{ "cells": [ { "cell_type": "markdown", "id": "84f1d2f2", "metadata": {}, "source": [ "# DCT Laboratory — Volume I, Chapter 4\n", "## Enterprise Modeling Principles\n", "**Seed `26104`** · Companion to the chapter and AXIOM Module **AXIOM-04**\n", "\n", "Two of the chapter's principles, made numerical: the **representation bound** — a\n", "richer model is not a better model once you leave the calibration window — and the\n", "**Hierarchical Consistency Theorem** — coarse dynamics are legitimate exactly when\n", "the fine rates agree. Mirrored in `DCT_V1_Ch04_Lab.xlsx`." ] }, { "cell_type": "code", "execution_count": null, "id": "0f412239", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T22:08:45.859678Z", "iopub.status.busy": "2026-07-13T22:08:45.859513Z", "iopub.status.idle": "2026-07-13T22:08:46.850020Z", "shell.execute_reply": "2026-07-13T22:08:46.848754Z" } }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "plt.rcParams['figure.dpi']=110\n", "\n", "import numpy as np\n", "SEED = 26104\n", "T = np.arange(21.0) # quarters 0..20\n", "TRUE = 50 + 30/(1 + np.exp(-0.5*(T - 8)))\n", "rng = np.random.default_rng(SEED)\n", "NOISE = np.round(rng.normal(0, 1.2, 21), 4) # frozen measurement noise (shipped as constants)\n", "OBS = TRUE + NOISE\n", "CAL, HOLD = slice(0, 13), slice(13, 21)\n", "\n", "def fit_poly(deg):\n", " c = np.polyfit(T[CAL], OBS[CAL], deg)\n", " return np.round(c, 6)\n", "\n", "C_LIN = fit_poly(1) # the simple model\n", "C_RICH = fit_poly(6) # the over-rich model\n", "\n", "def rmse(coeffs, window):\n", " pred = np.polyval(coeffs, T[window])\n", " return float(np.sqrt(np.mean((OBS[window] - pred)**2)))\n", "\n", "# Hierarchical consistency: three subsystems x_i' = a_i x_i, coarse X = sum(x)\n", "X0 = np.array([40.0, 35.0, 25.0])\n", "def agg_paths(a, n=20, dt=0.25):\n", " xs = np.empty((n+1, 3)); xs[0] = X0\n", " for k in range(n): xs[k+1] = xs[k] + dt*a*xs[k]\n", " fine = xs.sum(axis=1)\n", " a_bar = float((a*X0).sum()/X0.sum()) # naive coarse rate\n", " coarse = np.empty(n+1); coarse[0] = X0.sum()\n", " for k in range(n): coarse[k+1] = coarse[k] + dt*a_bar*coarse[k]\n", " return fine, coarse\n", "\n", "A_EQ = np.array([0.06, 0.06, 0.06]) # fiber-invariant: coarse law exact\n", "A_NEQ = np.array([0.12, 0.02, -0.04]) # not: coarse law drifts\n", "\n", "def reference_values():\n", " fe, ce = agg_paths(A_EQ); fn, cn = agg_paths(A_NEQ)\n", " return {\n", " \"rmse_simple_calib\": round(rmse(C_LIN, CAL), 4),\n", " \"rmse_rich_calib\": round(rmse(C_RICH, CAL), 4),\n", " \"rmse_simple_holdout\":round(rmse(C_LIN, HOLD), 4),\n", " \"rmse_rich_holdout\": round(rmse(C_RICH, HOLD), 4),\n", " \"agg_gap_equal_T\": round(abs(fe[-1]-ce[-1]), 4),\n", " \"agg_gap_unequal_T\": round(abs(fn[-1]-cn[-1]), 4),\n", " \"obs_calib_mean\": round(float(OBS[CAL].mean()), 4),\n", " }\n", "if __name__ == \"__main__\":\n", " [print(f\"{k:22s} {v}\") for k,v in reference_values().items()]" ] }, { "cell_type": "markdown", "id": "5acd108f", "metadata": {}, "source": [ "## Panel 1 — The validation lesson\n", "A logistic adoption curve observed with measurement noise. Two models fit on\n", "quarters 0–12: **simple** (linear) and **rich** (degree-6 polynomial). In-sample the\n", "rich model wins; on the holdout (quarters 13–20) it detonates. Validation on a\n", "declared record (Model Validation Theorem) is what catches this — in-sample fit\n", "never can." ] }, { "cell_type": "code", "execution_count": null, "id": "45ca30af", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T22:08:46.852070Z", "iopub.status.busy": "2026-07-13T22:08:46.851805Z", "iopub.status.idle": "2026-07-13T22:08:47.156938Z", "shell.execute_reply": "2026-07-13T22:08:47.155333Z" } }, "outputs": [], "source": [ "fig, ax = plt.subplots(figsize=(8.4,4.4))\n", "tt = np.linspace(0,20,300)\n", "ax.scatter(T[CAL], OBS[CAL], c=\"#0B3D2E\", s=28, label=\"calibration obs\")\n", "ax.scatter(T[HOLD], OBS[HOLD], facecolors=\"none\", edgecolors=\"#0B3D2E\", s=34, label=\"holdout obs\")\n", "ax.plot(tt, np.polyval(C_LIN, tt), c=\"#C8A24B\", lw=2.2, label=\"simple (linear)\")\n", "ax.plot(tt, np.polyval(C_RICH, tt), c=\"#1B6B52\", lw=2.2, ls=\"--\", label=\"rich (degree 6)\")\n", "ax.set(xlabel=\"quarter\", ylabel=\"enterprise output\", ylim=(30,110),\n", " title=\"The representation bound: richer ≠ better out of sample (seed 26104)\")\n", "ax.axvspan(12.5, 20.5, color=\"#C8A24B\", alpha=.08)\n", "ax.legend(frameon=False); ax.grid(alpha=.25); plt.tight_layout(); plt.show()\n", "for nm,c in [(\"simple\",C_LIN),(\"rich\",C_RICH)]:\n", " print(f\"{nm:7s} RMSE calib {rmse(c,CAL):8.4f} holdout {rmse(c,HOLD):10.4f}\")" ] }, { "cell_type": "markdown", "id": "e40c1fb6", "metadata": {}, "source": [ "## Panel 2 — Hierarchical consistency\n", "Three subsystems growing at rates $a_i$; the coarse state is their sum with a naive\n", "averaged rate. **Equal rates** (fiber-invariant): the coarse law is exact, gap = 0.\n", "**Unequal rates**: the coarse law drifts as composition shifts — the aggregate is\n", "not a state (Hierarchical Consistency Theorem)." ] }, { "cell_type": "code", "execution_count": null, "id": "7bb8006b", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T22:08:47.159532Z", "iopub.status.busy": "2026-07-13T22:08:47.158777Z", "iopub.status.idle": "2026-07-13T22:08:47.496221Z", "shell.execute_reply": "2026-07-13T22:08:47.495170Z" } }, "outputs": [], "source": [ "t = np.arange(21)*0.25\n", "fig, axes = plt.subplots(1,2, figsize=(9.6,3.9), sharey=False)\n", "for ax,(a,ttl) in zip(axes,[(A_EQ,\"equal rates — exact\"),(A_NEQ,\"unequal rates — drifting\")]):\n", " fine, coarse = agg_paths(a)\n", " ax.plot(t, fine, c=\"#0B3D2E\", lw=2.2, label=\"sum of fine states\")\n", " ax.plot(t, coarse, c=\"#C8A24B\", lw=2.2, ls=\"--\", label=\"coarse law\")\n", " ax.set(title=ttl, xlabel=\"years\"); ax.legend(frameon=False); ax.grid(alpha=.25)\n", "plt.tight_layout(); plt.show()\n", "fe,ce = agg_paths(A_EQ); fn,cn = agg_paths(A_NEQ)\n", "print(\"terminal gap, equal rates :\", round(abs(fe[-1]-ce[-1]),4))\n", "print(\"terminal gap, unequal rates:\", round(abs(fn[-1]-cn[-1]),4))" ] }, { "cell_type": "markdown", "id": "41716979", "metadata": {}, "source": [ "## Validation — agrees with `DCT_V1_Ch04_Lab.xlsx`" ] }, { "cell_type": "code", "execution_count": null, "id": "f89bba63", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T22:08:47.498082Z", "iopub.status.busy": "2026-07-13T22:08:47.497846Z", "iopub.status.idle": "2026-07-13T22:08:47.504244Z", "shell.execute_reply": "2026-07-13T22:08:47.503301Z" } }, "outputs": [], "source": [ "ref = reference_values()\n", "expected = {\"rmse_simple_calib\":2.2016,\"rmse_rich_calib\":1.2348,\"rmse_simple_holdout\":6.4713,\n", " \"rmse_rich_holdout\":241.9653,\"agg_gap_equal_T\":0.0,\"agg_gap_unequal_T\":6.2884,\"obs_calib_mean\":60.9148}\n", "for k,v in expected.items():\n", " assert abs(ref[k]-v)<5e-4, f\"MISMATCH {k}: {ref[k]} vs {v}\"\n", " print(f\"PASS {k:22s} {ref[k]}\")\n", "print(\"\\nAll checkpoints agree — seed 26104.\")" ] }, { "cell_type": "markdown", "id": "76a1075d", "metadata": {}, "source": [ "**Next**: Exercises 4.9–4.12 (Part C); AXIOM-04's model bench lets you slide the polynomial degree and watch the holdout error turn. Solutions: IM Ch. 4." ] } ], "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": "3.12.3" } }, "nbformat": 4, "nbformat_minor": 5 }