x=[0.0]*total_sample_number

y=[0.0]*total_sample_number

area=[0.0]*repeat_times

sample_mean=0.0

sample_variance=0.0

for repeat in range(0,repeat_times):

for i in range(0,total_sample_number):

x[i]=rd.uniform(0,1)

y[i]=rd.uniform(0,1)

area[repeat]+=((x[i]**2+y[i]**2)<=1)&(((x[i]-1)**2+(y[i]-1)**2)<=1)

area[repeat]/=total_sample_number

sample_mean+=area[repeat]/repeat_times

for i in range(0,repeat_times):

sample_variance+=(area[i]-sample_mean)**2/repeat_times

if plot==1:

plt.scatter(np.arange(0,repeat_times),area)

plt.title("Area estimation of the samples")

plt.xlabel("Samples")

plt.ylabel("Area Estimation")

plt.grid(True)

plt.show()

print "The average of this ",repeat_times," samples is: ",sample_mean

print "The sample variance of this ",repeat_times," samples is :",sample_variance

return sample_variance

variance=[0]*100

for i in range(1,101):

variance[i-1]=func_1a(50,500*i,0)

plt.scatter(np.arange(500,50500,500),variance,s=1)

plt.title("Variance-Number of Samples")

plt.xlabel("Number of Samples")

plt.ylabel("Variance")

plt.ylim(ymin=-0.00075)

plt.ylim(ymax=0.00075)

plt.grid(True)

result=[0]*repeat_times

result_mean=0

result_variance=0

for i in range(0,repeat_times):

for j in range(0,sample_number):

result[i]+=2.2*function1(rd.uniform(0.8,3))/sample_number

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 1 with standard Monte Carlo Estimation")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

result=[0]*repeat_times

result_mean=0

result_variance=0

for i in range(0,repeat_times):

for j in range(0,sample_number):

result[i]+=39.478*function2(rd.uniform(-math.pi,math.pi),rd.uniform(-math.pi,math.pi))/sample_number

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 2 with standard Monte Carlo Estimation")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

#use pdf of exponential distribution to determine the importance of each interval

#determine the parameter for exponential pdf

lamda=1.0/(np.log(function1(0.8)/function1(1.8)))

sample_allocation=[0]*10

for i in range(0,10):

x1=expon.cdf(0.22*i+0.8,loc=0,scale=lamda)

x2=expon.cdf(0.22*i+1.02,loc=0,scale=lamda)

sample_allocation[i]=sample_number*(x2-x1)

condition=expon.cdf(3,loc=0,scale=lamda)-expon.cdf(0.8,loc=0,scale=lamda)

sum_allocation=0

for i in range(0,9):

sample_allocation[9-i]=int(sample_allocation[9-i]/condition)

sum_allocation+=sample_allocation[9-i]

sample_allocation[0]=sample_number-sum_allocation

print "The allocation of Sample Numbers Derived from Exponential pdf with Lamda=",lamda,"is:\n"

print sample_allocation

result=[0]*repeat_times

result_mean=0

result_variance=0

for i in range(0,repeat_times):

for j in range(0,10):

for k in range(0,sample_allocation[j]):

result[i]+=0.22*function1(rd.uniform(0.22*j+0.8,0.22*j+1.02))/sample_allocation[j]

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 1 with Monte Carlo Estimation Imported with stratification")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

result=[0]*repeat_times

result_mean=0

result_variance=0

lamda=1.0/(np.log(function1(0.8)/function1(1.8)))

envelope_size=expon.cdf(3,loc=0,scale=lamda)-expon.cdf(0.8,loc=0,scale=lamda)

for i in range(0,repeat_times):

for j in range(0,sample_number):

x=rd.uniform(0.8,3)

result[i]+=(function1(x)*envelope_size)/(expon.pdf(x,loc=0,scale=lamda)*sample_number)

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 1 with Monte Carlo Estimation Imported with Importance Sampling")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

#use bivariate normal distribution of uncorrelated X & Y to determine the importance of each interval

#use normal distribution with mean=0, variance=0.8

sample_allocation=np.empty((5,5),dtype=float)

allocation=[0.0]*5

sum_allocation=0.0

for i in range(0,5):

x1=norm.cdf((2.0*i/5-1)*math.pi,loc=0,scale=0.8)

x2=norm.cdf((2.0*i/5-0.6)*math.pi,loc=0,scale=0.8)

allocation[i]=(x2-x1)

condition=(norm.cdf(-math.pi,loc=0,scale=1)-norm.cdf(math.pi,loc=0,scale=1))**2

for i in range(0,5):

for j in range(0,5):

sample_allocation[i][j]=int(allocation[i]*allocation[j]*sample_number/condition)

sum_allocation+=sample_allocation[i][j]

sample_allocation[2][2]+=(sample_number-sum_allocation)

print "The allocation of Sample Numbers Derived from Exponential pdf is:\n"

print sample_allocation

result=[0]*repeat_times

result_mean=0

result_variance=0

for i in range(0,repeat_times):

for j1 in range(0,5):

for j2 in range(0,5):

if sample_allocation[j1][j2]!=0:

for k in range(0,int(sample_allocation[j1][j2])):

x=rd.uniform((2.0*j1/5-1)*math.pi,(2.0*j1/5-0.6)*math.pi)

y=rd.uniform((2.0*j2/5-1)*math.pi,(2.0*j2/5-0.6)*math.pi)

result[i]+=1.579*function2(x,y)/sample_allocation[j1][j2]

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 2 with Monte Carlo Estimation Imported with stratification")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

result=[0.0]*repeat_times

result_mean=0.0

result_variance=0.0

envelope_size=np.sqrt(2)*math.pi*(norm.cdf(math.pi)-norm.cdf(-math.pi))**2

for i in range(0,repeat_times):

for j in range(0,sample_number):

x=rd.uniform(-math.pi,math.pi)

y=rd.uniform(-math.pi,math.pi)

result[i]+=(function2(x,y)*envelope_size)/(norm.pdf(x)*norm.pdf(y)*sample_number)

#((function2(x,y)*envelope_size)/(norm.pdf(x)*norm.pdf(y)*sample_number))

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 2 with Monte Carlo Estimation Imported with Importance Sampling")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

#Normal Monte Carlo Integration Estimate

result=[0]*repeat_times

result_mean=0

result_variance=0

for i in range(0,repeat_times):

for j in range(0,10000):

result[i]+=100*function3(rd.uniform(-5,5),rd.uniform(-5,5))/10000

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 3 with Standard Monte Carlo Estimation")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

#use bivariate beta distribution which are uncorrelated to determine the allocation of samples

#and make them even symmetric

#use beta distribution with a=3 b=1

sample_allocation=np.empty((5,5),dtype=float)

allocation=[0.0]*5

sum_allocation=0.0

condition=0

alpha_1=0.8

beta_1=0.7

allocation[4]=beta.cdf(1,alpha_1,beta_1)-beta.cdf(0.6,alpha_1,beta_1)

allocation[3]=beta.cdf(0.6,alpha_1,beta_1)-beta.cdf(0.2,alpha_1,beta_1)

allocation[2]=2*(beta.cdf(0.2,alpha_1,beta_1)-beta.cdf(0,alpha_1,beta_1))

allocation[1]=allocation[3]

allocation[0]=allocation[4]

for i in range(0,5):

condition+=allocation[i]

condition=condition**2

for i in range(0,5):

for j in range(0,5):

sample_allocation[i][j]=int(allocation[i]*allocation[j]*sample_number/condition)

sum_allocation+=sample_allocation[i][j]

sample_allocation[2][2]+=(sample_number-sum_allocation)

print "The allocation of Sample Numbers Derived from Exponential pdf is:\n"

print sample_allocation

result=[0.0]*repeat_times

result_mean=0.0

result_variance=0.0

for i in range(0,repeat_times):

for j1 in range(0,5):

for j2 in range(0,5):

if sample_allocation[j1][j2]!=0:

for k in range(0,int(sample_allocation[j1][j2])):

x=rd.uniform(2.0*j1-5,2.0*j1-3)

y=rd.uniform(2.0*j2-5,2.0*j2-3)

result[i]+=4*function3(x,y)/sample_allocation[j1][j2]

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 3 with Monte Carlo Estimation Imported with stratification")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()

#devide the interval into 5*5 bins

#Initial the number of samples in each bin equally

sample_allocation=np.full((5,5),400,dtype=float)

mean=[0.0]*1000

iteration_count=0

func_val_store=[0]*10000

#make sure to iterate at least once

while iteration_count<999:

iteration_count+=1

count=0

func_val_avg=np.empty((5,5))

for i in range(0,5):

for j in range(0,5):

step=int(math.sqrt(sample_allocation[i][j]))

for k1 in range(1,step+1):

for k2 in range(1,step+1):

x=2*i-5+float(2*k1)/(step+1)

y=2*j-5+float(2*k2)/(step+1)

func_val_store[count]=function3(x,y)

func_val_avg[i][j]+=func_val_store[count]

count+=1

if step==0:

func_val_avg[i][j]=0

else:

func_val_avg[i][j]/=(step**2)

#estimated integration in this iteration

for i in range(0,5):

for j in range(0,5):

mean[iteration_count]+=func_val_avg[i][j]*4

#Compare with the estimated mean in the last iteration

if abs(mean[iteration_count]-mean[iteration_count-1])<0.00001:

break

#re-distribute the number in each bins, with totol sample_number of samples

#The sample numbers in each bin is linearly positively propotional to abs of funv_val

func_val_total=0

number_in_bin_count=0

for i in range(0,5):

for j in range(0,5):

func_val_total+=abs(func_val_avg[i][j])

for i in range(0,5):

for j in range(0,5):

sample_allocation[i][j]=int(10000*abs(func_val_avg[i][j])/func_val_total)

number_in_bin_count+=sample_allocation[i][j]

#add the rest few sample to the interval that has the largest func_val

sample_allocation[2][2]+=(10000-number_in_bin_count)

#use the optimized sample_allocation to estimate the Integration

print "The allocation of Sample Numbers Derived from Exponential pdf is:\n"

print sample_allocation

result=[0]*repeat_times

result_mean=0

result_variance=0

for i in range(0,repeat_times):

for j1 in range(0,5):

for j2 in range(0,5):

if sample_allocation[j1][j2]!=0:

for k in range(0,int(sample_allocation[j1][j2])):

x=rd.uniform(2*j1-5,2*j1-3)

y=rd.uniform(2*j2-5,2*j2-3)

result[i]+=4*function3(x,y)/sample_allocation[j1][j2]

result_mean+=result[i]/repeat_times

for i in range(0,repeat_times):

result_variance+=(result[i]-result_mean)**2

result_variance/=repeat_times

print "The Variance of the 50 samples is ",result_variance

print "The average of this",repeat_times,"samples is: ",result_mean

plt.scatter(np.arange(0,repeat_times),result)

plt.title("Function 3 with Monte Carlo Estimation Imported with own sample allocation strategy")

plt.xlabel("Trial")

plt.ylabel("Estimation Result")

plt.grid(True)

plt.show()