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fix #8 profiling notebooks
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‎examples/profile-fractional-differentiation.ipynb‎

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Load Packages"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import sys\n",
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"sys.path.append('..')\n",
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"\n",
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"from fracdiff import fracdiff\n",
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"import numpy as np\n",
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"import scipy.special\n",
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"\n",
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"import matplotlib.pyplot as plt\n",
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"%matplotlib inline\n",
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"\n",
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"#!pip install memory_profiler\n",
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"import line_profiler\n",
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"%load_ext line_profiler\n",
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"\n",
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"#!pip install memory_profiler\n",
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"import memory_profiler\n",
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"%load_ext memory_profiler"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Load Demo Data"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"with np.load('data/demo1.npz') as data:\n",
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" x = data['px'][:, 0]"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Modeling\n",
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"Let $x_t$ a time series, \n",
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"$t\\in\\mathbb{N}$ the time step,\n",
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"$\\Delta^d$ the difference operator of fractional order $d\\in\\mathbb{R}^+$,\n",
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"and $m$ the truncation order"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Formula 1** (see Schoffer, 2003, p.50)\n",
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"\n",
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"$$\n",
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"(\\Delta^d x)_t = \\sum_{k=0}^\\infty \\frac{\\Gamma(k - d)}{\\Gamma(k + 1) \\, \\Gamma(-d)} \\, x_{t-k} \\\\\n",
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"(\\Delta^d x)_t \\approx \\sum_{k=0}^m \\frac{\\Gamma(k - d)}{\\Gamma(k + 1) \\, \\Gamma(-d)} \\, x_{t-k}\n",
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"$$\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Formula 2** (see Balisarsingh, 2013, pp.9737-9742)\n",
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"\n",
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"$$\n",
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"(\\Delta^d x)_t = \\sum_{k=0}^\\infty (-1)^k \\frac{\\Gamma(d + 1)}{k! \\, \\Gamma(d - k + 1)} \\, x_{t-k} \\\\\n",
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"(\\Delta^d x)_t \\approx \\sum_{k=0}^m (-1)^k \\frac{\\Gamma(d + 1)}{k! \\, \\Gamma(d - k + 1)} \\, x_{t-k}\n",
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"$$\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Formula 3/4** (see Lopez, 2018, p.78, from the 'iterative estimation' formula; Jensen and Nielsen, 2014)\n",
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"\n",
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"$$\n",
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"(\\Delta^d x)_t = x_t + \\sum_{k=1}^\\infty \\left(\\prod_{i=1}^k \\frac{d - i + 1}{i} \\right) x_{t-k} \\\\\n",
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"(\\Delta^d x)_t \\approx x_t + \\sum_{k=1}^m \\left(\\prod_{i=1}^k \\frac{d - i + 1}{i} \\right) x_{t-k}\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"def frac_weights_1(d: float, m: int) -> np.ndarray:\n",
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" w = np.empty((m + 1,))\n",
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" for k in range(m + 1):\n",
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" w[k] = scipy.special.gamma(k - d) \\\n",
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" / (scipy.special.gamma(k + 1) * scipy.special.gamma(-d))\n",
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" return w"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"def frac_weights_2(d: float, m: int) -> np.ndarray:\n",
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" w = np.empty((m + 1,))\n",
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" for k in range(m + 1):\n",
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" w[k] = np.power(-1, k) * scipy.special.gamma(d + 1) \\\n",
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" / (scipy.special.factorial(k) * scipy.special.gamma(d - k + 1))\n",
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" return w"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"def frac_weights_3(d: float, m: int) -> np.ndarray:\n",
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" w = np.empty((m + 1,))\n",
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" w[0] = 1\n",
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" for j in range(1, m + 1):\n",
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" w[j] = -w[j - 1] * ((d - j + 1) / j)\n",
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" return w"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"def frac_weights_4(d: float, m: int) -> np.ndarray:\n",
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" w = [1]\n",
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" for k in range(1, m + 1):\n",
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" w.append(-w[-1] * ((d - k + 1) / k))\n",
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" return np.array(w)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0.003395536381106888\n",
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"0.003395536381106581\n",
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"0.003395536381106833\n",
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"0.003395536381106833\n"
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]
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}
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],
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"source": [
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"d = 0.845\n",
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"m = 100 # truncation order\n",
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"\n",
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"w = frac_weights_1(d, m)\n",
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"print(np.nansum(w))\n",
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"\n",
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"w = frac_weights_2(d, m)\n",
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"print(np.nansum(w))\n",
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"\n",
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"w = frac_weights_3(d, m)\n",
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"print(np.nansum(w))\n",
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"\n",
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"w = frac_weights_4(d, m)\n",
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"print(np.nansum(w))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Speed\n",
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"The list version of the iterative estimation (formula 4) is by far the fastest"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"669 µs ± 104 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
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"2.73 ms ± 20 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n",
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"79 µs ± 6.34 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
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"38.2 µs ± 783 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
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]
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}
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],
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"source": [
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"%timeit w = frac_weights_1(d, m)\n",
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"%timeit w = frac_weights_2(d, m)\n",
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"%timeit w = frac_weights_3(d, m)\n",
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"%timeit w = frac_weights_4(d, m)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Memory"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"peak memory: 87.59 MiB, increment: -0.13 MiB\n",
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"peak memory: 87.59 MiB, increment: 0.00 MiB\n",
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"peak memory: 87.60 MiB, increment: 0.01 MiB\n",
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"peak memory: 87.61 MiB, increment: 0.02 MiB\n"
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]
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}
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],
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"source": [
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"%memit w = frac_weights_1(d, m)\n",
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"%memit w = frac_weights_2(d, m)\n",
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"%memit w = frac_weights_3(d, m)\n",
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"%memit w = frac_weights_4(d, m)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# References\n",
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"* Baliarsingh, P., 2013. Some new difference sequence spaces of fractional order and their dual spaces. Applied Mathematics and Computation 219, 9737–9742. https://doi.org/10.1016/j.amc.2013.03.073\n",
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"* Jensen, A.N., Nielsen, M.Ø., 2014. A Fast Fractional Difference Algorithm. Journal of Time Series Analysis 35, 428–436. https://doi.org/10.1111/jtsa.12074\n",
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"* Prado, M.L. de, 2018. Advances in Financial Machine Learning, 1st ed. Wiley.\n",
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"* Schoffer, O., n.d. Modellierung von Kapitalmarktrenditen mittels asymmetrischer GARCH-Modelle. Universitaet Dortmund.\n"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.1"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}

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