Estimate the Shift Between two Reference Signals

estimate_ts_shift.ipynb

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   "source": [
    "import polars as pl\n",
    "import polars_ds as pds\n",
    "import numpy as np\n",
    "import altair as alt\n",
    "\n",
    "alt.renderers.enable(\"svg\")\n",
    "\n",
    "\n",
    "def estimate_offset(ref1: pl.Expr | str, ref2: pl.Expr | str):\n",
    "    \"\"\"Estimate offset between two reference signals.\n",
    "    \"\"\"\n",
    "    if isinstance(ref1, str):\n",
    "        ref1 = pl.col(ref1)\n",
    "    \n",
    "    if isinstance(ref2, str):\n",
    "        ref2 = pl.col(ref2)\n",
    "    \n",
    "    return pds.convolve(\n",
    "        ref1 - ref1.mean(),\n",
    "        (ref2 - ref2.mean()).reverse(),\n",
    "        mode=\"full\",\n",
    "        method=\"fft\"\n",
    "    ).arg_max().cast(pl.Int64()) - pl.len() + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "alt.Chart(...)"
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display=\"none\"/></g></g><g class=\"mark-group role-axis\" role=\"graphics-symbol\" aria-roledescription=\"axis\" aria-label=\"Y-axis titled 'value' for a linear scale with values from −3 to 3\"><g transform=\"translate(0.5,0.5)\"><path class=\"background\" aria-hidden=\"true\" d=\"M0,0h0v0h0Z\" pointer-events=\"none\"/><g><g class=\"mark-rule role-axis-tick\" pointer-events=\"none\"><line transform=\"translate(0,300)\" x2=\"-5\" y2=\"0\" stroke=\"#888\" stroke-width=\"1\" opacity=\"1\"/><line transform=\"translate(0,270)\" x2=\"-5\" y2=\"0\" stroke=\"#888\" stroke-width=\"1\" opacity=\"1\"/><line transform=\"translate(0,240)\" x2=\"-5\" y2=\"0\" stroke=\"#888\" stroke-width=\"1\" opacity=\"1\"/><line transform=\"translate(0,210)\" x2=\"-5\" y2=\"0\" stroke=\"#888\" stroke-width=\"1\" opacity=\"1\"/><line transform=\"translate(0,180)\" x2=\"-5\" y2=\"0\" stroke=\"#888\" stroke-width=\"1\" opacity=\"1\"/><line transform=\"translate(0,150)\" x2=\"-5\" y2=\"0\" stroke=\"#888\" 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font-size=\"10px\" fill=\"#000\" opacity=\"1\">−1.5</text><text text-anchor=\"end\" transform=\"translate(-7,213)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">−1.0</text><text text-anchor=\"end\" transform=\"translate(-7,183)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">−0.5</text><text text-anchor=\"end\" transform=\"translate(-7,153)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">0.0</text><text text-anchor=\"end\" transform=\"translate(-7,123)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">0.5</text><text text-anchor=\"end\" transform=\"translate(-7,93.00000000000001)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">1.0</text><text text-anchor=\"end\" transform=\"translate(-7,62.999999999999986)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">1.5</text><text text-anchor=\"end\" 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stroke=\"#f58518\" stroke-width=\"2\"/></g></g><path class=\"foreground\" aria-hidden=\"true\" d=\"\" display=\"none\"/></g></g><g class=\"mark-group role-legend\" role=\"graphics-symbol\" aria-roledescription=\"legend\" aria-label=\"Symbol legend titled 'variable' for stroke color with 2 values: reference_1, reference_2\"><g transform=\"translate(618,0)\"><path class=\"background\" aria-hidden=\"true\" d=\"M0,0h70v40h-70Z\" pointer-events=\"none\"/><g><g class=\"mark-group role-legend-entry\"><g transform=\"translate(0,16)\"><path class=\"background\" aria-hidden=\"true\" d=\"M0,0h0v0h0Z\" pointer-events=\"none\"/><g><g class=\"mark-group role-scope\" role=\"graphics-object\" aria-roledescription=\"group mark container\"><g transform=\"translate(0,0)\"><path class=\"background\" aria-hidden=\"true\" d=\"M0,0h69.369140625v11h-69.369140625Z\" pointer-events=\"none\" opacity=\"1\"/><g><g class=\"mark-symbol role-legend-symbol\" pointer-events=\"none\"><path transform=\"translate(6,6)\" d=\"M-5,0L5,0\" stroke=\"#4c78a8\" stroke-width=\"1.5\" opacity=\"1\"/></g><g class=\"mark-text role-legend-label\" pointer-events=\"none\"><text text-anchor=\"start\" transform=\"translate(16,9)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">reference_1</text></g></g><path class=\"foreground\" aria-hidden=\"true\" d=\"\" pointer-events=\"none\" display=\"none\"/></g><g transform=\"translate(0,13)\"><path class=\"background\" aria-hidden=\"true\" d=\"M0,0h69.369140625v11h-69.369140625Z\" pointer-events=\"none\" opacity=\"1\"/><g><g class=\"mark-symbol role-legend-symbol\" pointer-events=\"none\"><path transform=\"translate(6,6)\" d=\"M-5,0L5,0\" stroke=\"#f58518\" stroke-width=\"1.5\" opacity=\"1\"/></g><g class=\"mark-text role-legend-label\" pointer-events=\"none\"><text text-anchor=\"start\" transform=\"translate(16,9)\" font-family=\"sans-serif\" font-size=\"10px\" fill=\"#000\" opacity=\"1\">reference_2</text></g></g><path class=\"foreground\" aria-hidden=\"true\" d=\"\" pointer-events=\"none\" display=\"none\"/></g></g></g><path class=\"foreground\" aria-hidden=\"true\" d=\"\" pointer-events=\"none\" display=\"none\"/></g></g><g class=\"mark-text role-legend-title\" pointer-events=\"none\"><text text-anchor=\"start\" transform=\"translate(0,9)\" font-family=\"sans-serif\" font-size=\"11px\" font-weight=\"bold\" fill=\"#000\" opacity=\"1\">variable</text></g></g><path class=\"foreground\" aria-hidden=\"true\" d=\"\" pointer-events=\"none\" display=\"none\"/></g></g></g><path class=\"foreground\" aria-hidden=\"true\" d=\"\" display=\"none\"/></g></g></g><defs><clipPath id=\"clip15\"><rect x=\"0\" y=\"0\" width=\"600\" height=\"300\"/></clipPath><clipPath id=\"clip16\"><rect x=\"0\" y=\"0\" width=\"600\" height=\"300\"/></clipPath></defs></svg>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 1000\n",
    "offset = 50\n",
    "\n",
    "base = pl.Series(\n",
    "    \"base\",\n",
    "    np.sin(np.linspace(0, 10*np.pi, N)) * -(np.linspace(-1,1, N)**2 - 2)\n",
    ")\n",
    "data = pl.DataFrame(\n",
    "    [\n",
    "        base[50:].alias(\"reference_1\") + 0.2 * np.random.randn(N - offset), \n",
    "        base[:-50].alias(\"reference_2\") + 0.2 * np.random.randn(N - offset)\n",
    "    ]\n",
    ").with_columns(pl.int_range(pl.len()).alias(\"index\"))\n",
    "\n",
    "\n",
    "display(\n",
    "    data\n",
    "    .unpivot(index=\"index\")\n",
    "    .plot\n",
    "    .line(x=\"index\", y=\"value\", color=\"variable\")\n",
    "    .properties(width=600)\n",
    ")\n",
    "\n",
    "synchronized = data.select(\n",
    "    \"index\",\n",
    "    \"reference_1\",\n",
    "    pl.col(\"reference_2\").shift(\n",
    "        estimate_offset(\"reference_1\", \"reference_2\")\n",
    "    )\n",
    ")\n",
    "\n",
    "display(\n",
    "    synchronized\n",
    "    .unpivot(index=\"index\")\n",
    "    .plot\n",
    "    .line(x=\"index\", y=\"value\", color=\"variable\")\n",
    "    .properties(width=600)\n",
    ")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "test-ocx4maxE-py3.13",
   "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": "3.13.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}

Guess it's on wikipedia: https://en.wikipedia.org/wiki/Cross-correlation#Time_delay_analysis