statistics - Nonlinear e^(-x) regression using scipy, python, numpy -

the code below giving me flat line line of best fit rather nice curve along model of e^(-x) fit data. can show me how fix code below fits data?

import numpy np   import matplotlib.pyplot plt import scipy.optimize  def _enegx_(p,x):     x0,y0,c,k=p       y = (c * np.exp(-k*(x-x0))) + y0     return y  def _enegx_residuals(p,x,y):     return y - _enegx_(p,x)  def get_enegx_coefficients(x,y):     print 'x is:  ',x       print 'y is:  ',y       # calculate p_guess vectors x,y.  note p_guess     # starting estimate minimization.     p_guess=(np.median(x),np.min(y),np.max(y),.01)      # calls leastsq() function, calls residuals function initial      # guess parameters , x , y vectors.  note residuals     # function calls _enegx_ function.  return parameters p     # minimize least squares error of _enegx_ function respect original     # x , y coordinate vectors sent it.     p, cov, infodict, mesg, ier = scipy.optimize.leastsq(           _enegx_residuals,p_guess,args=(x,y),full_output=1,warning=true)      # define optimal values each element of p returned leastsq() function.      x0,y0,c,k=p       print('''reference data:\       x0 = {x0}     y0 = {y0}     c = {c}     k = {k}     '''.format(x0=x0,y0=y0,c=c,k=k))        print 'x.min() is:  ',x.min()     print 'x.max() is:  ',x.max()     # create numpy array of x-values     numpoints = np.floor((x.max()-x.min())*100)     xp = np.linspace(x.min(), x.max(), numpoints)     print 'numpoints is:  ',numpoints     print 'xp is:  ',xp     print 'p is:  ',p     pxp=_enegx_(p,xp)     print 'pxp is:  ',pxp      # plot results       plt.plot(x, y, '>', xp, pxp, 'g-')     plt.xlabel('bpm%rest')      plt.ylabel('lvet/bpm',rotation='vertical')     plt.xlim(0,3)     plt.ylim(0,4)     plt.grid(true)      plt.show()      return p  # declare raw data use in creating regression equation  x = np.array([1,1.425,1.736,2.178,2.518],dtype='float')   y = np.array([3.489,2.256,1.640,1.043,0.853],dtype='float')    p=get_enegx_coefficients(x,y) 

it looks it's problem initial guesses; (1, 1, 1, 1) works fine:
have

p_guess=(np.median(x),np.min(y),np.max(y),.01) 

for function

def _enegx_(p,x):     x0,y0,c,k=p       y = (c * np.exp(-k*(x-x0))) + y0     return y 

so that's test_data_maxe^( -.01(x - test_data_median)) + test_data_min

i don't know art of choosing starting parameters, can few things. leastsq finding local minimum here - key in choosing these values find right mountain climb, not try cut down on work minimization algorithm has do. initial guess looks (green): (1.736, 0.85299999999999998, 3.4889999999999999, 0.01)

which results in flat line (blue): (-59.20295956, 1.8562 , 1.03477144, 0.69483784)

greater gains made in adjusting height of line in increasing k value. if know you're fitting kind of data, use larger k. if don't know, guess try find decent k value sampling data, or working slope between average of first half , second half, wouldn't know how go that.

edit: start several guesses, run minimization several times, , take line lowest residuals.