Optimization of a Voigt profile
===============================

.. code:: ipython3

    from exojax.spec.lpf import voigt
    import jax.numpy as jnp
    import matplotlib.pyplot as plt

Let’s optimize the Voigt function :math:`V(\nu, \beta, \gamma_L)` using
exojax! :math:`V(\nu, \beta, \gamma_L)` is a convolution of a Gaussian
with a STD of :math:`\beta` and a Lorentian with a gamma parameter of
:math:`\gamma_L`.

.. code:: ipython3

    nu=jnp.linspace(-10,10,100)
    plt.plot(nu, voigt(nu,1.0,2.0)) #beta=1.0, gamma_L=2.0




.. parsed-literal::

    [<matplotlib.lines.Line2D at 0x7f5dc8143100>]




.. image:: optimize_voigt_files/optimize_voigt_3_1.png


optimization of a simple absorption model
-----------------------------------------

Next, we try to fit a simple absorption model to mock data. The
absorption model is

$ f= e^{-a
V(:raw-latex:`\nu`,:raw-latex:`\beta`,:raw-latex:`\gamma`\_L)}$

.. code:: ipython3

    def absmodel(nu,a,beta,gamma_L):
        return jnp.exp(-a*voigt(nu,beta,gamma_L))

Adding a noise…

.. code:: ipython3

    from numpy.random import normal
    data=absmodel(nu,2.0,1.0,2.0)+normal(0.0,0.01,len(nu))
    plt.plot(nu,data,".")




.. parsed-literal::

    [<matplotlib.lines.Line2D at 0x7f5dc1f7ba30>]




.. image:: optimize_voigt_files/optimize_voigt_8_1.png


Let’s optimize the multiple parameters

.. code:: ipython3

    from jax import grad, vmap

We define the objective function as :math:`obj = |d - f|^2`

.. code:: ipython3

    # loss or objective function
    def obj(a,beta,gamma_L):
        f=data-absmodel(nu,a,beta,gamma_L)
        g=jnp.dot(f,f)
        return g


.. code:: ipython3

    #These are the derivative of the objective function
    h_a=grad(obj,argnums=0)
    h_beta=grad(obj,argnums=1)
    h_gamma_L=grad(obj,argnums=2)
    print(h_a(2.0,1.0,2.0),h_beta(2.0,1.0,2.0),h_gamma_L(2.0,1.0,2.0))


.. parsed-literal::

    0.0041426932 -0.00014259202 -0.0007440491


.. code:: ipython3

    from jax import jit
    
    @jit
    def step(t,opt_state):
        a,beta,gamma_L=get_params(opt_state)
        value=obj(a,beta,gamma_L)
        
        grads_a = h_a(a,beta,gamma_L)
        grads_beta = h_beta(a,beta,gamma_L)
        grads_gamma_L = h_gamma_L(a,beta,gamma_L)
    
        grads=jnp.array([grads_a,grads_beta,grads_gamma_L])
        
        opt_state = opt_update(t, grads, opt_state)
        return value, opt_state
    
    def doopt(r0,opt_init,get_params,Nstep):
        opt_state = opt_init(r0)
        traj=[r0]
        for t in range(Nstep):
            value, opt_state = step(t, opt_state)
            p=get_params(opt_state)
            traj.append(p)
        return traj, p

Here, we use the ADAM optimizer

.. code:: ipython3

    #adam
    #from jax.experimental import optimizers #for older versions of JAX
    from jax.example_libraries import optimizers
    opt_init, opt_update, get_params = optimizers.adam(1.e-1)
    r0 = jnp.array([1.5,1.5,1.5])
    trajadam, padam=doopt(r0,opt_init,get_params,1000)

Optimized values are given in padam

.. code:: ipython3

    padam




.. parsed-literal::

    DeviceArray([1.9842807, 1.0219767, 1.9737407], dtype=float32)



.. code:: ipython3

    traj=jnp.array(trajadam)
    plt.plot(traj[:,0],label="$\\alpha$")
    plt.plot(traj[:,1],ls="dashed",label="$\\beta$")
    plt.plot(traj[:,2],ls="dotted",label="$\\gamma_L$")
    plt.xscale("log")
    plt.legend()
    plt.show()



.. image:: optimize_voigt_files/optimize_voigt_19_0.png


.. code:: ipython3

    plt.plot(nu,data,".",label="data")
    plt.plot(nu,absmodel(nu,padam[0],padam[1],padam[2]),label="optimized")
    plt.show()



.. image:: optimize_voigt_files/optimize_voigt_20_0.png


Using SGD instead…, you need to increase the number of iteration for
convergence

.. code:: ipython3

    #sgd
    #from jax.experimental import optimizers #for older versions of JAX
    from jax.example_libraries import optimizers
    
    opt_init, opt_update, get_params = optimizers.sgd(1.e-1)
    r0 = jnp.array([1.5,1.5,1.5])
    trajsgd, psgd=doopt(r0,opt_init,get_params,10000)

.. code:: ipython3

    traj=jnp.array(trajsgd)
    plt.plot(traj[:,0],label="$\\alpha$")
    plt.plot(traj[:,1],ls="dashed",label="$\\beta$")
    plt.plot(traj[:,2],ls="dotted",label="$\\gamma_L$")
    plt.xscale("log")
    plt.legend()
    plt.show()



.. image:: optimize_voigt_files/optimize_voigt_23_0.png