arma.ipynb

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    "# Covariance Stationary Processes"
   ]
  },
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   "source": [
    "## Contents\n",
    "\n",
    "- [Covariance Stationary Processes](#Covariance-Stationary-Processes)  \n",
    "  - [Overview](#Overview)  \n",
    "  - [Introduction](#Introduction)  \n",
    "  - [Spectral Analysis](#Spectral-Analysis)  \n",
    "  - [Implementation](#Implementation)  "
   ]
  },
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   "source": [
    "## Overview\n",
    "\n",
    "In this lecture we study covariance stationary linear stochastic processes, a\n",
    "class of models routinely used to study economic and financial time series\n",
    "\n",
    "This class has the advantage of being\n",
    "\n",
    "1. simple enough to be described by an elegant and comprehensive theory  \n",
    "1. relatively broad in terms of the kinds of dynamics it can represent  \n",
    "\n",
    "\n",
    "We consider these models in both the time and frequency domain"
   ]
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   "source": [
    "### ARMA Processes\n",
    "\n",
    "We will focus much of our attention on linear covariance stationary models with a finite number of parameters\n",
    "\n",
    "In particular, we will study stationary ARMA processes, which form a cornerstone of the standard theory of time series analysis\n",
    "\n",
    "Every ARMA processes can be represented in [linear state space](https://lectures.quantecon.org/py/linear_models.html) form\n",
    "\n",
    "However, ARMA have some important structure that makes it valuable to study them separately"
   ]
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    "### Spectral Analysis\n",
    "\n",
    "Analysis in the frequency domain is also called spectral analysis\n",
    "\n",
    "In essence, spectral analysis provides an alternative representation of the\n",
    "autocovariance function of a covariance stationary process\n",
    "\n",
    "Having a second representation of this important object\n",
    "\n",
    "- shines light on the dynamics of the process in question  \n",
    "- allows for a simpler, more tractable representation in some important cases  \n",
    "\n",
    "\n",
    "The famous *Fourier transform* and its inverse are used to map between the two representations"
   ]
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   "source": [
    "### Other Reading\n",
    "\n",
    "For supplementary reading, see\n",
    "\n",
    "- [[LS18]](https://lectures.quantecon.org/py/zreferences.html#ljungqvist2012), chapter 2  \n",
    "- [[Sar87]](https://lectures.quantecon.org/py/zreferences.html#sargent1987), chapter 11  \n",
    "- John Cochrane’s <a href=/_static/pdfs/time_series_book.pdf download>notes on time series analysis</a>, chapter 8  \n",
    "- [[Shi95]](https://lectures.quantecon.org/py/zreferences.html#shiryaev1995), chapter 6  \n",
    "- [[CC08]](https://lectures.quantecon.org/py/zreferences.html#cryerchan2008), all  "
   ]
  },
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   "source": [
    "## Introduction\n",
    "\n",
    "Consider a sequence of random variables $ \\{ X_t \\} $ indexed by $ t \\in \\mathbb Z $ and taking values in $ \\mathbb R $\n",
    "\n",
    "Thus, $ \\{ X_t \\} $ begins in the infinite past and extends to the infinite future — a convenient and standard assumption\n",
    "\n",
    "As in other fields, successful economic modeling typically assumes the existence of features that are constant over time\n",
    "\n",
    "If these assumptions are correct, then each new observation $ X_t, X_{t+1},\\ldots $ can provide additional information about the time-invariant features, allowing us to  learn from as data arrive\n",
    "\n",
    "For this reason, we will focus in what follows on processes that are *stationary* — or become so after a transformation\n",
    "(see for example [this lecture](https://lectures.quantecon.org/py/additive_functionals.html) and [this lecture](https://lectures.quantecon.org/py/multiplicative_functionals.html))\n",
    "\n",
    "\n",
    "<a id='arma-defs'></a>"
   ]
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   "source": [
    "### Definitions\n",
    "\n",
    "\n",
    "<a id='index-3'></a>\n",
    "A real-valued stochastic process $ \\{ X_t \\} $ is called *covariance stationary* if\n",
    "\n",
    "1. Its mean $ \\mu := \\mathbb E X_t $ does not depend on $ t $  \n",
    "1. For all $ k $ in $ \\mathbb Z $, the $ k $-th autocovariance $ \\gamma(k) := \\mathbb E (X_t - \\mu)(X_{t + k} - \\mu) $ is finite and depends only on $ k $  \n",
    "\n",
    "\n",
    "The function $ \\gamma \\colon \\mathbb Z \\to \\mathbb R $ is called the *autocovariance function* of the process\n",
    "\n",
    "Throughout this lecture, we will work exclusively with zero-mean (i.e., $ \\mu = 0 $) covariance stationary processes\n",
    "\n",
    "The zero-mean assumption costs nothing in terms of generality, since working with non-zero-mean processes involves no more than adding a constant"
   ]
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    "### Example 1: White Noise\n",
    "\n",
    "Perhaps the simplest class of covariance stationary processes is the white noise processes\n",
    "\n",
    "A process $ \\{ \\epsilon_t \\} $ is called a *white noise process* if\n",
    "\n",
    "1. $ \\mathbb E \\epsilon_t = 0 $  \n",
    "1. $ \\gamma(k) = \\sigma^2 \\mathbf 1\\{k = 0\\} $ for some $ \\sigma > 0 $  \n",
    "\n",
    "\n",
    "(Here $ \\mathbf 1\\{k = 0\\} $ is defined to be 1 if $ k = 0 $ and zero otherwise)\n",
    "\n",
    "White noise processes play the role of **building blocks** for processes with more complicated dynamics\n",
    "\n",
    "\n",
    "<a id='generalized-lps'></a>"
   ]
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    "### Example 2: General Linear Processes\n",
    "\n",
    "From the simple building block provided by white noise, we can construct a very flexible family of covariance stationary processes — the *general linear processes*\n",
    "\n",
    "\n",
    "<a id='equation-ma-inf'></a>\n",
    "$$\n",
    "X_t = \\sum_{j=0}^{\\infty} \\psi_j \\epsilon_{t-j},\n",
    "\\qquad t \\in \\mathbb Z \\tag{1}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "- $ \\{\\epsilon_t\\} $ is white noise  \n",
    "- $ \\{\\psi_t\\} $ is a square summable sequence in $ \\mathbb R $ (that is, $ \\sum_{t=0}^{\\infty} \\psi_t^2 < \\infty $)  \n",
    "\n",
    "\n",
    "The sequence $ \\{\\psi_t\\} $ is often called a *linear filter*\n",
    "\n",
    "Equation [(1)](#equation-ma-inf) is said to present  a **moving average** process or a moving average representation\n",
    "\n",
    "With some manipulations it is possible to confirm that the autocovariance function for [(1)](#equation-ma-inf) is\n",
    "\n",
    "\n",
    "<a id='equation-ma-inf-ac'></a>\n",
    "$$\n",
    "\\gamma(k) = \\sigma^2 \\sum_{j=0}^{\\infty} \\psi_j \\psi_{j+k} \\tag{2}\n",
    "$$\n",
    "\n",
    "By the [Cauchy-Schwartz inequality](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality) one can show that $ \\gamma(k) $ satisfies equation [(2)](#equation-ma-inf-ac)\n",
    "\n",
    "Evidently, $ \\gamma(k) $ does not depend on $ t $"
   ]
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   "source": [
    "### Wold’s Decomposition\n",
    "\n",
    "Remarkably, the class of general linear processes goes a long way towards\n",
    "describing the entire class of zero-mean covariance stationary processes\n",
    "\n",
    "In particular, [Wold’s decomposition theorem](https://en.wikipedia.org/wiki/Wold%27s_theorem) states that every\n",
    "zero-mean covariance stationary process $ \\{X_t\\} $ can be written as\n",
    "\n",
    "$$\n",
    "X_t = \\sum_{j=0}^{\\infty} \\psi_j \\epsilon_{t-j} + \\eta_t\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "- $ \\{\\epsilon_t\\} $ is white noise  \n",
    "- $ \\{\\psi_t\\} $ is square summable  \n",
    "- $ \\eta_t $ can be expressed as a linear function of $ X_{t-1}, X_{t-2},\\ldots $ and is perfectly predictable over arbitrarily long horizons  \n",
    "\n",
    "\n",
    "For intuition and further discussion, see [[Sar87]](https://lectures.quantecon.org/py/zreferences.html#sargent1987), p. 286"
   ]
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    "### AR and MA\n",
    "\n",
    "\n",
    "<a id='index-8'></a>\n",
    "General linear processes are a very broad class of processes.\n",
    "\n",
    "It often pays to specialize to those for which there exists a representation having only finitely many parameters\n",
    "\n",
    "(Experience and theory combine to indicate that models with a relatively small number of parameters typically perform better than larger models, especially for forecasting)\n",
    "\n",
    "One very simple example of such a model is the first-order autoregressive or AR(1) process\n",
    "\n",
    "\n",
    "<a id='equation-ar1-rep'></a>\n",
    "$$\n",
    "X_t = \\phi X_{t-1} + \\epsilon_t\n",
    "\\quad \\text{where} \\quad\n",
    "| \\phi | < 1\n",
    "\\quad \\text{and } \\{ \\epsilon_t \\} \\text{ is white noise} \\tag{3}\n",
    "$$\n",
    "\n",
    "By direct substitution, it is easy to verify that $ X_t = \\sum_{j=0}^{\\infty} \\phi^j \\epsilon_{t-j} $\n",
    "\n",
    "Hence $ \\{X_t\\} $ is a general linear process\n",
    "\n",
    "Applying [(2)](#equation-ma-inf-ac) to the previous expression for $ X_t $, we get the AR(1) autocovariance function\n",
    "\n",
    "\n",
    "<a id='equation-ar-acov'></a>\n",
    "$$\n",
    "\\gamma(k) = \\phi^k \\frac{\\sigma^2}{1 - \\phi^2},\n",
    "\\qquad k = 0, 1, \\ldots \\tag{4}\n",
    "$$\n",
    "\n",
    "The next figure plots an example of this function for $ \\phi = 0.8 $ and $ \\phi = -0.8 $ with $ \\sigma = 1 $"
   ]
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\n",
      "text/plain": [
       "<Figure size 720x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "num_rows, num_cols = 2, 1\n",
    "fig, axes = plt.subplots(num_rows, num_cols, figsize=(10, 8))\n",
    "plt.subplots_adjust(hspace=0.4)\n",
    "\n",
    "for i, ϕ in enumerate((0.8, -0.8)):\n",
    "    ax = axes[i]\n",
    "    times = list(range(16))\n",
    "    acov = [ϕ**k / (1 - ϕ**2) for k in times]\n",
    "    ax.plot(times, acov, 'bo-', alpha=0.6, label=f'autocovariance, $\\phi = {ϕ:.2}$')\n",
    "    ax.legend(loc='upper right')\n",
    "    ax.set(xlabel='time', xlim=(0, 15))\n",
    "    ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another very simple process is the MA(1) process (here MA means “moving average”)\n",
    "\n",
    "$$\n",
    "X_t = \\epsilon_t + \\theta \\epsilon_{t-1}\n",
    "$$\n",
    "\n",
    "You will be able to verify that\n",
    "\n",
    "$$\n",
    "\\gamma(0) = \\sigma^2 (1 + \\theta^2),\n",
    "\\quad\n",
    "\\gamma(1) = \\sigma^2 \\theta,\n",
    "\\quad \\text{and} \\quad\n",
    "\\gamma(k) = 0 \\quad \\forall \\, k > 1\n",
    "$$\n",
    "\n",
    "The AR(1) can be generalized to an AR($ p $) and likewise for the MA(1)\n",
    "\n",
    "Putting all of this together, we get the"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ARMA Processes\n",
    "\n",
    "A stochastic process $ \\{X_t\\} $ is called an *autoregressive moving\n",
    "average process*, or ARMA($ p,q $), if it can be written as\n",
    "\n",
    "\n",
    "<a id='equation-arma'></a>\n",
    "$$\n",
    "X_t = \\phi_1 X_{t-1} + \\cdots + \\phi_p X_{t-p} +\n",
    "    \\epsilon_t + \\theta_1 \\epsilon_{t-1} + \\cdots + \\theta_q \\epsilon_{t-q} \\tag{5}\n",
    "$$\n",
    "\n",
    "where $ \\{ \\epsilon_t \\} $ is white noise\n",
    "\n",
    "An alternative notation for ARMA processes uses the *lag operator* $ L $\n",
    "\n",
    "**Def.** Given arbitrary variable $ Y_t $, let $ L^k Y_t := Y_{t-k} $\n",
    "\n",
    "It turns out that\n",
    "\n",
    "- lag operators facilitate  succinct representations for linear stochastic processes  \n",
    "- algebraic manipulations that treat the lag operator as an ordinary scalar  are legitimate  \n",
    "\n",
    "\n",
    "Using $ L $, we can rewrite [(5)](#equation-arma) as\n",
    "\n",
    "\n",
    "<a id='equation-arma-lag'></a>\n",
    "$$\n",
    "L^0 X_t - \\phi_1 L^1 X_t - \\cdots - \\phi_p L^p X_t\n",
    "= L^0 \\epsilon_t + \\theta_1 L^1 \\epsilon_t + \\cdots + \\theta_q L^q \\epsilon_t \\tag{6}\n",
    "$$\n",
    "\n",
    "If we let $ \\phi(z) $ and $ \\theta(z) $ be the polynomials\n",
    "\n",
    "\n",
    "<a id='equation-arma-poly'></a>\n",
    "$$\n",
    "\\phi(z) := 1 - \\phi_1 z - \\cdots - \\phi_p z^p\n",
    "\\quad \\text{and} \\quad\n",
    "\\theta(z) := 1 + \\theta_1 z + \\cdots + \\theta_q z^q \\tag{7}\n",
    "$$\n",
    "\n",
    "then [(6)](#equation-arma-lag)  becomes\n",
    "\n",
    "\n",
    "<a id='equation-arma-lag1'></a>\n",
    "$$\n",
    "\\phi(L) X_t = \\theta(L) \\epsilon_t \\tag{8}\n",
    "$$\n",
    "\n",
    "In what follows we **always assume** that the roots of the polynomial $ \\phi(z) $ lie outside the unit circle in the complex plane\n",
    "\n",
    "This condition is sufficient to guarantee that the ARMA($ p,q $) process is convariance stationary\n",
    "\n",
    "In fact it implies that the process falls within the class of general linear processes [described above](#generalized-lps)\n",
    "\n",
    "That is, given an ARMA($ p,q $) process $ \\{ X_t \\} $ satisfying the unit circle condition, there exists a square summable sequence $ \\{\\psi_t\\} $ with $ X_t = \\sum_{j=0}^{\\infty} \\psi_j \\epsilon_{t-j} $ for all $ t $\n",
    "\n",
    "The sequence $ \\{\\psi_t\\} $ can be obtained by a recursive procedure outlined on page 79 of [[CC08]](https://lectures.quantecon.org/py/zreferences.html#cryerchan2008)\n",
    "\n",
    "The function $ t \\mapsto \\psi_t $ is often called the *impulse response function*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Spectral Analysis\n",
    "\n",
    "Autocovariance functions provide a great deal of information about covariance stationary processes\n",
    "\n",
    "In fact, for zero-mean Gaussian processes, the autocovariance function characterizes the entire joint distribution\n",
    "\n",
    "Even for non-Gaussian processes, it provides a significant amount of information\n",
    "\n",
    "It turns out that there is an alternative representation of the autocovariance function of a covariance stationary process, called the *spectral density*\n",
    "\n",
    "At times, the spectral density is easier to derive, easier to manipulate, and provides additional intuition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Complex Numbers\n",
    "\n",
    "Before discussing the spectral density, we invite you to recall the main properties of complex numbers (or [skip to the next section](#arma-specd))\n",
    "\n",
    "It can be helpful to remember that, in a formal sense, complex numbers are just points $ (x, y) \\in \\mathbb R^2 $ endowed with a specific notion of multiplication\n",
    "\n",
    "When $ (x, y) $ is regarded as a complex number, $ x $ is called the *real part* and $ y $ is called the *imaginary part*\n",
    "\n",
    "The *modulus* or *absolute value* of a complex number $ z = (x, y) $ is just its Euclidean norm in $ \\mathbb R^2 $, but is usually written as $ |z| $ instead of $ \\|z\\| $\n",
    "\n",
    "The product of two complex numbers $ (x, y) $ and $ (u, v) $ is defined to be $ (xu - vy, xv + yu) $, while addition is standard pointwise vector addition\n",
    "\n",
    "When endowed with these notions of multiplication and addition, the set of complex numbers forms a [field](https://en.wikipedia.org/wiki/Field_%28mathematics%29) — addition and multiplication play well together, just as they do in $ \\mathbb R $\n",
    "\n",
    "The complex number $ (x, y) $ is often written as $ x + i y $, where $ i $ is called the *imaginary unit*, and is understood to obey $ i^2 = -1 $\n",
    "\n",
    "The $ x + i y $ notation provides an easy way to remember the definition of multiplication given above, because, proceeding naively,\n",
    "\n",
    "$$\n",
    "(x + i y) (u + i v) = xu - yv + i (xv + yu)\n",
    "$$\n",
    "\n",
    "Converted back to our first notation, this becomes $ (xu - vy, xv + yu) $ as promised\n",
    "\n",
    "Complex numbers can be represented in  the polar form $ r e^{i \\omega} $ where\n",
    "\n",
    "$$\n",
    "r e^{i \\omega} := r (\\cos(\\omega) + i \\sin(\\omega)) = x + i y\n",
    "$$\n",
    "\n",
    "where $ x = r \\cos(\\omega), y = r \\sin(\\omega) $, and $ \\omega = \\arctan(y/z) $ or $ \\tan(\\omega) = y/x $\n",
    "\n",
    "\n",
    "<a id='arma-specd'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Spectral Densities\n",
    "\n",
    "Let $ \\{ X_t \\} $ be a covariance stationary process with autocovariance function $ \\gamma $  satisfying $ \\sum_{k} \\gamma(k)^2 < \\infty $\n",
    "\n",
    "The *spectral density* $ f $ of $ \\{ X_t \\} $ is defined as the [discrete time Fourier transform](https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform) of its autocovariance function $ \\gamma $\n",
    "\n",
    "$$\n",
    "f(\\omega) := \\sum_{k \\in \\mathbb Z} \\gamma(k) e^{-i \\omega k},\n",
    "\\qquad \\omega \\in \\mathbb R\n",
    "$$\n",
    "\n",
    "(Some authors normalize the expression on the right by constants such as $ 1/\\pi $ — the convention chosen  makes little difference provided you are consistent)\n",
    "\n",
    "Using the fact that $ \\gamma $ is *even*, in the sense that $ \\gamma(t) = \\gamma(-t) $ for all $ t $, we can show that\n",
    "\n",
    "\n",
    "<a id='equation-arma-sd-cos'></a>\n",
    "$$\n",
    "f(\\omega) = \\gamma(0) + 2 \\sum_{k \\geq 1} \\gamma(k) \\cos(\\omega k) \\tag{9}\n",
    "$$\n",
    "\n",
    "It is not difficult to confirm that $ f $ is\n",
    "\n",
    "- real-valued  \n",
    "- even ($ f(\\omega) = f(-\\omega) $   ),  and  \n",
    "- $ 2\\pi $-periodic, in the sense that $ f(2\\pi + \\omega) = f(\\omega) $ for all $ \\omega $  \n",
    "\n",
    "\n",
    "It follows that the values of $ f $ on $ [0, \\pi] $ determine the values of $ f $ on\n",
    "all of $ \\mathbb R $ — the proof is an exercise\n",
    "\n",
    "For this reason it is standard to plot the spectral density only on the interval $ [0, \\pi] $\n",
    "\n",
    "\n",
    "<a id='arma-wnsd'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 1: White Noise\n",
    "\n",
    "Consider a white noise process $ \\{\\epsilon_t\\} $ with standard deviation $ \\sigma $\n",
    "\n",
    "It is easy to check that in  this case $ f(\\omega) = \\sigma^2 $.  So $ f $ is a constant function\n",
    "\n",
    "As we will see, this can be interpreted as meaning that “all frequencies are equally present”\n",
    "\n",
    "(White light has this property when frequency refers to the visible spectrum, a connection that provides the origins of the term “white noise”)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 2: AR and MA and ARMA\n",
    "\n",
    "It is an exercise to show that the MA(1) process $ X_t = \\theta \\epsilon_{t-1} + \\epsilon_t $ has spectral density\n",
    "\n",
    "\n",
    "<a id='equation-ma1-sd-ed'></a>\n",
    "$$\n",
    "f(\\omega)\n",
    "= \\sigma^2 ( 1 + 2 \\theta \\cos(\\omega) + \\theta^2 ) \\tag{10}\n",
    "$$\n",
    "\n",
    "With a bit more effort, it’s possible to show (see, e.g., p. 261 of [[Sar87]](https://lectures.quantecon.org/py/zreferences.html#sargent1987)) that the spectral density of the AR(1) process $ X_t = \\phi X_{t-1} + \\epsilon_t $ is\n",
    "\n",
    "\n",
    "<a id='equation-ar1-sd-ed'></a>\n",
    "$$\n",
    "f(\\omega)\n",
    "= \\frac{\\sigma^2}{ 1 - 2 \\phi \\cos(\\omega) + \\phi^2 } \\tag{11}\n",
    "$$\n",
    "\n",
    "More generally, it can be shown that the spectral density of the ARMA process [(5)](#equation-arma) is\n",
    "\n",
    "\n",
    "<a id='arma-spec-den'></a>\n",
    "\n",
    "<a id='equation-arma-sd'></a>\n",
    "$$\n",
    "f(\\omega) = \\left| \\frac{\\theta(e^{i\\omega})}{\\phi(e^{i\\omega})} \\right|^2 \\sigma^2 \\tag{12}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "- $ \\sigma $ is the standard deviation of the white noise process $ \\{\\epsilon_t\\} $  \n",
    "- the polynomials $ \\phi(\\cdot) $ and $ \\theta(\\cdot) $ are as defined in [(7)](#equation-arma-poly)  \n",
    "\n",
    "\n",
    "The derivation of [(12)](#equation-arma-sd) uses the fact that convolutions become products under Fourier transformations\n",
    "\n",
    "The proof is elegant and can be found in many places — see, for example, [[Sar87]](https://lectures.quantecon.org/py/zreferences.html#sargent1987), chapter 11, section 4\n",
    "\n",
    "It’s a nice exercise to verify that [(10)](#equation-ma1-sd-ed) and [(11)](#equation-ar1-sd-ed) are indeed special cases of [(12)](#equation-arma-sd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interpreting the Spectral Density\n",
    "\n",
    "\n",
    "<a id='index-18'></a>\n",
    "Plotting [(11)](#equation-ar1-sd-ed) reveals the shape of the spectral density for the AR(1) model when $ \\phi $ takes the values 0.8 and -0.8 respectively"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def ar1_sd(ϕ, ω):\n",
    "    return 1 / (1 - 2 * ϕ * np.cos(ω) + ϕ**2)\n",
    "\n",
    "ωs = np.linspace(0, np.pi, 180)\n",
    "num_rows, num_cols = 2, 1\n",
    "fig, axes = plt.subplots(num_rows, num_cols, figsize=(10, 8))\n",
    "plt.subplots_adjust(hspace=0.4)\n",
    "\n",
    "# Autocovariance when phi = 0.8\n",
    "for i, ϕ in enumerate((0.8, -0.8)):\n",
    "    ax = axes[i]\n",
    "    sd = ar1_sd(ϕ, ωs)\n",
    "    ax.plot(ωs, sd, 'b-', alpha=0.6, lw=2, label='spectral density, $\\phi = {ϕ:.2}$')\n",
    "    ax.legend(loc='upper center')\n",
    "    ax.set(xlabel='frequency', xlim=(0, np.pi))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These spectral densities correspond to the autocovariance functions for the\n",
    "AR(1) process shown above\n",
    "\n",
    "Informally, we think of the spectral density as being large at those $ \\omega \\in [0, \\pi] $ at which\n",
    "the autocovariance function seems approximately to exhibit big damped cycles\n",
    "\n",
    "To see the idea, let’s consider why, in the lower panel of the preceding figure, the spectral density for the case $ \\phi = -0.8 $ is large at $ \\omega = \\pi $\n",
    "\n",
    "Recall that the spectral density can be expressed as\n",
    "\n",
    "\n",
    "<a id='equation-sumpr'></a>\n",
    "$$\n",
    "f(\\omega)\n",
    "= \\gamma(0) + 2 \\sum_{k \\geq 1} \\gamma(k) \\cos(\\omega k)\n",
    "= \\gamma(0) + 2 \\sum_{k \\geq 1} (-0.8)^k \\cos(\\omega k) \\tag{13}\n",
    "$$\n",
    "\n",
    "When we evaluate this at $ \\omega = \\pi $, we get a large number because\n",
    "$ \\cos(\\pi k) $ is large and positive when $ (-0.8)^k $ is\n",
    "positive, and large in absolute value and negative when $ (-0.8)^k $ is negative\n",
    "\n",
    "Hence the product is always large and positive, and hence the sum of the\n",
    "products on the right-hand side of [(13)](#equation-sumpr) is large\n",
    "\n",
    "These ideas are illustrated in the next figure, which has $ k $ on the horizontal axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x576 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ϕ = -0.8\n",
    "times = list(range(16))\n",
    "y1 = [ϕ**k / (1 - ϕ**2) for k in times]\n",
    "y2 = [np.cos(np.pi * k) for k in times]\n",
    "y3 = [a * b for a, b in zip(y1, y2)]\n",
    "\n",
    "num_rows, num_cols = 3, 1\n",
    "fig, axes = plt.subplots(num_rows, num_cols, figsize=(10, 8))\n",
    "plt.subplots_adjust(hspace=0.25)\n",
    "\n",
    "# Autocovariance when ϕ = -0.8\n",
    "ax = axes[0]\n",
    "ax.plot(times, y1, 'bo-', alpha=0.6, label='$\\gamma(k)$')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set(xlim=(0, 15), yticks=(-2, 0, 2))\n",
    "ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)\n",
    "\n",
    "# Cycles at frequence π\n",
    "ax = axes[1]\n",
    "ax.plot(times, y2, 'bo-', alpha=0.6, label='$\\cos(\\pi k)$')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set(xlim=(0, 15), yticks=(-1, 0, 1))\n",
    "ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)\n",
    "\n",
    "# Product\n",
    "ax = axes[2]\n",
    "ax.stem(times, y3, label='$\\gamma(k) \\cos(\\pi k)$')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set(xlim=(0, 15), ylim=(-3, 3), yticks=(-1, 0, 1, 2, 3))\n",
    "ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)\n",
    "ax.set_xlabel(\"k\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On the other hand, if we evaluate $ f(\\omega) $ at $ \\omega = \\pi / 3 $, then the cycles are\n",
    "not matched, the sequence $ \\gamma(k) \\cos(\\omega k) $ contains\n",
    "both positive and negative terms, and hence the sum of these terms is much smaller"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x576 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ϕ = -0.8\n",
    "times = list(range(16))\n",
    "y1 = [ϕ**k / (1 - ϕ**2) for k in times]\n",
    "y2 = [np.cos(np.pi * k/3) for k in times]\n",
    "y3 = [a * b for a, b in zip(y1, y2)]\n",
    "\n",
    "num_rows, num_cols = 3, 1\n",
    "fig, axes = plt.subplots(num_rows, num_cols, figsize=(10, 8))\n",
    "plt.subplots_adjust(hspace=0.25)\n",
    "\n",
    "# Autocovariance when phi = -0.8\n",
    "ax = axes[0]\n",
    "ax.plot(times, y1, 'bo-', alpha=0.6, label='$\\gamma(k)$')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set(xlim=(0, 15), yticks=(-2, 0, 2))\n",
    "ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)\n",
    "\n",
    "# Cycles at frequence π\n",
    "ax = axes[1]\n",
    "ax.plot(times, y2, 'bo-', alpha=0.6, label='$\\cos(\\pi k/3)$')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set(xlim=(0, 15), yticks=(-1, 0, 1))\n",
    "ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)\n",
    "\n",
    "# Product\n",
    "ax = axes[2]\n",
    "ax.stem(times, y3, label='$\\gamma(k) \\cos(\\pi k/3)$')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set(xlim=(0, 15), ylim=(-3, 3), yticks=(-1, 0, 1, 2, 3))\n",
    "ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)\n",
    "ax.set_xlabel(\"$k$\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In summary, the spectral density is large at frequencies $ \\omega $ where the autocovariance function exhibits damped cycles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Inverting the Transformation\n",
    "\n",
    "\n",
    "<a id='index-19'></a>\n",
    "We have just seen that the spectral density is useful in the sense that it provides a frequency-based perspective on the autocovariance structure of a covariance stationary process\n",
    "\n",
    "Another reason that the spectral density is useful is that it can be “inverted” to recover the autocovariance function via the *inverse Fourier transform*\n",
    "\n",
    "In particular, for all $ k \\in \\mathbb Z $, we have\n",
    "\n",
    "\n",
    "<a id='equation-ift'></a>\n",
    "$$\n",
    "\\gamma(k) = \\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi} f(\\omega) e^{i \\omega k} d\\omega \\tag{14}\n",
    "$$\n",
    "\n",
    "This is convenient in situations where the spectral density is easier to calculate and manipulate than the autocovariance function\n",
    "\n",
    "(For example, the expression [(12)](#equation-arma-sd) for the ARMA spectral density is much easier to work with than the expression for the ARMA autocovariance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mathematical Theory\n",
    "\n",
    "\n",
    "<a id='index-20'></a>\n",
    "This section is loosely based on [[Sar87]](https://lectures.quantecon.org/py/zreferences.html#sargent1987), p. 249-253, and included for those who\n",
    "\n",
    "- would like a bit more insight into spectral densities  \n",
    "- and have at least some background in [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space) theory  \n",
    "\n",
    "\n",
    "Others should feel free to skip to the [next section](#arma-imp) — none of this material is necessary to progress to computation\n",
    "\n",
    "Recall that every [separable](https://en.wikipedia.org/wiki/Separable_space) Hilbert space $ H $ has a countable orthonormal basis $ \\{ h_k \\} $\n",
    "\n",
    "The nice thing about such a basis is that every $ f \\in H $ satisfies\n",
    "\n",
    "\n",
    "<a id='equation-arma-fc'></a>\n",
    "$$\n",
    "f = \\sum_k \\alpha_k h_k\n",
    "\\quad \\text{where} \\quad\n",
    "\\alpha_k := \\langle f, h_k \\rangle \\tag{15}\n",
    "$$\n",
    "\n",
    "where $ \\langle \\cdot, \\cdot \\rangle $ denotes the inner product in $ H $\n",
    "\n",
    "Thus, $ f $ can be represented to any degree of precision by linearly combining basis vectors\n",
    "\n",
    "The scalar sequence $ \\alpha = \\{\\alpha_k\\} $ is called the *Fourier coefficients* of $ f $, and satisfies $ \\sum_k |\\alpha_k|^2 < \\infty $\n",
    "\n",
    "In other words, $ \\alpha $ is in $ \\ell_2 $, the set of square summable sequences\n",
    "\n",
    "Consider an operator $ T $ that maps $ \\alpha \\in \\ell_2 $ into its expansion $ \\sum_k \\alpha_k h_k \\in H $\n",
    "\n",
    "The Fourier coefficients of $ T\\alpha $ are just $ \\alpha = \\{ \\alpha_k \\} $, as you can verify by confirming that $ \\langle T \\alpha, h_k \\rangle = \\alpha_k $\n",
    "\n",
    "Using elementary results from Hilbert space theory, it can be shown that\n",
    "\n",
    "- $ T $ is one-to-one — if $ \\alpha $ and $ \\beta $ are distinct in $ \\ell_2 $, then so are their expansions in $ H $  \n",
    "- $ T $ is onto — if $ f \\in H $ then its preimage in $ \\ell_2 $ is the sequence $ \\alpha $ given by $ \\alpha_k = \\langle f, h_k \\rangle $  \n",
    "- $ T $ is a linear isometry — in particular $ \\langle \\alpha, \\beta \\rangle = \\langle T\\alpha, T\\beta \\rangle $  \n",
    "\n",
    "\n",
    "Summarizing these results, we say that any separable Hilbert space is isometrically isomorphic to $ \\ell_2 $\n",
    "\n",
    "In essence, this says that each separable Hilbert space we consider is just a different way of looking at the fundamental space $ \\ell_2 $\n",
    "\n",
    "With this in mind, let’s specialize to a setting where\n",
    "\n",
    "- $ \\gamma \\in \\ell_2 $ is the autocovariance function of a covariance stationary process, and $ f $ is the spectral density  \n",
    "- $ H = L_2 $, where $ L_2 $ is the set of square summable functions on the interval $ [-\\pi, \\pi] $, with inner product $ \\langle g, h \\rangle = \\int_{-\\pi}^{\\pi} g(\\omega) h(\\omega) d \\omega $  \n",
    "- $ \\{h_k\\} = $ the orthonormal basis for $ L_2 $ given by the set of trigonometric functions  \n",
    "\n",
    "\n",
    "$$\n",
    "h_k(\\omega) = \\frac{e^{i \\omega k}}{\\sqrt{2 \\pi}},\n",
    "\\quad k \\in \\mathbb Z,\n",
    "\\quad \\omega \\in [-\\pi, \\pi]\n",
    "$$\n",
    "\n",
    "Using the definition of $ T $ from above and the fact that $ f $ is even, we now have\n",
    "\n",
    "\n",
    "<a id='equation-arma-it'></a>\n",
    "$$\n",
    "T \\gamma\n",
    "= \\sum_{k \\in \\mathbb Z}\n",
    "\\gamma(k) \\frac{e^{i \\omega k}}{\\sqrt{2 \\pi}} = \\frac{1}{\\sqrt{2 \\pi}} f(\\omega) \\tag{16}\n",
    "$$\n",
    "\n",
    "In other words, apart from a scalar multiple, the spectral density is just an transformation of $ \\gamma \\in \\ell_2 $ under a certain linear isometry — a different way to view $ \\gamma $\n",
    "\n",
    "In particular, it is an expansion of the autocovariance function with respect to the trigonometric basis functions in $ L_2 $\n",
    "\n",
    "As discussed above, the Fourier coefficients of $ T \\gamma $ are given by the sequence $ \\gamma $, and,\n",
    "in particular, $ \\gamma(k) = \\langle T \\gamma, h_k \\rangle $\n",
    "\n",
    "Transforming this inner product into its integral expression and using [(16)](#equation-arma-it) gives\n",
    "[(14)](#equation-ift), justifying our earlier expression for the inverse transform\n",
    "\n",
    "\n",
    "<a id='arma-imp'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementation\n",
    "\n",
    "Most code for working with covariance stationary models deals with ARMA models\n",
    "\n",
    "Python code for studying ARMA models can be found in the `tsa` submodule of [statsmodels](http://statsmodels.sourceforge.net/)\n",
    "\n",
    "Since this code doesn’t quite cover our needs — particularly vis-a-vis spectral analysis — we’ve put together the module [arma.py](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/arma.py), which is part of [QuantEcon.py](http://quantecon.org/python_index.html) package\n",
    "\n",
    "The module provides functions for mapping ARMA($ p,q $) models into their\n",
    "\n",
    "1. impulse response function  \n",
    "1. simulated time series  \n",
    "1. autocovariance function  \n",
    "1. spectral density  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Application\n",
    "\n",
    "Let’s use this code to replicate the plots on pages 68–69 of [[LS18]](https://lectures.quantecon.org/py/zreferences.html#ljungqvist2012)\n",
    "\n",
    "Here are some functions to generate the plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "def plot_impulse_response(arma, ax=None):\n",
    "    if ax is None:\n",
    "        ax = plt.gca()\n",
    "    yi = arma.impulse_response()\n",
    "    ax.stem(list(range(len(yi))), yi)\n",
    "    ax.set(xlim=(-0.5), ylim=(min(yi)-0.1, max(yi)+0.1),\n",
    "                 title='Impulse response', xlabel='time', ylabel='response')\n",
    "    return ax\n",
    "\n",
    "def plot_spectral_density(arma, ax=None):\n",
    "    if ax is None:\n",
    "        ax = plt.gca()\n",
    "    w, spect = arma.spectral_density(two_pi=False)\n",
    "    ax.semilogy(w, spect)\n",
    "    ax.set(xlim=(0, np.pi), ylim=(0, np.max(spect)),\n",
    "           title='Spectral density', xlabel='frequency', ylabel='spectrum')\n",
    "    return ax\n",
    "\n",
    "def plot_autocovariance(arma, ax=None):\n",
    "    if ax is None:\n",
    "        ax = plt.gca()\n",
    "    acov = arma.autocovariance()\n",
    "    ax.stem(list(range(len(acov))), acov)\n",
    "    ax.set(xlim=(-0.5, len(acov) - 0.5), title='Autocovariance',\n",
    "           xlabel='time', ylabel='autocovariance')\n",
    "    return ax\n",
    "\n",
    "def plot_simulation(arma, ax=None):\n",
    "    if ax is None:\n",
    "        ax = plt.gca()\n",
    "    x_out = arma.simulation()\n",
    "    ax.plot(x_out)\n",
    "    ax.set(title='Sample path', xlabel='time', ylabel='state space')\n",
    "    return ax\n",
    "\n",
    "def quad_plot(arma):\n",
    "    \"\"\"\n",
    "    Plots the impulse response, spectral_density, autocovariance,\n",
    "    and one realization of the process.\n",
    "\n",
    "    \"\"\"\n",
    "    num_rows, num_cols = 2, 2\n",
    "    fig, axes = plt.subplots(num_rows, num_cols, figsize=(12, 8))\n",
    "    plot_functions = [plot_impulse_response,\n",
    "                      plot_spectral_density,\n",
    "                      plot_autocovariance,\n",
    "                      plot_simulation]\n",
    "    for plot_func, ax in zip(plot_functions, axes.flatten()):\n",
    "        plot_func(arma, ax)\n",
    "    plt.tight_layout()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let’s call these functions to generate the plots\n",
    "\n",
    "We’ll use the model $ X_t = 0.5 X_{t-1} + \\epsilon_t - 0.8 \\epsilon_{t-2} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/quantecon/anaconda3/lib/python3.7/site-packages/numpy/core/numeric.py:501: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return array(a, dtype, copy=False, order=order)\n",
      "/home/quantecon/anaconda3/lib/python3.7/site-packages/matplotlib/axes/_base.py:3477: UserWarning: Attempted to set non-positive ylimits for log-scale axis; invalid limits will be ignored.\n",
      "  'Attempted to set non-positive ylimits for log-scale axis; '\n",
      "/home/quantecon/anaconda3/lib/python3.7/site-packages/matplotlib/transforms.py:968: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  self._points[:, 1] = interval\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import quantecon as qe\n",
    "\n",
    "ϕ = 0.5\n",
    "θ = 0, -0.8\n",
    "arma = qe.ARMA(ϕ, θ)\n",
    "quad_plot(arma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Explanation\n",
    "\n",
    "The call\n",
    "\n",
    "> `arma = ARMA(ϕ, θ, σ)`\n",
    "\n",
    "\n",
    "creates an instance `arma` that represents the ARMA($ p, q $) model\n",
    "\n",
    "$$\n",
    "X_t = \\phi_1 X_{t-1} + ... + \\phi_p X_{t-p} +\n",
    "    \\epsilon_t + \\theta_1 \\epsilon_{t-1} + ... + \\theta_q \\epsilon_{t-q}\n",
    "$$\n",
    "\n",
    "If `ϕ` and `θ` are arrays or sequences, then the interpretation will\n",
    "be\n",
    "\n",
    "- `ϕ` holds the vector of parameters $ (\\phi_1, \\phi_2,..., \\phi_p) $  \n",
    "- `θ` holds the vector of parameters $ (\\theta_1, \\theta_2,..., \\theta_q) $  \n",
    "\n",
    "\n",
    "The parameter `σ` is always a scalar, the standard deviation of the white noise\n",
    "\n",
    "We also permit `ϕ` and `θ` to be scalars, in which case the model will be interpreted as\n",
    "\n",
    "$$\n",
    "X_t = \\phi X_{t-1} + \\epsilon_t + \\theta \\epsilon_{t-1}\n",
    "$$\n",
    "\n",
    "The two numerical packages most useful for working with ARMA models are `scipy.signal` and `numpy.fft`\n",
    "\n",
    "The package `scipy.signal` expects the parameters to be passed into its functions in a manner consistent with the alternative ARMA notation [(8)](#equation-arma-lag1)\n",
    "\n",
    "For example, the impulse response sequence $ \\{\\psi_t\\} $ discussed above can be obtained using `scipy.signal.dimpulse`, and the function call should be of the form\n",
    "\n",
    "> `times, ψ = dimpulse((ma_poly, ar_poly, 1), n=impulse_length)`\n",
    "\n",
    "\n",
    "where `ma_poly` and `ar_poly` correspond to the polynomials in [(7)](#equation-arma-poly) — that is,\n",
    "\n",
    "- `ma_poly` is the vector $ (1, \\theta_1, \\theta_2, \\ldots, \\theta_q) $  \n",
    "- `ar_poly` is the vector $ (1, -\\phi_1, -\\phi_2, \\ldots, - \\phi_p) $  \n",
    "\n",
    "\n",
    "To this end, we also maintain the arrays `ma_poly` and `ar_poly` as instance data, with their values computed automatically from the values of `phi` and `theta` supplied by the user\n",
    "\n",
    "If the user decides to change the value of either `theta` or `phi` ex-post by assignments\n",
    "such as `arma.phi = (0.5, 0.2)` or `arma.theta = (0, -0.1)`\n",
    "\n",
    "then `ma_poly` and `ar_poly` should update automatically to reflect these new parameters\n",
    "\n",
    "This is achieved in our implementation by using [descriptors](https://lectures.quantecon.org/py/python_advanced_features.html#descriptors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing the Autocovariance Function\n",
    "\n",
    "As discussed above, for ARMA processes the spectral density has a [simple representation](#arma-spec-den) that is relatively easy to calculate\n",
    "\n",
    "Given this fact, the easiest way to obtain the autocovariance function is to recover it from the spectral\n",
    "density via the inverse Fourier transform\n",
    "\n",
    "Here we use NumPy’s Fourier transform package np.fft, which wraps a standard Fortran-based package called FFTPACK\n",
    "\n",
    "A look at [the np.fft documentation](http://docs.scipy.org/doc/numpy/reference/routines.fft.html) shows that the inverse transform np.fft.ifft takes a given sequence $ A_0, A_1, \\ldots, A_{n-1} $ and\n",
    "returns the sequence $ a_0, a_1, \\ldots, a_{n-1} $ defined by\n",
    "\n",
    "$$\n",
    "a_k = \\frac{1}{n} \\sum_{t=0}^{n-1} A_t e^{ik 2\\pi t / n}\n",
    "$$\n",
    "\n",
    "Thus, if we set $ A_t = f(\\omega_t) $, where $ f $ is the spectral density and\n",
    "$ \\omega_t := 2 \\pi t / n $, then\n",
    "\n",
    "$$\n",
    "a_k\n",
    "= \\frac{1}{n} \\sum_{t=0}^{n-1} f(\\omega_t) e^{i \\omega_t k}\n",
    "= \\frac{1}{2\\pi} \\frac{2 \\pi}{n} \\sum_{t=0}^{n-1} f(\\omega_t) e^{i \\omega_t k},\n",
    "\\qquad\n",
    "\\omega_t := 2 \\pi t / n\n",
    "$$\n",
    "\n",
    "For $ n $ sufficiently large, we then have\n",
    "\n",
    "$$\n",
    "a_k\n",
    "\\approx \\frac{1}{2\\pi} \\int_0^{2 \\pi} f(\\omega) e^{i \\omega k} d \\omega\n",
    "= \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(\\omega) e^{i \\omega k} d \\omega\n",
    "$$\n",
    "\n",
    "(You can check the last equality)\n",
    "\n",
    "In view of [(14)](#equation-ift) we have now shown that, for $ n $ sufficiently large, $ a_k \\approx \\gamma(k) $ — which is exactly what we want to compute"
   ]
  }
 ],
 "metadata": {
  "filename": "arma.rst",
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   "display_name": "Python",
   "language": "python3",
   "name": "python3"
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  "title": "Covariance Stationary Processes"
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}