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"source": [
"# Basic usage example #\n",
"This notebook demonstrates how to use the library to perform regression without delving into technical details. The interested reader is referred to the [Robust Local Polynomial Regression with Similarity Kernels draft paper](https://github.com/yaniv-shulman/rsklpr/tree/main/paper/rsklpr.pdf). It is assumed that the library is installed in the environment that you are using. You can install the library from [PyPI](https://pypi.org/project/rsklpr/) using pip: ```pip install rsklpr```\n",
"\n",
"**Note**, the plots are interactive, it is possible to pan, zoom, rotate as well as toggle viewing of elements by clicking / double-clicking the legend items. \n",
"\n",
"To begin lets define a function that generates some data to perform regression on."
],
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"cell_type": "code",
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"source": [
"from typing import Tuple\n",
"\n",
"import numpy as np\n",
"\n",
"\n",
"def benchmark_curve_1(noise_ratio: float, hetero: bool, num_points: int) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:\n",
" \"\"\"\n",
" Generates a dataset of points sampled from smooth curve with added noise. In the case of heteroscedastic noise the\n",
" noise scale is sampled from a uniform distribution up to noise_ratio of the response range.\n",
"\n",
" Args:\n",
" noise_ratio: The ratio of noise of the response range used to generate noise.\n",
" hetero: True to generate heteroscedastic noise, False for homoscedastic noise.\n",
" num_points: The number of points sampled from the curve.\n",
"\n",
" Returns:\n",
" The predictor, response and ground truth.\n",
" \"\"\"\n",
" generator: np.random.Generator = np.random.default_rng(seed=14)\n",
" x_: np.ndarray = np.linspace(start=0.0, stop=1.0, num=num_points)\n",
" x_ += generator.normal(scale=1 / np.sqrt(num_points), size=x_.shape[0])\n",
" x_ = np.sort(x_)\n",
"\n",
" y_true_: np.ndarray = np.sqrt(np.abs(x_**3 - 4 * x_**4 / 3)) + (\n",
" 0.31 * x_ * np.sin(x_ * 3 * np.pi) ** 2 / np.max(x_)\n",
" )\n",
"\n",
" scale: np.ndarray\n",
"\n",
" if hetero:\n",
" scale = generator.uniform(low=0.001, high=noise_ratio * (y_true_.max() - y_true_.min()), size=x_.shape[0])\n",
" else:\n",
" scale = np.full(shape=x_.shape[0], fill_value=noise_ratio * (y_true_.max() - y_true_.min()))\n",
"\n",
" y_: np.ndarray = y_true_ + generator.normal(scale=scale)\n",
"\n",
" return (\n",
" x_,\n",
" y_,\n",
" y_true_,\n",
" )"
],
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"end_time": "2024-03-23T15:13:53.718985Z",
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"id": "c23ce514f76767a1"
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{
"cell_type": "markdown",
"source": [
"Next, define a function to plot our data and regression results."
],
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"collapsed": false
},
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{
"cell_type": "code",
"execution_count": 2,
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"source": [
"import pandas as pd\n",
"from typing import Optional\n",
"import plotly\n",
"import plotly.graph_objects as go\n",
"\n",
"\n",
"def plot_results_1d(\n",
" x_: Optional[np.ndarray],\n",
" y_: Optional[np.ndarray],\n",
" y_true_: Optional[np.ndarray] = None,\n",
" estimates_: Optional[pd.DataFrame] = None,\n",
" title: str = \"\",\n",
" estimates_mode: str = \"lines+markers\",\n",
") -> None:\n",
" \"\"\"\n",
" Plots a scatter plot of all columns in the provided dataframe where each column is treated as a separate variable.\n",
" It is assumed estimates_.index is the same as x_ as both are used for x coordinates in the plot.\n",
"\n",
" Args:\n",
" x_: The predictors X for all observations with shape [N,K] where N is the number of observations and K is the\n",
" dimension of X.\n",
" y_: The corresponding response for all observations.\n",
" y_true_: The ground truth response at the locations x.\n",
" estimates_: The estimates y_hat given x. it is assumed each column in the dataframe is the results of a\n",
" different estimator.\n",
" title: The plot title.\n",
" estimates_mode: The style mode for estimates data.\n",
" \"\"\"\n",
" fig = go.Figure()\n",
" if x_ is not None and y_ is not None:\n",
" fig.add_trace(go.Scatter(x=x_, y=y_, mode=\"markers\", name=\"data\"))\n",
"\n",
" if y_true_ is not None:\n",
" fig.add_trace(\n",
" go.Scatter(x=x_ if x_ is not None else estimates_.index, y=y_true_, mode=\"markers\", name=\"y_true\")\n",
" )\n",
"\n",
" if estimates_ is not None:\n",
" for c in estimates_.columns:\n",
" fig.add_trace(go.Scatter(x=estimates_.index, y=estimates_[c], mode=estimates_mode, name=c))\n",
"\n",
" fig.update_layout(\n",
" go.Layout(\n",
" title=title,\n",
" autosize=True,\n",
" height=500,\n",
" )\n",
" )\n",
"\n",
" plotly.offline.iplot(fig)"
],
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"end_time": "2024-03-23T15:13:53.852433Z",
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"id": "5f700b91e93981d2"
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{
"cell_type": "markdown",
"source": [
"We can now use the function to generate some data as well as the ground truth and plot it."
],
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"collapsed": false
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"id": "f1e0d62ded816bef"
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{
"cell_type": "code",
"execution_count": 3,
"outputs": [],
"source": [
"x: np.ndarray\n",
"y: np.ndarray\n",
"y_true: np.ndarray\n",
"\n",
"x, y, y_true = benchmark_curve_1(noise_ratio=0.15, hetero=True, num_points=100)"
],
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"source": [
"from rsklpr.rsklpr import Rsklpr\n",
"\n",
"rsklpr: Rsklpr = Rsklpr(size_neighborhood=30)\n",
"y_hat: np.ndarray = rsklpr(x=x, y=y)\n",
"plot_results_1d(\n",
" x_=x,\n",
" y_=y,\n",
" y_true_=y_true,\n",
" estimates_=pd.DataFrame(data=y_hat, columns=[\"y_hat\"], index=x),\n",
" title=\"Regression with size_neighborhood=30\",\n",
")"
],
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"Note that calling the model directly as ```python rsklpr(x=x, y=y) ``` is a shortform for calling fit and predict on all locations given in the dataset. We can also make the curve smoother by increasing the number of datapoints in the neighborhood. "
],
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"source": [
"rsklpr = Rsklpr(size_neighborhood=60)\n",
"y_hat = rsklpr(x=x, y=y)\n",
"plot_results_1d(\n",
" x_=x,\n",
" y_=y,\n",
" y_true_=y_true,\n",
" estimates_=pd.DataFrame(data=y_hat, columns=[\"y_hat\"], index=x),\n",
" title=\"Regression with size_neighborhood=60\",\n",
")"
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"start_time": "2024-03-23T15:13:54.744270Z"
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"source": [
"rsklpr = Rsklpr(size_neighborhood=35)\n",
"y_hat = rsklpr(x=x, y=y, metrics=[\"residuals\", \"r_squared\"])\n",
"\n",
"plot_results_1d(\n",
" x_=x,\n",
" y_=y,\n",
" estimates_=pd.DataFrame(\n",
" data=np.concatenate([y_hat.reshape((-1, 1)), rsklpr.residuals.reshape((-1, 1))], axis=1),\n",
" columns=[\"y_hat\", \"residuals\"],\n",
" index=x,\n",
" ),\n",
" title=\"Regression residuals with size_neighborhood=35\",\n",
" estimates_mode=\"markers\",\n",
")"
],
"metadata": {
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")"
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"rsklpr = Rsklpr(size_neighborhood=35)\n",
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" x_=x,\n",
" y_=y,\n",
" y_true_=y_true,\n",
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" title=\"Regression in linearly spaced locations\",\n",
")"
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"## Confidence intervals ##\n",
"We can also use the library to calculate confidence intervals for the prediction. In this case we calculate the 95% confidence intervals at the linearly spaced locations."
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"import numpy as np\n",
"\n",
"rsklpr = Rsklpr(size_neighborhood=35)\n",
"rsklpr.fit(x=x, y=y)\n",
"y_hat = rsklpr.predict(x=x_linspace)\n",
"\n",
"y_mean_hat: np.ndarray\n",
"conf_low: np.ndarray\n",
"conf_high: np.ndarray\n",
"y_mean_hat, conf_low, conf_high, _ = rsklpr.predict_bootstrap(x=x_linspace)\n",
"\n",
"estimates: pd.DataFrame = pd.DataFrame(\n",
" data=np.concatenate([y_hat.reshape((-1, 1)), y_mean_hat.reshape((-1, 1))], axis=1),\n",
" columns=[\"y_hat\", \"y_mean_hat\"],\n",
" index=x_linspace,\n",
")\n",
"estimates[\"conf_0.025\"] = conf_low\n",
"estimates[\"conf_0.975\"] = conf_high\n",
"\n",
"plot_results_1d(\n",
" x_=x,\n",
" y_=y,\n",
" y_true_=y_true,\n",
" estimates_=estimates,\n",
" title=\"Regression in linearly spaced locations with 95% confidence intervals\",\n",
")"
],
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"cell_type": "markdown",
"source": [
"## Multivariate predictors ##\n",
"Now let's regress a multivariate predictor. In this case to be able to visualize we regress a 2D predictor but it is possible to input data of any dimension. As before start by generating some data and a function to visualize the results."
],
"metadata": {
"collapsed": false
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{
"cell_type": "code",
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"outputs": [],
"source": [
"from typing import List\n",
"\n",
"\n",
"def benchmark_plane_1(noise_ratio: float, hetero: bool, num_points: int) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:\n",
" \"\"\"\n",
" Generates a dataset of points sampled from a smooth function with added noise. In the case of heteroscedastic noise\n",
" the noise scale is sampled from a uniform distribution up to noise_ratio of the response range.\n",
"\n",
" Args:\n",
" noise_ratio: The ratio of noise of the response range used to generate noise.\n",
" hetero: True to generate heteroscedastic noise, False for homoscedastic noise.\n",
" num_points: The number of points sampled from the function.\n",
"\n",
" Returns:\n",
" The predictor, response and ground truth.\n",
" \"\"\"\n",
" generator: np.random.Generator = np.random.default_rng(seed=14)\n",
"\n",
" x_: np.ndarray = np.concatenate(\n",
" [\n",
" generator.uniform(low=0.0, high=0.7, size=(num_points, 1)),\n",
" generator.uniform(low=-0.5, high=0.8, size=(num_points, 1)),\n",
" ],\n",
" axis=1,\n",
" )\n",
"\n",
" y_true_: np.ndarray = np.power(x_[:, 0], 2) * np.cos(1.5 * np.pi * x_[:, 1]) + np.sin(np.square(x_.sum(axis=1)))\n",
" scale: np.ndarray\n",
"\n",
" if hetero:\n",
" scale = generator.uniform(\n",
" low=0.001,\n",
" high=noise_ratio * (y_true_.max() - y_true_.min()),\n",
" size=num_points,\n",
" )\n",
" else:\n",
" scale = np.full(shape=num_points, fill_value=noise_ratio * (y_true_.max() - y_true_.min()))\n",
"\n",
" y_: np.ndarray = y_true_ + generator.normal(scale=scale)\n",
"\n",
" return (\n",
" x_,\n",
" y_,\n",
" y_true_,\n",
" )\n",
"\n",
"\n",
"def plot_results_2d(\n",
" x_: np.ndarray,\n",
" y_: np.ndarray,\n",
" y_true_: Optional[np.ndarray],\n",
" estimates_: Optional[pd.DataFrame] = None,\n",
" title: str = \"\",\n",
") -> None:\n",
" \"\"\"\n",
" Plots a scatter plot of all columns in the provided dataframe where each column is treated as a separate variable.\n",
" It is assumed all plots share the same x coordinates.\n",
"\n",
" Args:\n",
" x_: The predictors X for all observations with shape [N,K] where N is the number of observations and K is the\n",
" dimension of X.\n",
" y_: The corresponding response for all observations.\n",
" y_true_: The ground truth response at the locations x.\n",
" estimates_: The estimates y_hat given x. it is assumed each column in the dataframe is the results of a\n",
" different estimator.\n",
" title: The plot title.\n",
" \"\"\"\n",
" fig = go.Figure()\n",
"\n",
" fig.add_trace(go.Scatter3d(x=x_[:, 0], y=x_[:, 1], z=y_, mode=\"markers\", name=\"data\"))\n",
"\n",
" if y_true_ is not None:\n",
" fig.add_trace(go.Scatter3d(x=x_[:, 0], y=x_[:, 1], z=y_true_, mode=\"markers\", name=\"y_true\"))\n",
"\n",
" if estimates_ is not None:\n",
" for c in estimates_.columns:\n",
" fig.add_trace(go.Scatter3d(x=x_[:, 0], y=x_[:, 1], z=estimates_[c], mode=\"markers\", name=c))\n",
"\n",
" x_lines: List[Optional[float]] = []\n",
" y_lines: List[Optional[float]] = []\n",
" z_lines: List[Optional[float]] = []\n",
" estimates_[\"y\"] = y_\n",
"\n",
" if y_true_ is not None:\n",
" estimates_[\"y_true\"] = y_true_\n",
"\n",
" min_z: np.ndarray = estimates_.values.min(axis=1)\n",
" max_z: np.ndarray = estimates_.values.max(axis=1)\n",
" estimates_.drop(columns=[\"y\", \"y_true\"], inplace=True)\n",
"\n",
" for i in range(x_.shape[0]):\n",
" x_lines.append(float(x_[i, 0]))\n",
" y_lines.append(float(x_[i, 1]))\n",
" z_lines.append(float(min_z[i]))\n",
"\n",
" x_lines.append(float(x_[i, 0]))\n",
" y_lines.append(float(x_[i, 1]))\n",
" z_lines.append(float(max_z[i]))\n",
"\n",
" x_lines.append(None)\n",
" y_lines.append(None)\n",
" z_lines.append(None)\n",
"\n",
" fig.add_trace(go.Scatter3d(x=x_lines, y=y_lines, z=z_lines, mode=\"lines\", name=\"lines\"))\n",
"\n",
" fig.update_traces(marker=dict(size=3))\n",
" fig.update_layout(\n",
" go.Layout(\n",
" title=title,\n",
" autosize=True,\n",
" height=800,\n",
" )\n",
" )\n",
"\n",
" plotly.offline.iplot(fig)"
],
"metadata": {
"collapsed": false,
"ExecuteTime": {
"end_time": "2024-03-23T15:13:56.663478Z",
"start_time": "2024-03-23T15:13:56.637252Z"
}
},
"id": "aa63350108b4ce26"
},
{
"cell_type": "raw",
"source": [
"Lets generate and plot the data."
],
"metadata": {
"collapsed": false
},
"id": "c8522fdec56607d"
},
{
"cell_type": "code",
"execution_count": 13,
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")"
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