![](\"https://img.shields.io/badge/R-v3.5.0-green.svg\"){kind=icon}
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"![](\"oil-spill.jpg\")
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"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Executive Summary of the BSTS Technique\n", "[[go back to the top]](#Table-of-contents)\n", "\n", "In general, structural time series (STS) models (either frequentist or Bayesian) can be written as a system of equations. The simplest possible example of STS model is the *local level model*, given by:\n", "\n", "$$\\begin{array}{l}\n", "{\\mu _{t + 1}} = {\\mu _t} + {\\xi _t},\\\\\n", "{y_t} = {\\mu _t} + {\\varepsilon _t}\n", "\\end{array}$$\n", "\n", "where ${\\xi _t} \\sim {\\cal N}(0,\\sigma _\\xi ^2)$ and ${\\varepsilon _t} \\sim {\\cal N}(0,\\sigma _\\varepsilon ^2)$.\n", "Here, the first equation is called *state equation* and ${\\mu _t}$ is an unobserved variable. Though ${\\mu _t}$ is not observed, the second equation, called *observation equation*, which depends on ${\\mu _t}$, contains the variable ${y_t}$ which is based on observed data (roughly speaking, ${\\mu _t}$ can be interpreted as a time-dependent version of the intercept of simple linear regressions).\n", "\n", "The general form of the BSTS is more convoluted so let us consider the following case, \n", "\n", "$$\\begin{array}{l}\n", "{y_t} = {\\mu _t} + {\\tau _t} + {\\beta ^T}{{\\vec x}_t} + {\\varepsilon _t}\\\\\n", "{\\mu _t} = {\\mu _{t - 1}} + {\\delta _{t - 1}} + {\\eta _t},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\n", "{\\delta _t} = {\\delta _{t - 1}} + {\\omega _t},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\n", "{\\tau _t} = - \\sum\\nolimits_{s = 1}^{S - 1} {{\\tau _{t - s}} + {\\gamma _t}} \n", "\\end{array}$$\n", "\n", "where the errors have similar properties as in the local level model. The variables have the following meanings:\n", "- ${\\tau _t}$ is the seasonal component\n", "- ${\\delta _t}$ is a random walk with trend\n", "- ${\\mu _t}$ is the trend component\n", "- ${{\\vec x}_t}$ is a vector of covariates.\n", "\n", "The last variable, namely, ${{\\vec x}_t}$ is of crucial importance in this discussion. It is a vector time series that can be used to predict ${y_t}$ and the coefficients $\\beta$ are estimated the [spike-and-slab](https://projecteuclid.org/download/pdfview_1/euclid.aos/1117114335) Bayesian method of feature selection. To use this system of equations to predict BP stock prices after the oil spill we need to identify seasonal effects, trends and choose ${{\\vec x}_t}$. An important condition that must be obeyed is that ${{\\vec x}_t}$ must (ideally) be highly correlated with ${y_t}$, but cannot have been impacted by the same factors that impacted $y_t$.\n", "\n", "Let us apply the `CausalImpact` package without going into much detail now. First let us build " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Installing packages\n", "\n", "[[go back to the top]](#Table-of-contents)\n", "\n", "Some of the packages we need to install are:\n", "\n", "- `devtools` to be able to download packages from Github\n", "- `magrittr` for code readability and maintainability\n", "- `dplyr` for data manipulation\n", "- `rga` to obtain data from the Google Analytics (GA) APIs\n", "\n", "We need also to provided the `id` corresponding to our view from GA." ] }, { "cell_type": "code", "execution_count": 107, "metadata": {}, "outputs": [], "source": [ "# # install.packages(\"devtools\")\n", "# # install.packages(\"magrittr\")\n", "# # install.packages(\"dplyr\")\n", "# # install.packages(\"curl\")\n", "# # install_github(\"skardhamar/rga\")\n", "# # install.packages(c(\"bitops\", \"jsonlite\", \"httr\"),repos='http://cran.us.r-project.org')\n", "# install.packages(\"tidyverse\")\n", "# install.packages(\"tidyr\")" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "library(devtools)\n", "library(magrittr)\n", "library(\"dplyr\")\n", "library(lubridate)\n", "library(curl)\n", "library(tidyverse)\n", "library(tidyr)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Reading the `csv` file containing the BP stock prices:" ] }, { "cell_type": "code", "execution_count": 187, "metadata": {}, "outputs": [ { "data": { "text/html": [ "