{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# **!!!_12_Training errors for different N on the same scale along the vertical axis**"
      ],
      "metadata": {
        "id": "Mhxm27_xOiZJ"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "YtVY7dH1OhDs",
        "outputId": "362d8dd1-3a41-430a-8f13-cad93a4155dd"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Collecting pyod\n",
            "  Downloading pyod-2.1.0-py3-none-any.whl.metadata (41 kB)\n",
            "\u001b[?25l     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/41.7 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m41.7/41.7 kB\u001b[0m \u001b[31m2.0 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25hRequirement already satisfied: joblib in /usr/local/lib/python3.12/dist-packages (from pyod) (1.5.3)\n",
            "Requirement already satisfied: matplotlib in /usr/local/lib/python3.12/dist-packages (from pyod) (3.10.0)\n",
            "Requirement already satisfied: numpy>=1.19 in /usr/local/lib/python3.12/dist-packages (from pyod) (2.0.2)\n",
            "Requirement already satisfied: numba>=0.51 in /usr/local/lib/python3.12/dist-packages (from pyod) (0.60.0)\n",
            "Requirement already satisfied: scipy>=1.5.1 in /usr/local/lib/python3.12/dist-packages (from pyod) (1.16.3)\n",
            "Requirement already satisfied: scikit-learn>=0.22.0 in /usr/local/lib/python3.12/dist-packages (from pyod) (1.6.1)\n",
            "Requirement already satisfied: llvmlite<0.44,>=0.43.0dev0 in /usr/local/lib/python3.12/dist-packages (from numba>=0.51->pyod) (0.43.0)\n",
            "Requirement already satisfied: threadpoolctl>=3.1.0 in /usr/local/lib/python3.12/dist-packages (from scikit-learn>=0.22.0->pyod) (3.6.0)\n",
            "Requirement already satisfied: contourpy>=1.0.1 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (1.3.3)\n",
            "Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (0.12.1)\n",
            "Requirement already satisfied: fonttools>=4.22.0 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (4.62.1)\n",
            "Requirement already satisfied: kiwisolver>=1.3.1 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (1.5.0)\n",
            "Requirement already satisfied: packaging>=20.0 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (26.0)\n",
            "Requirement already satisfied: pillow>=8 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (11.3.0)\n",
            "Requirement already satisfied: pyparsing>=2.3.1 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (3.3.2)\n",
            "Requirement already satisfied: python-dateutil>=2.7 in /usr/local/lib/python3.12/dist-packages (from matplotlib->pyod) (2.9.0.post0)\n",
            "Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.12/dist-packages (from python-dateutil>=2.7->matplotlib->pyod) (1.17.0)\n",
            "Downloading pyod-2.1.0-py3-none-any.whl (238 kB)\n",
            "\u001b[2K   \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m238.4/238.4 kB\u001b[0m \u001b[31m7.5 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25hInstalling collected packages: pyod\n",
            "Successfully installed pyod-2.1.0\n",
            "================================================================================\n",
            "FULL DATASET (441 points, one outlier)\n",
            "================================================================================\n",
            "Removed normal points: 396\n",
            "Remaining points: 45 (outliers: 1)\n",
            "Outlier percentage: 2.22%\n",
            "The true outlier is at index 24 (zero-based index 23)\n",
            "\n",
            "Q = 45, N_x = 2, N_y = 1\n",
            "N_min = 2.3106, N_max = 31.6667\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 2000x2800 with 28 Axes>"
            ],
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          },
          "metadata": {}
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\n",
            "\n",
            "\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1400x600 with 1 Axes>"
            ],
            "image/png": 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          },
          "metadata": {}
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\n",
            "Average error for the true outlier (sample 24): 0.043461\n",
            "Maximum average error: 0.043461 at sample 24\n"
          ]
        }
      ],
      "source": [
        "import sys\n",
        "!{sys.executable} -m pip install --upgrade pyod\n",
        "\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from sklearn.preprocessing import StandardScaler\n",
        "import torch\n",
        "import torch.nn as nn\n",
        "import torch.optim as optim\n",
        "import math\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 1. Generation of the full elliptic paraboloid dataset with an outlier\n",
        "#    Features: x1, x2 ; target variable: y = x1^2 - x2^2\n",
        "# ------------------------------------------------------------\n",
        "x1 = np.arange(-1.0, 1.05, 0.1)\n",
        "x2 = np.arange(-1.0, 1.05, 0.1)\n",
        "X1grid, X2grid = np.meshgrid(x1, x2)\n",
        "Ygrid = X1grid**2 - X2grid**2\n",
        "\n",
        "# Inject outlier at point (x1=0, x2=0) with value y = 0.45\n",
        "mask_out = (np.abs(X1grid) < 1e-10) & (np.abs(X2grid) < 1e-10)\n",
        "Ygrid[mask_out] = 0.45\n",
        "\n",
        "points_full = np.column_stack((X1grid.ravel(), X2grid.ravel(), Ygrid.ravel()))\n",
        "labels_full = np.zeros(len(points_full), dtype=int)\n",
        "labels_full[np.where(mask_out.ravel())[0]] = 1\n",
        "\n",
        "print(\"=\"*80)\n",
        "print(\"FULL DATASET (441 points, one outlier)\")\n",
        "print(\"=\"*80)\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 2. Random removal of 90% of normal points\n",
        "# ------------------------------------------------------------\n",
        "np.random.seed(42)\n",
        "normal_indices = np.where(labels_full == 0)[0]\n",
        "n_remove = int(0.90 * len(normal_indices))\n",
        "remove_idx = np.random.choice(normal_indices, size=n_remove, replace=False)\n",
        "keep_mask = np.ones(len(points_full), dtype=bool)\n",
        "keep_mask[remove_idx] = False\n",
        "points = points_full[keep_mask]\n",
        "labels_true = labels_full[keep_mask]\n",
        "\n",
        "print(f\"Removed normal points: {n_remove}\")\n",
        "print(f\"Remaining points: {len(points)} (outliers: {labels_true.sum()})\")\n",
        "print(f\"Outlier percentage: {labels_true.sum()/len(points)*100:.2f}%\")\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 3. Data preparation for NNF: inputs (x1,x2) -> output (y)\n",
        "# ------------------------------------------------------------\n",
        "X_nnf = points[:, :2]   # x1, x2\n",
        "y_nnf = points[:, 2]    # y\n",
        "\n",
        "Q = X_nnf.shape[0]\n",
        "N_x = X_nnf.shape[1]   # 2\n",
        "N_y = 1\n",
        "\n",
        "# Scaling to [-1, 1]\n",
        "def scale_to_minus1_1(data):\n",
        "    min_val = data.min(axis=0)\n",
        "    max_val = data.max(axis=0)\n",
        "    range_val = max_val - min_val\n",
        "    range_val[range_val == 0] = 1.0\n",
        "    scaled = 2.0 * (data - min_val) / range_val - 1.0\n",
        "    return scaled, min_val, max_val\n",
        "\n",
        "X_scaled_nnf, x_min, x_max = scale_to_minus1_1(X_nnf)\n",
        "y_scaled_nnf, y_min, y_max = scale_to_minus1_1(y_nnf.reshape(-1, 1))\n",
        "\n",
        "X_tensor = torch.tensor(X_scaled_nnf, dtype=torch.float32)\n",
        "y_tensor = torch.tensor(y_scaled_nnf, dtype=torch.float32)\n",
        "\n",
        "# Index of the true outlier (zero-based)\n",
        "true_outlier_idx = np.where(labels_true == 1)[0][0]\n",
        "print(f\"The true outlier is at index {true_outlier_idx+1} (zero-based index {true_outlier_idx})\")\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 4. Calculation of hidden layer neuron count bounds (empirical formulas)\n",
        "# ------------------------------------------------------------\n",
        "log2q = math.log2(Q)\n",
        "N_min = (N_y * Q) / ((1 + log2q) * (N_x + N_y))\n",
        "N_max = (N_y / (N_x + N_y)) * ((Q / N_x + 1) * (N_x + N_y + 1) + 1)\n",
        "print(f\"\\nQ = {Q}, N_x = {N_x}, N_y = {N_y}\")\n",
        "print(f\"N_min = {N_min:.4f}, N_max = {N_max:.4f}\")\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 5. Training for N from 3 to 30 and building all histograms\n",
        "# ------------------------------------------------------------\n",
        "N_range = range(3, 31)          # 3..30 inclusive\n",
        "n_N = len(N_range)\n",
        "sum_errors = np.zeros(Q)        # for accumulating error sums\n",
        "\n",
        "# First, compute all errors to find global max for consistent scaling\n",
        "all_errors = []\n",
        "for N in N_range:\n",
        "    torch.manual_seed(42)\n",
        "    model = nn.Sequential(\n",
        "        nn.Linear(N_x, N),\n",
        "        nn.Tanh(),\n",
        "        nn.Linear(N, N_y)\n",
        "    )\n",
        "    criterion = nn.MSELoss()\n",
        "    optimizer = optim.Rprop(model.parameters(), lr=0.01)\n",
        "    model.train()\n",
        "    for epoch in range(1000):\n",
        "        optimizer.zero_grad()\n",
        "        outputs = model(X_tensor)\n",
        "        loss = criterion(outputs, y_tensor)\n",
        "        loss.backward()\n",
        "        optimizer.step()\n",
        "    model.eval()\n",
        "    with torch.no_grad():\n",
        "        predictions = model(X_tensor).numpy().flatten()\n",
        "        errors = (predictions - y_scaled_nnf.flatten()) ** 2\n",
        "    all_errors.append(errors)\n",
        "    sum_errors += errors\n",
        "\n",
        "# Find global maximum across all N for consistent y-axis scaling\n",
        "global_max = np.max(all_errors)\n",
        "\n",
        "# Create a grid of subplots 7x4\n",
        "n_rows, n_cols = 7, 4\n",
        "fig_all, axes_all = plt.subplots(n_rows, n_cols, figsize=(20, 28))\n",
        "\n",
        "for idx, N in enumerate(N_range):\n",
        "    row = idx // n_cols\n",
        "    col = idx % n_cols\n",
        "    ax = axes_all[row, col]\n",
        "\n",
        "    errors = all_errors[idx]\n",
        "    examples = np.arange(1, Q+1)\n",
        "    bars = ax.bar(examples, errors, color='skyblue', edgecolor='black')\n",
        "    bars[true_outlier_idx].set_color('red')\n",
        "    ax.set_title(f'N = {N}')\n",
        "    ax.set_xlabel('Sample number')\n",
        "    ax.set_ylabel('MSE')\n",
        "    ax.set_xticks(examples[::5])\n",
        "    ax.set_ylim(0, global_max * 1.1)  # consistent vertical scale\n",
        "    ax.grid(axis='y', linestyle=':', alpha=0.7)\n",
        "\n",
        "# Hide extra subplots (if any)\n",
        "total_plots = n_rows * n_cols\n",
        "if n_N < total_plots:\n",
        "    for i in range(n_N, total_plots):\n",
        "        axes_all.flat[i].set_visible(False)\n",
        "\n",
        "plt.tight_layout()\n",
        "plt.show()\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 6. Calculation of average errors for each sample over all N\n",
        "# ------------------------------------------------------------\n",
        "mean_errors = sum_errors / n_N\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 7. Histogram of average errors\n",
        "# ------------------------------------------------------------\n",
        "print(\"\\n\\n\")\n",
        "plt.figure(figsize=(14, 6))\n",
        "examples = np.arange(1, Q+1)\n",
        "colors = ['red' if i == true_outlier_idx else 'skyblue' for i in range(Q)]\n",
        "bars = plt.bar(examples, mean_errors, color=colors, edgecolor='black')\n",
        "plt.xlabel('Sample number')\n",
        "plt.ylabel('Average MSE (over N = 3..30)')\n",
        "plt.title('Average MSE errors for each sample (averaged over N = 3..30)')\n",
        "plt.xticks(examples[::5])\n",
        "plt.grid(axis='y', linestyle=':', alpha=0.7)\n",
        "plt.tight_layout()\n",
        "plt.show()\n",
        "\n",
        "# ------------------------------------------------------------\n",
        "# 8. Additional information\n",
        "# ------------------------------------------------------------\n",
        "print(f\"\\nAverage error for the true outlier (sample {true_outlier_idx+1}): {mean_errors[true_outlier_idx]:.6f}\")\n",
        "print(f\"Maximum average error: {np.max(mean_errors):.6f} at sample {np.argmax(mean_errors)+1}\")"
      ]
    }
  ]
}