#!/usr/bin/env python

# DO NOT EDIT
# Autogenerated from the notebook statespace_custom_models.ipynb.
# Edit the notebook and then sync the output with this file.
#
# flake8: noqa
# DO NOT EDIT

# # Custom statespace models
#
# The true power of the state space model is to allow the creation and
# estimation of custom models. This notebook shows various statespace models
# that subclass `sm.tsa.statespace.MLEModel`.
#
# Remember the general state space model can be written in the following
# general way:
#
# $$
# \begin{aligned}
# y_t & = Z_t \alpha_{t} + d_t +  \varepsilon_t \\
# \alpha_{t+1} & = T_t \alpha_{t} + c_t + R_t \eta_{t}
# \end{aligned}
# $$
#
# You can check the details and the dimensions of the objects [in this
# link](https://www.statsmodels.org/stable/statespace.html#custom-state-
# space-models)
#
# Most models won't include all of these elements. For example, the design
# matrix $Z_t$ might not depend on time ($\forall t \;Z_t = Z$), or the
# model won't have an observation intercept $d_t$.
#
# We'll start with something relatively simple and then show how to extend
# it bit by bit to include more elements.
#
# + Model 1: time-varying coefficients. One observation equation with two
# state equations
# + Model 2: time-varying parameters with non identity transition matrix
# + Model 3: multiple observation and multiple state equations
# + Bonus: pymc3 for Bayesian estimation

from collections import OrderedDict

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import statsmodels.api as sm

plt.rc("figure", figsize=(16, 8))
plt.rc("font", size=15)

# ## Model 1: time-varying coefficients
#
# $$
# \begin{aligned}
# y_t & = d + x_t \beta_{x,t} + w_t \beta_{w,t} + \varepsilon_t
# \hspace{4em} \varepsilon_t \sim N(0, \sigma_\varepsilon^2)\\
# \begin{bmatrix} \beta_{x,t} \\ \beta_{w,t} \end{bmatrix} & =
# \begin{bmatrix} \beta_{x,t-1} \\ \beta_{w,t-1} \end{bmatrix} +
# \begin{bmatrix} \zeta_{x,t} \\ \zeta_{w,t} \end{bmatrix} \hspace{3.7em}
# \begin{bmatrix} \zeta_{x,t} \\ \zeta_{w,t} \end{bmatrix} \sim N \left (
# \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma_{\beta, x}^2
# & 0 \\ 0 & \sigma_{\beta, w}^2 \end{bmatrix} \right )
# \end{aligned}
# $$
#
# The observed data is $y_t, x_t, w_t$. With $x_t, w_t$ being the
# exogenous variables. Notice that the design matrix is time-varying, so it
# will have three dimensions (`k_endog x k_states x nobs`)
#
# The states are $\beta_{x,t}$ and $\beta_{w,t}$. The state equation tells
# us these states evolve with a random walk. Thus, in this case the
# transition matrix is a 2 by 2 identity matrix.
#
# We'll first simulate the data, the construct a model and finally
# estimate it.


def gen_data_for_model1():
    nobs = 1000

    rs = np.random.RandomState(seed=93572)

    d = 5
    var_y = 5
    var_coeff_x = 0.01
    var_coeff_w = 0.5

    x_t = rs.uniform(size=nobs)
    w_t = rs.uniform(size=nobs)
    eps = rs.normal(scale=var_y**0.5, size=nobs)

    beta_x = np.cumsum(rs.normal(size=nobs, scale=var_coeff_x**0.5))
    beta_w = np.cumsum(rs.normal(size=nobs, scale=var_coeff_w**0.5))

    y_t = d + beta_x * x_t + beta_w * w_t + eps
    return y_t, x_t, w_t, beta_x, beta_w


y_t, x_t, w_t, beta_x, beta_w = gen_data_for_model1()
_ = plt.plot(y_t)


class TVRegression(sm.tsa.statespace.MLEModel):
    def __init__(self, y_t, x_t, w_t):
        exog = np.c_[x_t, w_t]  # shaped nobs x 2

        super().__init__(endog=y_t,
                                           exog=exog,
                                           k_states=2,
                                           initialization="diffuse")

        # Since the design matrix is time-varying, it must be
        # shaped k_endog x k_states x nobs
        # Notice that exog.T is shaped k_states x nobs, so we
        # just need to add a new first axis with shape 1
        self.ssm["design"] = exog.T[np.newaxis, :, :]  # shaped 1 x 2 x nobs
        self.ssm["selection"] = np.eye(self.k_states)
        self.ssm["transition"] = np.eye(self.k_states)

        # Which parameters need to be positive?
        self.positive_parameters = slice(1, 4)

    @property
    def param_names(self):
        return ["intercept", "var.e", "var.x.coeff", "var.w.coeff"]

    @property
    def start_params(self):
        """
        Defines the starting values for the parameters
        The linear regression gives us reasonable starting values for the constant
        d and the variance of the epsilon error
        """
        exog = sm.add_constant(self.exog)
        res = sm.OLS(self.endog, exog).fit()
        params = np.r_[res.params[0], res.scale, 0.001, 0.001]
        return params

    def transform_params(self, unconstrained):
        """
        We constraint the last three parameters
        ('var.e', 'var.x.coeff', 'var.w.coeff') to be positive,
        because they are variances
        """
        constrained = unconstrained.copy()
        constrained[self.positive_parameters] = (
            constrained[self.positive_parameters]**2)
        return constrained

    def untransform_params(self, constrained):
        """
        Need to unstransform all the parameters you transformed
        in the `transform_params` function
        """
        unconstrained = constrained.copy()
        unconstrained[self.positive_parameters] = (
            unconstrained[self.positive_parameters]**0.5)
        return unconstrained

    def update(self, params, **kwargs):
        params = super().update(params, **kwargs)

        self["obs_intercept", 0, 0] = params[0]
        self["obs_cov", 0, 0] = params[1]
        self["state_cov"] = np.diag(params[2:4])


# ### And then estimate it with our custom model class

mod = TVRegression(y_t, x_t, w_t)
res = mod.fit()

print(res.summary())

# The values that generated the data were:
#
# + intercept = 5
# + var.e = 5
# + var.x.coeff = 0.01
# + var.w.coeff = 0.5
#
#
# As you can see, the estimation recovered the real parameters pretty
# well.
#
# We can also recover the estimated evolution of the underlying
# coefficients (or states in Kalman filter talk)

fig, axes = plt.subplots(2, figsize=(16, 8))

ss = pd.DataFrame(res.smoothed_state.T, columns=["x", "w"])

axes[0].plot(beta_x, label="True")
axes[0].plot(ss["x"], label="Smoothed estimate")
axes[0].set(title="Time-varying coefficient on x_t")
axes[0].legend()

axes[1].plot(beta_w, label="True")
axes[1].plot(ss["w"], label="Smoothed estimate")
axes[1].set(title="Time-varying coefficient on w_t")
axes[1].legend()

fig.tight_layout()

# ## Model 2: time-varying parameters with non identity transition matrix
#
# This is a small extension from Model 1. Instead of having an identity
# transition matrix, we'll have one with two parameters ($\rho_1, \rho_2$)
# that we need to estimate.
#
#
# $$
# \begin{aligned}
# y_t & = d + x_t \beta_{x,t} + w_t \beta_{w,t} + \varepsilon_t
# \hspace{4em} \varepsilon_t \sim N(0, \sigma_\varepsilon^2)\\
# \begin{bmatrix} \beta_{x,t} \\ \beta_{w,t} \end{bmatrix} & =
# \begin{bmatrix} \rho_1 & 0 \\ 0 & \rho_2 \end{bmatrix} \begin{bmatrix}
# \beta_{x,t-1} \\ \beta_{w,t-1} \end{bmatrix} + \begin{bmatrix} \zeta_{x,t}
# \\ \zeta_{w,t} \end{bmatrix} \hspace{3.7em} \begin{bmatrix} \zeta_{x,t} \\
# \zeta_{w,t} \end{bmatrix} \sim N \left ( \begin{bmatrix} 0 \\ 0
# \end{bmatrix}, \begin{bmatrix} \sigma_{\beta, x}^2 & 0 \\ 0 &
# \sigma_{\beta, w}^2 \end{bmatrix} \right )
# \end{aligned}
# $$
#
#
# What should we modify in our previous class to make things work?
# + Good news: not a lot!
# + Bad news: we need to be careful about a few things

# ### 1) Change the starting parameters function
#
# We need to add names for the new parameters $\rho_1, \rho_2$ and we need
# to start corresponding starting values.
#
# The `param_names` function goes from:
#
# ```python
# def param_names(self):
#     return ['intercept', 'var.e', 'var.x.coeff', 'var.w.coeff']
# ```
#
#
# to
#
# ```python
# def param_names(self):
#     return ['intercept', 'var.e', 'var.x.coeff', 'var.w.coeff',
#            'rho1', 'rho2']
# ```
#
# and we change the `start_params` function from
#
# ```python
# def start_params(self):
#     exog = sm.add_constant(self.exog)
#     res = sm.OLS(self.endog, exog).fit()
#     params = np.r_[res.params[0], res.scale, 0.001, 0.001]
#     return params
# ```
#
# to
#
# ```python
# def start_params(self):
#     exog = sm.add_constant(self.exog)
#     res = sm.OLS(self.endog, exog).fit()
#     params = np.r_[res.params[0], res.scale, 0.001, 0.001, 0.8, 0.8]
#     return params
# ```
#
# 2) Change the `update` function
#
# It goes from
#
# ```python
# def update(self, params, **kwargs):
#     params = super(TVRegression, self).update(params, **kwargs)
#
#     self['obs_intercept', 0, 0] = params[0]
#     self['obs_cov', 0, 0] = params[1]
#     self['state_cov'] = np.diag(params[2:4])
# ```
#
#
# to
#
# ```python
# def update(self, params, **kwargs):
#     params = super(TVRegression, self).update(params, **kwargs)
#
#     self['obs_intercept', 0, 0] = params[0]
#     self['obs_cov', 0, 0] = params[1]
#     self['state_cov'] = np.diag(params[2:4])
#     self['transition', 0, 0] = params[4]
#     self['transition', 1, 1] = params[5]
# ```
#
#
# 3) (optional) change `transform_params` and `untransform_params`
#
# This is not required, but you might wanna restrict $\rho_1, \rho_2$ to
# lie between -1 and 1.
# In that case, we first import two utility functions from `statsmodels`.
#
#
# ```python
# from statsmodels.tsa.statespace.tools import (
#     constrain_stationary_univariate, unconstrain_stationary_univariate)
# ```
#
# `constrain_stationary_univariate` constraint the value to be within -1
# and 1.
# `unconstrain_stationary_univariate` provides the inverse function.
# The transform and untransform parameters function would look like this
# (remember that $\rho_1, \rho_2$ are in the 4 and 5th index):
#
# ```python
# def transform_params(self, unconstrained):
#     constrained = unconstrained.copy()
#     constrained[self.positive_parameters] =
# constrained[self.positive_parameters]**2
#     constrained[4] = constrain_stationary_univariate(constrained[4:5])
#     constrained[5] = constrain_stationary_univariate(constrained[5:6])
#     return constrained
#
# def untransform_params(self, constrained):
#     unconstrained = constrained.copy()
#     unconstrained[self.positive_parameters] =
# unconstrained[self.positive_parameters]**0.5
#     unconstrained[4] =
# unconstrain_stationary_univariate(constrained[4:5])
#     unconstrained[5] =
# unconstrain_stationary_univariate(constrained[5:6])
#     return unconstrained
# ```
#
# I'll write the full class below (without the optional changes I have
# just discussed)


class TVRegressionExtended(sm.tsa.statespace.MLEModel):
    def __init__(self, y_t, x_t, w_t):
        exog = np.c_[x_t, w_t]  # shaped nobs x 2

        super().__init__(endog=y_t,
                                                   exog=exog,
                                                   k_states=2,
                                                   initialization="diffuse")

        # Since the design matrix is time-varying, it must be
        # shaped k_endog x k_states x nobs
        # Notice that exog.T is shaped k_states x nobs, so we
        # just need to add a new first axis with shape 1
        self.ssm["design"] = exog.T[np.newaxis, :, :]  # shaped 1 x 2 x nobs
        self.ssm["selection"] = np.eye(self.k_states)
        self.ssm["transition"] = np.eye(self.k_states)

        # Which parameters need to be positive?
        self.positive_parameters = slice(1, 4)

    @property
    def param_names(self):
        return [
            "intercept", "var.e", "var.x.coeff", "var.w.coeff", "rho1", "rho2"
        ]

    @property
    def start_params(self):
        """
        Defines the starting values for the parameters
        The linear regression gives us reasonable starting values for the constant
        d and the variance of the epsilon error
        """

        exog = sm.add_constant(self.exog)
        res = sm.OLS(self.endog, exog).fit()
        params = np.r_[res.params[0], res.scale, 0.001, 0.001, 0.7, 0.8]
        return params

    def transform_params(self, unconstrained):
        """
        We constraint the last three parameters
        ('var.e', 'var.x.coeff', 'var.w.coeff') to be positive,
        because they are variances
        """
        constrained = unconstrained.copy()
        constrained[self.positive_parameters] = (
            constrained[self.positive_parameters]**2)
        return constrained

    def untransform_params(self, constrained):
        """
        Need to unstransform all the parameters you transformed
        in the `transform_params` function
        """
        unconstrained = constrained.copy()
        unconstrained[self.positive_parameters] = (
            unconstrained[self.positive_parameters]**0.5)
        return unconstrained

    def update(self, params, **kwargs):
        params = super().update(params, **kwargs)

        self["obs_intercept", 0, 0] = params[0]
        self["obs_cov", 0, 0] = params[1]
        self["state_cov"] = np.diag(params[2:4])
        self["transition", 0, 0] = params[4]
        self["transition", 1, 1] = params[5]


# To estimate, we'll use the same data as in model 1 and expect the
# $\rho_1, \rho_2$ to be near 1.
#
# The results look pretty good!
# Note that this estimation can be quite sensitive to the starting value
# of $\rho_1, \rho_2$. If you try lower values, you'll see it fails to
# converge.

mod = TVRegressionExtended(y_t, x_t, w_t)
res = mod.fit(maxiter=2000)  # it doesn't converge with 50 iters
print(res.summary())

# ## Model 3: multiple observation and state equations
#
# We'll keep the time-varying parameters, but this time we'll also have
# two observation equations.
#
# ### Observation equations
#
# $\hat{i_t}, \hat{M_t},  \hat{s_t}$ are observed each period.
#
# The model for the observation equation has two equations:
#
# $$ \hat{i_t} = \alpha_1 * \hat{s_t} + \varepsilon_1 $$
#
# $$ \hat{M_t} = \alpha_2 + \varepsilon_2 $$
#
# Following the [general notation from state space
# models](https://www.statsmodels.org/stable/statespace.html), the
# endogenous part of the observation equation is $y_t = (\hat{i_t},
# \hat{M_t})$ and we only have one exogenous variable $\hat{s_t}$
#
#
# ### State equations
#
#
# $$ \alpha_{1, t+1} = \delta_1 \alpha_{1, t} + \delta_2 \alpha_{2, t} +
# W_1 $$
#
# $$ \alpha_{2, t+1} = \delta_3  \alpha_{2, t} + W_2 $$
#
#
# ### Matrix notation for the state space model
#
# $$
# \begin{aligned}
# \begin{bmatrix} \hat{i_t} \\ \hat{M_t} \end{bmatrix} &=
# \begin{bmatrix} \hat{s_t}  & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix}
# \alpha_{1, t} \\ \alpha_{2, t} \end{bmatrix} + \begin{bmatrix}
# \varepsilon_{1, t} \\ \varepsilon_{1, t} \end{bmatrix}  \hspace{6.5em}
# \varepsilon_t \sim N \left ( \begin{bmatrix} 0 \\ 0 \end{bmatrix},
# \begin{bmatrix} \sigma_{\varepsilon_1}^2 & 0 \\ 0 &
# \sigma_{\varepsilon_2}^2  \end{bmatrix} \right )
# \\
# \begin{bmatrix} \alpha_{1, t+1} \\ \alpha_{2, t+1} \end{bmatrix} & =
# \begin{bmatrix} \delta_1 & \delta_1 \\ 0 & \delta_3 \end{bmatrix}
# \begin{bmatrix} \alpha_{1, t} \\ \alpha_{2, t} \end{bmatrix} +
# \begin{bmatrix} W_1 \\ W_2 \end{bmatrix} \hspace{3.em} \begin{bmatrix} W_1
# \\ W_2 \end{bmatrix} \sim N \left ( \begin{bmatrix} 0 \\ 0 \end{bmatrix},
# \begin{bmatrix} \sigma_{W_1}^2 & 0 \\ 0 & \sigma_{W_2}^2 \end{bmatrix}
# \right )
# \end{aligned}
# $$
#
# I'll simulate some data, talk about what we need to modify and finally
# estimate the model to see if we're recovering something reasonable.
#

true_values = {
    "var_e1": 0.01,
    "var_e2": 0.01,
    "var_w1": 0.01,
    "var_w2": 0.01,
    "delta1": 0.8,
    "delta2": 0.5,
    "delta3": 0.7,
}


def gen_data_for_model3():
    # Starting values
    alpha1_0 = 2.1
    alpha2_0 = 1.1

    t_max = 500

    def gen_i(alpha1, s):
        return alpha1 * s + np.sqrt(true_values["var_e1"]) * np.random.randn()

    def gen_m_hat(alpha2):
        return 1 * alpha2 + np.sqrt(true_values["var_e2"]) * np.random.randn()

    def gen_alpha1(alpha1, alpha2):
        w1 = np.sqrt(true_values["var_w1"]) * np.random.randn()
        return true_values["delta1"] * alpha1 + true_values[
            "delta2"] * alpha2 + w1

    def gen_alpha2(alpha2):
        w2 = np.sqrt(true_values["var_w2"]) * np.random.randn()
        return true_values["delta3"] * alpha2 + w2

    s_t = 0.3 + np.sqrt(1.4) * np.random.randn(t_max)
    i_hat = np.empty(t_max)
    m_hat = np.empty(t_max)

    current_alpha1 = alpha1_0
    current_alpha2 = alpha2_0
    for t in range(t_max):
        # Obs eqns
        i_hat[t] = gen_i(current_alpha1, s_t[t])
        m_hat[t] = gen_m_hat(current_alpha2)

        # state eqns
        new_alpha1 = gen_alpha1(current_alpha1, current_alpha2)
        new_alpha2 = gen_alpha2(current_alpha2)

        # Update states for next period
        current_alpha1 = new_alpha1
        current_alpha2 = new_alpha2

    return i_hat, m_hat, s_t


i_hat, m_hat, s_t = gen_data_for_model3()

# ### What do we need to modify?
#
# Once again, we don't need to change much, but we need to be careful
# about the dimensions.
#
# #### 1) The `__init__` function changes from
#
#
# ```python
# def __init__(self, y_t, x_t, w_t):
#         exog = np.c_[x_t, w_t]
#
#         super(TVRegressionExtended, self).__init__(
#             endog=y_t, exog=exog, k_states=2,
#             initialization='diffuse')
#
#         self.ssm['design'] = exog.T[np.newaxis, :, :]  # shaped 1 x 2 x
# nobs
#         self.ssm['selection'] = np.eye(self.k_states)
#         self.ssm['transition'] = np.eye(self.k_states)
# ```
#
# to
#
#
# ```python
# def __init__(self, i_t: np.array, s_t: np.array, m_t: np.array):
#
#         exog = np.c_[s_t, np.repeat(1, len(s_t))]  # exog.shape =>
# (nobs, 2)
#
#         super(MultipleYsModel, self).__init__(
#             endog=np.c_[i_t, m_t], exog=exog, k_states=2,
#             initialization='diffuse')
#
#         self.ssm['design'] = np.zeros((self.k_endog, self.k_states,
# self.nobs))
#         self.ssm['design', 0, 0, :] = s_t
#         self.ssm['design', 1, 1, :] = 1
# ```
#
# Note that we did not have to specify `k_endog` anywhere. The
# initialization does this for us after checking the dimensions of the
# `endog` matrix.
#
#
# #### 2) The `update()` function
#
# changes from
#
# ```python
# def update(self, params, **kwargs):
#     params = super(TVRegressionExtended, self).update(params, **kwargs)
#
#     self['obs_intercept', 0, 0] = params[0]
#     self['obs_cov', 0, 0] = params[1]
#
#     self['state_cov'] = np.diag(params[2:4])
#     self['transition', 0, 0] = params[4]
#     self['transition', 1, 1] = params[5]
# ```
#
#
# to
#
#
# ```python
# def update(self, params, **kwargs):
#     params = super(MultipleYsModel, self).update(params, **kwargs)
#
#
#     #The following line is not needed (by default, this matrix is
# initialized by zeroes),
#     #But I leave it here so the dimensions are clearer
#     self['obs_intercept'] = np.repeat([np.array([0, 0])], self.nobs,
# axis=0).T
#     self['obs_cov', 0, 0] = params[0]
#     self['obs_cov', 1, 1] = params[1]
#
#     self['state_cov'] = np.diag(params[2:4])
#     #delta1, delta2, delta3
#     self['transition', 0, 0] = params[4]
#     self['transition', 0, 1] = params[5]
#     self['transition', 1, 1] = params[6]
# ```
#
# The rest of the methods change in pretty obvious ways (need to add
# parameter names, make sure the indexes work, etc). The full code for the
# function is right below

starting_values = {
    "var_e1": 0.2,
    "var_e2": 0.1,
    "var_w1": 0.15,
    "var_w2": 0.18,
    "delta1": 0.7,
    "delta2": 0.1,
    "delta3": 0.85,
}


class MultipleYsModel(sm.tsa.statespace.MLEModel):
    def __init__(self, i_t: np.array, s_t: np.array, m_t: np.array):

        exog = np.c_[s_t, np.repeat(1, len(s_t))]  # exog.shape => (nobs, 2)

        super().__init__(endog=np.c_[i_t, m_t],
                                              exog=exog,
                                              k_states=2,
                                              initialization="diffuse")

        self.ssm["design"] = np.zeros((self.k_endog, self.k_states, self.nobs))
        self.ssm["design", 0, 0, :] = s_t
        self.ssm["design", 1, 1, :] = 1

        # These have ok shape. Placeholders since I'm changing them
        # in the update() function
        self.ssm["selection"] = np.eye(self.k_states)
        self.ssm["transition"] = np.eye(self.k_states)

        # Dictionary of positions to names
        self.position_dict = OrderedDict(var_e1=1,
                                         var_e2=2,
                                         var_w1=3,
                                         var_w2=4,
                                         delta1=5,
                                         delta2=6,
                                         delta3=7)
        self.initial_values = starting_values
        self.positive_parameters = slice(0, 4)

    @property
    def param_names(self):
        return list(self.position_dict.keys())

    @property
    def start_params(self):
        """
        Initial values
        """
        # (optional) Use scale for var_e1 and var_e2 starting values
        params = np.r_[self.initial_values["var_e1"],
                       self.initial_values["var_e2"],
                       self.initial_values["var_w1"],
                       self.initial_values["var_w2"],
                       self.initial_values["delta1"],
                       self.initial_values["delta2"],
                       self.initial_values["delta3"], ]
        return params

    def transform_params(self, unconstrained):
        """
        If you need to restrict parameters
        For example, variances should be > 0
        Parameters maybe have to be within -1 and 1
        """
        constrained = unconstrained.copy()
        constrained[self.positive_parameters] = (
            constrained[self.positive_parameters]**2)
        return constrained

    def untransform_params(self, constrained):
        """
        Need to reverse what you did in transform_params()
        """
        unconstrained = constrained.copy()
        unconstrained[self.positive_parameters] = (
            unconstrained[self.positive_parameters]**0.5)
        return unconstrained

    def update(self, params, **kwargs):
        params = super().update(params, **kwargs)

        # The following line is not needed (by default, this matrix is initialized by zeroes),
        # But I leave it here so the dimensions are clearer
        self["obs_intercept"] = np.repeat([np.array([0, 0])],
                                          self.nobs,
                                          axis=0).T

        self["obs_cov", 0, 0] = params[0]
        self["obs_cov", 1, 1] = params[1]

        self["state_cov"] = np.diag(params[2:4])

        # delta1, delta2, delta3
        self["transition", 0, 0] = params[4]
        self["transition", 0, 1] = params[5]
        self["transition", 1, 1] = params[6]


mod = MultipleYsModel(i_hat, s_t, m_hat)
res = mod.fit()

print(res.summary())

# ## Bonus: pymc3 for fast Bayesian estimation
#
# In this section I'll show how you can take your custom state space model
# and easily plug it to `pymc3` and estimate it with Bayesian methods. In
# particular, this example will show you an estimation with a version of
# Hamiltonian Monte Carlo called the No-U-Turn Sampler (NUTS).
#
# I'm basically copying the ideas contained [in this notebook](https://www
# .statsmodels.org/dev/examples/notebooks/generated/statespace_sarimax_pymc3
# .html), so make sure to check that for more details.

# Extra requirements
import pymc3 as pm
import theano
import theano.tensor as tt

# We need to define some helper functions to connect theano to the
# likelihood function that is implied in our model


class Loglike(tt.Op):

    itypes = [tt.dvector]  # expects a vector of parameter values when called
    otypes = [tt.dscalar]  # outputs a single scalar value (the log likelihood)

    def __init__(self, model):
        self.model = model
        self.score = Score(self.model)

    def perform(self, node, inputs, outputs):
        (theta, ) = inputs  # contains the vector of parameters
        llf = self.model.loglike(theta)
        outputs[0][0] = np.array(llf)  # output the log-likelihood

    def grad(self, inputs, g):
        # the method that calculates the gradients - it actually returns the
        # vector-Jacobian product - g[0] is a vector of parameter values
        (theta, ) = inputs  # our parameters
        out = [g[0] * self.score(theta)]
        return out


class Score(tt.Op):
    itypes = [tt.dvector]
    otypes = [tt.dvector]

    def __init__(self, model):
        self.model = model

    def perform(self, node, inputs, outputs):
        (theta, ) = inputs
        outputs[0][0] = self.model.score(theta)


# We'll simulate again the data we used for model 1.
# We'll also `fit` it again and save the results to compare them to the
# Bayesian posterior we get.

y_t, x_t, w_t, beta_x, beta_w = gen_data_for_model1()
plt.plot(y_t)

mod = TVRegression(y_t, x_t, w_t)
res_mle = mod.fit(disp=False)
print(res_mle.summary())

# ### Bayesian estimation
#
# We need to define a prior for each parameter and the number of draws and
# burn-in points

# Set sampling params
ndraws = 3000  #  3000 number of draws from the distribution
nburn = 600  # 600 number of "burn-in points" (which will be discarded)

# Construct an instance of the Theano wrapper defined above, which
# will allow PyMC3 to compute the likelihood and Jacobian in a way
# that it can make use of. Here we are using the same model instance
# created earlier for MLE analysis (we could also create a new model
# instance if we preferred)
loglike = Loglike(mod)

with pm.Model():
    # Priors
    intercept = pm.Uniform("intercept", 1, 10)
    var_e = pm.InverseGamma("var.e", 2.3, 0.5)
    var_x_coeff = pm.InverseGamma("var.x.coeff", 2.3, 0.1)
    var_w_coeff = pm.InverseGamma("var.w.coeff", 2.3, 0.1)

    # convert variables to tensor vectors
    theta = tt.as_tensor_variable([intercept, var_e, var_x_coeff, var_w_coeff])

    # use a DensityDist (use a lamdba function to "call" the Op)
    pm.DensityDist("likelihood", loglike, observed=theta)

    # Draw samples
    trace = pm.sample(
        ndraws,
        tune=nburn,
        return_inferencedata=True,
        cores=1,
        compute_convergence_checks=False,
    )

# ### How does the posterior distribution compare with the MLE estimation?
#
# The clearly peak around the MLE estimate.

results_dict = {
    "intercept": res_mle.params[0],
    "var.e": res_mle.params[1],
    "var.x.coeff": res_mle.params[2],
    "var.w.coeff": res_mle.params[3],
}
plt.tight_layout()
_ = pm.plot_trace(
    trace,
    lines=[(k, {}, [v]) for k, v in dict(results_dict).items()],
    combined=True,
    figsize=(12, 12),
)
