importnumpyasnpimportmatplotlib.pyplotaspltimportmathN=100000y=np.random.rand(N)s1=0.5y1=y+np.random.normal(0.0,s1,(N))s2=0.8y2=y+np.random.normal(0.0,s2,(N))alphaOpti1=s2*s2/ (s1*s1+s2*s2)alphaOpti2=s1*s1/ (s1*s1+s2*s2)sOpti=math.sqrt(s1*s1*s2*s2/(s1*s1+s2*s2))print('Computed',alphaOpti1,alphaOpti2,sOpti)a1= []a2= []s= []sO= []foralphainrange(101):alphaX=alpha/100.0yy=alphaX*y1+ (1.0-alphaX)*y2a1.append(alphaX)a2.append(1.0-alphaX)s.append(math.sqrt(np.sum((yy-y)**2)/(N-1.0)))sO.append(sOpti)pos=np.argmin(s)print('Simulated',a1[pos],a2[pos],s[pos])plt.figure(figsize=(10,10))plt.plot(a1,s,'r')plt.plot(a1,sO,'g')plt.plot([alphaOpti1],[sOpti],'bo')plt.xlabel(r"$\alpha$")plt.ylabel(r"$\sigma_B^2$")plt.tick_params(axis='x',which='major',direction='in')plt.tick_params(axis='y',which='major',direction='in')plt.grid()plt.show()importnumpyasnpimportmatplotlib.pyplotaspltimportmathN=100000y=np.random.rand(N)s1=0.65y1=y+np.random.normal(0.0,s1,(N))s2=0.8y2=y+np.random.normal(0.0,s2,(N))s3=1.1y3=y+np.random.normal(0.0,s3,(N))alphaOpti1=s2*s2*s3*s3/ (s1*s1*s2*s2+s1*s1*s3*s3+s3*s3*s2*s2)alphaOpti2=s1*s1*s3*s3/ (s1*s1*s2*s2+s1*s1*s3*s3+s3*s3*s2*s2)alphaOpti3=s1*s1*s2*s2/ (s1*s1*s2*s2+s1*s1*s3*s3+s3*s3*s2*s2)sOpti=math.sqrt(s1*s1*s2*s2*s3*s3/(s1*s1*s2*s2+s1*s1*s3*s3+s3*s3*s2*s2))print('Computed:',alphaOpti1,alphaOpti2,alphaOpti3,sOpti)a1= []a2= []a3= []s= []sO= []foralpha1inrange(101):alpha1X=alpha1/100.0foralpha2inrange(101):alpha2X=alpha2/100.0alpha3X=1.0-alpha1X-alpha2Xifalpha3X>=0:yy=alpha1X*y1+alpha2X*y2+alpha3X*y3a1.append(alpha1X)a2.append(alpha2X)a3.append(alpha3X)s.append(math.sqrt(np.sum((yy-y)**2)/(N-1.0)))sO.append(sOpti)fig=plt.figure(figsize=(10,10))ax=fig.add_subplot(111,projection='3d')pos=np.argmin(s)print('Simulated:',a1[pos],a2[pos],a3[pos],s[pos])ax.scatter(a1,a2,s,c='r',alpha=0.1)ax.scatter(a1,a2,sO,c='g',alpha=0.1)ax.scatter([alphaOpti1],[alphaOpti2],[sOpti],c='b')ax.set_xlabel(r"$\alpha_1$")ax.set_ylabel(r"$\alpha_2$")ax.set_zlabel(r"$\sigma_B^2$")ax.tick_params(axis='x',which='major',direction='in')ax.tick_params(axis='y',which='major',direction='in')ax.tick_params(axis='z',which='major',direction='in')ax.w_xaxis.set_pane_color((1.0,1.0,1.0,1.0))ax.w_yaxis.set_pane_color((1.0,1.0,1.0,1.0))ax.w_zaxis.set_pane_color((1.0,1.0,1.0,1.0))plt.show()