 } ),\\ \lambda _{5}& =& - \log \Phi ( z_{5 } ),\\ p_{1} &:=&\Phi (\frac{z_{2}-\mu}{\sigma})-\Phi (\frac{z_{1}-\mu}{\sigma}), \\ p_{2} &:=&\Phi (\frac{z_{3}-\mu}{\sigma})-\Phi (\frac{z_{2}-\mu}{\sigma}), \\ p_{3} &:=&\Phi (\frac{z_{c4}-\mu}{\sigma})-\Phi (\frac{z_{3}-\mu}{\sigma}), \\ p_{4} &:=&\Phi (\frac{z_{5}-\mu}{\sigma})-\Phi (\frac{z_{4}-\mu}{\sigma}), \\ p_5 &:=& 1-\Phi (\frac{z_{5}-\mu}{\sigma}),\text{Be careful !!:'-D}\\ dz_c=z_{2}-z_{1} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{3}-z_{2} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{4}-z_{3} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{5}-z_{4} &\sim& \text{Uniform}(0,\infty), \\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).
\[\begin{eqnarray*} H_{c } & \sim &\text{Binomial} ( p_{c}, N_{L} ), \text{ for $c=1,2,...,C$.}\\ F_{c } & \sim &\text{Poisson}( (\lambda _{c} -\lambda _{c+1} )\times N_{L} ), \text{ for $c=1,2,...,C-1$.}\\ \lambda _{c}& =& - \log \Phi ( z_{c } ),\text{ for $c=1,2,...,C$.}\\ p_{c} &=&\Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}), \text{ for $c=1,2,...,C-1$.}\\ p_C & =& 1-\Phi (\frac{z_{C}-\mu}{\sigma}),\\ F_{C} & \sim & \text{Poisson}( (\lambda _{C} - 0)N_L),\\ dz_c=z_{c+1}-z_{c} &\sim& \text{Uniform}(0,\infty), \text{ for $c=1,2,...,C-1$.}\\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).
Or equivalently, if \(C=5\)
\[\begin{eqnarray*} H_{1 } & \sim &\text{Binomial} ( p_{1}, N_{L} ) \\ H_{2 } & \sim &\text{Binomial} ( p_{2}, N_{L} ) \\ H_{3 } & \sim &\text{Binomial} ( p_{3}, N_{L} ) \\ H_{4 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ H_{5 } & \sim &\text{Binomial} ( p_{4}, N_{L} ) \\ F_{1 } & \sim &\text{Poisson}( (\lambda _{1} -\lambda _{2} )\times N_{L} ), \\ F_{2 } & \sim &\text{Poisson}( (\lambda _{2} -\lambda _{3} )\times N_{L} ), \\ F_{3 } & \sim &\text{Poisson}( (\lambda _{3} -\lambda _{4} )\times N_{L} ), \\ F_{4 } & \sim &\text{Poisson}( (\lambda _{4} -\lambda _{5} )\times N_{L} ), \\ F_{5 } & \sim &\text{Poisson}( (\lambda _{5} -0 )\times N_{L} ), \text{Be careful !!:'-D}\\ \lambda _{1}& =& - \log \Phi ( z_{1 } ),\\ \lambda _{2}& =& - \log \Phi ( z_{2 } ),\\ \lambda _{3}& =& - \log \Phi ( z_{3 } ),\\ \lambda _{4}& =& - \log \Phi ( z_{4 } ),\\ \lambda _{5}& =& - \log \Phi ( z_{5 } ),\\ p_{1} &:=&\Phi (\frac{z_{2}-\mu}{\sigma})-\Phi (\frac{z_{1}-\mu}{\sigma}), \\ p_{2} &:=&\Phi (\frac{z_{3}-\mu}{\sigma})-\Phi (\frac{z_{2}-\mu}{\sigma}), \\ p_{3} &:=&\Phi (\frac{z_{c4}-\mu}{\sigma})-\Phi (\frac{z_{3}-\mu}{\sigma}), \\ p_{4} &:=&\Phi (\frac{z_{5}-\mu}{\sigma})-\Phi (\frac{z_{4}-\mu}{\sigma}), \\ p_5 &:=& 1-\Phi (\frac{z_{5}-\mu}{\sigma}),\text{Be careful !!:'-D}\\ dz_c=z_{2}-z_{1} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{3}-z_{2} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{4}-z_{3} &\sim& \text{Uniform}(0,\infty), \\ dz_c=z_{5}-z_{4} &\sim& \text{Uniform}(0,\infty), \\ \mu &\sim& \text{Uniform}(-\infty,\infty),\\ \sigma &\sim& \text{Uniform}(0,\infty),\\ \end{eqnarray*}\] Our model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C-1}\), \(\mu\), and \(\sigma\). Notation \(\text{Uniform}( -\infty,100000)\) means the improper uniform distribution of its support is the unbounded interval \(( -\infty,100000)\).
\({}^{\dagger}\) traditionally, case means modality in this context.
Two readers and two modalities and three kind of confidence levels.
Confidence Level | Modality ID | Reader ID | Number of Hits | Number of False alarms |
---|---|---|---|---|
3 = definitely present | 1 | 1 | \(H_{3,1,1}\) | \(F_{3,1,1}\) |
2 = equivocal | 1 | 1 | \(H_{2,1,1}\) | \(F_{2,1,1}\) |
1 = questionable | 1 | 1 | \(H_{1,1,1}\) | \(F_{1,1,1}\) |
3 = definitely present | 1 | 2 | \(H_{3,1,2}\) | \(F_{3,1,2}\) |
2 = equivocal | 1 | 2 | \(H_{2,1,2}\) | \(F_{2,1,2}\) |
1 = questionable | 1 | 2 | \(H_{1,1,2}\) | \(F_{1,1,2}\) |
3 = definitely present | 2 | 1 | \(H_{3,2,1}\) | \(F_{3,2,1}\) |
2 = equivocal | 2 | 1 | \(H_{2,2,1}\) | \(F_{2,2,1}\) |
1 = questionable | 2 | 1 | \(H_{1,2,1}\) | \(F_{1,2,1}\) |
3 = definitely present | 2 | 2 | \(H_{3,2,2}\) | \(F_{3,2,2}\) |
2 = equivocal | 2 | 2 | \(H_{2,2,2}\) | \(F_{2,2,2}\) |
1 = questionable | 2 | 2 | \(H_{1,2,2}\) | \(F_{1,2,2}\) |
\[\begin{eqnarray*} H_{c,m,r} & \sim &\text{Binomial }( p_{c,m,r}, N_L ),\\ F_{c,m,r} &\sim& \text{Poisson }( ( \lambda _{c} - \lambda _{c+1})N_L ),\\ \lambda _{c}& =& - \log \Phi (z_{c }),\\ p_{c,m,r} &:=&\Phi (\frac{z_{c +1}-\mu_{m,r}}{\sigma_{m,r}})-\Phi (\frac{z_{c}-\mu_{m,r}}{\sigma_{m,r}}), \\ p_C & =& 1-\Phi (\frac{z_{C}-\mu_{m,r}}{\sigma_{m,r}}),\\ F_{C,m,r} & \sim &\text{Poisson } ( (\lambda _{C} - 0)N_I),\\ A_{m,r}&:=&\Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \\ A_{m,r}&\sim&\text{Normal} (A_{m},\sigma_{r}^2), \\ dz_c&:=&z_{c+1}-z_{c},\\ dz_c, \sigma_{m,r} &\sim& \text{Uniform}(0,\infty),\\ z_{c} &\sim& \text{Uniform}( -\infty,100000),\\ A_{m} &\sim& \text{Uniform}(0,1).\\ \end{eqnarray*}\] Our new model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C}\), \(A_{m}\), \(\sigma_{r}\), \(\mu_{m,r}\), and \(\sigma_{m,r}\).
\[\begin{eqnarray*} H_{c,m,r} & \sim &\text{Binomial }( p_{c,m,r}, N_L ),\\ F_{c,m,r} &\sim& \text{Poisson }( ( \lambda _{c} - \lambda _{c+1})N_L ),\\ \lambda _{c}& =& - \log \Phi (z_{c }),\\ p_{c,m,r} &:=&\Phi (\frac{z_{c +1}-\mu_{m,r}}{\sigma_{m,r}})-\Phi (\frac{z_{c}-\mu_{m,r}}{\sigma_{m,r}}), \\ p_C & =& 1-\Phi (\frac{z_{C}-\mu_{m,r}}{\sigma_{m,r}}),\\ F_{C,m,r} & \sim &\text{Poisson } ( (\lambda _{C} - 0)N_I),\\ A_{m,r}&:=&\Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \\ A_{m,r}&\sim&\text{Normal} ( \color{red}{A},\sigma_{r}^2), \\ dz_c&:=&z_{c+1}-z_{c},\\ dz_c, \sigma_{m,r} &\sim& \text{Uniform}(0,\infty),\\ z_{c} &\sim& \text{Uniform}( -\infty,100000),\\ \color{red}{A} &\sim& \text{Uniform}(0,1).\\ \end{eqnarray*}\] Our new model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C}\), \(A\), \(\sigma_{r}\), \(\mu_{m,r}\), and \(\sigma_{m,r}\).