U25%(1,16) and U25%(1,168)on《C4.5:programs for machine learning》

when calculating
$U_{CF}$(e,N)
CF: Confidence Level(here is 25%)
e:misclassifying counts of current subtree we focus on
N:counts of sub-datasets relevant to current subtree who is under judgment whether to be pruned or not.

“Pr” is short for “Pessimistic error rate”.

$U_{CF}$(e,N)=Pr=$\frac{e+0.5+\frac{Coef{f}^2}{2}+\sqrt{Coef{f}^2\cdot \{(e+0.5）[1-\frac{e+0.5}{N}]+\frac{Coef{f}^2}{4}\}}}{N+Coef{f}^2}①$

The method to get $Coeff$ is as follows:

$\frac{Coeff-Deviation[i-1]}{Deviation[i]-Deviation[i-1]}=\frac{ConfidenceLevel-val[i-1]}{val[i]-val[i-1]}②$

Now Let’s analyse the meaning of ② and how to get “i” in formula ②.

What is Deviation and Val?
Val[] = { 0, 0.001, 0.005, 0.01, 0.05, 0.10, 0.20, 0.40, 1.00},
Deviation[] = {4.0, 3.09, 2.58, 2.33, 1.65, 1.28, 0.84, 0.25, 0.00};

for Normal Distribution：
P{X>Deviation[i]}=val[i],
for example:
P{X>2.58}=0.005
--------------------

Now let’s analyse the whole meaning of formula ②

the red line is $P(X>x)={\int }_x^{+\infty }\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}dx$

the blue line is $f(x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}$
and the two points are:
(deviation[i],val[i])=(0.25,0.4)
(deviation[i-1,val[i-1])=(0.84,0.2)

Let’s amplify the above figure to the following picture:

so tan C=$\frac{Val[i]-Val[i-1]}{Deviation[i-1]-Deviation[i]}=\frac{ConfidenceLevel-Val[i-1]}{Deviation[i-1]-Coeff}$

then we can get ②and compute $Coeff$ and $Coef{f}^2$

but how to get “i” in above formula?
when $Val[i-1]＜Confidencelevel\le Val[i]$
is satisfied,
we get “i”.
when Confidence Level=0.25
i=7
val[i-1]=0.2
val[i]=0.4
deviation[i-1]=0.84
deviation[i]=0.25
substitute the above values in ②
we get Coeff=0.6925
then,
$Coef{f}^2=0.479556$
--------------------

**Now Let’s calculate **
$U_{0.25}(1,16)$
CF=0.25
e=1
N=16
$Coef{f}^2=0.479556$
We’ll get :
Pr=

$\frac{1+0.5+\frac{0.479556}{2}+\sqrt{0.479556\cdot \{(1+0.5）[1-\frac{1+0.5}{16}]+\frac{0.479556}{4}\}}}{16+0.479556}$=0.156681

N·Pr=16×0.156681=2.507
Compared with Quinlan’s book page 42th:

Now Let’s calculate
$U_{0.25}(1,168)$
CF=0.25
e=1
N=168

Pr=$\frac{1+0.5+\frac{0.479556}{2}+\sqrt{0.479556\cdot \{(1+0.5）[1-\frac{1+0.5}{168}]+\frac{0.479556}{4}\}}}{168+0.479556}=0.015536$

$N\cdot U_{0.25}(1,168)=N\cdot Pr=168\cdot 0.015536=2.61$
Compared with Quinlan’s Book on Page 42th:

The code to veriy the above value:

/*************************************************************************/
/*									 */
/*  Statistical routines for C4.5					 */
/*  -----------------------------					 */
/*									 */
/*************************************************************************/
#include  
#include 

									
/*************************************************************************/
/*									 */
/*  Compute the additional errors if the error rate increases to the	 */
/*  upper limit of the confidence level.  The coefficient is the	 */
/*  square of the number of standard deviations corresponding to the	 */
/*  selected confidence level.  (Taken from Documenta Geigy Scientific	 */
/*  Tables (Sixth Edition), p185 (with modifications).)			 */
/*									 */
/*************************************************************************/
float CF=0.25;

float Val[] = {  0,  0.001, 0.005, 0.01, 0.05, 0.10, 0.20, 0.40, 1.00},//概率
      Dev[] = {4.0,  3.09,  2.58,  2.33, 1.65, 1.28, 0.84, 0.25, 0.00};//这个就是分位点
      // P{X>Dev[i]}=val[i],
      // according to standardized normal distribution N(0,1)
      // for example:
      // P{X>2.58}=0.005,


float AddErrs(float N, float e)//叶子的悲观错误
{
    static float Coeff=0;
    float Val0, Pr;

    if ( ! Coeff )//compute only once when pruning trees
    {
	/*  Compute and retain the coefficient value, interpolating from
	    the values in Val and Dev  */

	int i;

	i = 0;
	while ( CF > Val[i] ) i++;//这里可以看到，是根据置信度对悲观错误数量进行计算的。
    //但是呢，也没有像理论中直接计算“悲观错误率”

	Coeff = Dev[i-1] +
		  (Dev[i] - Dev[i-1]) * (CF - Val[i-1]) /(Val[i] - Val[i-1]);
	Coeff = Coeff * Coeff;
    printf("Coeff=%f\n",Coeff);//CF定下来以后，Coeff就是定的
    }

    if ( e < 1E-6 )
    {
	return N * (1 - exp(log(CF) / N));
    }
    else
    if ( e < 0.9999 )
    {
	Val0 = N * (1 - exp(log(CF) / N));
	return Val0 + e * (AddErrs(N, 1.0) - Val0);//这里在进行递归调用
    }
    else
    if ( e + 0.5 >= N )
    {
	return 0.67 * (N - e);
    }
    else
    {
        float fenzi;
        fenzi=(
        e + 0.5 + Coeff/2+ sqrt
        ( 
            Coeff * 
                (
                (e + 0.5) * (1 - (e + 0.5)/N) + Coeff/4
                )
        ) 
            );

        printf("fenzi=%f\n",fenzi);
	Pr = (
        e + 0.5 + Coeff/2+ sqrt
        ( 
            Coeff * 
                (
                (e + 0.5) * (1 - (e + 0.5)/N) + Coeff/4
                )
        ) 
            )
             / (N + Coeff);
       printf("Pr=%f\n",Pr);
	return (N * Pr - e);
    }
}




int main()
{

float result=AddErrs(16, 1);
printf("result=%f\n",result);
    exit(0);


}

running method:

gcc -o stats stats.c -lm
./stats

the result is:
Coeff=0.479556
fenzi=2.582031
Pr=0.156681
result=1.506894