demo_01.ipynb

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      "source": [
        "%pip install plotly\n",
        "import plotly.graph_objects as go\n",
        "import ipywidgets as widgets\n",
        "from math import sqrt\n",
        "\n",
        "# If Using Google Colab, enable these extensions\n",
        "# from google.colab import output\n",
        "# output.enable_custom_widget_manager()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "## Loss functions\n",
        "\n",
        "def avg(y_true, y_pred):\n",
        "    \"\"\"\n",
        "    Computes the average deviation between the true values (y_true) and the predicted values (y_pred).\n",
        "    \"\"\"\n",
        "    return sum((t - p) for t, p in zip(y_true, y_pred)) / len(y_true)\n",
        "\n",
        "\n",
        "def mse(y_true, y_pred):\n",
        "    \"\"\"\n",
        "    Computes the mean squared error (MSE) between the true values and the predicted values.\n",
        "    \"\"\"\n",
        "    return sum((t - p)**2 for t, p in zip(y_true, y_pred)) / len(y_true)\n",
        "\n",
        "\n",
        "def rmse(y_true, y_pred):\n",
        "    \"\"\"\n",
        "    Computes the root mean squared error (RMSE) between the true values and the predicted values.\n",
        "    \"\"\"\n",
        "    return sqrt(sum((t - p)**2 for t, p in zip(y_true, y_pred)) / len(y_true))\n",
        "\n",
        "\n",
        "def mae(y_true, y_pred):\n",
        "    \"\"\"\n",
        "    Computes the mean absolute error (MAE) between the true values and the predicted values.\n",
        "    \"\"\"\n",
        "    return sum(abs(t - p) for t, p in zip(y_true, y_pred)) / len(y_true)\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [],
      "source": [
        "def get_y_pred(model, x_data):\n",
        "    \"\"\"\n",
        "    Generates predictions from the model for the given input data.\n",
        "\n",
        "    Parameters:\n",
        "    model (object): The linear regression model, which has a forward() method signiture\n",
        "    x_data (list or array): A list or array of input data points.\n",
        "\n",
        "    Returns:\n",
        "    list: A list of predicted values.\n",
        "    \"\"\"\n",
        "    return [model.forward(x) for x in x_data]\n",
        "\n",
        "\n",
        "def create_figure(model, x_data, y_data, loss_fn_dropdown):\n",
        "    \"\"\"\n",
        "    Creates an interactive Plotly figure showing the data points, regression line, and error lines.\n",
        "\n",
        "    Parameters:\n",
        "    model (object): The linear regression model.\n",
        "    x_data (list or array): A list or array of input data points.\n",
        "    y_data (list or array): A list or array of true output data points.\n",
        "    loss_fn_dropdown (ipywidgets.Dropdown): A dropdown widget to select the loss function.\n",
        "\n",
        "    Returns:\n",
        "    plotly.graph_objs._figure.FigureWidget: A Plotly FigureWidget for visualization.\n",
        "    \"\"\"\n",
        "    # Determine initial axis limits\n",
        "    x_min, x_max = min(x_data), max(x_data)\n",
        "    y_min, y_max = min(y_data), max(y_data)\n",
        "    y_pred = get_y_pred(model, x_data)\n",
        "\n",
        "    # Create figure and initial trace\n",
        "    fig = go.FigureWidget(\n",
        "        data=[\n",
        "            go.Scatter(x=x_data, y=y_data, mode=\"markers\", name=\"Data\", marker=dict(opacity=0.5)),\n",
        "            go.Scatter(x=x_data, y=y_pred, mode=\"lines\", name=\"Regression Line\"),\n",
        "            go.Scatter(\n",
        "                x=[None], y=[None], mode=\"lines\", line=dict(color=\"gray\"), name=\"Error Lines\"\n",
        "            ),  # Dummy trace for error lines\n",
        "        ],\n",
        "        layout=go.Layout(\n",
        "            title=\"Interactive Linear Regression\",\n",
        "            xaxis_title=\"Input\",\n",
        "            yaxis_title=\"Output\",\n",
        "            xaxis_range=[x_min - 2, x_max + 2],\n",
        "            yaxis_range=[y_min - 2, y_max + 2],\n",
        "            xaxis=dict(dtick=5),  # Set tick frequency for x-axis\n",
        "            yaxis=dict(dtick=5),  # Set tick frequency for y-axis\n",
        "            autosize=False,  # Disable automatic resizing\n",
        "            width=16 * 50,  # Set width and height to the same value for square shape\n",
        "            height=9 * 50,\n",
        "            shapes=[\n",
        "                dict(\n",
        "                    type=\"line\",\n",
        "                    xref=\"x\",\n",
        "                    yref=\"y\",\n",
        "                    x0=x,\n",
        "                    y0=y_data[i],\n",
        "                    x1=x,\n",
        "                    y1=y_pred[i],\n",
        "                    line=dict(\n",
        "                        color=\"gray\",\n",
        "                        width=1,\n",
        "                    ),\n",
        "                )\n",
        "                for i, x in enumerate(x_data)\n",
        "            ],\n",
        "            annotations=[\n",
        "                dict(\n",
        "                    text=f\"{loss_fn_dropdown.label}: {loss_fn_dropdown.value[0](y_data, y_pred):.2f}\",\n",
        "                    x=0.5,\n",
        "                    y=0.98,\n",
        "                    xref=\"paper\",\n",
        "                    yref=\"paper\",\n",
        "                    showarrow=False,\n",
        "                )\n",
        "            ],\n",
        "        ),\n",
        "    )\n",
        "    return fig\n",
        "\n",
        "\n",
        "def create_loss_vs_param_figure(model, loss):\n",
        "    \"\"\"\n",
        "    Creates an interactive Plotly figure showing the loss vs. model parameters.\n",
        "\n",
        "    Parameters:\n",
        "    model (object): The linear regression model.\n",
        "    loss (float): The computed loss for the current model parameters.\n",
        "\n",
        "    Returns:\n",
        "    plotly.graph_objs._figure.FigureWidget: A Plotly FigureWidget for visualizing loss vs. model parameters.\n",
        "    \"\"\"\n",
        "    if len(vars(model)) == 1:\n",
        "        pk, pv = list(vars(model).items())[0]\n",
        "        return go.FigureWidget(\n",
        "            data=[\n",
        "                go.Scatter(x=[pv], y=[loss], mode=\"markers\", name=\"Loss\", marker=dict(color=[\"rgba(255, 0, 0, 1)\"], size=[10])),\n",
        "            ],\n",
        "            layout=go.Layout(\n",
        "                title=\"Loss vs Model Parameters\",\n",
        "                xaxis_title=pk + \"*\",\n",
        "                yaxis_title=\"Loss\",\n",
        "                xaxis=dict(dtick=5),\n",
        "                yaxis=dict(dtick=5),\n",
        "                xaxis_range=[-25, 25],\n",
        "                yaxis_range=[-25, 25],\n",
        "                autosize=False,\n",
        "                width=9 * 50,\n",
        "                height=9 * 50,\n",
        "            ),\n",
        "        )\n",
        "    elif len(vars(model)) == 2:\n",
        "        pk1, pv1 = list(vars(model).items())[0]\n",
        "        pk2, pv2 = list(vars(model).items())[1]\n",
        "        return go.FigureWidget(\n",
        "            data=[\n",
        "                go.Scatter3d(x=[pv1], y=[pv2], z=[loss], mode=\"markers\", name=\"Loss\", marker=dict(color=[\"rgba(255, 0, 0, 1)\"], size=[10])),\n",
        "            ],\n",
        "            layout=go.Layout(\n",
        "                title=\"Loss vs Model Parameters\",\n",
        "                scene=dict(\n",
        "                    xaxis=dict(title=pk1, dtick=5, range=[-25, 25]),\n",
        "                    yaxis=dict(title=pk2, dtick=5, range=[-25, 25]),\n",
        "                    zaxis=dict(title=\"Loss\", dtick=5, range=[-25, 25]),\n",
        "                ),\n",
        "                autosize=False,\n",
        "                width=16 * 50,\n",
        "                height=9 * 50,\n",
        "            ),\n",
        "        )\n",
        "    else:\n",
        "        return go.FigureWidget(\n",
        "            data=[],\n",
        "            layout=go.Layout(\n",
        "                title=\"Loss vs Model Parameters\",\n",
        "                autosize=False,\n",
        "                width=16 * 50,\n",
        "                height=9 * 50,\n",
        "                annotations=[\n",
        "                    dict(\n",
        "                        text=\"More than 2 model parameters not supported\",\n",
        "                        showarrow=False,\n",
        "                        xref=\"paper\",\n",
        "                        yref=\"paper\",\n",
        "                        x=0.5,\n",
        "                        y=0.5,\n",
        "                    )\n",
        "                ],\n",
        "            ),\n",
        "        )\n",
        "\n",
        "\n",
        "def create_sliders(model):\n",
        "    \"\"\"\n",
        "    Creates interactive sliders for each model parameter.\n",
        "\n",
        "    Parameters:\n",
        "    model (object): The linear regression model.\n",
        "\n",
        "    Returns:\n",
        "    list: A list of FloatSlider widgets for each model parameter.\n",
        "    \"\"\"\n",
        "    sliders = []\n",
        "    for k, v in vars(model).items():\n",
        "        if isinstance(v, float):\n",
        "            sliders.append(widgets.FloatSlider(min=-20, max=20, step=0.1, value=v, description=k, continuous_update=True))\n",
        "    return sliders\n",
        "\n",
        "\n",
        "def create_loss_fn_dropdown():\n",
        "    \"\"\"\n",
        "    Creates a dropdown widget to select the loss function.\n",
        "\n",
        "    Returns:\n",
        "    ipywidgets.Dropdown: A Dropdown widget with loss function options.\n",
        "    \"\"\"\n",
        "    return widgets.Dropdown(\n",
        "        options = {\n",
        "            \"Average Error\": [avg, \"rgba(0, 0, 255, 0.8)\"],  # Cyan\n",
        "            \"Mean Squared Error\": [mse, \"rgba(255, 0, 255, 0.8)\"],  # Magenta\n",
        "            \"Root Mean Squared Error\": [rmse, \"rgba(0, 255, 0, 0.8)\"],  # Lime Green\n",
        "            \"Mean Absolute Error\": [mae, \"rgba(255, 165, 0, 0.8)\"],  # Orange\n",
        "        },\n",
        "        value=[avg, \"rgba(0, 0, 255, 0.8)\"],\n",
        "        description=\"Losses:\",\n",
        "    )\n",
        "\n",
        "\n",
        "def plot(x_data, y_data, model):\n",
        "    \"\"\"\n",
        "    Creates the main interactive plot for the linear regression model.\n",
        "\n",
        "    Parameters:\n",
        "    x_data (list or array): A list or array of input data points.\n",
        "    y_data (list or array): A list or array of true output data points.\n",
        "    model (object): The linear regression model.\n",
        "\n",
        "    Returns:\n",
        "    ipywidgets.HBox: An HBox widget containing the main plot and sliders for interactivity.\n",
        "    \"\"\"\n",
        "    loss_fn_dropdown = create_loss_fn_dropdown()\n",
        "    fig = create_figure(model, x_data, y_data, loss_fn_dropdown)\n",
        "    loss_vs_param_fig = create_loss_vs_param_figure(model, loss_fn_dropdown.value[0](y_data, get_y_pred(model, x_data)))\n",
        "    sliders = create_sliders(model)\n",
        "\n",
        "    def update_plot(change):\n",
        "        \"\"\"\n",
        "        Updates the plot based on changes in the sliders or the loss function dropdown.\n",
        "\n",
        "        Parameters:\n",
        "        change (dict): A dictionary containing information about the change event.\n",
        "        \"\"\"\n",
        "        for i, slider in enumerate(sliders):\n",
        "            setattr(model, slider.description, slider.value)\n",
        "        y_pred = get_y_pred(model, x_data)\n",
        "        loss = loss_fn_dropdown.value[0](y_data, y_pred)\n",
        "        with fig.batch_update():\n",
        "            fig.data[1].y = y_pred  # Update the line\n",
        "            # Update error lines\n",
        "            fig.layout.shapes = [\n",
        "                dict(\n",
        "                    type=\"line\",\n",
        "                    xref=\"x\",\n",
        "                    yref=\"y\",\n",
        "                    x0=x,\n",
        "                    y0=y_data[i],\n",
        "                    x1=x,\n",
        "                    y1=y_pred[i],\n",
        "                    line=dict(\n",
        "                        color=\"gray\",\n",
        "                        width=2,\n",
        "                    ),\n",
        "                )\n",
        "                for i, x in enumerate(x_data)\n",
        "            ]\n",
        "            # Update loss function annotation\n",
        "            fig.layout.annotations[0].text = f\"{loss_fn_dropdown.label}: {loss:.2f}\"\n",
        "\n",
        "        # Update loss vs param figure\n",
        "        if len(vars(model)) == 1:\n",
        "            pv = list(vars(model).values())[0]\n",
        "            loss_vs_param_fig.data[0].x += (pv,)\n",
        "            loss_vs_param_fig.data[0].y += (loss,)\n",
        "        elif len(vars(model)) == 2:\n",
        "            pv1, pv2 = list(vars(model).values())\n",
        "            loss_vs_param_fig.data[0].x += (pv1,)\n",
        "            loss_vs_param_fig.data[0].y += (pv2,)\n",
        "            loss_vs_param_fig.data[0].z += (loss,)\n",
        "        \n",
        "        current_colors = list(loss_vs_param_fig.data[0].marker.color)\n",
        "        current_sizes = list(loss_vs_param_fig.data[0].marker.size)\n",
        "        current_colors[-1] = loss_fn_dropdown.value[1]\n",
        "        if change[\"owner\"] == loss_fn_dropdown:\n",
        "            current_colors[-1] = change[\"old\"][1]\n",
        "        current_colors += [\"rgba(255, 0, 0, 1)\"]\n",
        "        current_sizes[-1] = 6\n",
        "        current_sizes += [10]\n",
        "        loss_vs_param_fig.data[0].marker.color = current_colors\n",
        "        loss_vs_param_fig.data[0].marker.size = current_sizes\n",
        "\n",
        "    # Observe changes\n",
        "    for slider in sliders:\n",
        "        slider.observe(update_plot, names=\"value\")\n",
        "    loss_fn_dropdown.observe(update_plot, names=\"value\")\n",
        "    # Attach the update_plot function to each slider's value change event\n",
        "    # When a slider value changes, update_plot will be called to update the plot\n",
        "    # When a different loss function is selected, update_plot will be called to update the plot\n",
        "\n",
        "    return widgets.HBox([widgets.VBox([loss_fn_dropdown] + [fig] + sliders), loss_vs_param_fig])"
      ]
    },
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      "cell_type": "code",
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      "source": [
        "# Start with the constant model, first take a few data points where the x axis is the input and the y-axis is the output\n",
        "# With the constant model, any new input will always give the same output\n",
        "# So we are trying to find the best output we can get so as to minimize the loss function\n",
        "# Let's start with a simple loss function like average of how far all the outputs are from the best output\n",
        "# That is sum(yi-h)/n, but we will find that at the best h, the mean of the outputs is the minimizer and the error is always 0\n",
        "# But error of 0 is not correct because that tells that our model predicts perfectly when it is not\n",
        "\n",
        "# There will be two graphs for all the examples, one is the output vs input graph (with h* line shown)\n",
        "# The second graph will be the loss function vs h* (To show how the loss function changes when we change h*)\n",
        "# This tells them exactly the best h* where the loss function is minimum\n",
        "\n",
        "# So we go to a better loss function (say) the absolute error, i.e sum(|yi-h|) / n\n",
        "# Here the best h* would be the median and we can see how the loss function is piecewise\n",
        "\n",
        "# Then we show the MSE loss function i.e. sum((yi-h)^2) / n, same process and here the min h* would be the mean\n",
        "\n",
        "\n",
        "class ConstantModel:\n",
        "    def __init__(self, h=2.0):\n",
        "        self.h = h\n",
        "    \n",
        "    def forward(self, x):\n",
        "        return self.h\n",
        "\n",
        "x_data = [1, 2, 3, 4]\n",
        "y_data = [6, 8, 1, 3]\n",
        "\n",
        "model = ConstantModel()\n",
        "plot(x_data, y_data, model)\n"
      ]
    },
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      "source": [
        "# Simple linear regression\n",
        "# So rather than predicting a constant, the model will predict an output that is a function of the input\n",
        "# So in the simple linear regression model, the output function is a line with the output being mx + b\n",
        "# Now instead of a single parameter h that we have to calculate, we now have to calculate the parameters m and b\n",
        "# So the best way to do it is to still consider a loss function that gives you some error for your prediction and then\n",
        "# find the point where the loss function is minimum. So how do we find these parameters?\n",
        "\n",
        "class SimpleLinearRegression:\n",
        "    def __init__ (self, m=0.7, c=2.2):\n",
        "        self.m = m\n",
        "        self.c = c\n",
        "    \n",
        "    def forward(self, x):\n",
        "        return self.m * x + self.c\n",
        "\n",
        "# Initial data points\n",
        "x_data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n",
        "y_data = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]\n",
        "\n",
        "model = SimpleLinearRegression()\n",
        "plot(x_data, y_data, model)"
      ]
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          "model_module": "@jupyter-widgets/output",
          "model_module_version": "1.0.0",
          "model_name": "OutputModel",
          "state": {
            "_dom_classes": [],
            "_model_module": "@jupyter-widgets/output",
            "_model_module_version": "1.0.0",
            "_model_name": "OutputModel",
            "_view_count": null,
            "_view_module": "@jupyter-widgets/output",
            "_view_module_version": "1.0.0",
            "_view_name": "OutputView",
            "layout": "IPY_MODEL_58244c4d308c4fbd95635ccc3384dd46",
            "msg_id": "",
            "outputs": []
          }
        },
        "29a7c75c514d43f59271bbb9decf21f8": {
          "model_module": "@jupyter-widgets/controls",
          "model_module_version": "1.5.0",
          "model_name": "SliderStyleModel",
          "state": {
            "_model_module": "@jupyter-widgets/controls",
            "_model_module_version": "1.5.0",
            "_model_name": "SliderStyleModel",
            "_view_count": null,
            "_view_module": "@jupyter-widgets/base",
            "_view_module_version": "1.2.0",
            "_view_name": "StyleView",
            "description_width": "",
            "handle_color": null
          }
        },
        "3909e135a7d141d2ac59ba7ae12a2123": {
          "model_module": "@jupyter-widgets/output",
          "model_module_version": "1.0.0",
          "model_name": "OutputModel",
          "state": {
            "_dom_classes": [],
            "_model_module": "@jupyter-widgets/output",
            "_model_module_version": "1.0.0",
            "_model_name": "OutputModel",
            "_view_count": null,
            "_view_module": "@jupyter-widgets/output",
            "_view_module_version": "1.0.0",
            "_view_name": "OutputView",
            "layout": "IPY_MODEL_3a3ddafad1c54876886d25beab644828",
            "msg_id": "",
            "outputs": [
              {
                "data": {
                  "image/png": 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\n",
                  "text/plain": "<Figure size 640x480 with 1 Axes>"
                },
                "metadata": {},
                "output_type": "display_data"
              }
            ]
          }
        },
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