天文數(shù)據(jù)分析方法和計(jì)算

1、天文數(shù)據(jù)分析數(shù)據(jù)分析方法 累積概率分布任意分布的隨機(jī)數(shù)常用概率分布數(shù)據(jù)插值內(nèi)插和外推插值(erpolate)線性1d插值 (interp1d)1d樣條插值 (interpolate.splXXX)2d樣條插值 (bisplrep)RBF（radial basis function）插值/平滑import numpy as npfrom erpolate import Rbfimport matplotlib.pyplot as pltfrom matplotlib import cm# 2-d tests - setup scattered dataz =

2、x*np.exp(-x*2-y*2)ti = np.linspace(-2.0, 2.0, 100)XI, YI = np.meshgrid(ti, ti)# use RBFrbf = Rbf(x, y, z, epsilon=2)ZI = rbf(XI, YI)# plot the resultn = plt.Normalize(-2., 2.)plt.subplot(1, 1, 1)plt.pcolor(XI, YI, ZI, cmap=cm.jet)plt.scatter(x, y, 100, z, cmap=cm.jet)plt.title(RBF interpolation - mu

3、ltiquadrics)plt.xlim(-2, 2)plt.ylim(-2, 2)plt.colorbar()plt.show()數(shù)據(jù)擬合擬合模型O = T + e最小二乘法import numpy as npimport scipy.linalg as splinimport matplotlib.pyplot as plti = np.r_1:11xi = 0.1*iyi = c1*np.exp(-xi)+c2*xizi = yi + 0.05*np.max(yi)*np.random.randn(len(yi)A = np.c_np.exp(-xi):,np.newaxis,xi:,n

4、p.newaxisc,resid,rank,sigma = splin.lstsq(A,zi)print c,resid,rank,sigmaxi2 = np.r_0.1:1.0:100jyi2 = c0*np.exp(-xi2) + c1*xi2plt.plot(xi,zi,x,xi2,yi2)plt.axis(0,1.1,3.0,5.5)plt.xlabel($x_i$)plt.title(Data fitting with linalg.lstsq)plt.show()非線性擬合擬合系數(shù)有非線性函數(shù)關(guān)系from numpy import x = arange(0,6e-2,6e-2/30

5、)A,k,theta = 10, 1.0/3e-2, pi/6y_true = A*sin(2*pi*k*x+theta)y_meas = y_true + 2*random.randn(len(x)def residuals(p, y, x): A,k,theta = p err = y-A*sin(2*pi*k*x+theta) return errdef peval(x, p): return p0*sin(2*pi*p1*x+p2)p0 = 8, 1/2.3e-2, pi/3print array(p0)# 8. 43.4783 1.0472from scipy.optimize im

6、port leastsqplsq = leastsq(residuals, p0, args=(y_meas, x)print plsq0# 10.9437 33.3605 0.5834print array(A, k, theta)# 10. 33.3333 0.5236import matplotlib.pyplot as pltplt.plot(x,peval(x,plsq0),x,y_meas,o,x,y_true)plt.title(Least-squares fit to noisy data)plt.legend(Fit, Noisy, True)plt.show()Maximu

7、m Likelihood Estimation (MLE)最大似然估計(jì)一致收斂漸近于正態(tài)分布：有極值等精度高斯分布非等精度高斯分布上下限分布求極值A(chǔ) collection of general-purpose optimization routines. fmin - Nelder-Mead Simplex algorithm (uses only function calls) fmin_powell - Powells (modified) level set method (uses only function calls) fmin_cg - Non-linear (Polak-Rib

8、iere) conjugate gradient algorithm (can use function and gradient). fmin_bfgs - Quasi-Newton method (Broydon-Fletcher-Goldfarb-Shanno); (can use function and gradient) fmin_ncg - Line-search Newton Conjugate Gradient (can use function, gradient and Hessian). leastsq - Minimize the sum of squares of

9、M equations in N unknowns given a starting estimate. Constrained Optimizers (multivariate) fmin_l_bfgs_b - Zhu, Byrd, and Nocedals L-BFGS-B constrained optimizer (if you use this please quote their papers - see help) fmin_tnc - Truncated Newton Code originally written by Stephen Nash and adapted to

10、C by Jean-Sebastien Roy. fmin_cobyla - Constrained Optimization BY Linear ApproximationEM算法Expect-Maximum算法EM算法的目標(biāo)是找出有隱性變量的概率模型的最大可能性解分為2個(gè)步驟，E-step和M-stepE-step根據(jù)最初假設(shè)的模型參數(shù)值或者上一步的模型參數(shù)計(jì)算出隱性變量的后驗(yàn)概率，其實(shí)就是隱性變量的期望M-step根據(jù)這個(gè)E-step的后驗(yàn)概率重新計(jì)算出模型參數(shù)，然后再重復(fù)這兩個(gè)步驟，直至目標(biāo)函數(shù)收斂。擬合度好模型精度估計(jì)Bootstrap方法“杰克刀”方法Bootstrap方法用已知數(shù)據(jù)建立分布可有N!組新數(shù)據(jù)SN1987A中微子事件9個(gè)中微子“杰克刀”方法每次丟掉1或多個(gè)數(shù)據(jù)KolmogorovSmirnov (K-S) 檢驗(yàn)高斯分布？