assignment3.ipynb

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   "source": [
    "# Assignment: SVD Preprocessing on MNIST with Logistic Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "deffa444",
   "metadata": {},
   "source": [
    "## Instructions:\n",
    "In this assignment, you will apply **Singular Value Decomposition (SVD)** as a preprocessing step to the **MNIST dataset** and train a **logistic regression classifier**. You will compare the model performance and training time when using different levels of SVD for dimensionality reduction.\n",
    "\n",
    "In this assignment, you will need to:\n",
    "1. Load the MNIST dataset and normalize it.\n",
    "2. Perform SVD and reduce the dimensions of the data.\n",
    "3. Train a logistic regression model on the original and SVD-reduced data.\n",
    "4. Measure and compare the training time and accuracy of the model with varying SVD components.\n",
    "5. Plot the results and analyze how SVD impacts the performance and efficiency of the model.\n",
    "\n",
    "***\n",
    "Your tasks include:\n",
    "1. Implement SVD algorithm. You are not allowed to directly use SVD implemented by other packages, but you may use functions in NumPy. (Part 2)\n",
    "2. Explore the accuracy and time performance from different numbers of SVD components. (Part 4)\n",
    "3. Visualize the accuracy, time performance and top 5 singular vectors in the dataset, analyze and explain which number of SVD component looks best to you? (Part 4,5&6) Hint: singular vectors should be reshaped to 28x28 images for visualization.\n",
    "***\n",
    "**Note that you may not import any other function or package.** Let's get started!\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3e10cb3",
   "metadata": {},
   "source": [
    "## Part 1: Load the MNIST dataset and preprocess the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "79f2d907",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Loading MNIST dataset...\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import time\n",
    "from sklearn.linear_model import LogisticRegression\n",
    "from sklearn.model_selection import train_test_split\n",
    "from sklearn.datasets import fetch_openml\n",
    "from sklearn.metrics import accuracy_score, classification_report\n",
    "\n",
    "# Load MNIST dataset\n",
    "print(\"Loading MNIST dataset...\")\n",
    "mnist = fetch_openml('mnist_784', version=1)\n",
    "X = mnist.data\n",
    "y = mnist.target\n",
    "\n",
    "# Normalize the data\n",
    "X = X / 255.0\n",
    "\n",
    "# Split into training and test sets\n",
    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28b8572d",
   "metadata": {},
   "source": [
    "## Part 2: Implement SVD for Dimensionality Reduction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "647564f5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(56000, 784)\n",
      "(14000, 784)\n"
     ]
    }
   ],
   "source": [
    "print(X_train.shape)\n",
    "print(X_test.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b3bd59f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "def apply_svd_custom_optimized(X_train, X_test, n_components):\n",
    "    # Center the data by subtracting the mean\n",
    "    X_mean = np.mean(X_train, axis=0)\n",
    "    X_train_centered = X_train - X_mean\n",
    "    X_test_centered = X_test - X_mean\n",
    "    \n",
    "    # Compute the covariance matrix (X_train_centered.T @ X_train_centered) / n_samples\n",
    "    covariance_matrix = np.dot(X_train_centered.T, X_train_centered) / X_train_centered.shape[0]\n",
    "    \n",
    "    eigenvalues, eigenvectors = np.linalg.eig(covariance_matrix)\n",
    "    # Take only the real parts of eigenvalues and eigenvectors, otherwise get ComplexWarning on step 4\n",
    "    eigenvalues = np.real(eigenvalues)\n",
    "    eigenvectors = np.real(eigenvectors)\n",
    "    \n",
    "    # Use partial sorting to find the indices of the top n_components eigenvalues\n",
    "    top_indices = np.argpartition(eigenvalues, -n_components)[-n_components:]\n",
    "    top_indices = top_indices[np.argsort(eigenvalues[top_indices])[::-1]]\n",
    "    \n",
    "    # Get the top eigenvectors and corresponding eigenvalues\n",
    "    top_eigenvectors = eigenvectors[:, top_indices]\n",
    "    top_eigenvalues = eigenvalues[top_indices]\n",
    "    \n",
    "    # Project the train and test data onto the new lower-dimensional subspace\n",
    "    X_train_reduced = np.dot(X_train_centered, top_eigenvectors)\n",
    "    X_test_reduced = np.dot(X_test_centered, top_eigenvectors)\n",
    "    \n",
    "    return X_train_reduced, X_test_reduced, top_eigenvalues, top_eigenvectors"
   ]
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   "cell_type": "markdown",
   "id": "eb2a7a9d",
   "metadata": {},
   "source": [
    "## Part 3: Train Logistic Regression and Measure Performance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "dbfc0b10",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to train logistic regression and track training time\n",
    "def train_logistic_regression(X_train, y_train, X_test, y_test):\n",
    "    model = LogisticRegression(max_iter=1000, solver='saga', random_state=42)  # multi_class='multinomial'\n",
    "    \n",
    "    # Measure training time\n",
    "    start_time = time.time()\n",
    "    model.fit(X_train, y_train)\n",
    "    training_time = time.time() - start_time\n",
    "    \n",
    "    y_pred = model.predict(X_test)\n",
    "    accuracy = accuracy_score(y_test, y_pred)\n",
    "    \n",
    "    return accuracy, training_time\n",
    "\n",
    "\n",
    "# accuracy, training_time = train_logistic_regression(X_train, y_train, X_test, y_test)\n",
    "# print(f\"Accuracy: {accuracy:.4f}\")\n",
    "# print(f\"Training Time: {training_time:.2f} seconds\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e557a05",
   "metadata": {},
   "source": [
    "## Part 4: Experiment with Different Levels of SVD\n",
    "\n",
    "Now, apply SVD with varying numbers of components and observe how the dimensionality reduction impacts the model's performance. Record both the accuracy and training time for each number of components.\n"
   ]
  },
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   "cell_type": "code",
   "execution_count": 41,
   "id": "3a460725",
   "metadata": {},
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     "text": [
      "Training models with different levels of SVD preprocessing...\n",
      "Applying custom SVD with 1 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 1, Accuracy: 0.3069, Training time: 0.2951 seconds\n",
      "Applying custom SVD with 2 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 2, Accuracy: 0.4484, Training time: 0.4362 seconds\n",
      "Applying custom SVD with 3 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 3, Accuracy: 0.4718, Training time: 0.3563 seconds\n",
      "Applying custom SVD with 4 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 4, Accuracy: 0.5765, Training time: 0.4110 seconds\n",
      "Applying custom SVD with 5 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 5, Accuracy: 0.6796, Training time: 0.4444 seconds\n",
      "Applying custom SVD with 6 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 6, Accuracy: 0.7363, Training time: 0.4625 seconds\n",
      "Applying custom SVD with 7 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 7, Accuracy: 0.7619, Training time: 0.4541 seconds\n",
      "Applying custom SVD with 8 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 8, Accuracy: 0.7871, Training time: 0.5097 seconds\n",
      "Applying custom SVD with 9 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 9, Accuracy: 0.7893, Training time: 0.6632 seconds\n",
      "Applying custom SVD with 10 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 10, Accuracy: 0.8035, Training time: 0.7382 seconds\n",
      "Applying custom SVD with 12 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 12, Accuracy: 0.8267, Training time: 0.7891 seconds\n",
      "Applying custom SVD with 14 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 14, Accuracy: 0.8398, Training time: 1.3784 seconds\n",
      "Applying custom SVD with 17 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 17, Accuracy: 0.8617, Training time: 1.6096 seconds\n",
      "Applying custom SVD with 19 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 19, Accuracy: 0.8738, Training time: 1.6904 seconds\n",
      "Applying custom SVD with 23 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 23, Accuracy: 0.8857, Training time: 2.4764 seconds\n",
      "Applying custom SVD with 27 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 27, Accuracy: 0.8890, Training time: 3.3313 seconds\n",
      "Applying custom SVD with 32 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 32, Accuracy: 0.8972, Training time: 3.9381 seconds\n",
      "Applying custom SVD with 39 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 39, Accuracy: 0.9013, Training time: 5.4879 seconds\n",
      "Applying custom SVD with 46 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 46, Accuracy: 0.9062, Training time: 6.6538 seconds\n",
      "Applying custom SVD with 55 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 55, Accuracy: 0.9114, Training time: 9.6164 seconds\n",
      "Applying custom SVD with 67 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 67, Accuracy: 0.9129, Training time: 11.0167 seconds\n",
      "Applying custom SVD with 81 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 81, Accuracy: 0.9158, Training time: 18.1972 seconds\n",
      "Applying custom SVD with 98 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 98, Accuracy: 0.9176, Training time: 25.9773 seconds\n",
      "Applying custom SVD with 119 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 119, Accuracy: 0.9187, Training time: 36.9793 seconds\n",
      "Applying custom SVD with 145 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 145, Accuracy: 0.9199, Training time: 41.0914 seconds\n",
      "Applying custom SVD with 177 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 177, Accuracy: 0.9209, Training time: 53.8827 seconds\n",
      "Applying custom SVD with 217 components...\n",
      "Training logistic regresion...\n",
      "SVD components: 217, Accuracy: 0.9212, Training time: 67.8539 seconds\n"
     ]
    }
   ],
   "source": [
    "svd_components = sorted(list({int((1.07 + x/500)**x) for x in range(40)}))\n",
    "\n",
    "# Store the results\n",
    "results = []\n",
    "\n",
    "print(\"Training models with different levels of SVD preprocessing...\")\n",
    "for n_components in svd_components:\n",
    "    print(f\"Applying custom SVD with {n_components} components...\")\n",
    "    \n",
    "    # Apply SVD to the training and test sets using the optimized SVD function\n",
    "    X_train_svd, X_test_svd, singular_values, top_eigenvectors = apply_svd_custom_optimized(X_train, X_test, n_components)\n",
    "\n",
    "    print('Training logistic regresion...')\n",
    "    \n",
    "    # Train the logistic regression model and get accuracy and training time\n",
    "    accuracy, training_time = train_logistic_regression(X_train_svd, y_train, X_test_svd, y_test)\n",
    "    \n",
    "    # Store the result in the list\n",
    "    results.append({\n",
    "        \"n_components\": n_components,\n",
    "        \"accuracy\": accuracy,\n",
    "        \"training_time\": training_time\n",
    "    })\n",
    "    \n",
    "    print(f\"SVD components: {n_components}, Accuracy: {accuracy:.4f}, Training time: {training_time:.4f} seconds\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea7aca12",
   "metadata": {},
   "source": [
    "## Part 5: Visualize and Analyze the Results\n",
    "\n",
    "Finally, plot the accuracy, training time as a function of the number of SVD components, and top 5 singular vectors. This will help you understand the trade-off between dimensionality reduction, accuracy, and model training time, and how SVD generally works. Hint: singular vectors should be reshaped to 28x28 images for visualization.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "a32f9421",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "\n",
    "# Convert results to a DataFrame for better visualization\n",
    "results_df = pd.DataFrame(results)"
   ]
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   "source": [
    "import plotly.graph_objects as go\n",
    "import plotly.express as px\n",
    "\n",
    "def plot_performance(results_df):\n",
    "    \"\"\" Plots the accuracy and training time over number of SVD components with two y-axes\n",
    "    \"\"\"\n",
    "\n",
    "    fig = go.Figure()\n",
    "\n",
    "    # Plot accuracy on the left y-axis\n",
    "    fig.add_trace(go.Scatter(\n",
    "        x=results_df[\"n_components\"], \n",
    "        y=results_df[\"accuracy\"],\n",
    "        mode=\"lines+markers\",\n",
    "        name=\"Accuracy\",\n",
    "        line=dict(color='blue'),\n",
    "        yaxis=\"y1\"  # left side\n",
    "    ))\n",
    "\n",
    "    # Plot training time on the right y-axis\n",
    "    fig.add_trace(go.Scatter(\n",
    "        x=results_df[\"n_components\"], \n",
    "        y=results_df[\"training_time\"],\n",
    "        mode=\"lines+markers\",\n",
    "        name=\"Training Time (seconds)\",\n",
    "        line=dict(color='red', dash='dot'),\n",
    "        yaxis=\"y2\"  # right side\n",
    "    ))\n",
    "\n",
    "    # Update layout for dual y-axis\n",
    "    fig.update_layout(\n",
    "        title=\"Accuracy and Training Time vs Number of SVD Components\",\n",
    "        xaxis_title=\"Number of SVD Components\",        \n",
    "        yaxis=dict(\n",
    "            title=\"Accuracy\",\n",
    "            titlefont=dict(color=\"blue\"),\n",
    "            tickfont=dict(color=\"blue\"),\n",
    "            range=[0, 1]\n",
    "        ),\n",
    "        yaxis2=dict(\n",
    "            title=\"Training Time (seconds)\",  # Right side y-axis\n",
    "            titlefont=dict(color=\"red\"),\n",
    "            tickfont=dict(color=\"red\"),\n",
    "            anchor=\"x\",  # Link this y-axis to the x-axis\n",
    "            overlaying=\"y\",  # Overlay it on top of the left y-axis\n",
    "            side=\"right\",\n",
    "            range=[0, 100],\n",
    "        ),\n",
    "        legend=dict(\n",
    "            orientation=\"h\",\n",
    "            yanchor=\"bottom\",\n",
    "            y=1.02,\n",
    "            xanchor=\"right\",\n",
    "            x=1\n",
    "        )\n",
    "    )\n",
    "\n",
    "    fig.show()\n",
    "\n",
    "plot_performance(results_df)\n",
    "\n",
    "\n",
    "def plot_singular_vectors(singular_vectors):\n",
    "    \"\"\" Visualizes top 5 singular vectors as 28x28 images\n",
    "    \"\"\"\n",
    "    top_5_singular_vectors = singular_vectors[:, :5].T\n",
    "    reshaped_vectors = [vector.reshape(28, 28) for vector in top_5_singular_vectors]\n",
    "    \n",
    "    fig = px.imshow(np.concatenate(reshaped_vectors, axis=1), \n",
    "                    color_continuous_scale='gray', \n",
    "                    title=f\"Top 5 Singular Vectors\")\n",
    "    \n",
    "    fig.show()\n",
    "\n",
    "plot_singular_vectors(top_eigenvectors)"
   ]
  },
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   "id": "4210569a",
   "metadata": {},
   "source": [
    "## Part 6: Analyze / Conclusion \n",
    "\n",
    "YOUR ANSWER: \n",
    "\n",
    "In general, more components led to increased accuracy; however, there were clear diminishing returns, as accuracy plateued at very high number of components. Surprisingly, the model never exhibited overfitting; accuracy monotonically increased all the way up to the top tested value of 217 components even if ever so slightly. Training time was surprisingly noisy, but increased impressively close to linearly when running the below power law fit with a slope of 0.28 seconds per SVD component. Interestingly, using a reasonable ax/(b+x) + c fit suggests accuracy will asymptotically approach almost exactly 0.95 (1.04 - 0.09) for arbitrarily many components, but we achieve 0.92 accuracy in under a minute at 177 components and over 0.82 accuracy in under a second at 12 components, showing the dramatic tradeoff of increasing components and thereby increasing training time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "eec0df7e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "acc ~ 1.0440370364490723*x/(1.791869154266387 + x) + -0.09654091866991857\n",
      "time ~ 0.28232614996355965*x**0.99999999999737 + 9.035274860442381e-15\n"
     ]
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         "text": "Accuracy and Training Time vs Number of SVD Components with Logarithmic and Exponential Fits"
        },
        "xaxis": {
         "title": {
          "text": "Number of SVD Components"
         }
        },
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          0,
          1
         ],
         "tickfont": {
          "color": "blue"
         },
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          "font": {
           "color": "blue"
          },
          "text": "Accuracy"
         }
        },
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          0,
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         ],
         "side": "right",
         "tickfont": {
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         },
         "title": {
          "font": {
           "color": "red"
          },
          "text": "Training Time (seconds)"
         }
        }
       }
      }
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     "metadata": {},
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   ],
   "source": [
    "# best fits for above graph to sate my own curiosity\n",
    "\n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "def acc_fit(x, a, b, c):\n",
    "    return a * x / (b + x) + c\n",
    "\n",
    "def power_law_fit(x, a, b, c):\n",
    "    return a * x ** b + c\n",
    "\n",
    "def fit_curves_advanced(results_df):\n",
    "    n_components = results_df[\"n_components\"]\n",
    "    \n",
    "    # Exclude 0 values to avoid errors for logarithmic fit\n",
    "    n_components_non_zero = n_components[n_components > 0]\n",
    "    accuracy_non_zero = results_df[\"accuracy\"].iloc[n_components_non_zero.index]\n",
    "    time_non_zero = results_df[\"training_time\"].iloc[n_components_non_zero.index]\n",
    "    \n",
    "    acc_params, _ = curve_fit(acc_fit, n_components_non_zero, accuracy_non_zero)\n",
    "    a, b, c = acc_params\n",
    "    print(f'acc ~ {a}*x/({b} + x) + {c}')\n",
    "\n",
    "    initial_guess = [1, 0.01, 1]\n",
    "    exp_params, _ = curve_fit(power_law_fit, n_components_non_zero, time_non_zero, p0=initial_guess, bounds=(0, [1000, 1, 1000]))\n",
    "    a, b, c = exp_params\n",
    "    print(f'time ~ {a}*x**{b} + {c}')\n",
    "\n",
    "    # Generate fitted values\n",
    "    n_components_range = np.linspace(min(n_components), max(n_components), 100)\n",
    "    \n",
    "    fitted_accuracy_log = acc_fit(n_components_range, *acc_params)\n",
    "    fitted_time_exp = power_law_fit(n_components_range, *exp_params)\n",
    "\n",
    "    # Plot the data with fitted curves\n",
    "    fig = go.Figure()\n",
    "\n",
    "    # Plot original accuracy on the left y-axis (solid line)\n",
    "    fig.add_trace(go.Scatter(\n",
    "        x=n_components, \n",
    "        y=results_df[\"accuracy\"],\n",
    "        mode=\"lines+markers\",\n",
    "        name=\"Accuracy\",\n",
    "        line=dict(color='blue', dash='solid'),\n",
    "        yaxis=\"y1\"\n",
    "    ))\n",
    "\n",
    "    # Plot logarithmic fit for accuracy\n",
    "    fig.add_trace(go.Scatter(\n",
    "        x=n_components_range, \n",
    "        y=fitted_accuracy_log,\n",
    "        mode=\"lines\",\n",
    "        name=\"Accuracy 1/1+x Fit\",\n",
    "        line=dict(color='blue', dash='dash'),\n",
    "        yaxis=\"y1\"\n",
    "    ))\n",
    "\n",
    "    # Plot original training time on the right y-axis (dotted line)\n",
    "    fig.add_trace(go.Scatter(\n",
    "        x=n_components, \n",
    "        y=results_df[\"training_time\"],\n",
    "        mode=\"lines+markers\",\n",
    "        name=\"Training Time (seconds)\",\n",
    "        line=dict(color='red', dash='dot'),\n",
    "        yaxis=\"y2\"\n",
    "    ))\n",
    "\n",
    "    # Plot exponential fit for training time\n",
    "    fig.add_trace(go.Scatter(\n",
    "        x=n_components_range, \n",
    "        y=fitted_time_exp,\n",
    "        mode=\"lines\",\n",
    "        name=\"Training Time Exp Fit\",\n",
    "        line=dict(color='green', dash='dot'),\n",
    "        yaxis=\"y2\"\n",
    "    ))\n",
    "\n",
    "    # Update layout for dual y-axis with custom ranges\n",
    "    fig.update_layout(\n",
    "        title=\"Accuracy and Training Time vs Number of SVD Components with Logarithmic and Exponential Fits\",\n",
    "        xaxis_title=\"Number of SVD Components\",\n",
    "        \n",
    "        yaxis=dict(\n",
    "            title=\"Accuracy\",\n",
    "            titlefont=dict(color=\"blue\"),\n",
    "            tickfont=dict(color=\"blue\"),\n",
    "            range=[0, 1]\n",
    "        ),\n",
    "        \n",
    "        yaxis2=dict(\n",
    "            title=\"Training Time (seconds)\",\n",
    "            titlefont=dict(color=\"red\"),\n",
    "            tickfont=dict(color=\"red\"),\n",
    "            anchor=\"x\",\n",
    "            overlaying=\"y\",\n",
    "            side=\"right\",\n",
    "            range=[0, 100]\n",
    "        ),\n",
    "        \n",
    "        legend=dict(\n",
    "            orientation=\"h\",\n",
    "            yanchor=\"bottom\",\n",
    "            y=1.02,\n",
    "            xanchor=\"right\",\n",
    "            x=1\n",
    "        )\n",
    "    )\n",
    "\n",
    "    fig.show()\n",
    "\n",
    "# Call the advanced curve fitting function with the DataFrame results_df\n",
    "fit_curves_advanced(results_df)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "e051ff71",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    }\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>n_components</th>\n",
       "      <th>accuracy</th>\n",
       "      <th>training_time</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>0.306857</td>\n",
       "      <td>0.295077</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>0.448357</td>\n",
       "      <td>0.436165</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>0.471786</td>\n",
       "      <td>0.356321</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>0.576500</td>\n",
       "      <td>0.411016</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>0.679643</td>\n",
       "      <td>0.444396</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</td>\n",
       "      <td>0.736286</td>\n",
       "      <td>0.462523</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>7</td>\n",
       "      <td>0.761857</td>\n",
       "      <td>0.454060</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>8</td>\n",
       "      <td>0.787071</td>\n",
       "      <td>0.509744</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>9</td>\n",
       "      <td>0.789286</td>\n",
       "      <td>0.663166</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>10</td>\n",
       "      <td>0.803500</td>\n",
       "      <td>0.738182</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>12</td>\n",
       "      <td>0.826714</td>\n",
       "      <td>0.789084</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>14</td>\n",
       "      <td>0.839786</td>\n",
       "      <td>1.378411</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>17</td>\n",
       "      <td>0.861714</td>\n",
       "      <td>1.609638</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>19</td>\n",
       "      <td>0.873786</td>\n",
       "      <td>1.690372</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>23</td>\n",
       "      <td>0.885714</td>\n",
       "      <td>2.476373</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>27</td>\n",
       "      <td>0.889000</td>\n",
       "      <td>3.331303</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>32</td>\n",
       "      <td>0.897214</td>\n",
       "      <td>3.938132</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>39</td>\n",
       "      <td>0.901286</td>\n",
       "      <td>5.487883</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>46</td>\n",
       "      <td>0.906214</td>\n",
       "      <td>6.653816</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>55</td>\n",
       "      <td>0.911357</td>\n",
       "      <td>9.616442</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>67</td>\n",
       "      <td>0.912929</td>\n",
       "      <td>11.016707</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>81</td>\n",
       "      <td>0.915786</td>\n",
       "      <td>18.197224</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>98</td>\n",
       "      <td>0.917571</td>\n",
       "      <td>25.977342</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>119</td>\n",
       "      <td>0.918714</td>\n",
       "      <td>36.979312</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>145</td>\n",
       "      <td>0.919929</td>\n",
       "      <td>41.091435</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>177</td>\n",
       "      <td>0.920929</td>\n",
       "      <td>53.882709</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>217</td>\n",
       "      <td>0.921214</td>\n",
       "      <td>67.853903</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "    n_components  accuracy  training_time\n",
       "0              1  0.306857       0.295077\n",
       "1              2  0.448357       0.436165\n",
       "2              3  0.471786       0.356321\n",
       "3              4  0.576500       0.411016\n",
       "4              5  0.679643       0.444396\n",
       "5              6  0.736286       0.462523\n",
       "6              7  0.761857       0.454060\n",
       "7              8  0.787071       0.509744\n",
       "8              9  0.789286       0.663166\n",
       "9             10  0.803500       0.738182\n",
       "10            12  0.826714       0.789084\n",
       "11            14  0.839786       1.378411\n",
       "12            17  0.861714       1.609638\n",
       "13            19  0.873786       1.690372\n",
       "14            23  0.885714       2.476373\n",
       "15            27  0.889000       3.331303\n",
       "16            32  0.897214       3.938132\n",
       "17            39  0.901286       5.487883\n",
       "18            46  0.906214       6.653816\n",
       "19            55  0.911357       9.616442\n",
       "20            67  0.912929      11.016707\n",
       "21            81  0.915786      18.197224\n",
       "22            98  0.917571      25.977342\n",
       "23           119  0.918714      36.979312\n",
       "24           145  0.919929      41.091435\n",
       "25           177  0.920929      53.882709\n",
       "26           217  0.921214      67.853903"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results_df"
   ]
  }
 ],
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