Fitting a two-layer model to wind profile data in python

问题

I'm trying to fit a model to my dataset of wind profiles, i.e. wind speed values u(z) at different altitudes z.

The model consists of two parts, which I for now simplified to:

u(z) = ust/k * ln(z/z0)     for z < zsl

u(z) = a*z + b              for z > zsl

In the logarithmic model, ust and z0 are free parameters k is fixed. zsl is the height of the surface layer, which is also not known a priori.

I want to fit this model to my data, and I have already tried different approaches. The best result I'm getting so far is with:

def two_layer(z,hsl,ust,z0,a,b):
    return ust/0.4*(np.log(z/z0)) if z<hsl else a*z+b

two_layer = np.vectorize(two_layer)

def two_layer_err(p,z,u):
    return two_layer(z,*p)-u

popt, pcov ,infodict, mesg, ier = optimize.leastsq(two_layer_err,[150.,0.3,0.002,0.1,10.],(wspd_hgt,prof),full_output=1)

# wspd_hgt are my measurements heights and 
# prof are the corresponding wind speed values

This gives me reasonable estimates for all parameters, except for zsl which is not changed during the fitting procedure. I guess this has to do with the fact that is used as a threshold rather than a function parameter. Is there any way I could let zsl be varied during the optimization?

I tried something with numpy.piecewise, but that didn't work very well, perhaps because I don't really understand it very well, or I might be completely off here because it's not suitable for my cause.

For the idea, the wind profile looks like this if the axes are reversed (z plotted versus u):

回答1:

I think I finally have a solution for this type of problem, which I came across while answering a similar question.

The solution seems to be to implement a constraint saying that u1 == u2 at the switch between the two models. Because I cannot try it with your model because there is no data posted, I will instead show how it works for the other model, and you can adapt it to your situation. I solved this using a scipy wrapper I wrote to make fitting of such problems more pythonic, called symfit. But you could do the same using the SLSQP algorithm in scipy if you prefer.

from symfit import parameters, variables, Fit, Piecewise, exp, Eq
import numpy as np
import matplotlib.pyplot as plt

t, y = variables('t, y')
m, c, d, k, t0 = parameters('m, c, d, k, t0')

# Help the fit by bounding the switchpoint between the models
t0.min = 0.6
t0.max = 0.9

# Make a piecewise model
y1 = m * t + c
y2 = d * exp(- k * t)
model = {y: Piecewise((y1, t <= t0), (y2, t > t0))}

# As a constraint, we demand equality between the two models at the point t0
# Substitutes t in the components by t0
constraints = [Eq(y1.subs({t: t0}), y2.subs({t: t0}))]

# Read the data
tdata, ydata = np.genfromtxt('Experimental Data.csv', delimiter=',', skip_header=1).T

fit = Fit(model, t=tdata, y=ydata, constraints=constraints)
fit_result = fit.execute()
print(fit_result)

plt.scatter(tdata, ydata)
plt.plot(tdata, fit.model(t=tdata, **fit_result.params).y)
plt.show()

I think you should be able to adapt this example to your situation!

Edit: as requested in the comment, it is possible to demand matching derivatives as well. In order to do this, the following additional code is needed for the above example:

from symfit import Derivative

dy1dt = Derivative(y1, t)
dy2dt = Derivative(y2, t)
constraints = [
    Eq(y1.subs({t: t0}), y2.subs({t: t0})),
    Eq(dy1dt.subs({t: t0}), dy2dt.subs({t: t0}))
]

That should do the trick! So from a programming point of view it is very doable, although depending on the model this might not actually have a solution.

来源：https://stackoverflow.com/questions/35795341/fitting-a-two-layer-model-to-wind-profile-data-in-python

标签

python

model-fitting