Dynamic Linear Models and Kalman Filter.ipynb

{
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   "source": [
    "# Dynamic Linear Models and Kalman Filter\n",
    "### Luca Laringe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook I will show some examples of how to apply Kalman Filter to dynamic linear models (a subclass of state-space models) in R using the \"dlm\" package. The notation is based on Petris et al. \"Dynamic Linear models with R\" (2007)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "State-space models consider a time series as the output of a dynamic system perturbed by random disturbances. As we shall see, they allow a natural interpretation of a time series as the result of several components, such as trend, seasonal or\n",
    "regressive components. At the same time, they have an elegant and powerful probabilistic structure, offering a flexible framework for a very wide range of applications. Computations can be implemented by recursive algorithms. The problems of estimation and forecasting are solved by recursively computing the conditional distribution of the quantities of interest, given the available information. In this sense, they are quite naturally treated from a Bayesian approach. State-space models can be used for modeling univariate or multivariate time series, also in presence of non-stationarity, structural changes, irregular\n",
    "patterns. State-space models include ARMA models as special case, but as we shall see, they can be applied to nonstationary time series without requiring a preliminary transformation of the data to get stationarity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dynamic Linear Models"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first, important class of state-space models is given by Gaussian linear state-space models, also called dynamic linear models (DLM). These models are specified by means of two equations:\n",
    "\n",
    "The state equation, which describes the state vector $\\theta_t$ (rx1), which is a Markov Chain:\n",
    "\n",
    "$\\theta_t = G_t \\theta_{t-1} + w_t, w_t \\sim \\mathcal{N_r}(0,W_t)$\n",
    "\n",
    "The observation equation, which assumes th observation vector $Y_t$ (nx1) is a function of the state vector plus a random disturbance:\n",
    "\n",
    "$Y_t = F_t \\theta_t + v_t, v_t \\sim \\mathcal{N_n}(0,V_t)$"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "The random disturbances, ($v_T$) and ($w_T$), are two independent white noise sequences. Since the Kalman filter is a recursive algorithm, the first step must be usually specified. In particular, it is also assumed:\n",
    "\n",
    "$\\theta_0 \\sim \\mathcal{N_r}(m_0,C_0)$\n",
    "\n",
    "for some non-random vector $m_0$ and matrix $C_0$, and it is independent on ($v_T$) and ($w_T$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kalman Filter"
   ]
  },
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   "source": [
    "Let us know report the Kalman filter recursive formulas for the general dynamic linear model described earlier.\n",
    "\n",
    "If $\\theta_{t-1} \\mid \\mathcal{D}_{t-1} \\sim \\mathcal{N_r}(m_{t-1},C_{t-1})$,  where $t \\geq 1$, then:"
   ]
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   "source": [
    "(a) the one-step-ahead state predictive density of $\\theta_t \\mid \\mathcal{D}_{t-1}$ is Gaussian, with parameters:\n",
    "\n",
    "$a_t = \\mathbb{E}[\\theta_t \\mid \\mathcal{D}_{t-1}] = G_t m_{t-1}$\n",
    "\n",
    "$R_t = \\mathbb{V}[\\theta_t \\mid \\mathcal{D}_{t-1}] = G_t C_{t-1} G_t^\\intercal + W_t$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) the one-step-ahead predictive density of $Y_t \\mid \\mathcal{D}_{t-1}$ is Gaussian, with parameters:\n",
    "\n",
    "$f_t = \\mathbb{E}[\\theta_t \\mid \\mathcal{D}_{t-1}] = F_t a_{t}$\n",
    "\n",
    "$Q_t = \\mathbb{V}[\\theta_t \\mid \\mathcal{D}_{t-1}] = F_t R_{t} F_t^\\intercal + V_t$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) the filtering density of $\\theta_t \\mid \\mathcal{D}_{t}$ is Gaussian, with parameters:\n",
    "\n",
    "$m_t = \\mathbb{E}[\\theta_t \\mid \\mathcal{D}_{t}] = a_t + R_t F_{t}^\\intercal Q_t^{-1} (Y_t - f_t)$\n",
    "\n",
    "$C_t = \\mathbb{V}[\\theta_t \\mid \\mathcal{D}_{t}] = R_{t} - R_t F_{t}^\\intercal Q_t^{-1} F_t R_t$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the expression of $m_t$ has the intuitive estimation-correction form\n",
    "”filter mean equal to the prediction mean at plus a correction depending on\n",
    "how much the new observation differs from its prediction”. The weight of the\n",
    "correction term is given by the gain matrix:\n",
    "\n",
    "$ K_t = R_t F_{t}^\\intercal Q_t^{-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The evaluation of the posterior variances Ct (and consequently also of Rt and Qt) using the previous iterative updating formulae, as simple as it may appear, suffers from numerical instability that may lead to nonsymmetric and even negative definite calculated variance matrices. Alternative, stabler, algorithms have been developed to overcome this issue. Apparently,\n",
    "the most widely used, at least in the Statistics literature, is the square root filter, which provides formulae for the sequential update of a square root of Ct. References for the square root filter are Morf and Kailath (1975) and Anderson and Moore (1979, Ch. 6). A more robust algorithm is the one based on sequentially updating the singular value decomposition6 (SVD) of Ct. The details of the algorithm can be found in Oshman and Bar-Itzhack (1986) and Wang et al. (1992). The dlm package use this algorithm to update variance matrices."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example: Linear Regression"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linear regression can be thought as a DLM. Moreover, in the DLM structure, it can be extended to have time-varying coefficients. Let us give the state and observation equations that describe this process. For simplicity, let us assume that the $\\beta$ vector contains the constant as a first component and hence the $x_t$ vector contains 1 as first component."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\beta_t = \\beta_{t-1} + w_t, w_t \\sim \\mathcal{N_r}(0,W_t)$\n",
    "\n",
    "$y_t = x_t'\\beta_t + v_t, v_t \\sim \\mathcal{N_n}(0,V_t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also assume our prior distribution on $\\beta_0$ to be:\n",
    "\n",
    "$\\beta_0 \\sim \\mathcal{N_2}(b_0,C_0)$\n",
    "\n",
    "Let us know see an application of this model for the dynamic estimation of Google's $\\beta$."
   ]
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   "execution_count": 1,
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     "text": [
      "-- \u001b[1mAttaching packages\u001b[22m --------------------------------------- tidyverse 1.3.0 --\n",
      "\n",
      "\u001b[32mv\u001b[39m \u001b[34mggplot2\u001b[39m 3.3.2     \u001b[32mv\u001b[39m \u001b[34mpurrr  \u001b[39m 0.3.4\n",
      "\u001b[32mv\u001b[39m \u001b[34mtibble \u001b[39m 3.0.3     \u001b[32mv\u001b[39m \u001b[34mdplyr  \u001b[39m 1.0.2\n",
      "\u001b[32mv\u001b[39m \u001b[34mtidyr  \u001b[39m 1.1.2     \u001b[32mv\u001b[39m \u001b[34mstringr\u001b[39m 1.4.0\n",
      "\u001b[32mv\u001b[39m \u001b[34mreadr  \u001b[39m 1.3.1     \u001b[32mv\u001b[39m \u001b[34mforcats\u001b[39m 0.5.0\n",
      "\n",
      "-- \u001b[1mConflicts\u001b[22m ------------------------------------------ tidyverse_conflicts() --\n",
      "\u001b[31mx\u001b[39m \u001b[34mdplyr\u001b[39m::\u001b[32mfilter()\u001b[39m masks \u001b[34mstats\u001b[39m::filter()\n",
      "\u001b[31mx\u001b[39m \u001b[34mdplyr\u001b[39m::\u001b[32mlag()\u001b[39m    masks \u001b[34mstats\u001b[39m::lag()\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Loading useful libraries\n",
    "library(tidyverse)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Registered S3 method overwritten by 'quantmod':\n",
      "  method            from\n",
      "  as.zoo.data.frame zoo \n",
      "\n"
     ]
    }
   ],
   "source": [
    "library(tseries)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "Attaching package: 'dlm'\n",
      "\n",
      "\n",
      "The following object is masked from 'package:ggplot2':\n",
      "\n",
      "    %+%\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "library(dlm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(forecast)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: xts\n",
      "\n",
      "Loading required package: zoo\n",
      "\n",
      "\n",
      "Attaching package: 'zoo'\n",
      "\n",
      "\n",
      "The following objects are masked from 'package:base':\n",
      "\n",
      "    as.Date, as.Date.numeric\n",
      "\n",
      "\n",
      "\n",
      "Attaching package: 'xts'\n",
      "\n",
      "\n",
      "The following objects are masked from 'package:dplyr':\n",
      "\n",
      "    first, last\n",
      "\n",
      "\n",
      "Loading required package: TTR\n",
      "\n",
      "Version 0.4-0 included new data defaults. See ?getSymbols.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "library(quantmod)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(broom)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(timetk)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set ggplot theme\n",
    "th <- theme_minimal() + \n",
    "  theme(plot.title = element_text(size=12, face=\"bold\"),\n",
    "        axis.title.x = element_text(size=7),                \n",
    "        axis.title.y = element_text(size=7),\n",
    "        legend.text = element_text(size=5),\n",
    "        legend.title = element_text(size=7),\n",
    "        axis.text = element_text(size=9),\n",
    "        strip.text = element_text(size=8, face = \"italic\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "'getSymbols' currently uses auto.assign=TRUE by default, but will\n",
      "use auto.assign=FALSE in 0.5-0. You will still be able to use\n",
      "'loadSymbols' to automatically load data. getOption(\"getSymbols.env\")\n",
      "and getOption(\"getSymbols.auto.assign\") will still be checked for\n",
      "alternate defaults.\n",
      "\n",
      "This message is shown once per session and may be disabled by setting \n",
      "options(\"getSymbols.warning4.0\"=FALSE). See ?getSymbols for details.\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Downloading data\n",
    "y_raw <- getSymbols(\"GOOG\",src=\"yahoo\", auto.assign = FALSE)\n",
    "x_raw <- getSymbols(\"^GSPC\",src=\"yahoo\", auto.assign = FALSE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Focus on adjusted close price\n",
    "y <- y_raw[,\"GOOG.Adjusted\"]\n",
    "x <- x_raw[,\"GSPC.Adjusted\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Modify time window\n",
    "y <- window(y, start = \"2010-01-01\", end = \"2020-07-01\")\n",
    "x <- window(x, start = \"2010-01-01\", end = \"2020-07-01\")\n",
    "\n",
    "# interpolate na\n",
    "y <- na.approx(y)\n",
    "x <- na.approx(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Monthly frequency (each price corresponds to the first observed each month)\n",
    "x <- apply.monthly(x, first)\n",
    "index(x) <- as.yearmon(index(x))\n",
    "y <- apply.monthly(y, first)\n",
    "index(y) <- as.yearmon(index(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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j7nEcmZJ36zfi8qnO6KZXx+Gh0iheGteeK36jWa9lsbfKrDDqM/sZq//3L6iKTD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\nIRJFBRQiUVRAIRJFBRQiUVRAIRJFBdR/yRec+8eM8OMAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Make sure series have same length and plot\n",
    "length(x) == length(y)\n",
    "autoplot(merge(x,y)) + th"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Log return series\n",
    "x <- na.omit(diff(log(x)))\n",
    "y <- na.omit(diff(log(y)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A tibble: 2 × 5</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>term</th><th scope=col>estimate</th><th scope=col>std.error</th><th scope=col>statistic</th><th scope=col>p.value</th></tr>\n",
       "\t<tr><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>(Intercept)</td><td>0.003720893</td><td>0.004921827</td><td>0.7559983</td><td>4.510840e-01</td></tr>\n",
       "\t<tr><td>x          </td><td>1.046359624</td><td>0.105800235</td><td>9.8899556</td><td>2.339112e-17</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A tibble: 2 × 5\n",
       "\\begin{tabular}{lllll}\n",
       " term & estimate & std.error & statistic & p.value\\\\\n",
       " <chr> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t (Intercept) & 0.003720893 & 0.004921827 & 0.7559983 & 4.510840e-01\\\\\n",
       "\t x           & 1.046359624 & 0.105800235 & 9.8899556 & 2.339112e-17\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A tibble: 2 × 5\n",
       "\n",
       "| term &lt;chr&gt; | estimate &lt;dbl&gt; | std.error &lt;dbl&gt; | statistic &lt;dbl&gt; | p.value &lt;dbl&gt; |\n",
       "|---|---|---|---|---|\n",
       "| (Intercept) | 0.003720893 | 0.004921827 | 0.7559983 | 4.510840e-01 |\n",
       "| x           | 1.046359624 | 0.105800235 | 9.8899556 | 2.339112e-17 |\n",
       "\n"
      ],
      "text/plain": [
       "  term        estimate    std.error   statistic p.value     \n",
       "1 (Intercept) 0.003720893 0.004921827 0.7559983 4.510840e-01\n",
       "2 x           1.046359624 0.105800235 9.8899556 2.339112e-17"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Check beta estimate for clasical regression\n",
    "lin_reg <- lm(y ~ x)\n",
    "tidy(lin_reg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to the classical time-series regression, the beta of Google is estimated to be around 1.04. Let's now try to look at the beta estimate when we allow this parameter to vary through time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# In order to define the model, we only need the regressor, the regressand is going to enter the model once we use a filter or a smoother\n",
    "# We can use the estimates from the simple regression as initial guesses for the state parameters (alpha and beta)\n",
    "# All the other parameters are going to be left as default\n",
    "dlm_lin_reg <- dlmModReg(x, m0 = tidy(lin_reg)[\"estimate\"][[1]])\n",
    "\n",
    "# Create filter ans smoother objects\n",
    "dlm_lin_reg_filt <- dlmFilter(y, dlm_lin_reg)\n",
    "dlm_lin_reg_smooth <- dlmSmooth(dlm_lin_reg_filt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mSGH5GahEbegMwwpCahkTcgMzb91M5/\nH0ljcwMyY9uQMufXPc6QNCgzNv6IFMANSWNjgmhcvG9nH7UzJI1tCKKx4TdkDUljG4JobPio\nnSFpbEMQDUPS6BqQGYbUJDTyBmSGITUJjbwBmWFITUIjb0BmGFKT0MgbkBmG1CQ08gZkhiE1\nCY28AZlhSE1CI29AZhhSk9DIG5AZhtQkNPIGZIYhNQmNvAGZYUhNQiNvQGYYUpPQyBuQGYbU\nJDTyBmSGITUJjbwBmWFITUIjb0BmGFKT0MgbkBmG1CQ08gZkhiE1CY28AZlhSE1CI29AZhhS\nk9DIG5AZhtQkNPIGZIYhNQmNvAGZYUhNQiNvQGYYUpPQyBuQGYbUJDTyBmSGITUJjbwBmWFI\nTUIjb0BmGFKT0MgbkBmG1CQ08gZkhiE1CY28AZlhSE1CI29AZhhSk9DIG5AZhtQkNPIGZIYh\nNQmNvAGZYUhNQiNvQGYYUpPQyBuQGYbUJDTyBmSGITUJjbwBmWFITUIjb0BmGFKT0MgbkBmG\n1CQ08gZkhiE1CY28AZlhSE1CI29AZhhSk9DIG5AZhtQkNPIGZIYhNQmNvAGZYUhNQiNvQGYY\nUpPQyBuQGYbUJDTyBmSGITUJjbwBmWFITUIjb0BmGFKT0MgbkBmG1CQ08gZkhiE1CY28AZlh\nSE1CI29AZhhSk9DIG5AZhtQkNPIGZIYhNQmNvAGZYUhNQiNvQGYYUpPQyBuQGbsOaeLWvGJH\nNSAzDKlJaOQNyAxDahIaeQMyw5CahEbegMzohvTx96n/zhxvSBrbEETj4bv2WTprIzrjDUlj\nG4JoPHrX/vg0JI20AZnxpk/tZjoyJI1tCaKxNqTLL5F+rTmnVc/yePZ75kO6+N3g+BFJY1OC\naKx91O72d0PekDS2IYiGIWl0DciMdz7YYEgaCIJoTIS0+pE7Q9LYlCAaa0L6XdH6H2wwJI1t\nCaLhz9ppdA3IDENqEhp5AzLDkJqERt6AzDCkJqGRNyAzDKlJaOQNyAxDahIaeQMyw5CahEbe\ngMwwpCahkTcgMwypSWjkDcgMQ2oSGnkDMsOQmoRG3oDMMKQmoZE3IDMMqUlo5A3IDENqEhp5\nAzLDkJqERt6AzDCkJqGRNyAzDKlJaOQNyAxDahIaeQMyw5CahEbegMwwpCahkTcgMwypSWjk\nDcgMQ2oSGnkDMsOQmoRG3oDMMKQmoZE3IDMMqUlo5A3IDENqEhp5AzLDkJqERt6AzNhPSEuy\nIWlAZhhSk9DIG5AZhtQkNPIGZIYhNQmNvAGZYUhNQiNvQGYYUpPQyBuQGYbUJDTyBmSGITUJ\njbwBmWFITUIjb0BmGFKT0MgbkBlHDun8pV6xoxqQGYY0OUMDZkBmHDqksxd7xY5qQGYY0uQM\nDZgBmWFIkzM0YAZkxoFDOp2/3Ct2VAMyw5AmZ2jADMiMY4f09w94xY5qQGYY0uQMDZgBmfFT\nQjq9Pgx4b2tgZhw8pK8/cfr1ekrAe1sDM+NnhHT6bZz+nJdmvHY04gZkxo8I6XT5Zj6ZEvDe\n1sDMOHpIX18pXb6ZT6UEvLc1MDN+Qkinz9s384mUgPe2BmbGcUP6etn/ydy8mYZ0EAMy4weE\ndGasfd0HM147GnEDMuP4IZ0ba1/3wYzXjkbcgMz4uSFNTwXe2xqYGYY0OeO1oxE3IDMMaXLG\na0cjbkBm/OCQZrcC720NzAxDmpzx2tGIG5AZPzmkybHAe1sDM+OwIV29yJCOakBm/OiQ5tYC\n720NzAxDmpzx2tGIG5AZhjQ547WjETcgM352SFNzgfe2BmaGIU3OeO1oxA3IDEOanPHa0Ygb\nkBmGNDnjtaMRNyAzfnhIM3uB97YGZoYhTc547WjEDciMo4Z0/RJDOqoBmWFIkzNeOxpxAzLj\np4c0MRh4b2tgZhjS5IzXjkbcgMwwpMkZrx2NuAGZ8eNDWr8YeG9rYGYYkiHt24DMMCRD2rcB\nmWFIhrRvAzLDkFZPBt7bGpgZBw3p5gWGdFQDMsOQDGnfBmSGIRnSvg3IDENavRl4b2tgZhjS\n59r/eh/w3tbAzDCk3398TUvAe1sDM8OQvl5juBx4b2tgZhjS0FpPjI9G3IDMMKShtZ4YH424\nAZlhSENrPTE+GnEDMuOYId0+35COakBm7DukezdnSD/HgMwwpAE1Q4yPRtyAzDCkATVDjI9G\n3IDMMKQBNUOMj0bcgMwwpAE1Q4yPRtyAzDCkATVDjI9G3IDMMKSRNUMMj0bcgMwwpJE1QwyP\nRtyAzDCkkTVDDI9G3IDMMKSRNUMMj0bcgMxghPRrxTmtfubdZz9xAx5P8bwa0iCz+/LyzS08\n149IRzUgMxgfkdbwhqSxEUE0DEmja0BmGNLAmiKGRyNuQGYY0sCaIoZHI25AZhjSwJoihkcj\nbkBmGNLAmiKGRyNuQGYY0sCaIoZHI25AZhjSwJoihkcjbkBmGNLAmiKGRyNuQGYY0sCaIoZH\nI25AZhjSwJoihkcjbkBmHDGkpWca0lENyAxDemzNEcOjETcgMwzpsTVHDI9G3IDMMKTH1hwx\nPBpxAzLDkB5bc8TwaMQNyAxDGmBzxOhoxA3IDEMaYHPE6GjEDcgMQxpgc8ToaMQNyAxDGmBz\nxOhoxA3IDEMaYHPE6GjEDcgMQxpgc8ToaMQNyAxDGmBzxOhoxA3IjAOGtPgHDemoBmSGIQ1e\ncY4YHY24AZlhSINXnCNGRyNuQGYY0uAV54jR0YgbkBmGNHjFOWJ0NOIGZIYhDV5xjhgdjbgB\nmWFIg1ecI0ZHI25AZhjS4BXniNHRiBuQGbsJafVfc2JIP8uAzDCkwSvOEaOjETcgMwxp8Ipz\nxOhoxA3IDEMavOIcMToacQMy43ghLf9BQzqqAZlhSKPXnCJGRyNuQGYY0ug117541QFesb0b\nkBmGNHzVrxcaEtKAzDCk4at+vdCQkAZkhiENX/XrZb8Cbxvwiu3dgMwwpOGrfr3IkJAGZIYh\nDV/16yWGhDQgMwxp+KpfLzEkpAGZcaCQTo8WvBbS7xcYEtKAzDhSSKcHC14K6f/nGxLSgMww\npDvajfsv8fobB7xiezcgM44U0u/f3RnwfEjfzzQkpAGZcaiQ/vzfXWN0Hv8suSEhDcgMQ7qj\n3T7LkJAGZMaxQvrz6d09Y3QMaYcGZMbBQrr7E3GRkF5/64BXbO8GZIYh3dFun2VISAMy42gh\nhT+1O3+OISENyIzjhPT4lg3pqAZkhiEtawvPMSSkAZlhSMvawnPuhDT1FgOv2N4NyAxDWtYW\nnvNr+U8Y0nsNyAxDWtBON8/5Iq5v5HT3G1cvzNAoE0Rj1yHd/JsYF+ZCSKfxDT8zQ6NLEI0f\nFdLDB9xfmKHRJYjGEUI6LXyO93kb0uk6uxUHeMX2bkBmGNIt9yCksw9Wyx+2Bgd4xfZuQGYY\n0i23IqTT6ht+ZoZGlyAaPyOki+cb0lsNyAxDuuW+Q7ok/85Yeuxu3QFesb0bkBmGdMuNQrr5\nZtIqe2KGRpcgGgcI6W8rhrQDAzLDkG64hyF9LvwogyG904DMMKQbbhDS4x/IGxzgFdu7AZlh\nSDfc6CPSwxWDA7xiezcgMwzphjudP7GKMKQ3GpAZhnTN/W3IkPZgQGYcJqTBDU+GtPTonCEh\nDcgMQ7rmDGlfBmSGIV1zT4S0/q1++d4+nfxv2cYJomFIgRlLt/DnzNzYFjt4BmSGIV1z3yFd\ni7mQnviocvYK/nea0gTR2H1Ip+8nNwvpd663MT2q6/xFhpQmiIYhjWac/RuB5692WvgYuHwD\nhpQmiIYhLbz074uv/osW3y95+F8bvHq+/1WMDPH3q07Km3KckD5Pz4X08NZPZ18U3Xw0Obua\nDxxDyhOns18gb4ohPbr1rwt2WcwcZEh54nT2K+RNOVJIy1/ADImVjxOsmLEkXT7vV+BeBr7n\nlImLTwIgb8ohQ7oFnwvpJoLhhBWGIb1M/NiQlg8+pPXf1L1P3RqG9CpxungS8qYY0rpHCVbO\nuKEMKU6cLp/e8E05rfzi+No4TkhzRDCkm1da+Kj26t0MiaAY0uWVrYV09jX3imtmSGsebVs9\n4/Twt4Y0T/x+L7732caWIU1epwOF9Cyx+Iq3z1z9feEHiiGtJq4+r/r7xOWf2uxNmb1MhrTm\nYevZGfd+2PvBg5irDySCbUNa+ntqlr5gGYS08q42pGVvlhg+aj0543Qe061hSCNi/U/YPw5p\n7X9KbsGYvkiAkEa3u/kn49GQxl+rjd7ewQNFkAg2uSwzD5P9eY3BdzZWYbfGfAuGtLRgaVLu\nne/+d66+340eXP9nd5yTzJCe+ReIH4V0OvvnxI7n/k3m94V08RNTzxqBGUsLNg5p6ZP962fd\nvZhPh3T2MC4xpOf+PfxxSKP3r8s7/um/DsCQFhYsLkq/853/bODi1btzSZ8P6e9NAUN69r3v\n/uud3b+3L/t7/tvx/V9wfL4BQ1qYUAnp+1OPB+8KSy97csfZYzuYv4Ql8c63IqTzh38Wvwr7\n9f+feOluMaTbCdmPBA+M02l48eZ+kv0RdGEErnX0srzwkeDePXh1TR/9r8f/jwG9dp8Y0s2G\nO3s2+XRoxX2+9E3dJw46pBfm/LqXwIQZvrSG9Pup9Bf5Lxu3n4dcvXzVhbsMKXCxOSGdf7z5\nfmqGNKTEjMsN+YedXzcuNy08Znz+acudrwNujDVX+3S6+2VF9LJEvjY5/97ByZAmjcCMixGj\nb6ZuvmPxXH00WXrR7fv7IKQVl/v6q4zLl+Yuy0vveMsz5r7gMaTEjD8jht9Tf+dDxue7lh9C\nH7zSMyHdfnftQdBPncH3pyeMxI6Y8fgN+vj71L/nCf7moEKCGxd/Ldjf5z5+nbu/WfWjSovf\n2ro2XjurfmJqjZHYkTMevkV/2/n4/sccf3MMaeYs/FTC6I47LT7511j/8WzpuZnL8vK3tICX\n9tGb9PFpSO82rr8Kmvnc7M7XN3feje9/kpsOCfHg4Zs+tTOkNxurH98dh3T9wPH4K8XFTzFn\nzjkP+XYWI6RfT5/T1a+eted0WnevnRaeWuT+O2tve+WfO6Mvfvv9z1/rb3MX59WQHh8/Im1j\nTH4TNvTTERfW+Htz398bXXhs/vdvUHdpzjCkoxlvDGnVD8bu8C5dYxjS4Yx7d/FrD3p8Pv6C\navRlVmgG13hTSMP73JCeNjYJ6fPy32k6z2l9Q4EZWMOQjmfceYAvv2P4cN+YSMxAGGtC+u+f\n4Z9sMKQNjdODvxasuWMzgmi86WftDGlbY6Of3Pay3DUMSaNrQGYYUmSGxtsMyAxDiszQeJsB\nmWFIkRkabzMgMwwpMkPjbQZkhiFFZmi8zYDMMKTIDI23GZAZhhSZofE2AzLDkCIzNN5mQGYY\nUmSGxtsMyAxDiszQeJsBmWFIkRkabzMgMwwpMkPjbQZkhiFFZmi8zYDMMKTIDI23GZAZxwhp\nfKtesaMakBmGFJmh8TYDMsOQIjM03mZAZhhSZIbG2wzIDEOKzNB4mwGZYUiRGRpvMyAzDCky\nQ+NtBmSGIUVmaLzNgMwwpMgMjbcZkBmGFJmh8TYDMuMQIXX+4x/Ae1sDM+MIIa25Ta/YUQ3I\nDEOKzNB4mwGZcYCQVt2kV+yoBmTG/kOq/XfdgPe2BmbG7kNaeYNesaMakBmGFJmh8TYDMmP3\nIQWMHqGRNyAz9h5S0YDM0IgTRMOQNLoGZIYhNQmNvAGZYUhNQiNvQGYYUpPQyBuQGYbUJDTy\nBmSGITUJjbwBmWFITUIjb0BmGFKT0MgbkBmG1CQ08gZkhiE1CY28AZlhSE1CI29AZhhSk9DI\nG5AZhtQkNPIGZIYhNQmNvAGZYUhNQiNvQGYYUpPQyBuQGYbUJDTyBmSGITUJjbwBmWFITUIj\nb0BmGFKT0MgbkBmG1CQ08gZkhiE1CY28AZlhSE1CI29AZhhSk9DIG5AZhtQkNPIGZIYhNQmN\nvAGZYUhNQiNvQGYYUpPQyBuQGYbUJDTyBmSGITUJjbwBmWFITUIjb0BmGFKT0MgbkBmG1CQ0\n8gZkhiE1CY28AZmxh5A8nh93DMnjCRxD8ngCx5A8nsAxJI8ncAzJ4wkcQ/J4AseQPJ7AMSSP\nJ3AMyeMJnBdC+rj3/H/P+a9ff/Lv76MHMmN6x4NXqe7wsiz+OnvyIX38+cfH3998nD8/fSAz\nZnecP/HOHV6WxV+nTyGkj889OF8AAAH2SURBVE/EFdt4xuyOsyfeusPLggjpzwfCj8+rD4iX\nkza/YhMzrp56346PbWY8dVkgM957Wd4Z0nfHH1c33w1pZsbVU+/bsWFIc5dls6+RCO8dEzve\n+hHpzs1/Ly59RJqYsdGKp+4OxI6Nhsxelq0fbBjvuPn91Hk1pP/f/reHtH7GRitmd2z2v8A7\nvSxbfkRaueOtIf35/6ubv75rNr9iEzO2GTG94+P/8/YdlMuyaUiNHc+F9HVz17+evbRxVz0z\nY4vL9dyO/JIdX5YtQprbcbNr7rwe0s0HzY/LP7OwNnWembHV//pO79hgyo4vy9YhDXfc7po7\nT35qd/Z94OvWvz9nKXzv+okZm3xK9czd8fncBYvvgNwdW3yiO7FjYdfc8WftPJ7AMSSPJ3AM\nyeMJHEPyeALHkDyewDEkjydwDMnjCRxD8ngCx5A8nsAxpB2d04Pfed57vBo7OobEPV6NHZ3T\nf/93+n3J/v3n6fcvp/+vodfxzccLsKNz+h3Q5+n8qT91vXvajz9egR2d83Quf/Eyvvt4BXZ0\nbgs6/f7czqv4/uMl2NFZ/Ij06UckwvEK7Oj4qR33eAV2dM4eofv6nM5H7SDHC+DxBI4heTyB\nY0geT+AYkscTOIbk8QSOIXk8gWNIHk/gGJLHEziG5PEEjiF5PIHzD6FY7BIgAl/CAAAAAElF\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the filtered estimated of beta through time\n",
    "t <- time(x)\n",
    "filter_beta <- zoo(dlm_lin_reg_filt$m[-1,2])\n",
    "time(filter_beta) <- t\n",
    "smoother_beta <- zoo(dlm_lin_reg_smooth$s[-1,2])\n",
    "time(smoother_beta) <- t\n",
    "\n",
    "autoplot(filter_beta) + th"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Jul 2020 \n",
       " 1.04636 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# As of sample ending's estimate of beta:\n",
    "filter_beta[length(filter_beta)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see the beta estimate is very close to the classic regression model's one. Moreover, we can see that Google's beta has been pretty constant over time. Note that the initial variance is due to the very diffuse prior we had for the state variable at time 0. Note that for this simple example we kept all the parameters to default. In general, the parameters of the model (for example the white noise process covariance matrix) can and should be estimated via maximum likelihood or bayesian methods. Let's try to estimate the parameters of the model by mle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that in the dynamic regression model we just specified, using the terminology of the last section, the unknown parameters are $W_t, V_t$, the variance matrices (scalars in the $V_t$ case) of the error terms. These were set to the default:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     [,1]\n",
      "[1,]    1\n",
      "     [,1] [,2]\n",
      "[1,]    0    0\n",
      "[2,]    0    0\n"
     ]
    }
   ],
   "source": [
    "print(dlm_lin_reg$V)\n",
    "print(dlm_lin_reg$W)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the default value for W was actually a null matrix. This implied alpha and beta costant through time because their innovation process does not have variance. If we really wanted time-varying parameters, we should have specified this matrix to be non null. On the other hand, the default value for V was 1, which is totally unrelated with our problem, so we need to change this too. How to choose the right parameters? A very convenient solution is given by mle estimation. We first have to define the matrices as functions of some parameters, then the dlm library will automatically maximize the log likelohood of the model. The dlmModReg function assumes uncorrelated innovations for the state variables, alpha and beta, hence the matrix W is diagonal and we only need to estimate 3 parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n"
     ]
    }
   ],
   "source": [
    "# define a parmeter vector (you can also define it directly in the dlmMLE function)\n",
    "parm <- c(0,0,0)  # V, W_1_1, W_2_2\n",
    "\n",
    "# define a dlm as a function of the params (we taxe e^param so we impose variances to be positive)\n",
    "build <- function(parm) {\n",
    "dlmModReg(x, m0 = tidy(lin_reg)[\"estimate\"][[1]], dV = exp(parm[1]),\n",
    "          dW = c(exp(parm[2]),exp(parm[3])))\n",
    "}\n",
    "\n",
    "# MLE\n",
    "dlm_lin_reg_mle <- dlmMLE(y, parm, build)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$par</dt>\n",
       "\t\t<dd><style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>-5.8221050567299</li><li>-26.2387816316339</li><li>-19.8346640150441</li></ol>\n",
       "</dd>\n",
       "\t<dt>$value</dt>\n",
       "\t\t<dd>-281.09938828947</dd>\n",
       "\t<dt>$counts</dt>\n",
       "\t\t<dd><style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>function</dt><dd>37</dd><dt>gradient</dt><dd>37</dd></dl>\n",
       "</dd>\n",
       "\t<dt>$convergence</dt>\n",
       "\t\t<dd>0</dd>\n",
       "\t<dt>$message</dt>\n",
       "\t\t<dd>'CONVERGENCE: REL_REDUCTION_OF_F &lt;= FACTR*EPSMCH'</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$par] \\begin{enumerate*}\n",
       "\\item -5.8221050567299\n",
       "\\item -26.2387816316339\n",
       "\\item -19.8346640150441\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item[\\$value] -281.09938828947\n",
       "\\item[\\$counts] \\begin{description*}\n",
       "\\item[function] 37\n",
       "\\item[gradient] 37\n",
       "\\end{description*}\n",
       "\n",
       "\\item[\\$convergence] 0\n",
       "\\item[\\$message] 'CONVERGENCE: REL\\_REDUCTION\\_OF\\_F <= FACTR*EPSMCH'\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$par\n",
       ":   1. -5.8221050567299\n",
       "2. -26.2387816316339\n",
       "3. -19.8346640150441\n",
       "\n",
       "\n",
       "\n",
       "$value\n",
       ":   -281.09938828947\n",
       "$counts\n",
       ":   function\n",
       ":   37gradient\n",
       ":   37\n",
       "\n",
       "\n",
       "$convergence\n",
       ":   0\n",
       "$message\n",
       ":   'CONVERGENCE: REL_REDUCTION_OF_F &lt;= FACTR*EPSMCH'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$par\n",
       "[1]  -5.822105 -26.238782 -19.834664\n",
       "\n",
       "$value\n",
       "[1] -281.0994\n",
       "\n",
       "$counts\n",
       "function gradient \n",
       "      37       37 \n",
       "\n",
       "$convergence\n",
       "[1] 0\n",
       "\n",
       "$message\n",
       "[1] \"CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH\"\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dlm_lin_reg_mle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convergence equal to zero means that convergence to a log-likelihood maximum has been achieved. We can now buil the actual model and visualized filtered and smoothed estimates for the beta, out variable of intrest."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Build the model using the mle\n",
    "dlm_lin_reg <- build(dlm_lin_reg_mle$par)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "            [,1]\n",
      "[1,] 0.002961365\n",
      "             [,1]         [,2]\n",
      "[1,] 4.023851e-12 0.000000e+00\n",
      "[2,] 0.000000e+00 2.431727e-09\n"
     ]
    }
   ],
   "source": [
    "# Visualize estimated variance matrices:\n",
    "print(dlm_lin_reg$V)\n",
    "print(dlm_lin_reg$W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mJqjZHYkTMevkV/2/n4/sccf3MMaeYs/FTC6I47LT7511j/8WzpuZnL8vK3tICX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\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Bild a filter and visualize it\n",
    "# Create filter ans smoother objects\n",
    "dlm_lin_reg_filt <- dlmFilter(y, dlm_lin_reg)\n",
    "dlm_lin_reg_smooth <- dlmSmooth(dlm_lin_reg_filt)\n",
    "# Plot the filtered estimated of beta through time\n",
    "t <- time(x)\n",
    "filter_beta <- zoo(dlm_lin_reg_filt$m[-1,2])\n",
    "time(filter_beta) <- t\n",
    "smoother_beta <- zoo(dlm_lin_reg_smooth$s[-1,2])\n",
    "time(smoother_beta) <- t\n",
    "\n",
    "autoplot(filter_beta) + th"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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oS+3RFpwioAkUBoTUCkg7t2PLMBhJIE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\nRRkUIlGUQSESRRnUf4WB+ZMGdZ0tAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "autoplot(smoother_beta[2:length(t)]) + th"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the smoother chart indicates that Google's beta has slighlty increased over the last 10 years. Let us look at our final beta estimate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Jul 2020 \n",
       " 1.04636 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "filter_beta[length(t)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The beta estimate is the same as before. Meaning that the beta has really been stable over time, as the simple regression ponts us in the same direction of the dynamic regression."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Trend models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "DLMs have the very nice property, which allows us to split a time series into several components and model each of these components as a separate DLM. For example, instead of modelling the differenced series separately from the level series, one could specify a dlm that descibes the trend, one that describes the deviations from the trend (supposedly stationary) and add them up together to forecast directly the series of intrest. Let us look at how we could model a trend component. To model a trend component, one could draw from many models, one example of such models is the random walk plus noise, or local level model. Note that in such a model, if we set the noise variance to 0 the whole exercise we just described would just amount to model the differenced series using a dlm, as the forecast for the trend componend would simply be the value of the series at time t."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Random Walk plus Noise\n",
    "The random walk plus noise, or local level model, is defined by the following two equations. \n",
    "\n",
    "$\\mu_t = \\mu_{t-1} + w_t, w_t \\sim \\mathcal{N}(0,W)$\n",
    "\n",
    "$Y_t = \\mu_t + v_t, v_t \\sim \\mathcal{N}(0,V)$\n",
    "\n",
    "As noted there, the behavior of the process (Yt) is greatly influenced by the signal-to-noise ratio r = W/V , the ratio between the two error variances. This model is very helpful wher no clear upward or downward trend is detected. Let's look at the Nile river flows as an example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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42IhEhNIqFAG0ok0mit97NEGj7f1z7Hm8elOsVHIRIinUJBpET6FingI9IplIRI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\nIBIhDYJIhDQIIhHSIIhESIMgEiEN8g8MDsKagZIXiAAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's look at the Nile river flows\n",
    "y <- Nile\n",
    "autoplot(y, size = 1) + th"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The series doesn't show any trends and seems to fluctuate around some local level. The random walk plus noise is an appropriate model to describe this series."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define a parmeter vector (you can also define it directly in the dlmMLE function)\n",
    "parm <- c(0,0)  # V, W\n",
    "\n",
    "# define a dlm as a function of the params (we taxe e^param so we impose variances to be positive)\n",
    "# We set the initial estimate od the local level to be the average of the series\n",
    "build <- function(parm) {\n",
    "dlmModPoly(order = 1, m0 = mean(y), dV = exp(parm[1]),\n",
    "          dW = exp(parm[2]))\n",
    "}\n",
    "\n",
    "# MLE\n",
    "dlm_lin_reg_mle <- dlmMLE(y, parm, build)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$par</dt>\n",
       "\t\t<dd><style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>9.62238103176116</li><li>7.29225873058687</li></ol>\n",
       "</dd>\n",
       "\t<dt>$value</dt>\n",
       "\t\t<dd>549.631881419134</dd>\n",
       "\t<dt>$counts</dt>\n",
       "\t\t<dd><style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>function</dt><dd>33</dd><dt>gradient</dt><dd>33</dd></dl>\n",
       "</dd>\n",
       "\t<dt>$convergence</dt>\n",
       "\t\t<dd>0</dd>\n",
       "\t<dt>$message</dt>\n",
       "\t\t<dd>'CONVERGENCE: REL_REDUCTION_OF_F &lt;= FACTR*EPSMCH'</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$par] \\begin{enumerate*}\n",
       "\\item 9.62238103176116\n",
       "\\item 7.29225873058687\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item[\\$value] 549.631881419134\n",
       "\\item[\\$counts] \\begin{description*}\n",
       "\\item[function] 33\n",
       "\\item[gradient] 33\n",
       "\\end{description*}\n",
       "\n",
       "\\item[\\$convergence] 0\n",
       "\\item[\\$message] 'CONVERGENCE: REL\\_REDUCTION\\_OF\\_F <= FACTR*EPSMCH'\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$par\n",
       ":   1. 9.62238103176116\n",
       "2. 7.29225873058687\n",
       "\n",
       "\n",
       "\n",
       "$value\n",
       ":   549.631881419134\n",
       "$counts\n",
       ":   function\n",
       ":   33gradient\n",
       ":   33\n",
       "\n",
       "\n",
       "$convergence\n",
       ":   0\n",
       "$message\n",
       ":   'CONVERGENCE: REL_REDUCTION_OF_F &lt;= FACTR*EPSMCH'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$par\n",
       "[1] 9.622381 7.292259\n",
       "\n",
       "$value\n",
       "[1] 549.6319\n",
       "\n",
       "$counts\n",
       "function gradient \n",
       "      33       33 \n",
       "\n",
       "$convergence\n",
       "[1] 0\n",
       "\n",
       "$message\n",
       "[1] \"CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH\"\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dlm_lin_reg_mle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The log-likelihood maximization has worked. We can go on and plot the filtered series, which represents the local leves around which the series fluctuates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "loc_level <- build(dlm_lin_reg_mle$par)\n",
    "\n",
    "# Bild a filter and visualize it\n",
    "# Create filter ans smoother objects\n",
    "loc_level_filt <- dlmFilter(y, loc_level)\n",
    "loc_level_smooth <- dlmSmooth(loc_level_filt)\n",
    "# Plot the filtered estimated of beta through time\n",
    "t <- time(y)\n",
    "filter <- zoo(loc_level_filt$m[-1])\n",
    "time(filter) <- t\n",
    "smoother <- zoo(loc_level_smooth$s[-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Don't know how to automatically pick scale for object of type ts. Defaulting to continuous.\n",
      "\n"
     ]
    },
    {
     "data": {
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mxIRZKMwy3rIGVt72pInoMJvo6/YYC03apAukNSz86Q5oaQanZJxlFi94KUO/9O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\nAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the series, the filtered local level with 95% confidence bounds\n",
    "\n",
    "# Reconstruct filtering variances from SV Decomposition\n",
    "var_filter <- ts(with(loc_level_filt, unlist(dlmSvd2var(U.C, D.C))), start = start(filter))\n",
    "sd_filter <- sqrt(var_filter)\n",
    "\n",
    "#Confidence interval under normality of errors\n",
    "upper <- filter+qnorm(0.975)*sd_filter[-1]\n",
    "lower <- filter-qnorm(0.975)*sd_filter[-1]\n",
    "\n",
    "time = time(filter)\n",
    "data <- tibble(time, y, filter, upper, lower)\n",
    "\n",
    "ggplot(data, aes(x = time)) +\n",
    "    th +\n",
    "    ggtitle(\"Filtering Estimates and Credible Intervals\") +\n",
    "    scale_y_continuous(expand = c(0,0)) +\n",
    "    xlab(\"\") + ylab(\"\") +\n",
    "    geom_line(aes(y = filter),col = c(\"red\"), size = 2) +\n",
    "    geom_line(aes(y = y),col = c(\"black\")) +\n",
    "    geom_ribbon(aes(ymin = lower, ymax = upper), alpha=0.3, linetype=1, colour=\"lightgreen\", size=0.1,fill=\"green\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The series seems to show a breakpoint just before 1900. As a matter of fact, the annual flows drop in 1898 because the first Ashwan dam was built. Let us confirm this historical fact by testing for structural change."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: sandwich\n",
      "\n",
      "\n",
      "Attaching package: 'strucchange'\n",
      "\n",
      "\n",
      "The following object is masked from 'package:stringr':\n",
      "\n",
      "    boundary\n",
      "\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "\t Optimal (m+1)-segment partition: \n",
       "\n",
       "Call:\n",
       "breakpoints.formula(formula = y ~ 1)\n",
       "\n",
       "Breakpoints at observation number:\n",
       "                      \n",
       "m = 1      28         \n",
       "m = 2      28       83\n",
       "m = 3      28    68 83\n",
       "m = 4      28 45 68 83\n",
       "m = 5   15 30 45 68 83\n",
       "\n",
       "Corresponding to breakdates:\n",
       "                                \n",
       "m = 1        1898               \n",
       "m = 2        1898           1953\n",
       "m = 3        1898      1938 1953\n",
       "m = 4        1898 1915 1938 1953\n",
       "m = 5   1885 1900 1915 1938 1953\n",
       "\n",
       "Fit:\n",
       "                                                   \n",
       "m   0       1       2       3       4       5      \n",
       "RSS 2835157 1597457 1552924 1538097 1507888 1659994\n",
       "BIC    1318    1270    1276    1285    1292    1311"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ipCzSGYBPKsbYpXuIpQcwgigTwuiqTr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\ndFDvYGaLGgAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"BIC and Residual Sum of Squares\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Nile data with one breakpoint: the annual flows drop in 1898\n",
    "library(strucchange)\n",
    "bp.nile <- breakpoints(y ~ 1)\n",
    "summary(bp.nile)\n",
    "plot(bp.nile)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the lowes BIC happens when 1 breakpoint is considered, and that breakpoint actually corresponds to 1898, in line with our historical knowledge. How can we incorporate this information to improve the local level model? We should allow a structural break during 1898, so that the filter level can more quickly adapt to the abrupt change. This is usually done by modifying the W parameter and making it become time dependent, in particular we will set it to a cerain value, $W_0$ for all periods besided 1898, where it is gonna have a value of $W_0*k$, where k is a multiplier greater than 1. We will estimate $W_0$ and $k$ by maximum likelihood."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define a dlm as a function of the params (we taxe e^param so we impose variances to be positive)\n",
    "# We set the initial estimate od the local level to be the average of the series\n",
    "build_sb <- function(parm) {\n",
    "    m <- dlmModPoly(order = 1, m0 = mean(y), dV = exp(parm[1]))\n",
    "    m$JW <- matrix(1) # Makes W be time dependent\n",
    "    m$X <- matrix(exp(parm[2]), ncol=1, nrow=length(y))\n",
    "    j <- which(time(Nile) == 1898)\n",
    "    m$X[j,1] <- m$X[j,1] * (1 + exp(parm[3]))\n",
    "    return(m)\n",
    "}\n",
    "\n",
    "dlm_lin_reg_sb_mle <- dlmMLE(y, c(0,0,0), build_sb)\n",
    "loc_level_sb <- build_sb(dlm_lin_reg_sb_mle$par)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Bild a filter and visualize it\n",
    "# Create filter ans smoother objects\n",
    "loc_level_filt <- dlmFilter(y, loc_level_sb)\n",
    "loc_level_smooth <- dlmSmooth(loc_level_filt)\n",
    "# Plot the filtered estimated of beta through time\n",
    "t <- time(y)\n",
    "filter <- zoo(loc_level_filt$m[-1])\n",
    "time(filter) <- t\n",
    "smoother <- zoo(loc_level_smooth$s[-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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qvBbVokVwhEik3oBNUwMVvjIpJnwovK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\nEiEZgkiEZAgiEZIhiERIhiASIRnyf1ZarGiuutUQAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the series, the smoothed local level with 95% confidence bounds\n",
    "\n",
    "# Reconstruct smoothing variances from SV Decomposition\n",
    "var_smoother <- ts(with(loc_level_smooth, unlist(dlmSvd2var(U.S, D.S))), start = start(smoother))\n",
    "sd_smoother <- sqrt(var_smoother)\n",
    "\n",
    "#Confidence interval under normality of errors\n",
    "upper <- smoother+qnorm(0.975)*sd_smoother[-1]\n",
    "lower <- smoother-qnorm(0.975)*sd_smoother[-1]\n",
    "\n",
    "time <- time(smoother)\n",
    "data <- tibble(time, y, filter, smoother, upper, lower)\n",
    "\n",
    "ggplot(data, aes(x = time)) +\n",
    "    th +\n",
    "    ggtitle(\"Smoothing Estimates and Credible Intervals\") +\n",
    "    scale_y_continuous(expand = c(0,0)) +\n",
    "    xlab(\"\") + ylab(\"\") +\n",
    "    geom_line(aes(y = smoother),col = c(\"red\"), size = 2) +\n",
    "    geom_line(aes(y = y),col = c(\"black\")) +\n",
    "    geom_ribbon(aes(ymin = lower, ymax = upper), alpha=0.3, linetype=1, colour=\"lightgreen\", size=0.1,fill=\"green\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear Growth Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The local level model is a special case of as class of models defined al polynomial of order n. For instance, the random walk plus noise is a polynomial model of order 1. In general, polynomial models of order n have the following forecast function:\n",
    "\n",
    "$ f_t(k) = E(Y_{t+k} \\mid \\mathcal{D}_t) = a_{t,0} + a_{t,1}k + · · · + a_{t,n-1}k^{n−1}, k \\geq 0$ \n",
    "\n",
    "where $a_{t,0}, ..., a_{t,n-1}$ are linear functions of $ m_t = E(Y_{t} \\mid \\mathcal{D}_t)$ and are independent of k. Thus, the forecast function is a polynomial of order (n − 1) in k. In the random walk plus noise, the forecast function was simply:\n",
    "\n",
    "$f_t(k) = E(Y_{t+k} \\mid \\mathcal{D}_t) = a_{t,0} = m_t$\n",
    "\n",
    "A polynomial model of order 2 is defined linear growth model, or local linear trend model. The forecast function for a linear growth model has the following form:\n",
    "\n",
    "$f_t(k) = E(Y_{t+k} \\mid \\mathcal{D}_t) = a_{t,0} + a_{t,0} k$\n",
    "\n",
    "As we can see, this forecast function extrapolates and projects into the future a linear trend of the data. The local linear trend model is defined by the following equations:\n",
    "\n",
    "$Y_t = \\mu_t + v_t, v_t \\sim \\mathcal{N}(0,V)$\n",
    "\n",
    "$\\mu_t = \\mu_{t-1} + \\beta_{t-1} + w_{1,t}, w_{1,t} \\sim \\mathcal{N}(0,\\sigma^2_{w_1})$\n",
    "\n",
    "$\\beta_t = \\beta_{t-1} + w_{2,t}, w_{2,t} \\sim \\mathcal{N}(0,\\sigma^2_{w_2})$\n",
    "\n",
    "With uncorrelated errors. Let's try to fit this model to data that show a linear trend.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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KTOOaoFIlUFkdQ5R60Y5pHWFCCSOOe4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\nEiEGQSRCDPIPkJXsUjLyVw4AAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "data(\"USeconomic\")\n",
    "log_gnp <- USeconomic[,\"log(GNP)\"]\n",
    "autoplot(log_gnp, size = 1) + th"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The log og the GNP seems a very good time-series to fit a linear growth model as the trend component. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n"
     ]
    }
   ],
   "source": [
    "# Model builder\n",
    "build <- function(parm){\n",
    "    dlmModPoly(order = 2, dV = exp(parm[1]), dW = c(exp(parm[2]), exp(parm[3])))   \n",
    "}\n",
    "\n",
    "ll_mle <- dlmMLE(log_gnp, c(0,0,0), build)\n",
    "ll <- build(ll_mle$par)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Filter estimates\n",
    "ll_filt <- dlmFilter(log_gnp, ll) \n",
    "\n",
    "# Forecast\n",
    "ll_forecast <- dlmForecast(ll_filt, 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"Removed 136 row(s) containing missing values (geom_path).\"\n",
      "Warning message:\n",
      "\"Removed 20 row(s) containing missing values (geom_path).\"\n"
     ]
    },
    {
     "data": {
      "image/png": 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IfzQe9N7oJIu6EnjuAUx+F70LuTuyDSHui5Vy0ojsPtoG3IXRBpB/Tsq38Ux+F1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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "forecast <- ll_forecast$f\n",
    "var_forecast <- unlist(ll_forecast$Q)\n",
    "sd_forecast <- sqrt(var_forecast)\n",
    "\n",
    "#Confidence interval under normality of errors\n",
    "upper <- forecast+qnorm(0.975)*sd_forecast\n",
    "lower <- forecast-qnorm(0.975)*sd_forecast\n",
    "\n",
    "data <- ts.union(log_gnp, forecast, upper, lower)\n",
    "data <- tk_tbl(data, rename_index = \"time\")\n",
    "\n",
    "ggplot(data, aes(x = time)) +\n",
    "    th +\n",
    "    ggtitle(\"Local linear trend Forecast with credible intervals\") +\n",
    "    scale_y_continuous(expand = c(0,0)) +\n",
    "    xlab(\"\") + ylab(\"\") +\n",
    "    geom_line(aes(y = forecast),col = c(\"red\"), size = 1) +\n",
    "    geom_line(aes(y = log_gnp),col = c(\"black\"), size = 1) +\n",
    "    geom_ribbon(aes(ymin = lower, ymax = upper), alpha=0.3, linetype=1, colour=\"lightgreen\", size=0.1,fill=\"green\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Seasonality models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we understood how to model trend using DLMs, it is time to learn how to model seasonalities. One of the coolest things about DLMs is the fact that you can specify different DLMs to describe different components and then simply combine them together to get a compound model. This is done seamlessly in dlm with the use of the + operator.\n",
    "Suppose that we also want to model the seasonalities of the US log GNP series. This can be done in a varieties of ways but I will just focus on the main 2 ways. I will model the detrended log gnp series as an example, then I will sum up the two models."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get detrended series\n",
    "log_gnp_detrended <- log_gnp - ll_filt$m[,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] -6.447495e-11\n"
     ]
    },
    {
     "data": {
      "image/png": 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AQp8f8KBlImQCrNTpBe4WYgXQVIpQESkDKRLvNqWmWQnGGAlBsVSLczKKSrFEJK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\nSUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(mean(log_gnp_detrended))\n",
    "autoplot(log_gnp_detrended, size = 1) + th"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Seasonal Factor Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This model takes data with a certain frequency, say 1/n, and estimates the n deviations from the mean that repeat every cycle of n observations. In our example, we have quarterly data (frequency = 1/4), so the seasonal factor model will estimate 4 factors, each related to a specific quarter, that will add up to the mean (in this case 0, since the series is detrended). In the static seasonal model, wt is degenerate on a vector of zeros (i.e., Wt = 0) More generally, the seasonal effects might change in time, so that Wt is nonzero and has to be carefully specified."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n"
     ]
    }
   ],
   "source": [
    "# Model builder\n",
    "build <- function(parm){\n",
    "    dlmModSeas(frequency = 4, dV = exp(parm[1]), dW = rep(0,3))   \n",
    "}\n",
    "\n",
    "seas_mle <- dlmMLE(log_gnp_detrended, c(0,0,0), build)\n",
    "seas <- build(seas_mle$par)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in dlmFilter(log_gnp_detrended, seas):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n"
     ]
    }
   ],
   "source": [
    "seas_fil <- dlmFilter(log_gnp_detrended, seas)\n",
    "one_step_ahead_forecasts <- seas_fil$f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Don't know how to automatically pick scale for object of type ts. Defaulting to continuous.\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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qRpie+AJJZ5BEl0Lg+pV0AKrg/TKpCQ5o/XN0Lqwy5qILmDBSTXYHlKIAkvi5A+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\nAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "time = time(log_gnp_detrended)\n",
    "data <- tibble(time, log_gnp_detrended, one_step_ahead_forecasts)\n",
    "\n",
    "ggplot(data, aes(x = time)) +\n",
    "    th +\n",
    "    ggtitle(\"Seasonal Model with One step ahead forecasts\") +\n",
    "    scale_y_continuous(expand = c(0,0)) +\n",
    "    xlab(\"\") + ylab(\"\") +\n",
    "    geom_line(aes(y = one_step_ahead_forecasts),col = c(\"red\"), size = 1) +\n",
    "    geom_line(aes(y = log_gnp_detrended),col = c(\"black\"), size = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fourier Seasonality Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These kind of models are especially used when the frequency of the data is wery low, as instead of estimating n seasonal factors, using cyclical functions leads to a more parsimonious representation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n",
      "Warning message in dlmLL(y = y, mod = mod, debug = debug):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n"
     ]
    }
   ],
   "source": [
    "# Model builder\n",
    "# s = freq, q = number of periodic functions\n",
    "build <- function(parm){\n",
    "    dlmModTrig(s = 4, q = 2, dV = exp(parm[1]))   \n",
    "}\n",
    "\n",
    "seas_fourier_mle <- dlmMLE(log_gnp_detrended, c(0,0,0), build)\n",
    "seas_fourier <- build(seas_mle$par)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in dlmFilter(log_gnp_detrended, seas_fourier):\n",
      "\"a numerically singular 'V' has been slightly perturbed to make it nonsingular\"\n"
     ]
    },
    {
     "data": {
      "image/png": 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p5yP4+3QM+H9/+6kOxVp0E6Tikdkrboj900SNtDAiTtOw3SvrG+xTmkOR7SHAfJ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\nRK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "seas_fourier_fil <- dlmFilter(log_gnp_detrended, seas_fourier)\n",
    "one_step_ahead_forecasts <- seas_fourier_fil$f\n",
    "\n",
    "time = time(log_gnp_detrended)\n",
    "data <- tibble(time, log_gnp_detrended, one_step_ahead_forecasts)  %>% \n",
    "    filter(time > 1955)\n",
    "\n",
    "ggplot(data, aes(x = time)) +\n",
    "    th +\n",
    "    ggtitle(\"Seasonal Model with One step ahead forecasts\") +\n",
    "    scale_y_continuous(expand = c(0,0)) +\n",
    "    xlab(\"\") + ylab(\"\") +\n",
    "    geom_line(aes(y = one_step_ahead_forecasts),col = c(\"red\"), size = 1) +\n",
    "    geom_line(aes(y = log_gnp_detrended),col = c(\"black\"), size = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we said before, the dlm package seamlessy allows us to sum two models, for example, let's add the linear trend model to the Fourier seasonality model and plot forecasts with credible intervals. In order to do so, we are going to assume the parameters we estimated prevously."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "# New model as a sum of models\n",
    "\n",
    "build <- function(parm){\n",
    "    m1 <- dlmModPoly(order = 2, dV = exp(parm[1]), dW = c(exp(parm[2]), exp(parm[3])))\n",
    "    m2 <- dlmModSeas(frequency = 4, dV = exp(seas_mle$par[1]), dW = rep(0,3))\n",
    "    return(m1 + m2)\n",
    "}\n",
    "\n",
    "ll_seas <- build(ll_mle$par)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Filter estimates\n",
    "ll_seas_filt <- dlmFilter(log_gnp, ll_seas) \n",
    "\n",
    "# Forecast\n",
    "ll_seas_forecast <- dlmForecast(ll_seas_filt, 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"Removed 136 row(s) containing missing values (geom_path).\"\n",
      "Warning message:\n",
      "\"Removed 20 row(s) containing missing values (geom_path).\"\n"
     ]
    },
    {
     "data": {
      "image/png": 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rhzuezzCJE8kCVLK5Jj9GWQkuSiQSRr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\n1AAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "forecast <- ll_seas_forecast$f\n",
    "var_forecast <- unlist(ll_seas_forecast$Q)\n",
    "sd_forecast <- sqrt(var_forecast)\n",
    "\n",
    "#Confidence interval under normality of errors\n",
    "upper <- forecast+qnorm(0.975)*sd_forecast\n",
    "lower <- forecast-qnorm(0.975)*sd_forecast\n",
    "\n",
    "data <- ts.union(log_gnp, forecast, upper, lower)\n",
    "data <- tk_tbl(data, rename_index = \"time\")\n",
    "\n",
    "ggplot(data, aes(x = time)) +\n",
    "    th +\n",
    "    ggtitle(\"Local linear trend Forecast with credible intervals\") +\n",
    "    scale_y_continuous(expand = c(0,0)) +\n",
    "    xlab(\"\") + ylab(\"\") +\n",
    "    geom_line(aes(y = forecast),col = c(\"red\"), size = 1) +\n",
    "    geom_line(aes(y = log_gnp),col = c(\"black\"), size = 1) +\n",
    "    geom_ribbon(aes(ymin = lower, ymax = upper), alpha=0.3, linetype=1, colour=\"lightgreen\", size=0.1,fill=\"green\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in modeldf.default(object):\n",
      "\"Could not find appropriate degrees of freedom for this model.\"\n"
     ]
    },
    {
     "data": {
      "image/png": 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WZUHRoku7R7lLkalQSSztGxQbL2lIHk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\nQICEkID+D1YL3m2hyfD7AAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's run diagnostic on the one-step ahead forecasts residuals for this model\n",
    "e <- log_gnp - ll_seas_filt$f\n",
    "checkresiduals(e[30:length(e)])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "4.0.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}