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tolinternalfunc.py
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tolinternalfunc.py
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#internal functions, same code as nptolint.py
import numpy as np
import scipy.stats
import scipy.optimize as opt
import pandas as pd
#import tolinternalfunc as tif
def length(x):
if type(x) == float or type(x) == int or type(x) == np.int32 or type(x) == np.float64 or type(x) == np.float32 or type(x) == np.int64:
return 1
return len(x)
def f(n,alpha,P):
return alpha - (n * P**(n - 1) - (n - 1) * P**n)
def bisection(a,b,n,alpha,tol=1e-8):
xl = a
xr = b
while np.abs(xl-xr) >= tol:
c = (xl+xr)/2
prod = f(n=n,alpha=alpha,P=xl)*f(n=n,alpha=alpha,P=c)
if prod > tol:
xl = c
else:
if prod < tol:
xr = c
return c
def distfreeest2(n = None, alpha = None, P = None, side = 1):
temp = 0
if n == None:
temp += 1
if alpha == None:
temp +=1
if P == None:
temp += 1
if temp > 1:
return 'Must specify values for any two of n, alpha, and P'
if (side != 1 and side != 2):
return 'Must specify a 1-sided or 2-sided interval'
if side == 1:
if n == None:
ret = int(np.ceil(np.log(alpha)/np.log(P)))
if P == None:
ret = np.exp(np.log(alpha)/n)
ret = float(f'{ret:.4f}')
if alpha == None:
ret = 1-P**n
else:
if alpha == None:
ret = 1-(np.ceil((n*P**(n-1)-(n-1)*P**n)*10000))/10000
if n == None:
ret = int(np.ceil(opt.brentq(f,a=0,b=1e100,args=(alpha,P),maxiter=1000)))
if P == None:
ret = np.ceil(bisection(0,1,alpha =alpha, n = n, tol = 1e-8)*10000)/10000
return ret
def distfreeest(n = None, alpha = None, P = None, side = 1):
if n == None:
if type(alpha) == float:
alpha = [alpha]
if type(P) == float:
P = [P]
A = length(alpha)
B = length(P)
column_names = np.zeros(B)
row_names = np.zeros(A)
matrix = np.zeros((A,B))
for i in range(A):
row_names[i] = alpha[i]
for j in range(B):
column_names[j] = P[j]
matrix[i,j] = distfreeest2(alpha=alpha[i],P=P[j],side=side)
out = pd.DataFrame(matrix,columns = column_names, index = row_names)
if alpha == None:
if type(n) == float or type(n) == int:
n = [n]
if type(P) == float:
P = [P]
A = length(n)
B = length(P)
column_names = np.zeros(B)
row_names = np.zeros(A)
matrix = np.zeros((A,B))
for i in range(A):
row_names[i] = n[i]
for j in range(B):
column_names[j] = P[j]
matrix[i,j] = distfreeest2(n=n[i],P=P[j],side=side)
out = pd.DataFrame(matrix,columns = column_names, index = row_names)
if P == None:
if type(alpha) == float:
alpha = [alpha]
if type(n) == float or type(n) == int:
n = [n]
A = length(alpha)
B = length(n)
print(f'length of alpha = {A}',f'length of n = {B}')
column_names = np.zeros(B)
row_names = np.zeros(A)
matrix = np.zeros((A,B))
for i in range(A):
row_names[i] = alpha[i]
for j in range(B):
column_names[j] = n[j]
matrix[i,j] = distfreeest2(alpha=alpha[i],n=n[j],side=side)
out = pd.DataFrame(matrix,columns = column_names, index = row_names)
return out
def nptolint(x,alpha = 0.05, P = 0.99, side = 1, method = 'WILKS', upper = None, lower = None):
'''
nptolint(x,alpha = 0.05, P = 0.99, side = 1, method = ('WILKS','WALD','HM','YM'), upper = None, lower = None):
Parameters
----------
x: list
A vector of data which no distributional assumptions are made.
The data is only assumed to come from a continuous distribution.
alpha: float, optional
The level chosen such that 1-alpha is the confidence level.
The default is 0.05.
P: float, optional
The proportion of the population to be covered by this tolerance
interval. The default is 0.99.
side: 1 or 2, optional
Whether a 1-sided or 2-sided tolerance interval is required
(determined by side = 1 or side = 2, respectively). The default is 1.
method: string, optional
The method for determining which indices of the ordered observations will be used for the tolerance intervals.
"WILKS" is the Wilks method, which produces tolerance bounds symmetric
about the observed center of the data by using the beta distribution.
"WALD" is the Wald method, which produces (possibly) multiple tolerance
bounds for side = 2 (each having at least the specified confidence level),
but is the same as method = "WILKS" for side = 1.
"HM" is the Hahn-Meeker method, which is based on the binomial distribution,
but the upper and lower bounds may exceed the minimum and maximum of the sample data.
For side = 2, this method will yield two intervals if an odd number of
observations are to be trimmed from each side.
"YM" is the Young-Mathew method for performing interpolation or
extrapolation based on the order statistics.
See below for more information on this method.
The default is "WILKS"
upper: float, optional
The upper bound of the data. When None, then the maximum of x is used.
If method = "YM" and extrapolation is performed, then upper will be
greater than the maximum. The default value is None.
lower: float, optional
The lower bound of the data. When None, then the minimum of x is used.
If method = "YM" and extrapolation is performed, then lower will be
less than the minimum. The default value is None.
Details
For the Young-Mathew (YM) method, interpolation or extrapolation is performed.
When side = 1, two intervals are given: one based on linear
interpolation/extrapolation of order statistics (OS-Based) and one based
on fractional order statistics (FOS-Based). When side = 2, only an interval
based on linear interpolation/extrapolation of order statistics is given.
Returns
-------
nptolint returns a data frame with items:
alpha: The specified significance level.
P: The proportion of the population covered by this tolerance interval.
1-sided.lower: The 1-sided lower tolerance bound. This is given only if side = 1.
1-sided.upper: The 1-sided upper tolerance bound. This is given only if side = 1.
2-sided.lower: The 2-sided lower tolerance bound. This is given only if side = 2.
2-sided.upper: The 2-sided upper tolerance bound. This is given only if side = 2.
References
----------
Derek S. Young (2010). tolerance: An R Package for Estimating Tolerance Intervals.
Journal of Statistical Software, 36(5), 1-39. URL http://www.jstatsoft.org/v36/i05/.
Bury, K. (1999), Statistical Distributions in Engineering, Cambridge University Press.
Hahn, G. J. and Meeker, W. Q. (1991), Statistical Intervals: A Guide for
Practitioners, Wiley-Interscience.
Wald, A. (1943), An Extension of Wilks' Method for Setting Tolerance Limits,
The Annals of Mathematical Statistics, 14, 45–55.
Wilks, S. S. (1941), Determination of Sample Sizes for Setting Tolerance
Limits, The Annals of Mathematical Statistics, 12, 91–96.
Young, D. S. and Mathew, T. (2014), Improved Nonparametric Tolerance
Intervals Based on Interpolated and Extrapolated Order Statistics, Journal
of Nonparametric Statistics, 26, 415–432.
Examples
--------
## 90%/95% 2-sided nonparametric tolerance intervals for a sample of size 20.
nptol.int(x = x, alpha = 0.10, P = 0.95, side = 1, method = "WILKS", upper = NULL, lower = NULL)
'''
n = length(x)
if type(x) is list or type(x) is tuple:
x = np.array(x)
if n < 2:
return 'nptolint requires more than 1 datapoint.'
xsort = np.sort(x)
if(upper == None):
upper = np.max(x)
if(lower == None):
lower = np.min(x)
if(method == "WILKS"):
if(side == 2):
if(np.floor((n+1)/2) == (n+1)/2):
up = ((n + 1)/2) - 1
else:
up = np.floor((n + 1)/2)
r = np.arange(1,up+1)
#r = np.array([1,2,3,4,.001,.002])
out2 = 1-scipy.stats.beta.cdf(P, n - 2 * r + 1, 2 * r) - (1-alpha)
ti2 =pd.DataFrame([r,out2])
ti2 = ti2.T #transpose the dataframe to make it easier to work with
temp2 = ti2[ti2[1]>0] #Gets all rows where col2 > 0
if len(temp2) == 0:
lower = lower
upper = upper
else:
mins2 = min(temp2[1])
temp2 = temp2[temp2[1]==mins2]
r = int(temp2[0])
lower = xsort[r]
upper = xsort[n-r+1]
d = {'alpha': [alpha], 'P': [P], '2-sided lower':lower, '2-sided upper':upper}
temp = pd.DataFrame(data=d)
if(side ==1):
r = scipy.stats.binom.ppf(alpha, n= n, p=1-P)
s = n-r+1
if(r<1):
lower = lower
else:
lower = xsort[int(r)]
if (s > n):
upper = upper
else:
upper = xsort[int(s-1)]
d = {'alpha': [alpha], 'P': [P], '1-sided lower':lower, '1-sided upper':upper}
temp = pd.DataFrame(data=d)
if(method == "WALD"): #needs to be made more effient for side == 1 and side == 2
t = []
s = []
for i in range(2,n+1):
s.extend(list(range(1,i)))
t.extend((i,)*(i-1))
if side == 1: #Make this code more efficient
r = scipy.stats.binom.ppf(alpha, n = n, p = 1-P)
s = n-r+1
if r < 1:
lower = lower
else:
lower = xsort[int(r)]
if s > n:
upper = upper
else:
upper = xsort[int(s)]
d = {'alpha': [alpha], 'P': [P], '1-sided lower':lower, '1-sided upper':upper}
temp = pd.DataFrame(data = d)
else: #Make this code more efficient
out3 = []
for i in range(len(t)):
out3.append(1 - scipy.stats.beta.cdf(P, int(t[i]-s[i]), int(n-t[i]+s[i]+1))-(1-alpha))
#ti3 = pd.DataFrame({'s':s,'t':t,'out3':out3}).T
ti3 =pd.DataFrame([s,t,out3])
ti3 = ti3.T #transpose the dataframe to make it easier to work with
temp3 = ti3[ti3[2] > 0] #should be >
if len(temp3) == 0:
lower = lower
upper = upper
else:
mins3 = min(temp3[2])
out5 = temp3[temp3[2]==mins3]
s = out5[0]
t = out5[1]
s = s.tolist()
t = t.tolist()
for i in range(len(s)):
t[i] = t[i]-1
s[i] = s[i]-1
lower = np.zeros(len(s))
upper = np.zeros(len(s))
for i in range(len(t)):
lower[i] = xsort[int(s[i])]
upper[i] = xsort[int(t[i])]
if length(lower) == 1 and length(upper) == 1:
d = {'alpha': [alpha], 'P': [P], '2-sided lower':[lower], '2-sided upper':[upper]}
temp = pd.DataFrame(data = d)
else:
d = {'alpha': [alpha], 'P': [P], '2-sided lower':lower[0], '2-sided upper':upper[0]}
d = pd.DataFrame(data=d)
for i in range(1,len(lower)):
d.loc[len(d.index)] = [alpha,P,lower[i],upper[i]]
temp = d
if (method == 'HM'):
ind = range(n+1)
out = scipy.stats.binom.cdf(ind, n = n, p = P) - (1-alpha)
ti = pd.DataFrame([ind,out]).T
temp = ti[ti[1] > 0]
mins = min(temp[1])
HMind = int(temp[temp[1] == mins][0])
diff = n - HMind
if side == 2:
if diff == 0 or int(np.floor(diff/2)) == 0:
if lower != None:
xsort = np.insert(xsort, 0, lower)#pd.DataFrame([lower, xsort])
if upper != None:
xsort = np.insert(xsort, len(xsort), upper)#pd.DataFrame([xsort, upper]) #come back to this area when done
HM = [1] + [len(xsort)]
d = {'alpha': [alpha], 'P': [P], '2-sided lower': lower, '2-sided upper': upper}#xsort[int(HM.loc[0][1])]}
temp = pd.DataFrame(data=d)
else:
if np.floor(diff/2) == diff/2:
v1 = diff/2 #scalar
v2 = diff/2
else:
v1 = [np.floor(diff/2), np.ceil(diff/2)] #list
v2 = [sum(x) for x in zip(v1, [1,-1])] #add v1 to [1,-1] element-wise
if type(v1) == list:
#you can make this block more effient
data = {'v1': [v1[0]], 'v2*': [n- v2[0] +1]}
HM = pd.DataFrame(data = data)
for i in range(1,len(v1)):
HM.loc[len(HM.index)] = [v1[i], n-v2[i]+1]
d = {'alpha': [alpha], 'P': [P], '2-sided lower': xsort[int(HM.loc[0][0])], '2-sided upper': xsort[int(HM.loc[0][1]-1)]}#xsort[int(HM.loc[0][1])]}
d = pd.DataFrame(data = d)
for i in range(1,len(HM)):
d.loc[len(d.index)] = [alpha,P,xsort[int(HM.loc[i][0])],xsort[int(HM.loc[i][1]-1)]]
temp = d
else:
data = {'v1': [v1], 'v2*': [n-v2+1]}
HM = pd.DataFrame(data = data)
d = {'alpha': [alpha], 'P': [P], '2-sided lower': xsort[int(HM.loc[0][0])], '2-sided upper': xsort[int(HM.loc[0][1])-1]}
temp = pd.DataFrame(data = d)
if len(HM) == 2 and len(HM.loc[0]) == 2: #is the row dim 2 and col dim 2? T/F
if xsort[int(HM.loc[0][0])] == xsort[int(HM.loc[1][0])] and xsort[int(HM.loc[0][1])-1] == xsort[int(HM.loc[1][1])-1]:
temp = temp.loc[0]
temp = pd.DataFrame(temp).T
else:
l = pd.DataFrame(range(n+1),columns=['l'])
lp = pd.DataFrame((1-scipy.stats.binom.cdf(l-1,n,1-P))-(1-alpha),columns=[''])
lowtemp = pd.concat([l,lp],axis=1)
u = pd.DataFrame(range(1,n+2),columns=['u'])
up = pd.DataFrame((scipy.stats.binom.cdf(u-1,n,P))-(1-alpha),columns=[''])
uptemp = pd.concat([u,up],axis=1)
l = lowtemp[lowtemp.loc[:,'']>0]
l = max(l.loc[:,'l'])
if l > 0:
lower = xsort[l-1]
u = uptemp[uptemp.loc[:,'']>0]
u = min(u.loc[:,'u'])
if u < n+1:
upper = xsort[u-1]
d = {'alpha': [alpha], 'P': [P], '1-sided lower': lower, '1-sided upper': upper}
temp = pd.DataFrame(data=d)
if method == 'YM':
nmin = int(np.array(distfreeest(alpha = alpha, P=P, side = side).iloc[0]))
temp = None
if side == 1:
if n < nmin:
temp = extrap(x=x,alpha=alpha,P=P)
else:
temp = interp(x=x,alpha=alpha,P=P)
else:
temp = twosided(x=x,alpha=alpha,P=P)
return temp
def fl(u1,u,n,alpha):
#error of 1e-5 compared to R
return scipy.stats.beta.cdf(u,(n+1)*u1,(n+1)*(1-u1))-1+alpha
def fu(u2,u,n,alpha):
#error of 1e-5 compared to R
return scipy.stats.beta.cdf(u,(n+1)*u2,(n+1)*(1-u2))-alpha
def eps(u,n):
return (n+1)*u-np.floor((n+1)*u)
def neps(u,n):
return -((n+1)*u-np.floor((n+1)*u))
def LSReg(y,x,gamma):
xbar = sum(x)/len(x)
ybar = sum(y)/len(y)
sumx = []
sumx2 = []
sumy = []
sumxy = []
for i in range(len(x)):
sumx.append(x[i])
sumx2.append(x[i]**2)
sumy.append(y[i])
sumxy.append(y[i]*x[i])
ssxx = sum(sumx2) - (1/len(x))*sum(sumx)**2
ssxy = sum(sumxy) - (1/len(y))*sum(sumy)*sum(sumx)
B1 = ssxy/ssxx
B0 = ybar - B1*xbar
return B0+B1*gamma #regression equation with xi = gamma
def interp(x,alpha,P):
n = len(x)
x = sorted(x)
gamma = 1-alpha
out = list(nptolint(range(n+1),alpha=alpha,P=P,side=1)[['1-sided lower','1-sided upper']].loc[0])
s = out[0]
r = out[1]
###Beran-Hall
pil = (gamma-scipy.stats.binom.cdf(n-s-1,n,P))/scipy.stats.binom.pmf(n-s,n,P)
piu = (gamma-scipy.stats.binom.cdf(r-2,n,P))/scipy.stats.binom.pmf(r-1,n,P)
if s == n:
Ql = x[s]
else:
Ql = pil*x[s+1]+(1-pil)*x[s]
if r == 1:
Qu = x[r-1]
else:
Qu = piu*x[r-1] + (1-piu)*x[r-2]
###Hutson
u1 = opt.brentq(fl, a = 0.00001,b=0.99999,args=(1-P,n,alpha))
u2 = opt.brentq(fu, a = 0.00001,b=0.99999,args=(P,n,alpha))
if s == n:
Qhl = x[s]
else:
Qhl = (1 - eps(u1, n)) * x[s] + eps(u1, n) * x[s + 1]
if r == 1:
Qhu = x[r-1]
else:
Qhu = (1 - eps(u2, n)) * x[r - 2] + eps(u2, n) * x[r-1]
names = ['alpha','P','1-sided.lower','1-sided.upper']
temp = pd.DataFrame([[alpha,P,Ql,Qu],[alpha,P,Qhl,Qhu]],columns=names)
temp.index = ['OS-Based','FOS-Based']
#return value is slightly different than R due to rounding. and scipy.stats.beta.cdf() vs pbeta()
return temp
def extrap(x,alpha,P):
n = len(x)
x = sorted(x)
gamma = 1-alpha
out = list(nptolint(range(n+1),alpha=alpha,P=P,side=1)[['1-sided lower','1-sided upper']].loc[0])
pib = -(gamma-scipy.stats.binom.cdf(n-1,n,P))/(scipy.stats.binom.pmf(n-1,n,P))
Qexpl = pib*x[1]+(1-pib)*x[0]
Qexpu = pib*x[n-2]+(1-pib)*x[n-1]
u1b = opt.brentq(fl, a = 0.00001,b=0.99999,args=(1-P,n,alpha))
u2b = opt.brentq(fu, a = 0.00001,b=0.99999,args=(P,n,alpha))
Qhexpl = (1-neps(u1b,n))*x[0]+neps(u1b,n)*x[1]
Qhexpu = (1-neps(u2b,n))*x[n-1]+neps(u2b,n)*x[n-2]
names = ['alpha','P','1-sided.lower','1-sided.upper']
temp = pd.DataFrame([[alpha,P,Qexpl,Qexpu],[alpha,P,Qhexpl,Qhexpu]],columns=names)
temp.index = ['OS-Based','FOS-Based']
return temp
def twosided(x,alpha,P):
n = len(x)
x = sorted(x)
gamma = 1-alpha
out = nptolint(range(n+1),alpha=alpha,P=P,side=2,method='HM')[['2-sided lower','2-sided upper']]
r = np.ravel(np.array(out[['2-sided lower']]).T)
s = np.ravel(np.array(out[['2-sided upper']]).T)
r = [int(x) for x in r]
s = [int(x) for x in s]
if (len(out.index) == 2): #around 430,000 datapoints needed for this to be true
X1L = np.array([x[r[0]],x[r[0]+1]])
X2L = np.array([x[r[1]],x[r[1]+1]])
X1U = np.array([x[s[0]],x[s[0]-1]])
X2U = np.array([x[s[1]],x[s[1]-1]])
g = np.ravel(np.array([(scipy.stats.binom.cdf(s[0]-r[0]-1,n,P),(scipy.stats.binom.cdf(s[0]-(r[0]+1)-1,n,P)))]))
#predict using X1L and g, you are here
out1L = LSReg(X1L,g,gamma)
out2L = LSReg(X2L,g,gamma)
out1U = LSReg(X1U,g,gamma)
out2U = LSReg(X2U,g,gamma)
temp1 = pd.DataFrame({'0':[out1L,out2L,x[r[0]],x[r[1]]]})
temp2 = pd.DataFrame({'1':[x[s[0]],x[s[1]],out1U,out2U]})
temp3 = pd.DataFrame({'2':[x[s[0]]-out1L,x[s[1]]-out2L,out1U-x[r[0]],out2U-x[r[0]]]})
temp = pd.concat([temp1,temp2,temp3],axis=1)
if scipy.stats.binom.cdf(s[1]-r[1]-1,n,P) >= gamma:
indtemp = list(temp['2'])
ind = indtemp.index(max(indtemp))
temp = list(temp.iloc[ind,0:2])
if ind==1 or ind==3:
ord1 = 1
else:
indtemp = list(temp['2'])
ind = indtemp.index(max(indtemp))
temp = list(temp.iloc[ind,0:2])
if ind==1 or ind ==3:
ord1 = 1
else:
ord1 = 2
else:
XL = np.array([x[r[0]],x[r[0]+1]])
if s[0] == length(x):
XU = np.array([x[s[0]-1],x[s[0]-2]])
g = np.ravel(np.array([(scipy.stats.binom.cdf(s[0]-(r[0]+1)-1,n,P)),(scipy.stats.binom.cdf(s[0]-(r[0]+1)-2,n,P))]))
outL = LSReg(XL,g,gamma)
outU = LSReg(XU,g,gamma)
temp1 = pd.DataFrame({'0':[outL,x[r[0]]]})
temp2 = pd.DataFrame({'1':[x[s[0]-1],outU]})
temp3 = pd.DataFrame({'2':[x[s[0]-1]-outL,outU-x[r[0]]]})
temp = pd.concat([temp1,temp2,temp3],axis=1)
else:
XU = np.array([x[s[0]],x[s[0]+1]])
g = np.ravel(np.array([(scipy.stats.binom.cdf(s[0]-r[0]-1,n,P),(scipy.stats.binom.cdf(s[0]-(r[0]+1)-1,n,P)))]))
outL = LSReg(XL,g,gamma)
outU = LSReg(XU,g,gamma)
temp1 = pd.DataFrame({'0':[outL,x[r[0]]]})
temp2 = pd.DataFrame({'1':[x[s[0]],outU]})
temp3 = pd.DataFrame({'2':[x[s[0]]-outL,outU-x[r[0]]]})
temp = pd.concat([temp1,temp2,temp3],axis=1)
if scipy.stats.binom.cdf(s[0]-r[0]-1,n,P) >= gamma:
indtemp = list(temp['2'])
ind = indtemp.index(min(indtemp))
temp = list(temp.iloc[ind,0:2])
else:
temp = list([outL,outU])
temp = pd.DataFrame({'alpha':[alpha], 'P':[P],'2-sided.lower':temp[0],'2-sided.upper':temp[1]},['OS-Based'])
return temp
# def twosided(x,alpha,P):
# n = len(x)
# x = sorted(x)
# gamma = 1-alpha
# out = nptolint(range(n+1),alpha=alpha,P=P,side=2,method='HM')[['2-sided lower','2-sided upper']]
# r = np.ravel(np.array(out[['2-sided lower']]).T)
# s = np.ravel(np.array(out[['2-sided upper']]).T)
# r = [int(x) for x in r]
# s = [int(x) for x in s]
# if (len(out.index) == 2): #around 430,000 datapoints needed for this to be true
# X1L = np.array([x[r[0]],x[r[0]+1]])
# X2L = np.array([x[r[1]],x[r[1]+1]])
# X1U = np.array([x[s[0]],x[s[0]-1]])
# X2U = np.array([x[s[1]],x[s[1]-1]])
# g = np.ravel(np.array([(scipy.stats.binom.cdf(s[0]-r[0]-1,n,P),(scipy.stats.binom.cdf(s[0]-(r[0]+1)-1,n,P)))]))
# #predict using X1L and g, you are here
# out1L = LSReg(X1L,g,gamma)
# out2L = LSReg(X2L,g,gamma)
# out1U = LSReg(X1U,g,gamma)
# out2U = LSReg(X2U,g,gamma)
# temp1 = pd.DataFrame({'0':[out1L,out2L,x[r[0]],x[r[1]]]})
# temp2 = pd.DataFrame({'1':[x[s[0]],x[s[1]],out1U,out2U]})
# temp3 = pd.DataFrame({'2':[x[s[0]]-out1L,x[s[1]]-out2L,out1U-x[r[0]],out2U-x[r[0]]]})
# temp = pd.concat([temp1,temp2,temp3],axis=1)
# if scipy.stats.binom.cdf(s[1]-r[1]-1,n,P) >= gamma:
# indtemp = list(temp['2'])
# ind = indtemp.index(max(indtemp))
# temp = list(temp.iloc[ind,0:2])
# if ind==1 or ind==3:
# ord1 = 1
# else:
# indtemp = list(temp['2'])
# ind = indtemp.index(max(indtemp))
# temp = list(temp.iloc[ind,0:2])
# if ind==1 or ind ==3:
# ord1 = 1
# else:
# ord1 = 2
# else:
# XL = np.array([x[r[0]],x[r[0]+1]])
# XU = np.array([x[s[0]],x[s[0]+1]])
# g = np.ravel(np.array([(scipy.stats.binom.cdf(s[0]-r[0]-1,n,P),(scipy.stats.binom.cdf(s[0]-(r[0]+1)-1,n,P)))]))
# outL = LSReg(XL,g,gamma)
# outU = LSReg(XU,g,gamma)
# temp1 = pd.DataFrame({'0':[outL,x[r[0]]]})
# temp2 = pd.DataFrame({'1':[x[s[0]],outU]})
# temp3 = pd.DataFrame({'2':[x[s[0]]-outL,outU-x[r[0]]]})
# temp = pd.concat([temp1,temp2,temp3],axis=1)
# if scipy.stats.binom.cdf(s[0]-r[0]-1,n,P) >= gamma:
# indtemp = list(temp['2'])
# ind = indtemp.index(min(indtemp))
# temp = list(temp.iloc[ind,0:2])
# else:
# temp = list([outL,outU])
# temp = pd.DataFrame({'alpha':[alpha], 'P':[P],'2-sided.lower':temp[0],'2-sided.upper':temp[1]},['OS-Based'])
# return temp