BayesianFROC (version 1.0.0)

Draw_AUC: Draw the Region of AUC of AFROC

Description

An AFROC curve has two parameter denoted by \(a,b\). Specifying \(a,b\), we can draw an AFROC curve.

Def of AFROC

$$ (\xi(t),\eta(t) ) =(1-e^{-t}, \Phi( b\Phi^{-1}(\exp(-t) )- a ) ).$$

Def of AUC of AFROC

$$ AUC = \int \eta d \xi = \frac{ a }{ \sqrt{1+ b^2} }. $$

Usage

Draw_AUC(a = 0.13, b = 0.19, mesh.for.drawing.curve = 2222)

Arguments

a

One of the parameter of model which characterize AFROC curve

b

One of the parameter of model which characterize AFROC curve

mesh.for.drawing.curve

A positive large integer, indicating number of dots drawing the curves, Default =10000.

Value

none.

Details

We define the so-called FROC curve as a map from 1-dimensional Euclidean space to 2-dimensional Euclidean space, mapping each \(t>0\) to

$$ (x(t),y(t) ) =(t, \Phi(\frac{ \Phi^{-1}(\exp(-t)) - \mu}{\sigma}) ) $$

Sine \(x(t)=t,t>0\) is not bounded, the area under the FROC curve is infinity.

To calculates aleternative notion of AUC in the ordinal ROC theory, we define the so-called AFROC curve:

$$ (\xi(t),\eta(t) ) =(1-e^{-t}, \Phi(\frac{ \Phi^{-1}(\exp(-t)) - \mu}{\sigma}) ) $$

which contained in the rectangular space \([0,1]^2\). Introducing new parameter \(a:= \mu / \sigma\) and \(b:= 1 / \sigma\), we also write $$ (\xi(t),\eta(t) ) =(1-e^{-t}, \Phi( b\Phi^{-1}(\exp(-t) )- a ) )$$

The area Under the (AFROC) curve (breifly, we call it AUC) represents the observer performance. For example, if radiologist detects more lesions with small False Positives (FPs), then AUC would be high.

Using the parameter of the signal distribution, we express AUC as follows,

$$ AUC = \frac{ \mu / \sigma}{ \sqrt{1+ 1/\sigma^2} }.$$

Using new parameter \(a:= \mu / \sigma\) and \(b:= 1 / \sigma\), we also write

$$ AUC = \frac{ a }{ \sqrt{1+ b^2} }. $$

Examples

# NOT RUN {


        Draw_AUC()

     Close_all_graphic_devices() # 2020 August
# }