scaler_demo.ipynb

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    "## Custom scaler methods: DerivativeRank and logscaler\n",
    "\n",
    "two custom scaler methods are defined and tested\n",
    "\n",
    "#### DerivativeRank\n",
    "allows to set a suitable tradeoff between fixing the distribution of a feature and preserving information from original data.\n",
    "It is basically a discrete derivative of rank scaling (e.g. like [pandas rank](https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.rank.html) or similar to sklearn [QuantileTransformer](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.QuantileTransformer.html#sklearn.preprocessing.QuantileTransformer)). compared to those it preserves more of the information from original data. while rank or quantile transformer produce uniform distribution (or close to uniform in case of repeated values), DerivativeRank produces a distribution which is somewhere between uniform and original. The parameter $d$, the order of derivative, controls how close the resulting distribution is to the original versus the uniform distribution. \n",
    "\n",
    "Implementation as recursive funtion:\n",
    "1. array of values is ordered and consecutive deltas taken, repeated recursively $d$ times\n",
    "2. differences are set to constant value $1$\n",
    "3. integrating back $d$ times, each time inversing the permutation of ordering applied in delta calculation, to obtain scaled array of values\n",
    "\n",
    "\n",
    "#### logscaler\n",
    "an extension of logarithm to whole real numbers for feature scaling purpose. log-scaler has one parameter to control the behaviour around zero\n",
    "\n",
    "### this notebook\n",
    "1. defines and plots an extension of logarithm to whole real numbers\n",
    "\n",
    "2. imports and demos the DerivativeRank on two example datasets and observe the correlation\n"
   ]
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   "source": [
    "import numpy as np\n",
    "import  pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import plotly.express as px\n",
    "from sklearn.preprocessing import QuantileTransformer\n",
    "from scipy.spatial.distance import pdist\n",
    "\n",
    "\n",
    "# import own derivative rank scaler\n",
    "from derivative_rank import DerivativeRankScaler"
   ]
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    "#### Logscaler to extend log scaling on whole real numbers\n",
    "is it not possible to continuously combine the functions $\\log(x)$ and $-\\log(-x)$. to obtain at least a monotone function, between $-1$ and $1$ a bridge is added, and it can be extended to allow a smoother transition: bridge parameter close to one gives a sharp transition. close to 0 it gives a smoother alignment to the log function and will differ from log function on a wider interval around zero\n"
   ]
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   "source": [
    "def logscaler(X, bridge_param=0.3):\n",
    "    \"\"\" scale log-like, extended on all real numbers\n",
    "    \n",
    "    X: array of real numbers to scale\n",
    "    bridge_param: from (0,1) determines shape around zero, \n",
    "    the closer to 1 the smaller the interval around 0 where the logscaler differs from log\n",
    "    \"\"\"\n",
    "    assert (0 < bridge_param) and (bridge_param < 1), \"0 < bridge_param < 1 must be fulfilled\"\n",
    "    return np.sign(X)*np.log(np.abs(X)+np.maximum(1-bridge_param*np.abs(X),0))"
   ]
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   "source": [
    "#### plotting logscaler with different bridge parameters\n",
    "- 0.1 as bridge parameter: transition between positive and negative part is rather smooth (visually) and only at large x values above $10$ is the logscaler equal to logarithm\n",
    "- 0.9 on the other side achieves a quick equality with logarithm, and has abrupt angles\n",
    "- 0.5 is a cpmpromise between the two"
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     "metadata": {},
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   ],
   "source": [
    "X = np.linspace(-10,10,200)\n",
    "px.line(pd.DataFrame({\"x\": np.tile(X,4), \"y\": np.hstack((logscaler(X,bridge_param=0.1),\n",
    "                                                           logscaler(X,bridge_param=0.5),\n",
    "                                                            logscaler(X,bridge_param=0.9),\n",
    "                                                        np.sign(X)*np.log(np.abs(X)))),\n",
    "                                                  \"info\": np.hstack((np.repeat(\"bridge_param=0.1\",200),\n",
    "                                                                     np.repeat(\"bridge_param=0.5\",200),\n",
    "                                                                     np.repeat(\"bridge_param=0.9\",200),\n",
    "                                                                    np.repeat(\"mirrored log\",200))),}).iloc[::-1],\n",
    "                                                           x=\"x\", y=\"y\", \n",
    "                                                           color=\"info\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derivative Rank Scaler\n",
    "#### example distribution 1\n",
    "increasing the order of derivative is weakening the effect of scaling, i.e. distribution gets closed to original"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([<AxesSubplot:title={'center':'Quantile Tr'}>], dtype=object)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1152x288 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(nrows=1, ncols=6, figsize= (16,4))  \n",
    "k = 2000\n",
    "example_1 = np.exp( sorted(np.abs(np.random.randn(k))) *np.arange(k)/k)\n",
    "pd.DataFrame({\"original\": example_1}).hist(ax=axes[0])\n",
    "\n",
    "log_ex_1 = np.log(example_1)\n",
    "pd.DataFrame({\"log-scaled\": log_ex_1}).hist(ax=axes[1])\n",
    "\n",
    "drs = DerivativeRankScaler(d=1)\n",
    "drscaled_ex_1 = drs.fit_transform(example_1)\n",
    "pd.DataFrame({\"derivative rank, d=1\": drscaled_ex_1}).hist(ax=axes[2])\n",
    "\n",
    "drs = DerivativeRankScaler(d=10)\n",
    "drscaled_ex_1 = drs.fit_transform(example_1)\n",
    "pd.DataFrame({\"derivative rank, d=10\": drscaled_ex_1}).hist(ax=axes[3])\n",
    "\n",
    "drs = DerivativeRankScaler(d=20)\n",
    "drscaled_ex_1 = drs.fit_transform(example_1)\n",
    "pd.DataFrame({\"derivative rank, d=20\": drscaled_ex_1}).hist(ax=axes[4])\n",
    "\n",
    "qnTr = QuantileTransformer()\n",
    "pd.DataFrame({\"Quantile Tr\": qnTr.fit_transform(example_1[:,np.newaxis]).flatten()}).hist(ax=axes[5])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### example distribution 2\n",
    "increasing the order of derivative is weakening the effect of scaling, i.e. distribution gets closed to original"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([<AxesSubplot:title={'center':'Quantile Transform'}>], dtype=object)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1152x288 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(nrows=1, ncols=6, figsize= (16,4))  \n",
    "example_2 = (np.concatenate((np.random.randn(20000),(np.random.rand(4000)+10))) )\n",
    "pd.DataFrame({\"original\": example_2}).hist(ax=axes[0])\n",
    "\n",
    "log_ex_2 = logscaler(example_2)\n",
    "pd.DataFrame({\"log-scaled\": log_ex_2}).hist(ax=axes[1])\n",
    "\n",
    "drs = DerivativeRankScaler(d=1)\n",
    "drscaled_ex_2 = drs.fit_transform(example_2)\n",
    "pd.DataFrame({\"derivative rank, d=1\": drscaled_ex_2}).hist(ax=axes[2])\n",
    "\n",
    "drs = DerivativeRankScaler(d=10)\n",
    "drscaled_ex_2 = drs.fit_transform(example_2)\n",
    "pd.DataFrame({\"derivative rank, d=10\": drscaled_ex_2}).hist(ax=axes[3])\n",
    "\n",
    "d = 20\n",
    "drs = DerivativeRankScaler(d=d)\n",
    "drscaled_ex_2 = drs.fit_transform(example_2)\n",
    "pd.DataFrame({\"derivative rank, d={0}\".format(d): drscaled_ex_2}).hist(ax=axes[4])\n",
    "\n",
    "qnTr = QuantileTransformer()\n",
    "pd.DataFrame({\"Quantile Transform\": qnTr.fit_transform(example_2[:,np.newaxis]).flatten()}).hist(ax=axes[5])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### example distribution 3\n",
    "increasing the order of derivative is weakening the effect of scaling, i.e. distribution gets closed to original"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([<AxesSubplot:title={'center':'Quantile Tr'}>], dtype=object)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1152x288 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(nrows=1, ncols=6, figsize= (16,4))  \n",
    "example_3 = (np.concatenate((np.random.randn(8000),(np.random.rand(8000)+10), (10*np.random.rand(1000)+30))) )\n",
    "pd.DataFrame({\"original\": example_3}).hist(ax=axes[0])\n",
    "\n",
    "\n",
    "log_ex_3 = logscaler(example_3)\n",
    "pd.DataFrame({\"log-scaled\": log_ex_3}).hist(ax=axes[1])\n",
    "\n",
    "drs = DerivativeRankScaler(d=1)\n",
    "drscaled_ex_3 = drs.fit_transform(example_3)\n",
    "pd.DataFrame({\"derivative rank, d=1\": drscaled_ex_3}).hist(ax=axes[2])\n",
    "\n",
    "drs = DerivativeRankScaler(d=10)\n",
    "drscaled_ex_3 = drs.fit_transform(example_3)\n",
    "pd.DataFrame({\"derivative rank, d=10\": drscaled_ex_3}).hist(ax=axes[3])\n",
    "\n",
    "drs = DerivativeRankScaler(d=20)\n",
    "drscaled_ex_3 = drs.fit_transform(example_3)\n",
    "pd.DataFrame({\"derivative rank, d=20\": drscaled_ex_3}).hist(ax=axes[4])\n",
    "\n",
    "qnTr = QuantileTransformer()\n",
    "pd.DataFrame({\"Quantile Tr\": qnTr.fit_transform(example_3[:,np.newaxis]).flatten()}).hist(ax=axes[5])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### compare how much the pairwise differences of points are correlated between scaled and original data \n",
    "as a measure of how much of the information from the original distribution is preserved the correlation of pairwise differences between values is used.  \n",
    "this value is compared for derivative scaled distribution, the log-scaled distribution and the rank-scaled distribution\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "derivative rank (d=1) scaled 0.5773692747684686\n",
      "derivative rank (d=10) scaled 0.8900006903867509\n",
      "log-scaler 0.902846407714289\n",
      "QuantileTransformer 0.3967337511246382\n"
     ]
    }
   ],
   "source": [
    "# select the example (from above) on which to calculate the measure\n",
    "example = example_1\n",
    "\n",
    "n = example.shape[0]\n",
    "# calculate only on a sample of points (since num of all pairs can get too large)\n",
    "sample = np.random.choice(np.arange(n),size=1000,replace=False).astype(int)\n",
    "\n",
    "drs = DerivativeRankScaler(d=1)\n",
    "drscaled_1_dist_corr = np.corrcoef(np.stack((pdist(example[sample,np.newaxis]),\n",
    "             pdist(drs.fit_transform(example)[sample,np.newaxis])), axis=0))[0,1]\n",
    "\n",
    "drs = DerivativeRankScaler(d=10)\n",
    "drscaled_10_dist_corr = np.corrcoef(np.stack((pdist(example[sample,np.newaxis]),\n",
    "             pdist(drs.fit_transform(example)[sample,np.newaxis])), axis=0))[0,1]\n",
    "\n",
    "\n",
    "logscaled_dist_corr =  np.corrcoef(np.stack((pdist(example[sample,np.newaxis]),\n",
    "             pdist(logscaler(example)[sample,np.newaxis])), axis=0))[0,1]\n",
    "\n",
    "qnTr = QuantileTransformer()\n",
    "quantile_dist_corr = np.corrcoef(np.stack((pdist(example[sample,np.newaxis]),\n",
    "             pdist(qnTr.fit_transform(example[sample,np.newaxis]))), axis=0))[0,1]\n",
    "\n",
    "\n",
    "print(\"derivative rank (d=1) scaled\", drscaled_1_dist_corr)\n",
    "print(\"derivative rank (d=10) scaled\", drscaled_10_dist_corr)\n",
    "print(\"log-scaler\", logscaled_dist_corr)\n",
    "print(\"QuantileTransformer\", quantile_dist_corr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}