.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/6-Poisson-fLDS.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_auto_examples_6-Poisson-fLDS.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_6-Poisson-fLDS.py:


Poisson fLDS
============

.. GENERATED FROM PYTHON SOURCE LINES 19-147



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_001.png
         :alt: Simulated latent trajectories
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_002.png
         :alt: Simulated firing rates
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_002.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_003.png
         :alt: Neuron 1, Neuron 2, Neuron 3, Neuron 4, Neuron 5, Neuron 6, Neuron 7, Neuron 8, Neuron 9
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_003.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_004.png
         :alt: Batch 1
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_004.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_005.png
         :alt: Batch 2
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_005.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_006.png
         :alt: Batch 3
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_006.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_007.png
         :alt: Batch 4
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_007.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_008.png
         :alt: Batch 5
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_008.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_009.png
         :alt: 6 Poisson fLDS
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_009.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_6-Poisson-fLDS_010.png
         :alt: 6 Poisson fLDS
         :srcset: /auto_examples/images/sphx_glr_6-Poisson-fLDS_010.png
         :class: sphx-glr-multi-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Fitting LDS with SVI

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    ELBO: -17741.2: : 10178it [10:38, 15.09it/s]
    ELBO: -17730.4: : 10179it [10:38, 15.09it/s]
    ELBO: -17730.4: : 10180it [10:38, 16.62it/s]
    ELBO: -17757.1: : 10180it [10:38, 16.62it/s]
    ELBO: -17785.7: : 10181it [10:38, 16.62it/s]
    ELBO: -17785.7: : 10182it [10:38, 15.15it/s]
    ELBO: -17786.0: : 10182it [10:38, 15.15it/s]
    ELBO: -17772.1: : 10183it [10:38, 15.15it/s]
    ELBO: -17778.3: : 10184it [10:38, 15.15it/s]
    ELBO: -17778.3: : 10185it [10:38, 16.68it/s]
    ELBO: -17754.9: : 10185it [10:39, 16.68it/s]
    ELBO: -17753.1: : 10186it [10:39, 16.68it/s]
    ELBO: -17753.1: : 10187it [10:39, 15.16it/s]
    ELBO: -17764.5: : 10187it [10:39, 15.16it/s]
    ELBO: -17788.7: : 10188it [10:39, 15.16it/s]
    ELBO: -17737.1: : 10189it [10:39, 15.16it/s]
    ELBO: -17737.1: : 10190it [10:39, 14.69it/s]
    ELBO: -17791.0: : 10190it [10:39, 14.69it/s]
    ELBO: -17757.9: : 10191it [10:39, 14.69it/s]
    ELBO: -17805.9: : 10192it [10:39, 14.69it/s]
    ELBO: -17805.9: : 10193it [10:39, 16.14it/s]
    ELBO: -17771.8: : 10193it [10:39, 16.14it/s]
    ELBO: -17749.5: : 10194it [10:39, 16.14it/s]
    ELBO: -17749.5: : 10195it [10:39, 14.90it/s]
    ELBO: -17795.5: : 10195it [10:39, 14.90it/s]
    ELBO: -17740.7: : 10196it [10:39, 14.90it/s]
    ELBO: -17740.7: : 10197it [10:39, 15.75it/s]
    ELBO: -17818.5: : 10197it [10:39, 15.75it/s]
    ELBO: -17787.0: : 10198it [10:39, 15.75it/s]
    ELBO: -17787.0: : 10199it [10:39, 14.52it/s]
    ELBO: -17813.8: : 10199it [10:40, 14.52it/s]
    ELBO: -17762.5: : 10200it [10:40, 14.52it/s]
    ELBO: -17762.5: : 10201it [10:40, 15.53it/s]
    ELBO: -17762.5: 100%|##########| 10000/10000 [10:40<00:00, 15.62it/s]

    (0.0, 50.0)





|

.. code-block:: default


    # +
    import autograd.numpy as np
    import autograd.numpy.random as npr
    npr.seed(0)

    import matplotlib.pyplot as plt
    from mpl_toolkits.axes_grid1 import make_axes_locatable
    # %matplotlib inline

    import seaborn as sns
    sns.set_style("white")
    sns.set_context("talk")

    import ssm
    from ssm.util import random_rotation, find_permutation
    # -

    # Set the parameters of the LDS
    T = 50      # number of time bins per batch
    B = 20      # number of batches
    D = 2       # number of latent dimensions
    N = 10      # number of observed dimensions

    # +
    # Make an SLDS with the true parameters
    true_lds = ssm.LDS(N, D, emissions="poisson_nn", 
                   emission_kwargs=dict(link="softplus", 
                                        hidden_layer_sizes=(50, 50))
                  )
    true_lds.dynamics.As[0] = .95 * random_rotation(D, theta=(1) * np.pi/20)

    # Sample a bunch of short trajectories 
    # (they all converge so we only learn from the initial condition)
    xs, ys = list(zip(*[true_lds.sample(T) for _ in range(B)]))
    # -

    for x in xs:
        plt.plot(x[:, 0], x[:, 1])
    plt.xlabel("$x_1$")
    plt.ylabel("$x_2$")
    plt.title("Simulated latent trajectories")

    # +
    # Compute the firing rates
    rates = [true_lds.smooth(x, y) for x, y in zip(xs, ys)]

    plt.figure(figsize=(12, 8))
    for n in range(N):
        plt.subplot(N, 1, n+1)
        for lmbda in rates:
            plt.plot(lmbda[:, n])
        plt.xlabel("time")
        plt.ylabel("$\\lambda_{}(t)$".format(n+1))
    plt.suptitle("Simulated firing rates")
    # -

    # Plot the nonlinear firing rate map for neuron 1
    xmin, xmax = np.concatenate(xs).min(), np.concatenate(xs).max()
    npts = 50
    xx = np.linspace(xmin-1, xmax+1, npts)
    XX, YY = np.meshgrid(xx, xx)
    XY = np.column_stack((XX.ravel(), YY.ravel()))
    tuning_curves = true_lds.smooth(XY, np.zeros((npts**2, N)))
    assert np.all(tuning_curves > 0 )

    vmax = 1.1 * tuning_curves.max()
    plt.figure(figsize=(12, 12))
    splt = 3
    for i in range(splt):
        for j in range(splt):
            n = i * splt + j 
            if n < N:
                ax = plt.subplot(splt, splt, n+1)
                im = plt.imshow(tuning_curves[:, n].reshape((npts, npts)), vmin=0, vmax=vmax, cmap="Greys")
                if i == splt-1:
                    plt.xlabel("$x_1$")
                if j == 0:
                    plt.ylabel("$x_2$")
                
                plt.title("Neuron {}".format(n+1))
            
                divider = make_axes_locatable(ax)
                cax = divider.append_axes("right", size="5%", pad=0.05)
                plt.colorbar(im, cax=cax)
    plt.tight_layout()

    for b in range(5):
        plt.figure()
        plt.imshow(ys[b].T, aspect="auto", interpolation="none")
        plt.xlabel("time")
        plt.ylabel("neuron")
        plt.title("Batch {}".format(b+1))
        plt.colorbar()

    # +
    print("Fitting LDS with SVI")
    lds = ssm.LDS(N, D, emissions="poisson_nn", 
              emission_kwargs=dict(link="softplus", 
                                   hidden_layer_sizes=(50, 50))
             )
    lds.initialize(ys)

    lds_elbos, q = lds.fit(ys, method="bbvi", variational_posterior="mf",
                           num_iters=10000, print_intvl=100, initialize=False)
    lds_xs = q.mean
    # -

    plt.plot(lds_elbos)
    plt.xlabel("SVI Iteration")
    plt.ylabel("ELBO")

    # Smooth the observations
    lds_ys = [lds.smooth(x, y) for x, y in zip(lds_xs, ys)]

    plt.figure(figsize=(8,4))
    plt.plot(xs[0], '-k')
    plt.plot(lds_xs[0], '-')
    plt.ylabel("$x$")
    plt.xlim(0, T)

    # Plot the smoothed observations
    plt.figure(figsize=(8,4))
    plt.plot(rates[0] + 10 * np.arange(N), '-k', lw=2)
    plt.plot(lds_ys[0] + 10 * np.arange(N), '-', lw=2)
    plt.ylabel("$y$")
    plt.xlabel("time")
    plt.xlim(0, T)


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 10 minutes  41.698 seconds)


.. _sphx_glr_download_auto_examples_6-Poisson-fLDS.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: 6-Poisson-fLDS.py <6-Poisson-fLDS.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: 6-Poisson-fLDS.ipynb <6-Poisson-fLDS.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_