notebooks/Systems_Identification_Fitting.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Systems Identification Model Fitting\n",
    "\n",
    "Fit a Systems Identification model off based off of this [specification](https://hackmd.io/w-vfdZIMTDKwdEupeS3qxQ) and [spec](https://hackmd.io/XVaejEw-QaCghV1Tkv3eVQ) with data obtained in [data_acquisition.ipynb](data/data_acquisition.ipynb). \n",
    "\n",
    "#### Process changes and decision points\n",
    "* Create differenced linear regressor model for refining data formatting\n",
    "* Fit VAR model off of differenced states with Yeo-Johnson power transformation\n",
    "* Implemented coordinate transformations \n",
    "* Created inverse transformations\n",
    "* Fit one step forward VAR model that takes the difference between local arbitrager values and observed values and forcasts the errors within the coordinate transformation state.\n",
    "* Fit VARMAX model with exogenous signal - error between redemption price and rai market price - retrain after every timestep \n",
    "* Compare VARMAX vs VAR model (we chose VARMAX with an exogenous signal)\n",
    "* VARMAX is too slow to retrain at each time step (25x slower than VAR). To determine which model performs better, we created a [validation notebook](VAR_vs_VARMAX_evaluation.ipynb)\n",
    "* Refactor to functions for deployment\n",
    "* Add back Yeo-Johnson power transformation\n",
    "* Move from arbitrageur to exponentially weighted moving average of actual data\n",
    "* Swept alpha of exponentially weighted moving average and found that a VAR(15) with an alpha of 0.8 performed best. \n",
    "\n",
    "\n",
    "## Analyze and Prepare Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>marketPriceEth</th>\n",
       "      <th>marketPriceUsd</th>\n",
       "      <th>block_number</th>\n",
       "      <th>debtAvailableToSettle</th>\n",
       "      <th>globalDebt</th>\n",
       "      <th>globalDebtCeiling</th>\n",
       "      <th>systemSurplus</th>\n",
       "      <th>totalActiveSafeCount</th>\n",
       "      <th>RedemptionRateAnnualizedRate</th>\n",
       "      <th>RedemptionRateHourlyRate</th>\n",
       "      <th>RedemptionRateEightHourlyRate</th>\n",
       "      <th>RedemptionPrice</th>\n",
       "      <th>RaiDrawnFromSAFEs</th>\n",
       "      <th>collateral</th>\n",
       "      <th>debt</th>\n",
       "      <th>EthInUniswap</th>\n",
       "      <th>RaiInUniswap</th>\n",
       "      <th>ETH Price (OSM)</th>\n",
       "      <th>block_timestamp</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.362594</td>\n",
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       "      <td>0.0</td>\n",
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       "      <td>0.068533</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>2788.440433</td>\n",
       "      <td>14.89236</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>676.933727</td>\n",
       "      <td>1803.656430</td>\n",
       "      <td>2021-02-15 10:16:16+00:00</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.419266</td>\n",
       "      <td>11861008</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2788.657935</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.108751</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>2788.440433</td>\n",
       "      <td>14.89236</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>676.933727</td>\n",
       "      <td>1763.974936</td>\n",
       "      <td>2021-02-15 11:13:28+00:00</td>\n",
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       "      <th>2</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.419266</td>\n",
       "      <td>11861237</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2788.657935</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.108751</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>2788.440433</td>\n",
       "      <td>14.89236</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>676.933727</td>\n",
       "      <td>1763.974936</td>\n",
       "      <td>2021-02-15 12:07:36+00:00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.001769</td>\n",
       "      <td>3.215446</td>\n",
       "      <td>11861502</td>\n",
       "      <td>0.0</td>\n",
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       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.116162</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
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       "      <td>1.418569</td>\n",
       "      <td>801.933727</td>\n",
       "      <td>1775.977923</td>\n",
       "      <td>2021-02-15 13:07:59+00:00</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.001769</td>\n",
       "      <td>3.215446</td>\n",
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       "      <td>0.0</td>\n",
       "      <td>3048.645955</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
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       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>3048.440433</td>\n",
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       "      <td>3048.358490</td>\n",
       "      <td>1.418569</td>\n",
       "      <td>801.933727</td>\n",
       "      <td>1805.792735</td>\n",
       "      <td>2021-02-15 14:07:59+00:00</td>\n",
       "    </tr>\n",
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      "text/plain": [
       "   marketPriceEth  marketPriceUsd  block_number  debtAvailableToSettle  \\\n",
       "0        0.002481        4.362594      11860755                    0.0   \n",
       "1        0.002481        4.419266      11861008                    0.0   \n",
       "2        0.002481        4.419266      11861237                    0.0   \n",
       "3        0.001769        3.215446      11861502                    0.0   \n",
       "4        0.001769        3.215446      11861791                    0.0   \n",
       "\n",
       "    globalDebt                    globalDebtCeiling  systemSurplus  \\\n",
       "0  2788.522358  115792089237316195423570985008687.9       0.068533   \n",
       "1  2788.657935  115792089237316195423570985008687.9       0.108751   \n",
       "2  2788.657935  115792089237316195423570985008687.9       0.108751   \n",
       "3  3048.645955  115792089237316195423570985008687.9       0.116162   \n",
       "4  3048.645955  115792089237316195423570985008687.9       0.116162   \n",
       "\n",
       "   totalActiveSafeCount  RedemptionRateAnnualizedRate  \\\n",
       "0                     3                           1.0   \n",
       "1                     3                           1.0   \n",
       "2                     3                           1.0   \n",
       "3                     3                           1.0   \n",
       "4                     3                           1.0   \n",
       "\n",
       "   RedemptionRateHourlyRate  RedemptionRateEightHourlyRate  RedemptionPrice  \\\n",
       "0                       1.0                            1.0             3.14   \n",
       "1                       1.0                            1.0             3.14   \n",
       "2                       1.0                            1.0             3.14   \n",
       "3                       1.0                            1.0             3.14   \n",
       "4                       1.0                            1.0             3.14   \n",
       "\n",
       "   RaiDrawnFromSAFEs  collateral         debt  EthInUniswap  RaiInUniswap  \\\n",
       "0        2788.440433    14.89236  2788.385291      1.679713    676.933727   \n",
       "1        2788.440433    14.89236  2788.385291      1.679713    676.933727   \n",
       "2        2788.440433    14.89236  2788.385291      1.679713    676.933727   \n",
       "3        3048.440433    16.89236  3048.358490      1.418569    801.933727   \n",
       "4        3048.440433    16.89236  3048.358490      1.418569    801.933727   \n",
       "\n",
       "   ETH Price (OSM)            block_timestamp  \n",
       "0      1803.656430  2021-02-15 10:16:16+00:00  \n",
       "1      1763.974936  2021-02-15 11:13:28+00:00  \n",
       "2      1763.974936  2021-02-15 12:07:36+00:00  \n",
       "3      1775.977923  2021-02-15 13:07:59+00:00  \n",
       "4      1805.792735  2021-02-15 14:07:59+00:00  "
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# import libraries\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "import math\n",
    "import statsmodels.api as sm\n",
    "from statsmodels.tsa.api import VAR, VARMAX\n",
    "from sklearn.preprocessing import PowerTransformer\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "import os\n",
    "warnings.filterwarnings(\"ignore\")\n",
    "\n",
    "os.chdir('..')\n",
    "states = pd.read_csv('data/states.csv')\n",
    "del states['Unnamed: 0']\n",
    "states.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# add additional state variables\n",
    "states['RedemptionPriceinEth'] = states['RedemptionPrice'] / states['ETH Price (OSM)']\n",
    "states['RedemptionPriceError'] = states['RedemptionPrice'] - states['marketPriceUsd']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Systems identification steps:\n",
    "1. Calculate optimal state from APT model (updated to exponential weighted moving average of the real data)\n",
    "\n",
    "2. Perform a coordinate transformation of data \n",
    "\n",
    "3. Difference the local coordinate from the observed to get error\n",
    "\n",
    "4. Perform a Yeo-Johnson power transformation\n",
    "<!-- 4. Train VARMAX the errors + exogenous signal[s]\n",
    " -->\n",
    "5. Train a VAR(15) model\n",
    "\n",
    "6. One step forecast\n",
    "\n",
    "7. Invert the Yeo-Johnson power transformation\n",
    "\n",
    "8. Convert forecasted values back from coordinate system \n",
    "\n",
    "9. Add forecasted values to previous state to get new state \n",
    "\n",
    "\n",
    "### Mapping of specification states to data\n",
    "\n",
    "#### Initial vector \n",
    "The quantity state variables of the system are as value, mathematical notation, and Graph and Big Query field names from [data_acquisition.ipynb](data/data_acquisition.ipynb).\n",
    "* ETH in collateral = $Q$ = collateral \n",
    "* ETH in Uniswap = $R_{ETH}$ = EthInUniswap\n",
    "* RAI in Uniswap = $R_{RAI}$ = RaiInUniswap \n",
    "* RAI drawn from SAFEs = $D$ = RaiDrawnFromSAFEs\n",
    "\n",
    "\n",
    "<!-- (GlobalDebt won't equal total supply (create graphics around?)) -->\n",
    "\n",
    "The metric state variables of the system are:\n",
    "\n",
    "* Market Price of RAI in ETH = $p_{E/R} > 0$   = marketPriceEth\n",
    "* Market Price of RAI in USD =  $p_{U/R} > 0$ =  marketPriceUsd\n",
    "* Market Price of ETH in USD = $p_{U/E} > 0$ = ETH Price (OSM)\n",
    "\n",
    "The metric control variables of the system are:\n",
    "\n",
    "* Redemption Price of RAI in USD = $p^r_{U/R} > 0$ = RedemptionPrice\n",
    "* Redemption Price of RAI in ETH = $p^r_{E/R} > 0$ = RedemptionPriceinEth \n",
    "\n",
    "The system parameters are:\n",
    "* Liquidation Ratio = $\\bar{L} > 0$ = 1.45\n",
    "* SAFE Debt Ceiling = $\\bar{D} > 0$ = globalDebtCeiling\n",
    "* Uniswap Fee =  $\\phi_U \\in (0,1)$  = 0.003 \n",
    "* Gas Costs = $\\bar{C}_{gas} \\geq 0$ = 100e-9, # 100 gwei\n",
    "\n",
    "The aggregate flow variables are:\n",
    "* Collateral added or removed =  $q \\in \\mathbb{R}$ (ETH)\n",
    "* SAFE Debt drawn or repaid = $d \\in \\mathbb{R}$   (RAI)\n",
    "* Uniswap RAI bought or sold = $r \\in \\mathbb{R}$   (RAI)\n",
    "* Uniswap ETH bought or sold = $z \\in \\mathbb{R}$   (ETH)\n",
    "\n",
    "\n",
    "### Model Formulation\n",
    "\n",
    "\n",
    "There is an admissible action set of vectors:\n",
    "(Graph values)\n",
    "* ETH in collateral = $Q$ = collateral\n",
    "* ETH in Uniswap = $R_{ETH}$ reserve1\n",
    "* RAI in Uniswap = $R_{RAI}$ = reserve0 \n",
    "* RAI drawn from SAFEs = $D$ = erc20CoinTotalSupply\n",
    "\n",
    "\n",
    "\n",
    "Action vector: $\\vec{u} = (\\Delta Q, \\Delta R_{ETH}, \\Delta R_{RAI}, \\Delta D)$\n",
    "\n",
    "Admissible action set: $\\vec{u} \\in \\mathcal{U}$\n",
    "\n",
    "Optimal Action Vector: $\\vec{u^*} = (Q^*, R_{ETH}^*, \\Delta R_{RAI}^*, \\Delta D^*)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define constants (will come from cadCAD model but added here for calculations)\n",
    "params = {\n",
    "    'liquidation_ratio': 1.45,\n",
    "    'debt_ceiling': 1e9,\n",
    "    'uniswap_fee': 0.003,\n",
    "    'arbitrageur_considers_liquidation_ratio': True,\n",
    "}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create Arbtrageur data vector $u^*$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_aggregated_arbitrageur_decision(params, state):\n",
    "\n",
    "    # This Boolean indicates whether or not the arbitrageur is rationally considering \n",
    "    # borrowing to the liquidation ratio limit. If TRUE, arbitrage opportunities are less\n",
    "    # frequent when RAI is expensive and more frequent when RAI is cheap. If FALSE, only \n",
    "    # the difference in market and redemption prices (net of Uniswap fee) matters for trading, \n",
    "    # which may conform more to individual trader expectations and behavior.\n",
    "    consider_liquidation_ratio = params['arbitrageur_considers_liquidation_ratio']\n",
    "    \n",
    "    # These are the states of the SAFE balances in aggregate & its fixed parameters\n",
    "    total_borrowed = state['SAFE_Debt'] # D\n",
    "    total_collateral = state['SAFE_Collateral'] # Q\n",
    "    liquidation_ratio = params['liquidation_ratio']\n",
    "    debt_ceiling = params['debt_ceiling']\n",
    "\n",
    "\n",
    "    # These are the states of the Uniswap secondary market balances and its fee\n",
    "    RAI_balance = state['RAI_balance'] # R_Rai\n",
    "    ETH_balance = state['ETH_balance'] # R_Eth\n",
    "    uniswap_fee = params['uniswap_fee']\n",
    "\n",
    "    # These are the prices of RAI in USD/RAI for SAFE redemption and the market price oracle, resp.\n",
    "    redemption_price = state['target_price'] # $p^r_{U/R}\n",
    "    market_price = state['market_price'] # p_{U/R} > 0\n",
    "    \n",
    "    # This is the price of ETH in USD/ETH\n",
    "    eth_price = state['eth_price'] # p_{U/E}\n",
    "\n",
    "        \n",
    "    # These functions define the optimal borrowing/repayment decisions of the aggregated arbitrageur\n",
    "    def g1(RAI_balance, ETH_balance, uniswap_fee, liquidation_ratio, redemption_price):\n",
    "        return ((eth_price * RAI_balance * ETH_balance * (1 - uniswap_fee)) / (liquidation_ratio * redemption_price)) ** 0.5\n",
    "\n",
    "    def g2(RAI_balance, ETH_balance, uniswap_fee, liquidation_ratio, redemption_price):\n",
    "        return (RAI_balance * ETH_balance * (1 - uniswap_fee) * liquidation_ratio * (redemption_price / eth_price)) ** 0.5\n",
    "\n",
    "    # This Boolean resolves to TRUE if the agg. arb. acts this timestep when RAI is expensive \n",
    "    # on the secondary market\n",
    "    expensive_RAI_on_secondary_market = \\\n",
    "        redemption_price < ((1 - uniswap_fee) / liquidation_ratio) * market_price  \\\n",
    "        if consider_liquidation_ratio \\\n",
    "        else redemption_price < (1 - uniswap_fee) * market_price\n",
    "        \n",
    "    # This Boolean resolves to TRUE if the agg. arb. acts this timestep when RAI is cheap \n",
    "    # on the secondary market\n",
    "    cheap_RAI_on_secondary_market = \\\n",
    "        redemption_price > (1 / ((1 - uniswap_fee) * liquidation_ratio)) * market_price  \\\n",
    "        if consider_liquidation_ratio \\\n",
    "        else redemption_price > (1 / (1 - uniswap_fee)) * market_price\n",
    "\n",
    "    if expensive_RAI_on_secondary_market:\n",
    "        '''\n",
    "        Expensive RAI on Uni:\n",
    "        (put ETH from pocket into additional collateral in SAFE)\n",
    "        draw RAI from SAFE -> Uni\n",
    "        ETH from Uni -> into pocket\n",
    "        '''\n",
    "\n",
    "        _g1 = g1(RAI_balance, ETH_balance, uniswap_fee,\n",
    "                 liquidation_ratio, redemption_price)\n",
    "        d = (_g1 - RAI_balance) / (1 - uniswap_fee) # should be \\geq 0\n",
    "        q = ((liquidation_ratio * redemption_price) /\n",
    "                     eth_price) * (total_borrowed + d) - total_collateral # should be \\geq 0\n",
    "        z = -(ETH_balance * d * (1 - uniswap_fee)) / \\\n",
    "            (RAI_balance + d * (1 - uniswap_fee)) # should be leq 0\n",
    "        r = d # should be \\geq 0\n",
    "        \n",
    "    elif cheap_RAI_on_secondary_market:\n",
    "        '''\n",
    "        Cheap RAI on Uni:\n",
    "        ETH out of pocket -> Uni\n",
    "        RAI from UNI -> SAFE to wipe debt\n",
    "        (and collect collateral ETH from SAFE into pocket)\n",
    "        '''\n",
    "\n",
    "        _g2 = g2(RAI_balance, ETH_balance, uniswap_fee,\n",
    "                 liquidation_ratio, redemption_price)\n",
    "        z = (_g2 - ETH_balance) / (1 - uniswap_fee) # should be \\geq 0\n",
    "        r = -(RAI_balance * z * (1 - uniswap_fee)) / \\\n",
    "            (ETH_balance + z * (1 - uniswap_fee)) # should be \\leq 0\n",
    "        d = r # should be \\leq 0\n",
    "        q = ((liquidation_ratio * redemption_price /\n",
    "             eth_price) * (total_borrowed + d) - total_collateral) # should be \\leq 0\n",
    "    else:\n",
    "        pass\n",
    "\n",
    "    return {\n",
    "        'q' : q,\n",
    "        'd' : d,\n",
    "        'r' : r,\n",
    "        'z' : z\n",
    "    }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# UPDATED: We will use an exponentially weighted moving average instead of this arbitrageur logic\n",
    "\n",
    "# # subset state variables for arbitrageur vector\n",
    "# state_subset = states[['marketPriceUsd','RedemptionPrice','ETH Price (OSM)','collateral',\n",
    "#                        'EthInUniswap','RaiInUniswap','RaiDrawnFromSAFEs']]\n",
    "# # map state data to arbitrageur vector fields \n",
    "# state_subset.columns = ['market_price','target_price','eth_price','SAFE_Collateral',\n",
    "#                        'ETH_balance','RAI_balance','SAFE_Debt']\n",
    "\n",
    "# # create list of u^* vectors\n",
    "# values = []\n",
    "\n",
    "# # iterate through real data to create u^* and save to values\n",
    "# for i in range(0,len(state_subset)):\n",
    "#     values.append(get_aggregated_arbitrageur_decision(params,state_subset.loc[i]))\n",
    "    \n",
    "# # create historic u^* dataframe\n",
    "# local = pd.DataFrame(values)\n",
    "# local.columns = ['Q','D','Rrai','Reth']\n",
    "# local.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>marketPriceEth</th>\n",
       "      <th>marketPriceUsd</th>\n",
       "      <th>block_number</th>\n",
       "      <th>debtAvailableToSettle</th>\n",
       "      <th>globalDebt</th>\n",
       "      <th>globalDebtCeiling</th>\n",
       "      <th>systemSurplus</th>\n",
       "      <th>totalActiveSafeCount</th>\n",
       "      <th>RedemptionRateAnnualizedRate</th>\n",
       "      <th>RedemptionRateHourlyRate</th>\n",
       "      <th>...</th>\n",
       "      <th>RedemptionPrice</th>\n",
       "      <th>RaiDrawnFromSAFEs</th>\n",
       "      <th>collateral</th>\n",
       "      <th>debt</th>\n",
       "      <th>EthInUniswap</th>\n",
       "      <th>RaiInUniswap</th>\n",
       "      <th>ETH Price (OSM)</th>\n",
       "      <th>block_timestamp</th>\n",
       "      <th>RedemptionPriceinEth</th>\n",
       "      <th>RedemptionPriceError</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.362594</td>\n",
       "      <td>11860755</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.788522e+03</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.068533</td>\n",
       "      <td>3</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>...</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>14.892360</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1803.656430</td>\n",
       "      <td>2021-02-15 10:16:16+00:00</td>\n",
       "      <td>0.001741</td>\n",
       "      <td>-1.222594</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.419266</td>\n",
       "      <td>11861008</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.788658e+03</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.108751</td>\n",
       "      <td>3</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>...</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>14.892360</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1763.974936</td>\n",
       "      <td>2021-02-15 11:13:28+00:00</td>\n",
       "      <td>0.001780</td>\n",
       "      <td>-1.279266</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.419266</td>\n",
       "      <td>11861237</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.788658e+03</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.108751</td>\n",
       "      <td>3</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>...</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>14.892360</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1763.974936</td>\n",
       "      <td>2021-02-15 12:07:36+00:00</td>\n",
       "      <td>0.001780</td>\n",
       "      <td>-1.279266</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.001769</td>\n",
       "      <td>3.215446</td>\n",
       "      <td>11861502</td>\n",
       "      <td>0.0</td>\n",
       "      <td>3.048646e+03</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.116162</td>\n",
       "      <td>3</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>...</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>3.048440e+03</td>\n",
       "      <td>16.892360</td>\n",
       "      <td>3048.358490</td>\n",
       "      <td>1.418569</td>\n",
       "      <td>8.019337e+02</td>\n",
       "      <td>1775.977923</td>\n",
       "      <td>2021-02-15 13:07:59+00:00</td>\n",
       "      <td>0.001768</td>\n",
       "      <td>-0.075446</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.001769</td>\n",
       "      <td>3.215446</td>\n",
       "      <td>11861791</td>\n",
       "      <td>0.0</td>\n",
       "      <td>3.048646e+03</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.116162</td>\n",
       "      <td>3</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>...</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>3.048440e+03</td>\n",
       "      <td>16.892360</td>\n",
       "      <td>3048.358490</td>\n",
       "      <td>1.418569</td>\n",
       "      <td>8.019337e+02</td>\n",
       "      <td>1805.792735</td>\n",
       "      <td>2021-02-15 14:07:59+00:00</td>\n",
       "      <td>0.001739</td>\n",
       "      <td>-0.075446</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
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       "    <tr>\n",
       "      <th>2314</th>\n",
       "      <td>0.001082</td>\n",
       "      <td>3.017312</td>\n",
       "      <td>12584395</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.127334e+07</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>334713.030602</td>\n",
       "      <td>397</td>\n",
       "      <td>1.004843</td>\n",
       "      <td>1.000001</td>\n",
       "      <td>...</td>\n",
       "      <td>3.007422</td>\n",
       "      <td>1.084780e+07</td>\n",
       "      <td>1083.900454</td>\n",
       "      <td>250083.541093</td>\n",
       "      <td>6564.157346</td>\n",
       "      <td>6.014012e+06</td>\n",
       "      <td>2702.280000</td>\n",
       "      <td>2021-06-07 01:05:54+00:00</td>\n",
       "      <td>0.001113</td>\n",
       "      <td>-0.009891</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2315</th>\n",
       "      <td>0.001079</td>\n",
       "      <td>3.010809</td>\n",
       "      <td>12585445</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.127334e+07</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>334713.030602</td>\n",
       "      <td>397</td>\n",
       "      <td>1.004843</td>\n",
       "      <td>1.000001</td>\n",
       "      <td>...</td>\n",
       "      <td>3.007430</td>\n",
       "      <td>1.084780e+07</td>\n",
       "      <td>1083.900454</td>\n",
       "      <td>250083.541093</td>\n",
       "      <td>6527.685380</td>\n",
       "      <td>6.047051e+06</td>\n",
       "      <td>2757.389589</td>\n",
       "      <td>2021-06-07 05:07:45+00:00</td>\n",
       "      <td>0.001091</td>\n",
       "      <td>-0.003379</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2316</th>\n",
       "      <td>0.001086</td>\n",
       "      <td>3.013381</td>\n",
       "      <td>12586509</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.127175e+07</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>334713.030602</td>\n",
       "      <td>395</td>\n",
       "      <td>0.973379</td>\n",
       "      <td>0.999997</td>\n",
       "      <td>...</td>\n",
       "      <td>3.007431</td>\n",
       "      <td>1.084620e+07</td>\n",
       "      <td>1083.900454</td>\n",
       "      <td>250083.541093</td>\n",
       "      <td>6565.000885</td>\n",
       "      <td>6.039329e+06</td>\n",
       "      <td>2773.590000</td>\n",
       "      <td>2021-06-07 09:06:20+00:00</td>\n",
       "      <td>0.001084</td>\n",
       "      <td>-0.005950</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2317</th>\n",
       "      <td>0.001076</td>\n",
       "      <td>2.994558</td>\n",
       "      <td>12587589</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.128063e+07</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>334865.769477</td>\n",
       "      <td>395</td>\n",
       "      <td>0.973379</td>\n",
       "      <td>0.999997</td>\n",
       "      <td>...</td>\n",
       "      <td>3.007393</td>\n",
       "      <td>1.085480e+07</td>\n",
       "      <td>1083.900454</td>\n",
       "      <td>250083.541093</td>\n",
       "      <td>6495.694866</td>\n",
       "      <td>6.107233e+06</td>\n",
       "      <td>2761.044990</td>\n",
       "      <td>2021-06-07 13:15:51+00:00</td>\n",
       "      <td>0.001089</td>\n",
       "      <td>0.012835</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2318</th>\n",
       "      <td>0.001110</td>\n",
       "      <td>2.986134</td>\n",
       "      <td>12588614</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.128063e+07</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>334865.769477</td>\n",
       "      <td>395</td>\n",
       "      <td>0.970124</td>\n",
       "      <td>0.999997</td>\n",
       "      <td>...</td>\n",
       "      <td>3.007355</td>\n",
       "      <td>1.085480e+07</td>\n",
       "      <td>1083.900454</td>\n",
       "      <td>250083.541093</td>\n",
       "      <td>6541.018950</td>\n",
       "      <td>6.065035e+06</td>\n",
       "      <td>2828.150972</td>\n",
       "      <td>2021-06-07 17:06:11+00:00</td>\n",
       "      <td>0.001063</td>\n",
       "      <td>0.021221</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2319 rows × 21 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "      marketPriceEth  marketPriceUsd  block_number  debtAvailableToSettle  \\\n",
       "0           0.002481        4.362594      11860755                    0.0   \n",
       "1           0.002481        4.419266      11861008                    0.0   \n",
       "2           0.002481        4.419266      11861237                    0.0   \n",
       "3           0.001769        3.215446      11861502                    0.0   \n",
       "4           0.001769        3.215446      11861791                    0.0   \n",
       "...              ...             ...           ...                    ...   \n",
       "2314        0.001082        3.017312      12584395                    0.0   \n",
       "2315        0.001079        3.010809      12585445                    0.0   \n",
       "2316        0.001086        3.013381      12586509                    0.0   \n",
       "2317        0.001076        2.994558      12587589                    0.0   \n",
       "2318        0.001110        2.986134      12588614                    0.0   \n",
       "\n",
       "        globalDebt                    globalDebtCeiling  systemSurplus  \\\n",
       "0     2.788522e+03  115792089237316195423570985008687.9       0.068533   \n",
       "1     2.788658e+03  115792089237316195423570985008687.9       0.108751   \n",
       "2     2.788658e+03  115792089237316195423570985008687.9       0.108751   \n",
       "3     3.048646e+03  115792089237316195423570985008687.9       0.116162   \n",
       "4     3.048646e+03  115792089237316195423570985008687.9       0.116162   \n",
       "...            ...                                  ...            ...   \n",
       "2314  1.127334e+07  115792089237316195423570985008687.9  334713.030602   \n",
       "2315  1.127334e+07  115792089237316195423570985008687.9  334713.030602   \n",
       "2316  1.127175e+07  115792089237316195423570985008687.9  334713.030602   \n",
       "2317  1.128063e+07  115792089237316195423570985008687.9  334865.769477   \n",
       "2318  1.128063e+07  115792089237316195423570985008687.9  334865.769477   \n",
       "\n",
       "      totalActiveSafeCount  RedemptionRateAnnualizedRate  \\\n",
       "0                        3                      1.000000   \n",
       "1                        3                      1.000000   \n",
       "2                        3                      1.000000   \n",
       "3                        3                      1.000000   \n",
       "4                        3                      1.000000   \n",
       "...                    ...                           ...   \n",
       "2314                   397                      1.004843   \n",
       "2315                   397                      1.004843   \n",
       "2316                   395                      0.973379   \n",
       "2317                   395                      0.973379   \n",
       "2318                   395                      0.970124   \n",
       "\n",
       "      RedemptionRateHourlyRate  ...  RedemptionPrice  RaiDrawnFromSAFEs  \\\n",
       "0                     1.000000  ...         3.140000       2.788440e+03   \n",
       "1                     1.000000  ...         3.140000       2.788440e+03   \n",
       "2                     1.000000  ...         3.140000       2.788440e+03   \n",
       "3                     1.000000  ...         3.140000       3.048440e+03   \n",
       "4                     1.000000  ...         3.140000       3.048440e+03   \n",
       "...                        ...  ...              ...                ...   \n",
       "2314                  1.000001  ...         3.007422       1.084780e+07   \n",
       "2315                  1.000001  ...         3.007430       1.084780e+07   \n",
       "2316                  0.999997  ...         3.007431       1.084620e+07   \n",
       "2317                  0.999997  ...         3.007393       1.085480e+07   \n",
       "2318                  0.999997  ...         3.007355       1.085480e+07   \n",
       "\n",
       "       collateral           debt  EthInUniswap  RaiInUniswap  ETH Price (OSM)  \\\n",
       "0       14.892360    2788.385291      1.679713  6.769337e+02      1803.656430   \n",
       "1       14.892360    2788.385291      1.679713  6.769337e+02      1763.974936   \n",
       "2       14.892360    2788.385291      1.679713  6.769337e+02      1763.974936   \n",
       "3       16.892360    3048.358490      1.418569  8.019337e+02      1775.977923   \n",
       "4       16.892360    3048.358490      1.418569  8.019337e+02      1805.792735   \n",
       "...           ...            ...           ...           ...              ...   \n",
       "2314  1083.900454  250083.541093   6564.157346  6.014012e+06      2702.280000   \n",
       "2315  1083.900454  250083.541093   6527.685380  6.047051e+06      2757.389589   \n",
       "2316  1083.900454  250083.541093   6565.000885  6.039329e+06      2773.590000   \n",
       "2317  1083.900454  250083.541093   6495.694866  6.107233e+06      2761.044990   \n",
       "2318  1083.900454  250083.541093   6541.018950  6.065035e+06      2828.150972   \n",
       "\n",
       "                block_timestamp RedemptionPriceinEth  RedemptionPriceError  \n",
       "0     2021-02-15 10:16:16+00:00             0.001741             -1.222594  \n",
       "1     2021-02-15 11:13:28+00:00             0.001780             -1.279266  \n",
       "2     2021-02-15 12:07:36+00:00             0.001780             -1.279266  \n",
       "3     2021-02-15 13:07:59+00:00             0.001768             -0.075446  \n",
       "4     2021-02-15 14:07:59+00:00             0.001739             -0.075446  \n",
       "...                         ...                  ...                   ...  \n",
       "2314  2021-06-07 01:05:54+00:00             0.001113             -0.009891  \n",
       "2315  2021-06-07 05:07:45+00:00             0.001091             -0.003379  \n",
       "2316  2021-06-07 09:06:20+00:00             0.001084             -0.005950  \n",
       "2317  2021-06-07 13:15:51+00:00             0.001089              0.012835  \n",
       "2318  2021-06-07 17:06:11+00:00             0.001063              0.021221  \n",
       "\n",
       "[2319 rows x 21 columns]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Q</th>\n",
       "      <th>D</th>\n",
       "      <th>Rrai</th>\n",
       "      <th>Reth</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>14.892360</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1.679713</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>14.892360</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1.679713</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>14.892360</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1.679713</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>16.494925</td>\n",
       "      <td>2.996774e+03</td>\n",
       "      <td>7.770940e+02</td>\n",
       "      <td>1.470463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>16.812975</td>\n",
       "      <td>3.038120e+03</td>\n",
       "      <td>7.969721e+02</td>\n",
       "      <td>1.428934</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2314</th>\n",
       "      <td>1083.900534</td>\n",
       "      <td>1.084647e+07</td>\n",
       "      <td>6.003190e+06</td>\n",
       "      <td>6575.363667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2315</th>\n",
       "      <td>1083.900470</td>\n",
       "      <td>1.084753e+07</td>\n",
       "      <td>6.038279e+06</td>\n",
       "      <td>6537.221037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2316</th>\n",
       "      <td>1083.900457</td>\n",
       "      <td>1.084646e+07</td>\n",
       "      <td>6.039119e+06</td>\n",
       "      <td>6559.444915</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2317</th>\n",
       "      <td>1083.900455</td>\n",
       "      <td>1.085313e+07</td>\n",
       "      <td>6.093610e+06</td>\n",
       "      <td>6508.444876</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2318</th>\n",
       "      <td>1083.900454</td>\n",
       "      <td>1.085446e+07</td>\n",
       "      <td>6.070750e+06</td>\n",
       "      <td>6534.504135</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2319 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                Q             D          Rrai         Reth\n",
       "0       14.892360  2.788440e+03  6.769337e+02     1.679713\n",
       "1       14.892360  2.788440e+03  6.769337e+02     1.679713\n",
       "2       14.892360  2.788440e+03  6.769337e+02     1.679713\n",
       "3       16.494925  2.996774e+03  7.770940e+02     1.470463\n",
       "4       16.812975  3.038120e+03  7.969721e+02     1.428934\n",
       "...           ...           ...           ...          ...\n",
       "2314  1083.900534  1.084647e+07  6.003190e+06  6575.363667\n",
       "2315  1083.900470  1.084753e+07  6.038279e+06  6537.221037\n",
       "2316  1083.900457  1.084646e+07  6.039119e+06  6559.444915\n",
       "2317  1083.900455  1.085313e+07  6.093610e+06  6508.444876\n",
       "2318  1083.900454  1.085446e+07  6.070750e+06  6534.504135\n",
       "\n",
       "[2319 rows x 4 columns]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# subset state variables for arbitrageur vector\n",
    "state_subset = states[['collateral','RaiDrawnFromSAFEs','RaiInUniswap','EthInUniswap']]\n",
    "\n",
    "# map state data to vector fields \n",
    "state_subset.columns = ['Q','D','Rrai','Reth']\n",
    "\n",
    "# alpha is the smoothing factor\n",
    "local = state_subset.ewm(alpha=0.8).mean()\n",
    "local"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coordinate Transformations \n",
    "\n",
    "1. $\\alpha := \\frac{d}{\\bar{D}}$\n",
    "\n",
    "Constraint: $\\bar{D} \\geq D + d$\n",
    "\n",
    "$ C_0 := \\frac{p^r_{U/R}}{p_{U/E}}\\bar{L} > 0$\n",
    "\n",
    "$ C_0 D - Q =: C_1.$\n",
    "\n",
    "2. $\\beta := \\frac{q - C_0 d}{C_1}$\n",
    "\n",
    "3. $\\gamma := \\frac{r}{R_{RAI}}$\n",
    "\n",
    "4. $\\delta := \\frac{z}{R_{ETH}}$\n",
    "\n",
    "\n",
    "## Inverse Transformations\n",
    "\n",
    "1. $d^* = \\alpha * \\bar{D}$.\n",
    "\n",
    "2. $q^* = C_0 * \\bar{D} * \\alpha + C_1 * \\beta$\n",
    "\n",
    "3. $r^* = \\gamma * {R_{RAI}}$\n",
    "\n",
    "4. $z^* = \\delta * {R_{ETH}}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# function to create coordinate transformations\n",
    "def coordinate_transformations(params,df,Q,R_eth,R_rai,D,RedemptionPrice,EthPrice):\n",
    "    '''\n",
    "    Description:\n",
    "    Function that takes in pandas dataframe and the names of columns\n",
    "    \n",
    "    Parameters:\n",
    "    df: pandas dataframe containing states information\n",
    "    Q: dataframe column name\n",
    "    R_eth: dataframe column name\n",
    "    R_rai: dataframe column name\n",
    "    D: dataframe column name\n",
    "    RedemptionPrice: dataframe column name\n",
    "    EthPrice: dataframe column name\n",
    "\n",
    "    Returns: Pandas dataframe with alpha, beta, gamma, delta transformed values\n",
    "    \n",
    "    Example:\n",
    "    \n",
    "    coordinate_transformations(params,states,'collateral','EthInUniswap','RaiInUniswap',\n",
    "                           'RaiDrawnFromSAFEs','RedemptionPrice','ETH Price (OSM)')[['alpha','beta','gamma','delta']]\n",
    "    '''\n",
    "    \n",
    "    # Calculate alpha\n",
    "    d = df[D].diff()\n",
    "    d.fillna(0,inplace=True)\n",
    "    df['d'] = d\n",
    "    \n",
    "    df['alpha'] = df['d'] / params['debt_ceiling']\n",
    "    \n",
    "    # alpha constraint check\n",
    "    for i, row in df.iterrows():\n",
    "        #constraint\n",
    "        constraint = params['debt_ceiling'] >= row[D] + row['d']\n",
    "        if constraint == False:\n",
    "            print('For row index {}'.format(i))\n",
    "            print('Alpha constraint is not passed')\n",
    "\n",
    "    # calculate beta\n",
    "    df['C_o'] = (df[RedemptionPrice]/states[EthPrice]) * params['liquidation_ratio']\n",
    "    \n",
    "    # C_0 constraint check\n",
    "    for i, row in df.iterrows():\n",
    "        #constraint\n",
    "        constraint = row['C_o'] > 0\n",
    "        if constraint == False:\n",
    "            print('For row index {}'.format(i))\n",
    "            print('C_0 constraint is not passed')\n",
    "\n",
    "    q = df[Q].diff()\n",
    "    q.fillna(0,inplace=True)\n",
    "    df['q'] = q\n",
    "\n",
    "\n",
    "    df['C_1'] = (df['C_o'] * df[D]) - df[Q]\n",
    "\n",
    "    df['beta'] = (df['q'] - (df['C_o']*df['d']))/ df['C_1']\n",
    "\n",
    "    # calculate gamma\n",
    "    r = df[R_rai].diff()\n",
    "    r.fillna(0,inplace=True)\n",
    "    df['r'] = r\n",
    "\n",
    "    df['gamma'] = df['r']/df[R_rai]\n",
    "\n",
    "    # calculate delta\n",
    "    z = df[R_eth].diff()\n",
    "    z.fillna(0,inplace=True)\n",
    "    df['z'] = z\n",
    "\n",
    "    df['delta'] = df['z']/df[R_eth]\n",
    "    \n",
    "    return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>gamma</th>\n",
       "      <th>delta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2.600000e-07</td>\n",
       "      <td>-0.146901</td>\n",
       "      <td>0.155873</td>\n",
       "      <td>-0.184090</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2314</th>\n",
       "      <td>3.654143e-06</td>\n",
       "      <td>-0.000359</td>\n",
       "      <td>0.009173</td>\n",
       "      <td>-0.009223</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2315</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.005464</td>\n",
       "      <td>-0.005587</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2316</th>\n",
       "      <td>-1.604192e-06</td>\n",
       "      <td>0.000158</td>\n",
       "      <td>-0.001279</td>\n",
       "      <td>0.005684</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2317</th>\n",
       "      <td>8.600000e-06</td>\n",
       "      <td>-0.000846</td>\n",
       "      <td>0.011119</td>\n",
       "      <td>-0.010670</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2318</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.006958</td>\n",
       "      <td>0.006929</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2319 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "             alpha      beta     gamma     delta\n",
       "0     0.000000e+00 -0.000000  0.000000  0.000000\n",
       "1     0.000000e+00 -0.000000  0.000000  0.000000\n",
       "2     0.000000e+00 -0.000000  0.000000  0.000000\n",
       "3     2.600000e-07 -0.146901  0.155873 -0.184090\n",
       "4     0.000000e+00 -0.000000  0.000000  0.000000\n",
       "...            ...       ...       ...       ...\n",
       "2314  3.654143e-06 -0.000359  0.009173 -0.009223\n",
       "2315  0.000000e+00  0.000000  0.005464 -0.005587\n",
       "2316 -1.604192e-06  0.000158 -0.001279  0.005684\n",
       "2317  8.600000e-06 -0.000846  0.011119 -0.010670\n",
       "2318  0.000000e+00  0.000000 -0.006958  0.006929\n",
       "\n",
       "[2319 rows x 4 columns]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# transform historical data \n",
    "transformed = coordinate_transformations(params,states,'collateral','EthInUniswap','RaiInUniswap',\n",
    "                           'RaiDrawnFromSAFEs','RedemptionPrice','ETH Price (OSM)')[['alpha','beta','gamma','delta']]\n",
    "\n",
    "transformed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Q</th>\n",
       "      <th>D</th>\n",
       "      <th>Rrai</th>\n",
       "      <th>Reth</th>\n",
       "      <th>RedemptionPrice</th>\n",
       "      <th>ETH Price (OSM)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>14.892360</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1803.656430</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>14.892360</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1763.974936</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>14.892360</td>\n",
       "      <td>2.788440e+03</td>\n",
       "      <td>6.769337e+02</td>\n",
       "      <td>1.679713</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1763.974936</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>16.494925</td>\n",
       "      <td>2.996774e+03</td>\n",
       "      <td>7.770940e+02</td>\n",
       "      <td>1.470463</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1775.977923</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>16.812975</td>\n",
       "      <td>3.038120e+03</td>\n",
       "      <td>7.969721e+02</td>\n",
       "      <td>1.428934</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1805.792735</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2314</th>\n",
       "      <td>1083.900534</td>\n",
       "      <td>1.084647e+07</td>\n",
       "      <td>6.003190e+06</td>\n",
       "      <td>6575.363667</td>\n",
       "      <td>3.007422</td>\n",
       "      <td>2702.280000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2315</th>\n",
       "      <td>1083.900470</td>\n",
       "      <td>1.084753e+07</td>\n",
       "      <td>6.038279e+06</td>\n",
       "      <td>6537.221037</td>\n",
       "      <td>3.007430</td>\n",
       "      <td>2757.389589</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2316</th>\n",
       "      <td>1083.900457</td>\n",
       "      <td>1.084646e+07</td>\n",
       "      <td>6.039119e+06</td>\n",
       "      <td>6559.444915</td>\n",
       "      <td>3.007431</td>\n",
       "      <td>2773.590000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2317</th>\n",
       "      <td>1083.900455</td>\n",
       "      <td>1.085313e+07</td>\n",
       "      <td>6.093610e+06</td>\n",
       "      <td>6508.444876</td>\n",
       "      <td>3.007393</td>\n",
       "      <td>2761.044990</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2318</th>\n",
       "      <td>1083.900454</td>\n",
       "      <td>1.085446e+07</td>\n",
       "      <td>6.070750e+06</td>\n",
       "      <td>6534.504135</td>\n",
       "      <td>3.007355</td>\n",
       "      <td>2828.150972</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2319 rows × 6 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                Q             D          Rrai         Reth  RedemptionPrice  \\\n",
       "0       14.892360  2.788440e+03  6.769337e+02     1.679713         3.140000   \n",
       "1       14.892360  2.788440e+03  6.769337e+02     1.679713         3.140000   \n",
       "2       14.892360  2.788440e+03  6.769337e+02     1.679713         3.140000   \n",
       "3       16.494925  2.996774e+03  7.770940e+02     1.470463         3.140000   \n",
       "4       16.812975  3.038120e+03  7.969721e+02     1.428934         3.140000   \n",
       "...           ...           ...           ...          ...              ...   \n",
       "2314  1083.900534  1.084647e+07  6.003190e+06  6575.363667         3.007422   \n",
       "2315  1083.900470  1.084753e+07  6.038279e+06  6537.221037         3.007430   \n",
       "2316  1083.900457  1.084646e+07  6.039119e+06  6559.444915         3.007431   \n",
       "2317  1083.900455  1.085313e+07  6.093610e+06  6508.444876         3.007393   \n",
       "2318  1083.900454  1.085446e+07  6.070750e+06  6534.504135         3.007355   \n",
       "\n",
       "      ETH Price (OSM)  \n",
       "0         1803.656430  \n",
       "1         1763.974936  \n",
       "2         1763.974936  \n",
       "3         1775.977923  \n",
       "4         1805.792735  \n",
       "...               ...  \n",
       "2314      2702.280000  \n",
       "2315      2757.389589  \n",
       "2316      2773.590000  \n",
       "2317      2761.044990  \n",
       "2318      2828.150972  \n",
       "\n",
       "[2319 rows x 6 columns]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# add additional signals to arbitrageur state\n",
    "local['RedemptionPrice'] = states['RedemptionPrice']\n",
    "local['ETH Price (OSM)'] = states['ETH Price (OSM)']\n",
    "\n",
    "local"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>gamma</th>\n",
       "      <th>delta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2.083333e-07</td>\n",
       "      <td>-0.121249</td>\n",
       "      <td>0.128891</td>\n",
       "      <td>-0.142303</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4.134656e-08</td>\n",
       "      <td>-0.023359</td>\n",
       "      <td>0.024942</td>\n",
       "      <td>-0.029063</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2314</th>\n",
       "      <td>5.311448e-06</td>\n",
       "      <td>-0.000522</td>\n",
       "      <td>0.007211</td>\n",
       "      <td>-0.006817</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2315</th>\n",
       "      <td>1.062290e-06</td>\n",
       "      <td>-0.000105</td>\n",
       "      <td>0.005811</td>\n",
       "      <td>-0.005835</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2316</th>\n",
       "      <td>-1.070896e-06</td>\n",
       "      <td>0.000105</td>\n",
       "      <td>0.000139</td>\n",
       "      <td>0.003388</td>\n",
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       "    <tr>\n",
       "      <th>2317</th>\n",
       "      <td>6.665821e-06</td>\n",
       "      <td>-0.000656</td>\n",
       "      <td>0.008942</td>\n",
       "      <td>-0.007836</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2318</th>\n",
       "      <td>1.333164e-06</td>\n",
       "      <td>-0.000131</td>\n",
       "      <td>-0.003766</td>\n",
       "      <td>0.003988</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2319 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "             alpha      beta     gamma     delta\n",
       "0     0.000000e+00 -0.000000  0.000000  0.000000\n",
       "1     0.000000e+00 -0.000000  0.000000  0.000000\n",
       "2     0.000000e+00 -0.000000  0.000000  0.000000\n",
       "3     2.083333e-07 -0.121249  0.128891 -0.142303\n",
       "4     4.134656e-08 -0.023359  0.024942 -0.029063\n",
       "...            ...       ...       ...       ...\n",
       "2314  5.311448e-06 -0.000522  0.007211 -0.006817\n",
       "2315  1.062290e-06 -0.000105  0.005811 -0.005835\n",
       "2316 -1.070896e-06  0.000105  0.000139  0.003388\n",
       "2317  6.665821e-06 -0.000656  0.008942 -0.007836\n",
       "2318  1.333164e-06 -0.000131 -0.003766  0.003988\n",
       "\n",
       "[2319 rows x 4 columns]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# transform u*\n",
    "transformed_arbitrageur = coordinate_transformations(params,local,'Q','Reth','Rrai',\n",
    "                           'D','RedemptionPrice','ETH Price (OSM)')[['alpha','beta','gamma','delta']]\n",
    "\n",
    "transformed_arbitrageur"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>gamma</th>\n",
       "      <th>delta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>5.166667e-08</td>\n",
       "      <td>-0.025652</td>\n",
       "      <td>0.026982</td>\n",
       "      <td>-0.041788</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-4.134656e-08</td>\n",
       "      <td>0.023359</td>\n",
       "      <td>-0.024942</td>\n",
       "      <td>0.029063</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          alpha      beta     gamma     delta\n",
       "0  0.000000e+00  0.000000  0.000000  0.000000\n",
       "1  0.000000e+00  0.000000  0.000000  0.000000\n",
       "2  0.000000e+00  0.000000  0.000000  0.000000\n",
       "3  5.166667e-08 -0.025652  0.026982 -0.041788\n",
       "4 -4.134656e-08  0.023359 -0.024942  0.029063"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def create_transformed_errors(transformed_states,transformed_arbitrageur):\n",
    "    '''\n",
    "    Description:\n",
    "    Function for taking two pandas dataframes of transformed states and taking the difference\n",
    "    to produce an error dataframe. \n",
    "    \n",
    "    Parameters:\n",
    "    transformed_states: pandas dataframe with alpha, beta, gamma, and delta features\n",
    "    transformed_arbitrageur: pandas dataframe with alpha, beta, gamma, and delta features\n",
    "\n",
    "    Returns:\n",
    "    error pandas dataframe and transformation object\n",
    "    \n",
    "    '''\n",
    "    alpha_diff = transformed_states['alpha'] - transformed_arbitrageur['alpha']\n",
    "    beta_diff = transformed_states['beta'] - transformed_arbitrageur['beta']\n",
    "    gamma_diff = transformed_states['gamma'] - transformed_arbitrageur['gamma']\n",
    "    delta_diff = transformed_states['delta'] - transformed_arbitrageur['delta']\n",
    "\n",
    "\n",
    "    e_u = pd.DataFrame(alpha_diff)\n",
    "    e_u['beta'] = beta_diff\n",
    "    e_u['gamma'] = gamma_diff\n",
    "    e_u['delta'] = delta_diff\n",
    "\n",
    "    e_u = e_u.astype(float)\n",
    "    \n",
    "    return e_u\n",
    "\n",
    "e_u = create_transformed_errors(transformed,transformed_arbitrageur)\n",
    "e_u.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>gamma</th>\n",
       "      <th>delta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>2.319000e+03</td>\n",
       "      <td>2.319000e+03</td>\n",
       "      <td>2319.000000</td>\n",
       "      <td>2319.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>1.437219e-10</td>\n",
       "      <td>4.655444e-02</td>\n",
       "      <td>-0.000254</td>\n",
       "      <td>-0.000241</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>8.144377e-05</td>\n",
       "      <td>2.065773e+00</td>\n",
       "      <td>0.007700</td>\n",
       "      <td>0.009004</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>-7.476813e-04</td>\n",
       "      <td>-8.036261e-02</td>\n",
       "      <td>-0.153277</td>\n",
       "      <td>-0.190718</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>-1.165382e-05</td>\n",
       "      <td>-4.446707e-04</td>\n",
       "      <td>-0.000759</td>\n",
       "      <td>-0.000743</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>-2.382897e-12</td>\n",
       "      <td>7.991362e-07</td>\n",
       "      <td>-0.000015</td>\n",
       "      <td>0.000014</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>1.241447e-05</td>\n",
       "      <td>4.459252e-04</td>\n",
       "      <td>0.000704</td>\n",
       "      <td>0.000769</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>5.989221e-04</td>\n",
       "      <td>9.924303e+01</td>\n",
       "      <td>0.064828</td>\n",
       "      <td>0.093804</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              alpha          beta        gamma        delta\n",
       "count  2.319000e+03  2.319000e+03  2319.000000  2319.000000\n",
       "mean   1.437219e-10  4.655444e-02    -0.000254    -0.000241\n",
       "std    8.144377e-05  2.065773e+00     0.007700     0.009004\n",
       "min   -7.476813e-04 -8.036261e-02    -0.153277    -0.190718\n",
       "25%   -1.165382e-05 -4.446707e-04    -0.000759    -0.000743\n",
       "50%   -2.382897e-12  7.991362e-07    -0.000015     0.000014\n",
       "75%    1.241447e-05  4.459252e-04     0.000704     0.000769\n",
       "max    5.989221e-04  9.924303e+01     0.064828     0.093804"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_u.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[<AxesSubplot:title={'center':'alpha'}>,\n",
       "        <AxesSubplot:title={'center':'beta'}>],\n",
       "       [<AxesSubplot:title={'center':'gamma'}>,\n",
       "        <AxesSubplot:title={'center':'delta'}>]], dtype=object)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "e_u.hist()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When data isn't normal (as is shown above), it is best practice to do a transformation. For our initial transformation, we will use the Yeo-Johnson power transformation. The Yeo-Johnson power transformation is used to stabilize variance, and make data more Gausian. The Yeo-Johnson is an extension of Box-Cox that allows for both zero and negative values(https://en.wikipedia.org/wiki/Power_transform). You could use any other type of normalization tranformation as well, whichever fits the data the best.\n",
    "\n",
    "Scikit-learn has a great implementation of the transformer, which we will use below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[<AxesSubplot:title={'center':'alpha'}>,\n",
       "        <AxesSubplot:title={'center':'beta'}>],\n",
       "       [<AxesSubplot:title={'center':'gamma'}>,\n",
       "        <AxesSubplot:title={'center':'delta'}>]], dtype=object)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pt = PowerTransformer()\n",
    "yeo= pd.DataFrame(pt.fit_transform(e_u),columns=e_u.columns)\n",
    "yeo.hist()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,\n",
       "         0.00000000e+00],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,\n",
       "         0.00000000e+00],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,\n",
       "         0.00000000e+00],\n",
       "       ...,\n",
       "       [-5.33296342e-07,  5.25099073e-05, -1.41771657e-03,\n",
       "         2.29593308e-03],\n",
       "       [ 1.93417915e-06, -1.90105025e-04,  2.17625580e-03,\n",
       "        -2.83355183e-03],\n",
       "       [-1.33316417e-06,  1.31326999e-04, -3.19196860e-03,\n",
       "         2.94126100e-03]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# transform back into coordinate system\n",
    "pt.inverse_transform(yeo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data looks a little better, but we can always experiment with additional techniques"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def power_transformation(e_u):\n",
    "    '''\n",
    "    Definition:\n",
    "    Function to perform a power transformation on the coordinate \n",
    "    transformed differenced data\n",
    "    \n",
    "    Parameters:\n",
    "    e_u: Dataframe of coordinated transformed differenced data\n",
    "    \n",
    "    Required:\n",
    "    import pandas as pd\n",
    "    from sklearn.preprocessing import PowerTransformer\n",
    "    \n",
    "    Returns:\n",
    "    Transformed dataframe and transformation object\n",
    "    \n",
    "    Example:\n",
    "    transformed_df, pt = power_transformation(e_u)\n",
    "    '''\n",
    "    pt = PowerTransformer()\n",
    "    yeo= pd.DataFrame(pt.fit_transform(e_u),columns=e_u.columns)\n",
    "\n",
    "    return yeo, pt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "e_u,pt = power_transformation(e_u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "# split data between train and test (in production deployment, can remove)\n",
    "split_point = int(len(e_u) * .8)\n",
    "train = e_u.iloc[0:split_point]\n",
    "test = e_u.iloc[split_point:]\n",
    "\n",
    "\n",
    "states_train = states.iloc[0:split_point]\n",
    "states_test = states.iloc[split_point:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- Potential alternative transformations are as follows:\n",
    "\n",
    "* sin\n",
    "* log of the Yeo-Johnson\n",
    "\n",
    "Both of which provide a better fit than the Yeo-Johnson (as seen below).\n",
    "\n",
    "\n",
    "For the rest of this notebook, we will implement the model training, forecasting, and evaluation process which will allow us to iterate over different transformations until we find one that fits our use case the best. -->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- ### Autogressive lag selection -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "aic = []\n",
    "for i in range(1,25):\n",
    "    model = VAR(train)\n",
    "    results = model.fit(i,ic='aic')\n",
    "    aic.append(results.aic)\n",
    "    \n",
    "    \n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.plot(aic, 'r+')\n",
    "plt.legend(['AIC'])\n",
    "plt.xlabel('Autocorrelation Lag')\n",
    "plt.ylabel('AIC')\n",
    "plt.title('Plot of sweeps over lag depths over AIC Loss functions')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# aic = []\n",
    "# for i in range(1,16):\n",
    "#     model = VARMAX(endog=train.values,exog=states_train['RedemptionPriceError'].values,initialization='approximate_diffuse')\n",
    "#     results = model.fit(order=(i,0))\n",
    "#     aic.append(results.aic)\n",
    "    \n",
    "    \n",
    "# plt.figure(figsize=(10, 8))\n",
    "# plt.plot(aic, 'r+')\n",
    "# plt.legend(['AIC'])\n",
    "# plt.xlabel('Autocorrelation Lag')\n",
    "# plt.ylabel('AIC')\n",
    "# plt.title('Plot of sweeps over lag depths over AIC Loss functions')\n",
    "# plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a set of candidate models for the data, **the preferred model is the one with the minimum AIC value, the sign of the data does not matter**. AIC optimizes for goodness of fit but also includes a penalty for each additional parameter, which discourages overfitting. In our case, this appears that a lag of ***15*** is optimal.\n",
    "\n",
    "\n",
    "For a VARMAX model, which we have decided to use, an order of 1 is selected. To determine which model performs better overall for predictions, given the computational constraints that VARMAX is too slow to be retrained at each timestep, a [validation notebook](VAR_vs_VARMAX_evaluation.ipynb) was created to test if a VAR retrained every timestep vs a VARMAX retrained very 20 predictions. The result over 20 predictions was that VAR performed best for alpha, gamma, and delta but VARMAX performed better with beta by a higher magnitude than VAR. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def VARMAX_prediction(e_u,RedemptionPriceError,newRedemptionPriceError,steps=1,lag=1):\n",
    "    '''\n",
    "    Description:\n",
    "    Function to train and forecast a VARMAX model one step into the future\n",
    "    \n",
    "    Parameters:\n",
    "    e_u: errors pandas dataframe\n",
    "    RedemptionPriceErrorPrevious: 1d Numpy array of RedemptionPriceError values\n",
    "    newRedemptionPriceError: exogenous latest redemption price error signal - float\n",
    "    steps: Number of forecast steps. Default is 1\n",
    "    lag: number of autoregressive lags. Default is 1\n",
    "    \n",
    "    Returns:\n",
    "    Numpy array of transformed state changes\n",
    "    \n",
    "    Example\n",
    "    Y_pred = VARMAX_prediction(train,states_train['RedemptionPriceError'],\n",
    "                  states_test['RedemptionPriceError'][0:5],steps=5,lag=1)\n",
    "    '''\n",
    "    # instantiate the VARMAX model object from statsmodels \n",
    "    model = VARMAX(endog=e_u.values,exog=RedemptionPriceError,\n",
    "                   initialization='approximate_diffuse',measurement_error=True)\n",
    "\n",
    "    # fit model with determined lag values\n",
    "    results = model.fit(order=(lag,0))\n",
    "    \n",
    "    Y_pred = results.forecast(steps = steps, exog=newRedemptionPriceError)\n",
    "    \n",
    "    return Y_pred.values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def VAR_prediction(e_u,lag=1):\n",
    "    '''\n",
    "    Description:\n",
    "    Function to train and forecast a VAR model one step into the future\n",
    "    \n",
    "    Parameters:\n",
    "    e_u: errors pandas dataframe\n",
    "    lag: number of autoregressive lags. Default is 1\n",
    "    \n",
    "    Returns:\n",
    "    Numpy array of transformed state changes\n",
    "    \n",
    "    Example\n",
    "    VAR_prediction(e_u,6)    \n",
    "    '''\n",
    "    # instantiate the VAR model object from statsmodels \n",
    "    model = VAR(e_u.values)\n",
    "\n",
    "    # fit model with determined lag values\n",
    "    results = model.fit(lag)\n",
    "    \n",
    "    lag_order = results.k_ar\n",
    "    \n",
    "    Y_pred = results.forecast(e_u.values[-lag_order:],1)\n",
    "\n",
    "    \n",
    "    return Y_pred[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y_pred = VAR_prediction(e_u,15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.00085067, -0.20262886,  0.57261877, -0.2041078 ])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y_pred"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 7.15408313e-07, -1.36809014e-03,  3.86209722e-03,\n",
       "        -1.46021110e-03]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def invert_power_transformation(pt,prediction):\n",
    "    '''\n",
    "    Definition:\n",
    "    Function to invert power transformation\n",
    "    \n",
    "    Parameters:\n",
    "    pt: transformation object\n",
    "    prediction: Numpy array of model state coordinate transformed percentage changes\n",
    "    \n",
    "    Required:\n",
    "    import pandas as pd\n",
    "    from sklearn.preprocessing import PowerTransformer\n",
    "    \n",
    "    Returns:\n",
    "    inverted transformation numpy array\n",
    "    \n",
    "    Example:\n",
    "    inverted_array = invert_power_transformation(pt,prediction)\n",
    "    \n",
    "    '''\n",
    "    # transform back into coordinate system\n",
    "    inverted = pt.inverse_transform(prediction.reshape(1,-1))\n",
    "    \n",
    "    return inverted\n",
    "\n",
    "\n",
    "Y_pred = invert_power_transformation(pt,Y_pred)\n",
    "Y_pred"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# New states"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inverse Transformations\n",
    "\n",
    "1. $d^* = \\alpha * \\bar{D}$\n",
    "\n",
    "2. $q^* = C_0 * \\bar{D} * \\alpha + C_1 * \\beta$.\n",
    "\n",
    "3. $r^* = \\gamma * {R_{RAI}}$\n",
    "\n",
    "4. $z^* = \\delta * {R_{ETH}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "715.4083132565603"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y_pred[0][0]*params['debt_ceiling']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def inverse_transformation_and_state_update(Y_pred,previous_state,params):\n",
    "    '''\n",
    "    Description:\n",
    "    Function to take system identification model prediction and invert transfrom and create new state\n",
    "    \n",
    "    Parameters:\n",
    "    y_pred: numpy array of transformed state changes\n",
    "    previous_state: pandas dataframe of previous state or 'current' state\n",
    "    params: dictionary of system parameters\n",
    "    \n",
    "    Returns:\n",
    "    pandas dataframe of new states \n",
    "    \n",
    "    Example:\n",
    "    inverse_transformation_and_state_update(Y_pred,previous_state,params)\n",
    "    '''\n",
    "    \n",
    "    d_star = Y_pred[0] * params['debt_ceiling']\n",
    "    \n",
    "    q_star = previous_state['C_o'] * params['debt_ceiling'] * Y_pred[0] + previous_state['C_1'] * Y_pred[1]\n",
    "    \n",
    "    r_star = Y_pred[2] * previous_state['gamma'] * previous_state['RaiInUniswap']\n",
    "\n",
    "    z_star = Y_pred[3] * previous_state['delta'] * previous_state['EthInUniswap']\n",
    "\n",
    "\n",
    "    new_state = pd.DataFrame(previous_state[['collateral','EthInUniswap','RaiInUniswap','RaiDrawnFromSAFEs']].to_dict(),index=[0]) \n",
    "    new_state['Q'] = new_state['collateral'] + q_star\n",
    "    new_state['D'] = new_state['RaiDrawnFromSAFEs'] + d_star\n",
    "    new_state['R_Rai'] = new_state['RaiInUniswap'] + r_star\n",
    "    new_state['R_Eth'] = new_state['EthInUniswap'] + z_star\n",
    "\n",
    "    return new_state[['Q','D','R_Rai','R_Eth']]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Previous state:\n",
      "{'collateral': 1463.4226, 'RaiDrawnFromSAFEs': 29412888.07431956, 'RaiInUniswap': 19664591.95781854, 'EthInUniswap': 19142.982052762283}\n",
      "\n",
      " New state:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Q</th>\n",
       "      <th>D</th>\n",
       "      <th>R_Rai</th>\n",
       "      <th>R_Eth</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1410.927207</td>\n",
       "      <td>2.941360e+07</td>\n",
       "      <td>1.966456e+07</td>\n",
       "      <td>19143.070369</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             Q             D         R_Rai         R_Eth\n",
       "0  1410.927207  2.941360e+07  1.966456e+07  19143.070369"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "previous_state = states.iloc[train.index[-1]]\n",
    "print('Previous state:')\n",
    "print(previous_state[['collateral','RaiDrawnFromSAFEs','RaiInUniswap','EthInUniswap']].to_dict())\n",
    "print('\\n New state:')\n",
    "inverse_transformation_and_state_update(Y_pred[0],previous_state,params)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "In this notebook, we have iterated through several different models and decided on a VAR(15) model for us in the Rai Digital Twin."
   ]
  }
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