Halton_Sequence_Numbers.md

Halton Sequence

To generate random numbers using Halton’s method, I have implemented a function from scratch and then compared the results with the Halton method that is included in the scipy package. We will see the results together.

import numpy as np

def get_halton_num(base, num):
    base_list = []
    i = 0
    while num > 0:
        if num // (base ** i) >= base:
             i += 1
        else:
            if num // (base ** i) == 0:
                base_list.append(0)
                i -= 1
            else:
                base_list.append(num // (base ** i))
                num = num % (base ** i)
                if num == 0 and i != 0:
                    for j in range(i):
                        base_list.append(0)
                    break
                else: 
                    i -= 1
    base_list2 = np.array(base_list)
    base_list2 = base_list2[::-1]
    nn = len(base_list2)
    temp_init = (np.arange(nn) + 1)
    temp_init2 = base ** temp_init
    temp_init2 = 1 / temp_init2
    halton_num = base_list2 * temp_init2
    halton_num = halton_num.sum()
    return halton_num

def get_seq_halton(base, n):
    mylist = np.arange(n) + 1
    mylist = mylist[:(n-1)]
    result = list(map(lambda x: get_halton_num(base = base, num = x), mylist))
    result.insert(0, 0)
    return result

def get_seq_halton2(base1, base2, n, dim = 2):
    xx = get_seq_halton(base = base1, n = n)
    yy = get_seq_halton(base = base2, n = n)
    final_result = np.stack((xx, yy), axis = 1)
    return final_result
get_seq_halton2(base1 = 2, base2 = 3, n = 4, dim = 2)





my_mat = get_seq_halton2(base1 = 2, base2 = 3, n = 10 ** 5, dim = 2)
def f(x):
    res = x[0] ** 2 + x[1] ** 2
    return res
int_result = np.apply_along_axis(f, 1, my_mat)
result_scratch = int_result.mean()




## with package

from scipy.stats.qmc import Halton    
samp = Halton(d = 2, scramble = False, seed = 1)
my_mat2 = samp.random(n = 10 **5)

def f(x):
    res = x[0] ** 2 + x[1] ** 2
    return res
int_result2 = np.apply_along_axis(f, 1, my_mat2)
result_python_package = int_result2.mean()
print("""
Result of my function that implement from scratch: {}, \n\n
Result of python package: {}
""".format(result_scratch, result_python_package))

Result of my function that implement from scratch: 0.6666154594225222, 


Result of python package: 0.6666154594225222