《人工智能數(shù)學基礎》程序代碼

#【例1-1-1】importnumpyasnpa=np.arange(100)b=a[2:100:8]#從索引2開始到索引100停止，步長為8print(b)#【例1-1-2】importnumpyasnpa=np.arange(100)s=slice(2,100,8)#從索引2開始到索引100停止，間隔為8print(a[s])#【例1-2】importnumpyasnpx=np.array([[1,2,3],[4,5,6],[7,8,9]])y=x[[0,1,2],[0,1,2]]#訪問(0,0),(1,1),(2,2)位置元素print(x)print(y)#【例1-3】importnumpyasnpx=np.array([[0,1,2],[3,4,5],[6,7,8],[9,10,11]])print('數(shù)組是：')print(x)print('大于6的元素是：')#輸出大于6的元素print(x[x>6])#【例1-4】importnumpyasnpa=[[1,2],[3,4]]b=[[5,6],[7,8]]print(np.matmul(a,b))#【例1-5】fromscipy.specialimportcbrtres=cbrt([1000,27,8,125])print(res)#【例1-6】fromscipy.cluster.vqimportkmeans,vq,whitenfromnumpyimportvstack,arrayfromnumpy.randomimportrand#具有3個特征值的樣本數(shù)據(jù)生成data=vstack((rand(100,3)+array([.5,.5,.5]),rand(100,3)))#計算K=4時的中心點centroids,_=kmeans(data,4)print(centroids)#將樣本數(shù)據(jù)中的每個值分配給一個中心點，形成3個聚類。#返回值clx標出了對應索引樣本的聚類，dist表示對應索引樣本與聚類中心的距離。clx,dist=vq(data,centroids)#打印聚類print(clx)#【例1-7】importegratefromnumpyimportexpf=lambdax:x**3i=egrate.quad(f,0,1)print(i)#【例1-8】fromsympyimport*x,y=symbols('x,y')z=x**2+y**2+x*y+2dx=diff(z,x)#對x求偏導print(dx)result=dx.subs({x:1,y:2})print(result)dy=diff(z,y)#對y求偏導print(dy)result=dy.subs({x:1,y:2})print(result)#【例1-9】importnumpyasnpfromscipyimportinterpolateasintpimportmatplotlib.pyplotaspltx=np.linspace(0,4,12)y=np.cos(x**2+4)print(x)print(y)f1=erp1d(x,y,kind='linear')#使用interp1d類創(chuàng)建擬合函數(shù)f2=erp1d(x,y,kind='cubic')#使用interp1d類創(chuàng)建擬合函數(shù)xnew=np.linspace(0,4,30)plt.plot(x,y,'o',xnew,f1(xnew),'-',xnew,f2(xnew),'--')plt.legend(['data','linear','cubic','nearest'],loc='best')plt.show()#【例1-10】importnumpyasnpsamples=np.random.normal(size=500)bins=np.arange(-10,10)histogram=np.histogram(samples,bins=bins,density=True)[0]bins=0.5*(bins[1:]+bins[:-1])fromscipyimportstatspdf=stats.norm.pdf(bins)#norm是一個分布對象importmatplotlib.pyplotaspltplt.plot(bins,histogram)plt.plot(bins,pdf)plt.show()#【例1-11】importnumpyasnpsamples=[65,23,57,92,54,38,87,78,58,92,57,85,82,73,92]print(np.mean(samples))print(stats.scoreatpercentile(samples,40))print(stats.scoreatpercentile(samples,80))#【例1-12】plt.rcParams["font.sans-serif"]=["SimHei"]#設置字體plt.rcParams["axes.unicode_minus"]=False#該語句解決圖像中的“-”負號的亂碼問題importmatplotlib.pyplotasplt#創(chuàng)建圖形對象fig=plt.figure()#添加子圖區(qū)域，參數(shù)值表示[left,bottom,width,height]ax=fig.add_axes([0,0,1,1])#準備數(shù)據(jù)langs=['數(shù)學','程序設計','外語','數(shù)據(jù)庫','機器人']students=[24,45,27,35,39]#繪制柱狀圖ax.bar(langs,students)plt.show()#【例1-13】frommatplotlibimportpyplotaspltimportnumpyasnpplt.rcParams["font.sans-serif"]=["SimHei"]plt.rcParams["axes.unicode_minus"]=Falsefig=plt.figure()ax=fig.add_axes([0,0,1,1])#使得X/Y軸的間距相等ax.axis('equal')#準備數(shù)據(jù)langs=['數(shù)學','程序設計','外語','數(shù)據(jù)庫','機器人']students=[24,45,27,35,39]#繪制餅狀圖ax.pie(students,labels=langs,autopct='%1.2f%%')plt.show()#【例1-14】importmatplotlib.pyplotaspltplt.rcParams["font.sans-serif"]=["SimHei"]plt.rcParams["axes.unicode_minus"]=Falsex=["星期一","星期二","星期三","星期四","星期五","星期六","星期天"]y=[46,57,74,69,72,33,62]plt.plot(x,y,"g",marker='D',markersize=5,label="人數(shù)")plt.xlabel("星期")plt.ylabel("晚自習人數(shù)")plt.title("晚自習情況統(tǒng)計")#顯示圖例plt.legend(loc="best")plt.show()#【例1-15】frommpl_toolkitsimportmplot3dimportnumpyasnpimportmatplotlib.pyplotaspltfig=plt.figure()ax=plt.axes(projection='3d')z=np.linspace(0,1,100)x=z*np.sin(20*z)y=z*np.cos(20*z)c=x**2+y**2ax.scatter3D(x,y,z,c=c)ax.set_title('3維散點圖')plt.show()#【例1-16】importmatplotlib.pyplotaspltnp.random.seed(10)x1=np.random.normal(100,50,500)x2=np.random.normal(80,50,500)x3=np.random.normal(90,10,500)x4=np.random.normal(70,30,500)x5=np.random.normal(80,20,500)#創(chuàng)建繪制小提琴圖的數(shù)據(jù)序列data_to_plot=[x1,x2,x3,x4,x5]#創(chuàng)建一個畫布fig=plt.figure()#創(chuàng)建一個繪圖區(qū)域ax=fig.add_axes([0,0,1,1])#創(chuàng)建一個小提琴圖bp=ax.violinplot(data_to_plot)plt.show()#【實驗一數(shù)據(jù)可視化-1】importmatplotlib.pyplotaspltimportnumpyasnpx=np.arange(-np.pi,np.pi,0.1)y=np.sin(x)plt.plot(x,y)plt.show()#【實驗一數(shù)據(jù)可視化-2】importmatplotlib.pyplotaspltimportnumpyasnpx=np.arange(-np.pi,np.pi,0.1)y=np.sin(x)plt.plot(x,y,color="green",linewidth=2.0,linestyle='-.')plt.show()#【實驗一數(shù)據(jù)可視化-3】importmatplotlib.pyplotaspltimportnumpyasnpx=np.arange(-np.pi,np.pi,0.1)y=np.sin(x)sin,cos=np.sin(x),np.cos(x)plt.plot(x,sin,label='sin(x)')plt.plot(x,cos,color='red',linewidth=2.0,linestyle=':',label='cos(x)')plt.grid(True)plt.legend()plt.show()#【實驗一數(shù)據(jù)可視化-4】importmatplotlib.pyplotaspltimportnumpyasnpx=np.arange(-np.pi,np.pi,0.1)y=np.sin(x)sin,cos=np.sin(x),np.cos(x)plt.plot(x,sin,label='sin(x)')plt.plot(x,cos,color='red',linewidth=2.0,linestyle=':',label='cos(x)')plt.xlim(-4,4)plt.ylim(-1.5,1.5)plt.xlabel('x')plt.ylabel('y')plt.title('sinx')plt.plot(x,y)plt.show()#【例2-1】fromsympyimport*n=symbols('n')a=limit((2*n**2-n+2)/(4*n**2+3*n-2),n,oo)print(a)#【例2-2】fromsympyimport*x=symbols('x')a=limit(((3-x)/(2-x))**(2*x),x,oo)print(a)#【例2-3】fromsympyimport*x=Symbol('x')#定義符號表達式f=sin(x)/x#定義函數(shù)式result=limit(f,x,0)#求x趨于0時極限print('x-->0,limit:',result)result1=limit(f,x,oo)#求x趨于無窮時極限print('x-->oo,limit:',result1)plot(f,(x,-100,100))#【例2-4】importsympyfromsympyimport*x=symbols('x')f=sin(x)**2print(f.diff())#【例2-5】fromsympyimport*x=symbols('x')f=5*x**4+sin(x)print(f.diff(x,2))print(f.diff(x,2).evalf(subs={x:1}))#【例2-6】fromsympyimport*x,y=symbols('xy')F=exp(y)+x*y-y**2dydx=idiff(F,y,x)print(dydx)#【例2-7】fromsympyimport*t=symbols('t')x=exp(t)*cos(t)y=exp(t)*sin(t)f=diff(y,t)/diff(x,t)print(f)#【例2-8】fromsympyimport*x,y=symbols('xy')F=exp(-2*x)*cos(x+2*y)dFdx=F.diff(x)print(dFdx)print(dFdx.evalf(subs={x:0,y:pi/4}))dFdy=F.diff(y)print(dFdy)print(dFdy.evalf(subs={x:0,y:pi/4}))#【例2-10】fromsympyimport*x,y=symbols('xy')F=2*x**3*y-3*x*y**3dFdx=F.diff(x)dFdy=F.diff(y)print("[",dFdx,",",dFdy,"]")#【例2-11】fromsympyimport*x=Symbol('x')f=2*(x**3)-6*(x**2)-18*x+7plot(f,(x,-10,10))dfdx=f.diff(x)print(dfdx)plot(dfdx,(x,-10,10))root=solve(dfdx,x)print('導數(shù)為0的x值為',root)#【例2-12】fromsympyimport*x=Symbol('x')f=x**3-3*x**2-1plot(f,(x,-2,2))dfdx2=f.diff(x,2)print(dfdx2)plot(dfdx2,(x,-2,2))root=solve(dfdx2,x)print('拐點橫坐標為：',root)#【例2-13】fromscipyimportoptimizefun1=lambdax:x**3-6*x**2+9*x+5fun2=lambdax:-(x**3-6*x**2+9*x+5)xmin=optimize.minimize(fun1,0)xmax=optimize.minimize(fun2,0)print('極小值點和極小值為',xmin.x[0],xmin.fun)print('極大值點和極大值為',xmax.x[0],xmax.fun)print('最小值為',min(fun1(-2),xmin.x[0],xmax.x[0],fun1(3)))print('最大值為',max(fun1(-2),xmin.x[0],xmax.x[0],fun1(3)))#【例2-17】importsympyfromsympyimport*x=symbols('x')fx=4*x**3-2**x+3*sin(x)-2/(x**2)-1/sqrt(x)Fx=integrate(fx,x)Fx=simplify(Fx)print(Fx)#【例2-18】importsympyfromsympyimport*x=symbols('x')fx=exp(x)*sin(2*x)Fx=integrate(fx,x)Fx=simplify(Fx)print(Fx)#【例2-19】importsympyfromsympyimport*x=symbols('x')fx=(x+1)*log(x)Fx=integrate(fx,x)Fx=simplify(Fx)print(Fx)#【例2-21】importsympyfromsympyimport*x=symbols('x')fx=(x+2)/sqrt(2*x+1)Fx=integrate(fx,(x,0,4))print(Fx)#【例2-22】importsympyfromsympyimport*x=symbols('x')fx=x*exp(-x)Fx=integrate(fx,(x,0,oo))print(Fx)#【例2-24】importsympyfromsympyimport*x,y,z=symbols('xyz')f=x*yFx=integrate(f,(z,0,1-x-y),(y,0,1-x),(x,0,1))print(Fx)#【例2-28】importsympyfromsympyimport*n=Symbol('n',integer=True)x=Symbol('x')expr=(-1)**n*(x**n)/nan=(-1)**n/n#計算收斂半徑r=limit(abs(an.subs(n,n+1)/an),n,oo)print('r=',r)#判定兩端點是否收斂c1=Sum(expr.subs(x,r),(n,1,oo)).is_convergent()c2=Sum(expr.subs(x,-r),(n,1,oo)).is_convergent()print('x=r:',c1)print('x=-r:',c2)#【實驗二定積分近似值】defsum_fun_xi(xi,f):returnsum([f(each)foreachinxi])defintegral(a,b,n,func):h=(b-a)/float(n)xk=[a+i*hforiinrange(1,n)]returnh/2*(func(a)+2*sum_fun_xi(xk,func)+func(b))if__name__=="__main__":func=lambdax:x**2a,b=1,2n=100print(integral(a,b,n,func))importmatplotlib.pyplotaspltplt.figure("play")ax1=plt.subplot(111)plt.sca(ax1)tmpx=[2+float(8-2)/50*eachforeachinrange(50+1)]plt.plot(tmpx,[func(each)foreachintmpx],linestyle='-',color='black')forranginrange(n):tmpx=[a+float(8-2)/n*rang,a+float(8-2)/n*rang,a+float(8-2)/n*(rang+1),a+float(8-2)/n*(rang+1)]tmpy=[0,func(tmpx[1]),func(tmpx[2]),0]c=['r','y','b','g']plt.fill(tmpx,tmpy,color=c[rang%4])plt.grid(True)plt.show()#【例3-1】importnumpyasnpD=np.array([[1,2,4],[3,1,5],[-2,2,7]])print(D)print(np.linalg.det(D))#【例3-3】importnumpyasnpD=np.array([[5,4,2,1],[2,1,4,2],[3,5,8,6],[1,-1,0,1]])print(D)print(np.linalg.det(D))#【例3-4】importnumpyasnpimportcopydefcramer(d,b):ifd.shape[0]!=d.shape[1]:print('此矩陣不是方陣！')returnifnp.linalg.det(d)==0:print('系數(shù)方陣為0')returnd_i=[]foriinrange(b.shape[0]):d_i.append(copy.deepcopy(d))d_i[i][:,i]=bx=[]foriinrange(b.shape[0]):x.append(np.linalg.det(d_i[i])/np.linalg.det(d))print(x)d=np.array([[2,1,1],[1,2,1],[1,1,2]])b=np.array([15,16,17])cramer(d,b)#【例3-5】importnumpyasnpa=np.array([[1,1,1,0],[2,1,-1,1],[1,2,-1,1],[0,1,2,3]])b=np.array([5,1,2,3])print(np.linalg.solve(a,b))#【例3-7-1】fromnumpyimport*a1=mat([[1,0,3,-1],[2,1,0,2]])a2=mat([[4,1,0],[-1,1,3],[2,0,1],[1,3,4]])a3=(3*a1)*a2print(a3)#【例3-7-2】importnumpyasnpa1=np.mat([[1,0,3,-1],[2,1,0,2]])a2=np.mat([[4,1,0],[-1,1,3],[2,0,1],[1,3,4]])a3=np.dot(3*a1,a2)print(a3)#【例3-8】importnumpyasnpinput=\np.array([[2,1,1,3,6],[12,3,3,6,7],[7,2,3,1,5],[2,3,5,1,2],[0,3,2,2,1]])kernel=np.array([[2,1,3],[2,0,2],[4,2,1]])print(input.shape,kernel.shape)defmy_conv(input,kernel):output_size=(len(input)-len(kernel)+1)res=np.zeros([output_size,output_size],np.float32)foriinrange(len(res)):forjinrange(len(res)):res[i][j]=compute_conv(input,kernel,i,j)returnresdefcompute_conv(input,kernel,i,j):res=0forkkinrange(3):forkinrange(3):res+=input[i+kk][j+k]*kernel[kk][k]returnresprint("卷積后的矩陣為：\n",my_conv(input,kernel))#【例3-9】importnumpyasnpa=np.array([[1,2,3],[4,5,6],[7,8,9]])print(a.T)#【例3-11-1】fromnumpyimport*a=mat([[1,2,-2],[2,-3,2],[-2,-1,1]]);a1=a.I#求逆a2=a1.T#求轉置print(a2)#【例3-11-2】importnumpyasnpa=mat([[1,2,-2],[2,-3,2],[-2,-1,1]]);a1=np.linalg.inv(a)#求逆a2=a1.T#求轉置print(a2)#【例3-12】importnumpyasnpa=np.mat([[1,3,-2,2],[0,2,-1,3],[-2,0,1,5]])r=np.linalg.matrix_rank(a)print("矩陣的秩為：",r)#【例3-17】importnumpyasnpA=np.array([[-1,1,0],[-4,3,0],[1,0,2]])print('打印A：\n{}'.format(A))a,b=np.linalg.eig(A)print('特征值a：\n{}'.format(a))print('特征向量b：\n{}'.format(b))#【例3-21】importnumpyasnpA=np.array([[5,2,-4],[2,1,-2],[-4,-2,5]])B=np.linalg.eigvals(A)#求A的特征根print(A)print(B)ifnp.all(B>0):print('是正定矩陣')else:print('不是正定矩陣')#【實驗三矩陣相乘-1】importnumpyasnpa1=np.mat([[4,-1,2,1],[1,1,0,3],[0,3,1,4]])a2=np.mat([[1,2],[0,1],[3,0],[-1,2]])b1=a1*a2print(b1)#【實驗三矩陣相乘-2】importnumpyasnpa1=np.mat([[4,-1,2,1],[1,1,0,3],[0,3,1,4]])a2=np.mat([[1,2],[0,1],[3,0],[-1,2]])b2=np.dot(a1,a2)print(b2)#【實驗三矩陣相乘-3】a1=[[4,-1,2,1],[1,1,0,3],[0,3,1,4]]a2=[[1,2],[0,1],[3,0],[-1,2]]b3=[[0,0],[0,0],[0,0]]foriinrange(3):forjinrange(2):forkinrange(4):b3[i][j]=b3[i][j]+a1[i][k]*a2[k][j]print(b3)#【例4-1】importitertoolsprint("任取1個組合：")fori,valinenumerate(list(binations('ABCD',1))):print("序號：%s值：%s"%(i+1,''.join(val)))print("任取2個組合：")fori,valinenumerate(list(binations('ABCD',2))):print("序號：%s值：%s"%(i+1,''.join(val)))print("任取3個組合：")fori,valinenumerate(list(binations('ABCD',3))):print("序號：%s值：%s"%(i+1,''.join(val)))print("任取4個組合：")fori,valinenumerate(list(binations('ABCD',4))):print("序號：%s值：%s"%(i+1,''.join(val)))print("任取2個排列：")fori,valinenumerate(list(itertools.permutations('ABCD',2))):print("序號：%s值：%s"%(i+1,''.join(val)))#【例4-2】fromscipy.specialimport*print(comb(5,2))#5個中取2個的組合數(shù)print(perm(5,2))#5個中取2個的排序數(shù)#【例4-3】importnumpyasnpa=np.array([[3,7,8,3],[9,1,2,37]])print('數(shù)組是：')print(a)print('調用sort()函數(shù)：')print(np.sort(a))print('按列排序：')print(np.sort(a,axis=0))#在sort函數(shù)中排序字段dt=np.dtype([('name','S10'),('age',int)])a=np.array([("yhw",21),("yl",25),("wm",17),("ysb",27)],dtype=dt)print('數(shù)組是：')print(a)print('按name排序：')print(np.sort(a,order='name'))#【例4-4】importnumpyasnpx=np.array([[2,4,5],[6,7,4],[7,3,9]])print(x)print('大于5的元素的索引：')y=np.where(x>5)print(y)print('使用這些索引來獲取滿足條件的元素：')print(x[y])#定義條件,選擇偶數(shù)元素condition=np.mod(x,2)==0print('按元素的條件值：')print(condition)print('使用條件提取元素：')print(np.extract(condition,x))#【例4-5】fromscipyimportstatsasstsimportnumpyasnpimportmatplotlib.pyplotaspltdata=[3850,3950,4050,3880,3755,3710,3890,4130,3940,4225,3920,3880]month=['一月','二月','三月','四月','五月','六月','七月','八月','九月','十月','十一月','十二月']print('眾數(shù)為：',sts.mode(data))#print('中位數(shù)為：',np.median(data))plt.rcParams["font.sans-serif"]=["SimHei"]#設置字體plt.rcParams["axes.unicode_minus"]=False#該語句解決圖像中的“-”負號的亂碼問題plt.plot(month,data)plt.title("楊老師的工資")plt.show()#【例4-6】fromscipyimportstatsasstsimportnumpyasnpimportmatplotlib.pyplotaspltdata=[3850,3950,4050,3880,3755,3710,3890,4130,3940,4225,3920,3880]month=['一月','二月','三月','四月','五月','六月','七月','八月','九月','十月','十一月','十二月']print('平均工資為：',sts.tmean(data))print('工資極差為：',np.ptp(data))#【例4-7】fromscipyimportstatsasstsimportnumpyasnpdata=[3850,3950,4050,3880,3755,3710,3890,4130,3940,4225,3920,3880]month=['一月','二月','三月','四月','五月','六月','七月','八月','九月','十月','十一月','十二月']print('平均工資標準差為：',sts.tstd(data))#【例4-8】importnumpyasnpimportmatplotlib.pyplotaspltimportrandom#中文顯示plt.rcParams['font.sans-serif']=['simhei']#用代碼實現(xiàn)重復50次拋硬幣的試驗，觀察每次正面朝上的概率#定義做50次試驗，每次拋500次(可以設置不同次數(shù))batch=int(input("請輸入試驗次數(shù)："))samples=500*np.ones(batch,dtype=32)result=[]result_mean=[]#統(tǒng)計每次試驗正面朝上的概率forkinrange(batch):foriinrange(samples[k]):result.append(random.randint(0,1))result_mean.append(np.mean(result))xaxis=list(range(batch))plt.plot(xaxis,result_mean)plt.xlabel('拋硬幣數(shù)')plt.ylabel('正面朝上概率')plt.show()#【例4-11】#A為"發(fā)送信號為0",B為"發(fā)送信號為1",C為"接收信號為0",D為"接收信號為1"P_A=0.5P_B=0.5P_C_A=0.8P_C_B=0.1P_D_A=0.2P_D_B=0.9#正確傳輸信號0、1的概率分別為：P_A_C=(P_C_A*P_A)/(P_C_A*P_A+P_C_B*P_B)P_B_D=(P_D_B*P_B)/(P_D_B*P_B+P_D_A*P_A)print("正確傳輸信號0的概率為：%.3f"%P_A_C)print("正確傳輸信號1的概率為：%.3f"%P_B_D)#接收機正確獲取信源信息的概率P(R)為：print("接收機正確獲取信源信息的概率為：%.2f"%((P_A_C**3)*(P_B_D**2)))#【例4-17】importnumpyasnpimportmatplotlib.pyplotaspltimportmathfromscipyimportstatsn=20p=0.3k=np.arange(0,41)print(k)print("*"*20)binomial=stats.binom.pmf(k,n,p)print(binomial)plt.plot(k,binomial,'o-')plt.title('binomial:n=%i,p=%.2f'%(n,p),fontsize=15)plt.xlabel('numberofsuccess')plt.ylabel('probalilityofsuccess',fontsize=15)plt.grid(True)plt.show()#【例4-19】importnumpyasnpimportmatplotlib.pyplotaspltimportmathx=np.random.poisson(lam=5,size=10000)#lam為λsize為kpillar=20a=plt.hist(x,bins=pillar,density=True,range=[0,pillar],alpha=0.5)plt.plot(a[1][0:pillar],a[0])plt.grid()plt.show()#【例4-24】importnumpyasnpimportmatplotlib.pyplotaspltmu=1#期望為1sigma=3#標準差為3num=10000#個數(shù)為10000rand_data=np.random.normal(mu,sigma,num)count,bins,ignored=plt.hist(rand_data,30,density=True)plt.plot(bins,1/(sigma*np.sqrt(2*np.pi))*np.exp(-(bins-mu)**2/(2*sigma**2)),linewidth=2,color='r')plt.show()#【例4-25】fromscipy.statsimportnormfromscipy.optimizeimportfsolveprint("p=",norm.cdf(6,3,5)-norm.cdf(2,3,5))f=lambdac:norm.cdf(2*c,3,5)-norm.cdf(-3*c,3,5)-0.6print("c=",fsolve(f,0))#【例4-30】importnumpyasnpnum=[88,90,70,75,93,85,85,88,86,85]#求眾數(shù)c=np.bincount(num)#用在numpy中建立元素出現(xiàn)次數(shù)的索引的方法求眾數(shù)num_mod=np.argmax(c)#求中位數(shù)num_med=np.median(num)#求均值num_mea=np.mean(num)#求極差num_ptp=np.ptp(num)#求方差num_var=np.var(num,ddof=1)#用n-1計算方差#求標準差num_std=np.std(num,ddof=1)print("眾數(shù)：",num_mod)print("中位數(shù)：",num_med)print("均值：",num_mea)print("極差：",num_ptp)print("方差：%5.2f"%num_var)print("標準差：%5.2f"%num_std)#【例4-31-1】importnumpyasnpfromscipy.statsimportpearsonrx=np.array([1,3,8,10,16,18,20])y=np.array([4,5,6,9,11,15,17])r,p=pearsonr(x,y)print(r)print(p)#【例4-31-2】importnumpyasnpfromscipy.statsimportpearsonrx=np.array([1,3,8,10,16,18,20])y=np.array([4,5,6,9,11,15,17])n=len(x)s_xy=np.sum(np.sum(x*y))s_x=np.sum(np.sum(x))s_y=np.sum(np.sum(y))s_x1=np.sum(np.sum(x*x))s_y1=np.sum(np.sum(y*y))r=(n*s_xy-s_x*s_y)/np.sqrt((n*s_x1-s_x*s_x)*(n*s_y1-s_y*s_y))print(r)#【例4-37】fromscipyimportstatsasstsCI=erval(0.95,loc=8.34,scale=0.03)print('置信區(qū)間為：',CI)#【實驗四數(shù)據(jù)統(tǒng)計與分析-1】fromnumpyimportreshape,hstack,mean,median,ptp,var,std,cov,corrcoefimportpandasaspdimportxlrddf=pd.read_excel("d:\shuju.xlsx")#統(tǒng)計數(shù)據(jù)存放在d:\shuju.xlsx中a=df.values#將二維數(shù)據(jù)轉化為一維數(shù)據(jù)h=a[:,::2]#提取奇數(shù)列身高數(shù)據(jù)w=a[:,1::2]#提取偶數(shù)列體重數(shù)據(jù)h=reshape(h,(-1,1))#轉換成列向量w=reshape(w,(-1,1))hw=hstack([h,w])print([mean(h),median(h),ptp(h),var(h),std(h)])#計算均值，中位數(shù)，極差，方差，標準差print("協(xié)方差為：%f\n相關系數(shù)為：%f"%(cov(hw.T)[0,1],corrcoef(hw.T)[0,1]))#【實驗四數(shù)據(jù)統(tǒng)計與分析-2】fromnumpyimportreshape,c_importpandasaspdimportxlrddf=pd.read_excel("d:\shuju.xlsx")a=df.values#將二維數(shù)據(jù)轉化為一維數(shù)據(jù)h=a[:,::2]#提取奇數(shù)列身高數(shù)據(jù)w=a[:,1::2]#提取偶數(shù)列體重數(shù)據(jù)h=reshape(h,(-1,1))#轉換成列向量w=reshape(w,(-1,1))df=pd.DataFrame(c_[h,w],columns=["身高","體重"])print("求得的描述統(tǒng)計量如下：\n",df.describe())print("偏度為：\n",df.skew())print("峰度為：\n",df.kurt())print("25%分位數(shù)為：\n",df.quantile(0.25))print("50%分位數(shù)為：\n",df.quantile(0.5))print("90%分位數(shù)為：\n",df.quantile(0.9))#【實驗四數(shù)據(jù)統(tǒng)計與分析-3】fromnumpyimportreshape,c_importpandasaspdimportmatplotlib.pyplotaspltimportxlrdplt.rcParams["font.sans-serif"]=["SimHei"]#設置字體plt.rcParams["axes.unicode_minus"]=Falsedf=pd.read_excel("d:\shuju.xlsx")a=df.values#將二維數(shù)據(jù)轉化為一維數(shù)據(jù)h=a[:,::2]#提取奇數(shù)列身高數(shù)據(jù)w=a[:,1::2]#提取偶數(shù)列體重數(shù)據(jù)h=reshape(h,(-1,1))#轉換成列向量w=reshape(w,(-1,1))#plt.rc('font',size=16)#plt.rc("font",family="SimHei")plt.subplot(121)plt.xlabel("身高")plt.hist(h,10)plt.subplot(122)ps=plt.hist(w,6)plt.xlabel("體重")print("體重的頻數(shù)表為：",ps)plt.savefig("figure4_8.png",dpi=500)plt.show()#【例5-1】importnumpyasnpfromscipy.optimizeimportfminimportmathdeff(x):return3*math.pow(x,4)-8*math.pow(x,3)+6*math.pow(x,2)+7print(fmin(func=f,x0=1))#【例5-2】importnumpyasnpfromscipy.optimizeimportfminimportmathdeff(x):returnmath.pow(x[0],3)+\math.pow(x[1],3)+3*math.pow(x[0],2)+3*math.pow(x[1],2)-9*x[0]#func為函數(shù)名，x0為函數(shù)參數(shù)的起始點print(fmin(func=f,x0=np.array([0,0])))#【例5-3】importsympyassyimportnumpyasnpfromsympy.functionsimportsin,cosimportmatplotlib.pyplotaspltplt.style.use("ggplot")x=sy.Symbol('x')f=cos(x)#求階乘deffactorial(n):ifn<=0:return1else:returnn*factorial(n-1)deftaylor(function,x0,n):i=0p=0whilei<=n:p=p+(function.diff(x,i).subs(x,x0))/(factorial(i))*(x-x0)**ii+=1returnpdefplot():x_lims=[-5,5]x1=np.linspace(x_lims[0],x_lims[1],800)y1=[]forjinrange(1,10,2):func=taylor(f,0,j)print('Taylorexpansionatn='+str(j),func)forkinx1:y1.append(func.subs(x,k))plt.plot(x1,y1,label='order'+str(j))y1=[]plt.plot(x1,np.sin(x1),label='sinofx')plt.xlim(x_lims)plt.ylim([-5,5])plt.xlabel('x')plt.ylabel('y')plt.legend()plt.grid(True)plt.title('Taylorseriesapproximation')plt.show()plot()#【例5-5】importnumpyasnpfromscipy.optimizeimportfminimportmathf=lambdax:math.pow(x[0],2)-4*x[0]+math.pow(x[1],2)-2*x[1]+5#func為函數(shù)名，x0為函數(shù)參數(shù)的起始點print(fmin(func=f,x0=np.array([0,0])))#【例5-6】importmathimportnumpyasnpdeffunc_2d(x):#定義目標函數(shù)：x為二維自變量return-math.exp(-(x[0]**2+x[1]**2))defgrad_2d(x):#定義梯度，由目標函數(shù)求偏導而得，返回二維向量deriv0=2*x[0]*math.exp(-(x[0]**2+x[1]**2))deriv1=2*x[1]*math.exp(-(x[0]**2+x[1]**2))returnnp.array([deriv0,deriv1])defgradient_descent_2d(grad,cur_x=np.array([0.1,0.1]),\learning_rate=0.01,precision=0.0001,max_iters=10000):print(f"{cur_x}作為初始值開始迭代...")foriinrange(max_iters):grad_cur=grad(cur_x)ifnp.linalg.norm(grad_cur,ord=2)<precision:break#當梯度趨近為0時，視為收斂cur_x=cur_x-grad_cur*learning_rateprint("第",i,"次迭代：x值為",cur_x)print("最小值點x=",cur_x)returncur_xif__name__=='__main__':gradient_descent_2d(grad_2d,cur_x=np.array([1,-1]),\learning_rate=0.2,precision=0.000001,max_iters=10000)#【例5-9】importnumpyasnpfromscipyimportoptimizez=np.array([1,1,1])a=np.array([[1,4,2],[3,2,0]])b=np.array([8,6])x1_bound=x2_bound=x3_bound=(0,None)res=optimize.linprog(z,A_ub=-a,b_ub=-b,bounds=(x1_bound,x2_bound,x3_bound))print(res)#【例5-10】#導入sympy包，用于求導，方程組求解等等fromsympyimport*#設置變量x1=symbols("x1")x2=symbols("x2")alpha=symbols("alpha")#構造拉格朗日函數(shù)L=60-10*x1-4*x2+x1*x1+x2*x2-x1*x2-alpha*(x1+x2-8)#求導difyL_x1=diff(L,x1)#對變量x1求導difyL_x2=diff(L,x2)#對變量x2求導difyL_alpha=diff(L,alpha)#對alpha求導#求解aa=solve([difyL_x1,difyL_x2,difyL_alpha],[x1,x2,alpha])print(aa)x1=aa.get(x1)x2=aa.get(x2)alpha=aa.get(alpha)print("最優(yōu)解為：",60-10*x1-4*x2+x1*x1+x2*x2-x1*x2-alpha*(x1+x2-8))#【例5-11】fromscipy.optimizeimportminimizeimportnumpyasnpdeffun(args):a,b,c,d=argsv=lambdax:(a+x[0])/(b+x[1])-c*x[0]+d*x[2]returnvdefcon(args):#約束條件分為eq和ineq#eq表示函數(shù)結果等于0；ineq表示表達式大于等于0x1min,x1max,x2min,x2max,x3min,x3max=argscons=({'type':'ineq','fun':lambdax:x[0]-x1min},\{'type':'ineq','fun':lambdax:-x[0]+x1max},\{'type':'ineq','fun':lambdax:x[1]-x2min},\{'type':'ineq','fun':lambdax:-x[1]+x2max},\{'type':'ineq','fun':lambdax:x[2]-x3min},\{'type':'ineq','fun':lambdax:-x[2]+x3max})returnconsif__name__=="__main__":#定義常量值args=(2,1,3,4)#a,b,c,d#設置參數(shù)范圍/約束條件args1=(0.1,0.9,0.1,0.9,0.1,0.9)#x1min,x1max,x2min,x2maxcons=con(args1)#設置初始猜測值x0=np.asarray((0.5,0.5,0.5))res=minimize(fun(args),x0,method='SLSQP',constraints=cons)print(res.fun)print(res.success)print(res.x)#【實驗五利用牛頓迭代法法求解方程的根】importmath#函數(shù)式deff1(x,f):iff==0:returnx**2-3*x+2-math.exp(x)else:return2*x-3-math.exp(x)#函數(shù)式的求導#計算循環(huán)次數(shù)count=0#設初始值為2x1=2print("迭代初始值為：x=2")i=x1-f1(x1,0)/f1(x1,1)while(abs(x1-i)>=1e-8)&(f1(x1,1)!=0):#print的格式化輸出，count和x1去補前面兩個空位print("迭代次數(shù)：%f,迭代值為:%f"%(count,x1))x2=x1-f1(x1,0)/f1(x1,1)i=x1#暫時存儲x1的值x1=x2count+=1#【例6-4】importnumpyasnpA=np.array([[-0.2,0.07,0.083],[0.1,-0.1,0.067],[1,1,1]])b=np.array([0,0,1])x=np.linalg.solve(A,b)print(x)#【例6-8-1】fromnumpyimport*importnumpyasnpQ=mat([[0.3,0.3,0],[0.15,0.25,0.3],[0.1,0.1,0.3]])A=np.eye(3)-QN=A.Iprint(N)#【例6-8-2】fromnumpyimport*importnumpyasnpN=mat([[1.618,0.686,0.294],[0.441,1.601,0.686],[0.294,0.327,1.569]])R=mat([[0,0.4],[0,0.3],[0.35,0.15]])B=N*Rprint(B)#【例6--8-3】fromnumpyimport*importnumpyasnpB=mat([[0.1029,0.8971],[0.2401,0.7596],[0.54915,0.45105]])A=mat([400,120,280])C=A*Bprint(C)#【例6-9】importjiebaseg_list=jieba.cut("我來到南京信息職業(yè)技術學院",cut_all=True)print("全模式:"+"/".join(seg_list))#全模式seg_list=jieba.cut("我來到南京信息職業(yè)技術學院",cut_all=False)print("精確模式:"+"/".join(seg_list))#精確模式seg_list=jieba.cut("他來到了南京紫金山山頂")#默認是精確模式print("默認模式："+"/".join(seg_list))seg_list=jieba.cut_for_search("小明碩士畢業(yè)于北京清華大學")#搜索引擎模式print("搜索引擎模式："+"/".join(seg_list))#【實驗六馬爾科夫分析】importnumpyasnpA=np.array([[-0.1,0.1,0.1],[0.2,-1,0.2],[1,1,1]])b=np.array([0,0,1])x=np.linalg.solve(A,b)print(x)#【例7-1】importpandasaspdimportnumpyasnp#一數(shù)值中缺兩個值da=pd.DataFrame(data=[0,np.nan,np.nan,1])#利用線性插值函數(shù)進行插值補全erpolate()#【例7-2】importnumpyasnpfromscipyimportinterpolateimportpylabasplx=np.linspace(0,10,11)y=np.sin(x)xnew=np.linspace(0,10,101)pl.plot(x,y,"ro")f=erp1d(x,y,kind="quadratic")ynew=f(xnew)pl.plot(xnew,ynew)pl.show()#【例7-3】fromerpolateimportlagrangex=[1,2,3,4,5,0]y=[1,2,2,6,-1,-1]a=lagrange(x,y)#輸出拉格朗日多項式print(a)#輸出拉格朗日多項式在1,2,3處的值print(a(1),a(2),a(3))#【例7-4】importnumpyasnpfromscipyimportinterpolateimportpylabasplx=np.linspace(0,10,11)y=np.cos(x)xnew=np.linspace(0,10,101)pl.plot(x,y,"ro")pl.rcParams['font.sans-serif']=['SimHei']#指定默認字體pl.rcParams['axes.unicode_minus']=False#解決負數(shù)坐標顯示問題#x值a=["nearest","slinear","quadratic","cubic"]b=["階梯插值","線性插值","2階樣條插值","3階樣條插值"]fork,vinzip(a,b):#插值方式f=erp1d(x,y,kind=k)ynew=f(xnew)pl.plot(xnew,ynew,label=str(v))pl.legend()pl.show()#【例7-5】importnumpyasnpfrommpl_toolkits.mplot3dimportAxes3Dimportmatplotlibasmplfromscipyimportinterpolateimportmatplotlib.cmascmimportmatplotlib.pyplotaspltdeffunc(x,y):return(x+y)*np.exp(-5.0*(x**2+y**2))x=np.linspace(-1,1,20)y=np.linspace(-1,1,20)x,y=np.meshgrid(x,y)fvals=func(x,y)fig=plt.figure(figsize=(9,6))#Drawsub-graph1ax=plt.subplot(1,2,1,projection='3d')surf=ax.plot_surface(x,y,fvals,rstride=2,cstride=2)ax.set_xlabel('x')ax.set_ylabel('y')ax.set_zlabel('f(x,y)')#二維插值newfunc=erp2d(x,y,fvals,kind='cubic')#計算100*100的網(wǎng)格上的插值xnew=np.linspace(-1,1,100)ynew=np.linspace(-1,1,100)fnew=newfunc(xnew,ynew)xnew,ynew=np.meshgrid(xnew,ynew)ax2=plt.subplot(1,2,2,projection='3d')surf2=ax2.plot_surface(xnew,ynew,fne