{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c03a7d37",
   "metadata": {},
   "source": [
    "# Mathematische und statistische Methoden\n",
    "\n",
    "Eine Reihe von mathematischen Funktionen, die Statistiken über ein ganzes Array oder über die Daten entlang einer Achse berechnen, sind als Methoden der Array-Klasse zugänglich. So könnt ihr Aggregationen verwenden wie Summe, Mittelwert und Standardabweichung, indem ihr entweder die Array-Instanzmethode aufruft oder die NumPy-Funktion der obersten Ebene verwendet."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4013204e",
   "metadata": {},
   "source": [
    "Im folgenden generiere ich einige normalverteilte Zufallsdaten und berechne einige aggregierte Statistiken:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b413c7b0",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.653175Z",
     "iopub.status.busy": "2026-05-21T16:57:07.652785Z",
     "iopub.status.idle": "2026-05-21T16:57:07.698255Z",
     "shell.execute_reply": "2026-05-21T16:57:07.697969Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.653153Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.79440596, 0.10972775, 0.35435383],\n",
       "       [0.74358108, 0.43438519, 0.83743312],\n",
       "       [0.94298041, 0.39124331, 0.08939286],\n",
       "       [0.18968082, 0.60070664, 0.5113487 ],\n",
       "       [0.76300762, 0.88587544, 0.6838369 ],\n",
       "       [0.714206  , 0.17504049, 0.58097537],\n",
       "       [0.22871752, 0.95094866, 0.99379828]])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "rng = np.random.default_rng()\n",
    "data = rng.random((7, 3))\n",
    "\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "456e9376",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.698752Z",
     "iopub.status.busy": "2026-05-21T16:57:07.698616Z",
     "iopub.status.idle": "2026-05-21T16:57:07.701331Z",
     "shell.execute_reply": "2026-05-21T16:57:07.701091Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.698742Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.5702688544567261)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c8dc9d4e",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.701873Z",
     "iopub.status.busy": "2026-05-21T16:57:07.701764Z",
     "iopub.status.idle": "2026-05-21T16:57:07.703966Z",
     "shell.execute_reply": "2026-05-21T16:57:07.703688Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.701862Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.5702688544567261)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "5092c04d",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.704489Z",
     "iopub.status.busy": "2026-05-21T16:57:07.704386Z",
     "iopub.status.idle": "2026-05-21T16:57:07.707083Z",
     "shell.execute_reply": "2026-05-21T16:57:07.706829Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.704481Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(11.97564594359125)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b67e9edd",
   "metadata": {},
   "source": [
    "Funktionen wie `mean` und `sum` benötigen ein optionales Achsenargument, das die Statistik über die angegebene Achse berechnet, was zu einem Array mit einer Dimension weniger führt:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "d84656ba",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.707743Z",
     "iopub.status.busy": "2026-05-21T16:57:07.707633Z",
     "iopub.status.idle": "2026-05-21T16:57:07.710074Z",
     "shell.execute_reply": "2026-05-21T16:57:07.709787Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.707735Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.62522563, 0.50684678, 0.57873415])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.mean(axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b7657887",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.711579Z",
     "iopub.status.busy": "2026-05-21T16:57:07.711477Z",
     "iopub.status.idle": "2026-05-21T16:57:07.713919Z",
     "shell.execute_reply": "2026-05-21T16:57:07.713703Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.711572Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.3765794 , 3.54792749, 4.05113905])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.sum(axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca7e65c0",
   "metadata": {},
   "source": [
    "Mit `data.mean(0)`, was dasselbe ist wie `data.mean(axis=0)`, wird der Mittelwert über die Zeilen berechnet, während `data.sum(0)` die Summe über die Zeilen berechnet."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84530009",
   "metadata": {},
   "source": [
    "Andere Methoden wie `cumsum` und `cumprod` aggregieren hingegen nicht, sondern erzeugen ein neues Array mit den Zwischenergebnissen.\n",
    "\n",
    "In mehrdimensionalen Arrays geben Akkumulationsfunktionen wie `cumsum` und `cumprodm` ein Array derselben Größe zurück, aber mit den entlang der angegebenen Achse berechneten Teilaggregaten:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "d50cfc21",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.714237Z",
     "iopub.status.busy": "2026-05-21T16:57:07.714182Z",
     "iopub.status.idle": "2026-05-21T16:57:07.716485Z",
     "shell.execute_reply": "2026-05-21T16:57:07.716267Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.714231Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.79440596,  0.90413371,  1.25848754,  2.00206862,  2.43645381,\n",
       "        3.27388693,  4.21686734,  4.60811065,  4.69750351,  4.88718433,\n",
       "        5.48789098,  5.99923968,  6.76224729,  7.64812274,  8.33195963,\n",
       "        9.04616563,  9.22120611,  9.80218148, 10.030899  , 10.98184767,\n",
       "       11.97564594])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.cumsum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a6524b2b",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:57:07.716804Z",
     "iopub.status.busy": "2026-05-21T16:57:07.716734Z",
     "iopub.status.idle": "2026-05-21T16:57:07.718850Z",
     "shell.execute_reply": "2026-05-21T16:57:07.718631Z",
     "shell.execute_reply.started": "2026-05-21T16:57:07.716796Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.94405957e-01, 8.71683779e-02, 3.08884486e-02, 2.29680659e-02,\n",
       "       9.97698767e-03, 8.35505991e-03, 7.87865783e-03, 3.08247220e-03,\n",
       "       2.75551012e-04, 5.22667412e-05, 3.13969787e-05, 1.60548043e-05,\n",
       "       1.22499380e-05, 1.08519192e-05, 7.42094275e-06, 5.30008181e-06,\n",
       "       9.27728893e-07, 5.38987633e-07, 1.23275916e-07, 1.17229068e-07,\n",
       "       1.16502046e-07])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.cumprod()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b73111f2",
   "metadata": {},
   "source": [
    "Grundlegende statistische Methoden für Arrays:\n",
    "\n",
    "Methode | Beschreibung\n",
    ":------ | :-----------\n",
    "`sum` | Summe aller Elemente im Array oder entlang einer Achse\n",
    "`mean` | Arithmetisches Mittel; bei Arrays mit der Länge Null wird`NaN` zurückgegeben \n",
    "`std`, `var` | Standardabweichung bzw. Varianz\n",
    "`min`, `max` | Minimum und Maximum\n",
    "`argmin`, `argmax` | Indizes der minimalen bzw. maximalen Elemente\n",
    "`cumsum` | Kumulative Summe der Elemente, beginnend mit `0`\n",
    "`cumprod` | Kumulatives Produkt der Elemente, beginnend mit `1`"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.13 Kernel",
   "language": "python",
   "name": "python313"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.13.0"
  },
  "widgets": {
   "application/vnd.jupyter.widget-state+json": {
    "state": {},
    "version_major": 2,
    "version_minor": 0
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}