局部近似

2024年12月1日

训练深度神经网络本质上是一个压缩任务。我们希望将训练数据分布表示为由一组矩阵参数化的函数。分布越复杂，所需的参数就越多。近似整个分布的理由是，我们可以在推理时使用相同的模型和权重，对任何有效点进行前向传播。

但是，如果我们的模型在推理时即时训练呢？那么，在传播 $x$ 时，我们只需要对 $x$ 周围的局部分布进行建模。由于局部区域的维度应该比整个训练集低得多，因此一个更简单的模型就足够了！

这就是局部近似或局部回归背后的思想。让我们考虑一个简单的回归任务。

任务

我们得到了以下数据的 $100$ 个样本：

Y = sin (4 X) + ϵ

其中

ϵ \sim N (0, \frac{1}{3})

绘图代码

from pathlib import Path

import numpy as np
import plotly.graph_objects as go

# Generate data
np.random.seed(42)
n_points = 100
X = np.random.uniform(0, 1, n_points)
epsilon = np.random.normal(0, 1 / 3, n_points)
Y = np.sin(4 * X) + epsilon

# True function
x_true = np.linspace(0, 1, 500)
y_true = np.sin(4 * x_true)

# Create the plot
fig = go.Figure()

# Add scatter points for noisy data
fig.add_trace(
    go.Scatter(
        x=X,
        y=Y,
        mode="markers",
        name="Noisy Data",
        marker=dict(color="gray"),
    )
)

# Add true function
fig.add_trace(
    go.Scatter(
        x=x_true,
        y=y_true,
        mode="lines",
        name="True Function",
        line=dict(color="red"),
    )
)

# Update layout shared across themes
fig.update_layout(
    autosize=True,
    title="Data",
    xaxis_title="X",
    yaxis_title="Y",
)

# Theme configuration aligning with dario.css colors
themes = [
    {
        "name": "light",
        "template": "plotly_white",
        "font_color": "#141413",
        "background": "#f0efea",
        "axis_color": "#141413",
        "gridcolor": "rgba(20, 20, 19, 0.2)",
    },
    {
        "name": "dark",
        "template": "plotly_dark",
        "font_color": "#f0efea",
        "background": "#141413",
        "axis_color": "#f0efea",
        "gridcolor": "rgba(240, 239, 234, 0.2)",
    },
]

output_dir = Path(__file__).resolve().parents[3] / "static"
output_dir.mkdir(parents=True, exist_ok=True)

for theme in themes:
    themed_fig = go.Figure(fig)
    themed_fig.update_layout(
        template=theme["template"],
        font=dict(color=theme["font_color"]),
        paper_bgcolor=theme["background"],
        plot_bgcolor=theme["background"],
    )
    themed_fig.update_xaxes(
        showline=True,
        linecolor=theme["axis_color"],
        tickcolor=theme["axis_color"],
        tickfont=dict(color=theme["axis_color"]),
        title_font=dict(color=theme["axis_color"]),
        gridcolor=theme["gridcolor"],
        zeroline=False,
    )
    themed_fig.update_yaxes(
        showline=True,
        linecolor=theme["axis_color"],
        tickcolor=theme["axis_color"],
        tickfont=dict(color=theme["axis_color"]),
        title_font=dict(color=theme["axis_color"]),
        gridcolor=theme["gridcolor"],
        zeroline=False,
    )

    filename = output_dir / f"local_approximation_data_{theme['name']}.html"
    themed_fig.write_html(filename)
    print(f"Saved plot to {filename}")

# Show the plot
fig.show()

我们将数据集记为 $D$ ，其中包含样本 $(x_{i}, y_{i}) \in D$ 。

我们的任务是通过数据拟合一条合理的曲线，使其近似匹配真实函数。我们将这条曲线记为 $\hat{f}$ 。

K近邻算法

给定某个 $x$ ，一种方法是取 $x$ 的 $k$ 个最近值 $x_{i}$ ，并将它们的 $y_{i}$ 值平均作为估计值。即，

\hat{f} (x) = Ave (y_{i} ∣ x_{i} \in N_{k} (x))

其中 $N_{k} (x)$ 表示 $x$ 的 $k$ 个最近点。

绘图代码

from pathlib import Path

import numpy as np
import plotly.graph_objects as go

# Generate data
np.random.seed(42)
n_points = 100
X = np.random.uniform(0, 1, n_points)
epsilon = np.random.normal(0, 1 / 3, n_points)
Y = np.sin(4 * X) + epsilon

# True function
x_true = np.linspace(0, 1, 500)
y_true = np.sin(4 * x_true)

# k-NN for a range of k
x_curve = np.arange(0, 1, 0.01)
k_range = range(1, 21)
y_curves_knn = {}

for k in k_range:
    y_curve = []
    for x in x_curve:
        distances = np.square(X - x)
        nearest_indices = np.argsort(distances)[:k]
        y_curve.append(np.mean(Y[nearest_indices]))
    y_curves_knn[k] = y_curve

# Create the Plotly figure
fig = go.Figure()

# Add static traces
fig.add_trace(
    go.Scatter(x=X, y=Y, mode="markers", name="Noisy Data", marker=dict(color="gray"))
)

fig.add_trace(
    go.Scatter(
        x=x_true, y=y_true, mode="lines", name="True Function", line=dict(color="red")
    )
)

# Add the first k-NN curve (k=13, the default slider position)
initial_k = 13
fig.add_trace(
    go.Scatter(
        x=x_curve,
        y=y_curves_knn[initial_k],
        mode="lines",
        name="k-NN Curve",
        line=dict(color="yellow"),
    )
)

# Define slider steps
steps = []
for k in k_range:
    step = dict(
        method="update",
        args=[
            {"y": [Y, y_true, y_curves_knn[k]]},  # Update y-data for the traces
            {
                "title": f"Interactive k-NN Curve with Slider for k = {k}"
            },  # Update the title dynamically
        ],
        label=f"{k}",
    )
    steps.append(step)

# Add slider to the layout
sliders = [
    dict(
        active=initial_k - 1,
        currentvalue={"prefix": "k = "},
        pad={"t": 50},
        steps=steps,
    )
]

fig.update_layout(
    sliders=sliders,
    autosize=True,
    title=f"Interactive k-NN Curve with Slider for k = {initial_k}",
    xaxis_title="X",
    yaxis_title="Y",
)

themes = [
    {
        "name": "light",
        "template": "plotly_white",
        "font_color": "#141413",
        "background": "#f0efea",
        "axis_color": "#141413",
        "gridcolor": "rgba(20, 20, 19, 0.2)",
    },
    {
        "name": "dark",
        "template": "plotly_dark",
        "font_color": "#f0efea",
        "background": "#141413",
        "axis_color": "#f0efea",
        "gridcolor": "rgba(240, 239, 234, 0.2)",
    },
]

output_dir = Path(__file__).resolve().parents[3] / "static"
output_dir.mkdir(parents=True, exist_ok=True)

for theme in themes:
    themed_fig = go.Figure(fig)
    themed_fig.update_layout(
        template=theme["template"],
        font=dict(color=theme["font_color"]),
        paper_bgcolor=theme["background"],
        plot_bgcolor=theme["background"],
    )
    themed_fig.update_xaxes(
        showline=True,
        linecolor=theme["axis_color"],
        tickcolor=theme["axis_color"],
        tickfont=dict(color=theme["axis_color"]),
        title_font=dict(color=theme["axis_color"]),
        gridcolor=theme["gridcolor"],
        zeroline=False,
    )
    themed_fig.update_yaxes(
        showline=True,
        linecolor=theme["axis_color"],
        tickcolor=theme["axis_color"],
        tickfont=dict(color=theme["axis_color"]),
        title_font=dict(color=theme["axis_color"]),
        gridcolor=theme["gridcolor"],
        zeroline=False,
    )

    html_path = output_dir / f"knn_slider_{theme['name']}.html"
    themed_fig.write_html(html_path)
    print(f"Saved interactive plot to {html_path}")

# Show the plot
fig.show()

通过使用滑块，你可以看到较大的 $k$ 值会导致曲线更平滑，但较小的 $k$ 值曲线会包含一些噪声。在极端情况下， $k = 1$ 会完全追踪训练数据，而 $k = 100$ 会给出一个平坦的全局平均值。

纳达拉亚-沃森核回归

与其将数据子集限制为 $k$ 个点，不如考虑集合中的所有点，但根据每个点与 $x$ 的接近程度来加权其贡献。考虑以下模型

\hat{f} (x) = \frac{\sum _{i = 1}^{∣ D ∣} K _{λ} ( x , x _{i} ) y _{i}}{\sum _{i = 1}^{∣ D ∣} K _{λ} ( x , x _{i} )}

其中 $K_{λ}$ 是一个核函数，我们将用它作为接近度的度量。

K_{λ} (x_{0}, x) = D (\frac{∣ x - x _{0} ∣}{λ})

该函数由 $λ$ 参数化，称为带宽，它控制数据中哪些范围的 $x$ 值会影响 $\hat{f}$ 的输出。如果我们绘制这些函数，这一点就会变得清晰。

核函数

下图绘制的是

f (x) = α K_{λ, D} (0, x)

其中 $α$ 使得 $f$ 在其支持域上积分为 $1$ 。

D (u) = ⎩ ⎨ ⎧ \frac{3}{4} (1 - u^{2}) 0 如果 ∣ u ∣ \leq 1, 如果 ∣ u ∣ > 1.

D (u) = ⎩ ⎨ ⎧ (1 - ∣ u ∣^{3})^{3} 0 如果 ∣ u ∣ \leq 1, 如果 ∣ u ∣ > 1.

D (u) = \frac{1}{2 π} e^{- \frac{1}{2} u^{2}} .

绘图代码

from pathlib import Path

import numpy as np
import plotly.graph_objects as go
from scipy.integrate import quad


# Define kernel functions
def epanechnikov_kernel(u):
    return np.maximum(0, 0.75 * (1 - u**2))


def tricube_kernel(u):
    return np.maximum(0, (1 - np.abs(u) ** 3) ** 3)


def gaussian_kernel(u):
    return np.exp(-0.5 * u**2) / np.sqrt(2 * np.pi)


def renormalized_kernel(kernel_func, u_range, bandwidth):
    def kernel_with_lambda(u):
        scaled_u = u / bandwidth
        normalization_factor, _ = quad(lambda v: kernel_func(v / bandwidth), *u_range)
        return kernel_func(scaled_u) / normalization_factor

    return kernel_with_lambda


# Kernel function plot generator
def generate_kernel_plot(
    kernel_name, kernel_func, x_range, u_range, lambda_values, y_range
):
    fig = go.Figure()

    # Initial lambda
    initial_lambda = lambda_values[len(lambda_values) // 2]

    # Generate initial kernel curve
    x = np.linspace(*x_range, 500)
    kernel_with_lambda = renormalized_kernel(kernel_func, u_range, initial_lambda)
    y = kernel_with_lambda(x)
    fig.add_trace(
        go.Scatter(
            x=x,
            y=y,
            mode="lines",
            name=f"{kernel_name} Kernel (λ={initial_lambda:.2f})",
            line=dict(color="green"),
        )
    )

    # Create frames for the slider
    frames = []
    for bandwidth in lambda_values:
        kernel_with_lambda = renormalized_kernel(kernel_func, u_range, bandwidth)
        y = kernel_with_lambda(x)
        frames.append(
            go.Frame(
                data=[
                    go.Scatter(
                        x=x,
                        y=y,
                        mode="lines",
                        name=f"{kernel_name} Kernel (λ={bandwidth:.2f})",
                        line=dict(color="green"),
                    )
                ],
                name=f"{bandwidth:.2f}",
            )
        )

    # Add frames to the figure
    fig.frames = frames

    # Add slider
    sliders = [
        {
            "active": len(lambda_values) // 2,
            "currentvalue": {"prefix": "Bandwidth λ: "},
            "steps": [
                {
                    "args": [
                        [f"{bandwidth:.2f}"],
                        {"frame": {"duration": 0, "redraw": True}, "mode": "immediate"},
                    ],
                    "label": f"{bandwidth:.2f}",
                    "method": "animate",
                }
                for bandwidth in lambda_values
            ],
        }
    ]

    # Update layout
    fig.update_layout(
        title=f"{kernel_name} Kernel Function",
        xaxis_title="u",
        yaxis_title="K(u)",
        yaxis_range=y_range,
        sliders=sliders,
        autosize=True,
        updatemenus=[
            {
                "buttons": [
                    # {
                    #     "args": [
                    #         None,
                    #         {
                    #             "frame": {"duration": 500, "redraw": True},
                    #             "fromcurrent": True,
                    #         },
                    #     ],
                    #     "label": "Play",
                    #     "method": "animate",
                    # },
                    # {
                    #     "args": [
                    #         [None],
                    #         {
                    #             "frame": {"duration": 0, "redraw": True},
                    #             "mode": "immediate",
                    #         },
                    #     ],
                    #     "label": "Pause",
                    #     "method": "animate",
                    # },
                ],
                "direction": "left",
                "pad": {"r": 10, "t": 87},
                "showactive": False,
                "type": "buttons",
                "x": 0.1,
                "xanchor": "right",
                "y": 0,
                "yanchor": "top",
            }
        ],
    )

    return fig


# Kernel functions
kernels = {
    "Epanechnikov": epanechnikov_kernel,
    "Tricube": tricube_kernel,
    "Gaussian": gaussian_kernel,
}

# Parameters
x_range_plot = (-3, 3)  # Range of u values for the plot
u_range_integration = (-3, 3)  # Range for normalization
lambda_values = np.linspace(0.01, 2, 20)  # Linear lambda values from 0.01 to 2
y_range_plot = (0, 1.5)  # Adjusted range to fit the normalized functions

# Generate and display plots for each kernel
themes = [
    {
        "name": "light",
        "template": "plotly_white",
        "font_color": "#141413",
        "background": "#f0efea",
        "axis_color": "#141413",
        "gridcolor": "rgba(20, 20, 19, 0.2)",
    },
    {
        "name": "dark",
        "template": "plotly_dark",
        "font_color": "#f0efea",
        "background": "#141413",
        "axis_color": "#f0efea",
        "gridcolor": "rgba(240, 239, 234, 0.2)",
    },
]

output_dir = Path(__file__).resolve().parents[3] / "static"
output_dir.mkdir(parents=True, exist_ok=True)

for kernel_name, kernel_func in kernels.items():
    fig = generate_kernel_plot(
        kernel_name,
        kernel_func,
        x_range_plot,
        u_range_integration,
        lambda_values,
        y_range_plot,
    )

    # Save themed figures to HTML files
    for theme in themes:
        themed_fig = go.Figure(fig)
        themed_fig.update_layout(
            template=theme["template"],
            font=dict(color=theme["font_color"]),
            paper_bgcolor=theme["background"],
            plot_bgcolor=theme["background"],
        )
        themed_fig.update_xaxes(
            showline=True,
            linecolor=theme["axis_color"],
            tickcolor=theme["axis_color"],
            tickfont=dict(color=theme["axis_color"]),
            title_font=dict(color=theme["axis_color"]),
            gridcolor=theme["gridcolor"],
            zeroline=False,
        )
        themed_fig.update_yaxes(
            showline=True,
            linecolor=theme["axis_color"],
            tickcolor=theme["axis_color"],
            tickfont=dict(color=theme["axis_color"]),
            title_font=dict(color=theme["axis_color"]),
            gridcolor=theme["gridcolor"],
            zeroline=False,
        )

        filename = (
            output_dir
            / f"{kernel_name}_dynamic_normalization_kernel_function_{theme['name']}.html"
        )
        themed_fig.write_html(filename, auto_play=False)
        print(f"Saved {kernel_name} kernel plot to {filename}")

    # Show the figure
    fig.show()

结果

我们现在绘制每个核函数的结果。每个图都有一个 $λ$ 滑块，可以实时控制输出。

绘图代码

from pathlib import Path

import numpy as np
import plotly.graph_objects as go


# Define kernel functions
def epanechnikov_kernel(u):
    return np.maximum(0, 0.75 * (1 - u**2))


def tricube_kernel(u):
    return np.maximum(0, (1 - np.abs(u) ** 3) ** 3)


def gaussian_kernel(u):
    return np.exp(-0.5 * u**2) / np.sqrt(2 * np.pi)


# Kernel regression function
def kernel_regression(X, Y, x_curve, kernel_func, bandwidth):
    y_curve = []
    for x in x_curve:
        distances = np.abs(X - x) / bandwidth
        weights = kernel_func(distances)
        weighted_average = (
            np.sum(weights * Y) / np.sum(weights) if np.sum(weights) > 0 else 0
        )
        y_curve.append(weighted_average)
    return y_curve


# Generate data
np.random.seed(42)
n_points = 100
X = np.random.uniform(0, 1, n_points)
epsilon = np.random.normal(0, 1 / 3, n_points)
Y = np.sin(4 * X) + epsilon

# True curve
x_true = np.linspace(0, 1, 500)
y_true = np.sin(4 * x_true)

# Points for kernel estimation
x_curve = x_true

# Kernel functions
kernels = {
    "Epanechnikov": epanechnikov_kernel,
    "Tricube": tricube_kernel,
    "Gaussian": gaussian_kernel,
}

# Range of bandwidths for the slider in logspace
lambda_values = np.logspace(-2, 0, 20)  # From 0.01 to 1

# Generate separate plots for each kernel
themes = [
    {
        "name": "light",
        "template": "plotly_white",
        "font_color": "#141413",
        "background": "#f0efea",
        "axis_color": "#141413",
        "gridcolor": "rgba(20, 20, 19, 0.2)",
    },
    {
        "name": "dark",
        "template": "plotly_dark",
        "font_color": "#f0efea",
        "background": "#141413",
        "axis_color": "#f0efea",
        "gridcolor": "rgba(240, 239, 234, 0.2)",
    },
]

output_dir = Path(__file__).resolve().parents[3] / "static"
output_dir.mkdir(parents=True, exist_ok=True)

# Generate separate plots for each kernel
for kernel_name, kernel_func in kernels.items():
    fig = go.Figure()

    # Add scatter points for noisy data
    fig.add_trace(
        go.Scatter(
            x=X, y=Y, mode="markers", name="Noisy Data", marker=dict(color="gray")
        )
    )

    # Add true function
    fig.add_trace(
        go.Scatter(
            x=x_true,
            y=y_true,
            mode="lines",
            name="True Function",
            line=dict(color="red"),
        )
    )

    # Add initial kernel curve
    initial_bandwidth = lambda_values[0]
    y_curve = kernel_regression(X, Y, x_curve, kernel_func, initial_bandwidth)
    fig.add_trace(
        go.Scatter(
            x=x_curve,
            y=y_curve,
            mode="lines",
            name=f"Nadaraya-Watson ({kernel_name})",
            line=dict(color="green"),
        )
    )

    # Create frames for the slider
    frames = []
    for bandwidth in lambda_values:
        y_curve = kernel_regression(X, Y, x_curve, kernel_func, bandwidth)
        frames.append(
            go.Frame(
                data=[
                    go.Scatter(
                        x=X,
                        y=Y,
                        mode="markers",
                        name="Noisy Data",
                        marker=dict(color="gray"),
                    ),
                    go.Scatter(
                        x=x_true,
                        y=y_true,
                        mode="lines",
                        name="True Function",
                        line=dict(color="red"),
                    ),
                    go.Scatter(
                        x=x_curve,
                        y=y_curve,
                        mode="lines",
                        name=f"Nadaraya-Watson ({kernel_name})",
                        line=dict(color="green"),
                    ),
                ],
                name=f"{bandwidth:.2f}",
            )
        )

    # Add frames to the figure
    fig.frames = frames

    # Add slider
    sliders = [
        {
            "active": 0,
            "currentvalue": {"prefix": "Bandwidth λ: "},
            "steps": [
                {
                    "args": [
                        [f"{bandwidth:.2f}"],
                        {"frame": {"duration": 0, "redraw": True}, "mode": "immediate"},
                    ],
                    "label": f"{bandwidth:.2f}",
                    "method": "animate",
                }
                for bandwidth in lambda_values
            ],
        }
    ]

    # Update layout
    fig.update_layout(
        autosize=True,
        title=f"Nadaraya-Watson Kernel Regression ({kernel_name} Kernel)",
        xaxis_title="X",
        yaxis_title="Y",
        sliders=sliders,
        updatemenus=[
            {
                "buttons": [
                    {
                        "args": [
                            None,
                            {
                                "frame": {"duration": 500, "redraw": True},
                                "fromcurrent": True,
                            },
                        ],
                        "label": "Play",
                        "method": "animate",
                    },
                    {
                        "args": [
                            [None],
                            {
                                "frame": {"duration": 0, "redraw": True},
                                "mode": "immediate",
                            },
                        ],
                        "label": "Pause",
                        "method": "animate",
                    },
                ],
                "direction": "left",
                "pad": {"r": 10, "t": 87},
                "showactive": False,
                "type": "buttons",
                "x": 0.1,
                "xanchor": "right",
                "y": 0,
                "yanchor": "top",
            }
        ],
    )

    # Save the figure to an HTML file per theme
    for theme in themes:
        themed_fig = go.Figure(fig)
        themed_fig.update_layout(
            template=theme["template"],
            font=dict(color=theme["font_color"]),
            paper_bgcolor=theme["background"],
            plot_bgcolor=theme["background"],
        )
        themed_fig.update_xaxes(
            showline=True,
            linecolor=theme["axis_color"],
            tickcolor=theme["axis_color"],
            tickfont=dict(color=theme["axis_color"]),
            title_font=dict(color=theme["axis_color"]),
            gridcolor=theme["gridcolor"],
            zeroline=False,
        )
        themed_fig.update_yaxes(
            showline=True,
            linecolor=theme["axis_color"],
            tickcolor=theme["axis_color"],
            tickfont=dict(color=theme["axis_color"]),
            title_font=dict(color=theme["axis_color"]),
            gridcolor=theme["gridcolor"],
            zeroline=False,
        )

        filename = output_dir / f"{kernel_name}_kernel_regression_{theme['name']}.html"
        themed_fig.write_html(filename, auto_play=False)
        print(f"Saved {kernel_name} kernel plot to {filename}")

    # Show the figure
    fig.show()

我们可以看到，数据的简单加权平均值能够很好地模拟正弦曲线。

局部线性回归

在Nadaraya-Watson核回归中，我们通过核函数 $K_{λ}$ 定义的邻域内进行加权平均。这种方法的一个潜在问题是局部邻域内的平滑插值，因为我们实际上并没有假设该区域遵循任何模型。

如果我们假设每个区域都是局部线性的呢？那么，我们可以求解最小二乘拟合并自由插值！

区域：$k$-近邻

让我们将局部区域定义为输入点的 $k$ 个最近邻。设 $X = [N_{k} (x_{0}) 1]$ ， $Y$ 为对应的 $y$ 值。最小二乘拟合系数为

β = (X^{⊤} X)^{- 1} XY

绘图代码

from pathlib import Path

import numpy as np
import plotly.graph_objects as go

# Generate data
np.random.seed(42)
n_points = 100
X = np.random.uniform(0, 1, n_points)
epsilon = np.random.normal(0, 1 / 3, n_points)
Y = np.sin(4 * X) + epsilon

# True function
x_true = np.linspace(0, 1, 500)
y_true = np.sin(4 * x_true)


# k-NN Local Linear Regression
def knn_linear_regression(X, Y, x_curve, k_range):
    y_curves = {}
    for k in k_range:
        y_curve = []
        for x in x_curve:
            # Find k nearest neighbors
            distances = np.abs(X - x)
            nearest_indices = np.argsort(distances)[:k]

            # Select k nearest neighbors
            X_knn = X[nearest_indices]
            Y_knn = Y[nearest_indices]

            # Create design matrix for k-nearest neighbors
            X_design = np.vstack((np.ones_like(X_knn), X_knn)).T

            # Solve for beta using ordinary least squares
            beta = np.linalg.pinv(X_design.T @ X_design) @ X_design.T @ Y_knn

            # Predict y-value
            y_curve.append(beta[0] + beta[1] * x)
        y_curves[k] = y_curve
    return y_curves


# Common variables
x_curve = np.arange(0, 1, 0.01)
k_range = range(1, 21)  # Values of k from 1 to 20
initial_k = 10  # Default value of k

# Compute LLR using k-NN
y_curves_knn = knn_linear_regression(X, Y, x_curve, k_range)

# Create the Plotly figure
fig = go.Figure()

# Add static traces
fig.add_trace(
    go.Scatter(x=X, y=Y, mode="markers", name="Noisy Data", marker=dict(color="gray"))
)

fig.add_trace(
    go.Scatter(
        x=x_true, y=y_true, mode="lines", name="True Function", line=dict(color="red")
    )
)

# Add the first k-NN curve (k=initial_k)
fig.add_trace(
    go.Scatter(
        x=x_curve,
        y=y_curves_knn[initial_k],
        mode="lines",
        name="k-NN Curve",
        line=dict(color="yellow"),
    )
)

# Define slider steps
steps = []
for k in k_range:
    step = dict(
        method="update",
        args=[
            {"y": [Y, y_true, y_curves_knn[k]]},  # Update y-data for the traces
            {
                "title": f"k-NN Local Linear Regression Curve with k = {k}"
            },  # Update the title dynamically
        ],
        label=f"{k}",
    )
    steps.append(step)

# Add slider to the layout
sliders = [
    dict(
        active=k_range.index(initial_k),  # Use the index of initial_k
        currentvalue={"prefix": "k = "},
        pad={"t": 50},
        steps=steps,
    )
]

fig.update_layout(
    autosize=True,
    sliders=sliders,
    title=f"k-NN Local Linear Regression Curve with k = {initial_k}",
    xaxis_title="X",
    yaxis_title="Y",
)

themes = [
    {
        "name": "light",
        "template": "plotly_white",
        "font_color": "#141413",
        "background": "#f0efea",
        "axis_color": "#141413",
        "gridcolor": "rgba(20, 20, 19, 0.2)",
    },
    {
        "name": "dark",
        "template": "plotly_dark",
        "font_color": "#f0efea",
        "background": "#141413",
        "axis_color": "#f0efea",
        "gridcolor": "rgba(240, 239, 234, 0.2)",
    },
]

output_dir = Path(__file__).resolve().parents[3] / "static"
output_dir.mkdir(parents=True, exist_ok=True)

for theme in themes:
    themed_fig = go.Figure(fig)
    themed_fig.update_layout(
        template=theme["template"],
        font=dict(color=theme["font_color"]),
        paper_bgcolor=theme["background"],
        plot_bgcolor=theme["background"],
    )
    themed_fig.update_xaxes(
        showline=True,
        linecolor=theme["axis_color"],
        tickcolor=theme["axis_color"],
        tickfont=dict(color=theme["axis_color"]),
        title_font=dict(color=theme["axis_color"]),
        gridcolor=theme["gridcolor"],
        zeroline=False,
    )
    themed_fig.update_yaxes(
        showline=True,
        linecolor=theme["axis_color"],
        tickcolor=theme["axis_color"],
        tickfont=dict(color=theme["axis_color"]),
        title_font=dict(color=theme["axis_color"]),
        gridcolor=theme["gridcolor"],
        zeroline=False,
    )

    html_path = output_dir / f"knn_slider_llr_{theme['name']}.html"
    themed_fig.write_html(html_path)
    print(f"Saved interactive k-NN plot to {html_path}")

# Show the plot
fig.show()

我们可以看到，当 $k$ 较小时，输出可能会显得相当粗糙。

区域：核函数

或许我们可以借鉴 Nadaraya-Watson 核函数的一些思想。我们希望不同程度地考虑训练集中的所有点，局部区域内的点赋予较高权重，区域外的点赋予较低权重。

为此，我们可以使用加权最小二乘目标函数，权重为 $W (x_{0}) = diag (K_{λ} (x_{0}, x_{i}))$ 。其解为

β = (X^{⊤} WX)^{- 1} X^{⊤} WY

绘制不同核函数 $D$ 的结果：

绘图代码

from pathlib import Path

import numpy as np
import plotly.graph_objects as go

# Generate data
np.random.seed(42)
n_points = 100
X = np.random.uniform(0, 1, n_points)
epsilon = np.random.normal(0, 1 / 3, n_points)
Y = np.sin(4 * X) + epsilon

# True function
x_true = np.linspace(0, 1, 500)
y_true = np.sin(4 * x_true)


# Kernels
def gaussian_kernel(u):
    return np.exp(-0.5 * u**2)


def epanechnikov_kernel(u):
    return np.maximum(0, 1 - u**2)


def tricube_kernel(u):
    return np.maximum(0, (1 - np.abs(u) ** 3) ** 3)


# Local Linear Regression for a specific kernel
def local_linear_regression(X, Y, x_curve, bandwidths, kernel):
    y_curves = {}
    for λ in bandwidths:
        λ_rounded = round(λ, 2)
        y_curve = []
        for x in x_curve:
            # Calculate weights using the specified kernel
            distances = (X - x) / λ
            weights = kernel(distances)
            W = np.diag(weights)

            # Create design matrix
            X_design = np.vstack((np.ones_like(X), X)).T

            # Solve for beta using weighted least squares
            beta = np.linalg.pinv(X_design.T @ W @ X_design) @ X_design.T @ W @ Y

            # Predict y-value
            y_curve.append(beta[0] + beta[1] * x)
        y_curves[λ_rounded] = y_curve
    return y_curves


# Common variables
x_curve = np.arange(0, 1, 0.01)
bandwidths = np.linspace(0.05, 0.5, 20)
initial_λ = bandwidths[len(bandwidths) // 2]

# Generate plots for each kernel
kernels = {
    "Gaussian Kernel": gaussian_kernel,
    "Epanechnikov Kernel": epanechnikov_kernel,
    "Tricube Kernel": tricube_kernel,
}
plots = []

for kernel_name, kernel_func in kernels.items():
    # Compute LLR with the specified kernel
    y_curves = local_linear_regression(X, Y, x_curve, bandwidths, kernel_func)

    # Create the Plotly figure
    fig = go.Figure()

    # Add static traces
    fig.add_trace(
        go.Scatter(
            x=X, y=Y, mode="markers", name="Noisy Data", marker=dict(color="gray")
        )
    )

    fig.add_trace(
        go.Scatter(
            x=x_true,
            y=y_true,
            mode="lines",
            name="True Function",
            line=dict(color="red"),
        )
    )

    # Add the first LLR curve (using the middle value of bandwidths)
    fig.add_trace(
        go.Scatter(
            x=x_curve,
            y=y_curves[round(initial_λ, 2)],
            mode="lines",
            name=f"{kernel_name} Curve",
            line=dict(color="yellow"),
        )
    )

    # Define slider steps
    steps = []
    for λ in bandwidths:
        λ_rounded = round(λ, 2)
        step = dict(
            method="update",
            args=[
                {"y": [Y, y_true, y_curves[λ_rounded]]},  # Update y-data for the traces
                {
                    "title": f"LLR: {kernel_name} with Bandwidth λ = {λ_rounded}"
                },  # Update the title dynamically
            ],
            label=f"{λ_rounded}",
        )
        steps.append(step)

    # Add slider to the layout
    sliders = [
        dict(
            active=len(bandwidths) // 2,  # Use the index of the middle bandwidth
            currentvalue={"prefix": "λ = "},
            pad={"t": 50},
            steps=steps,
        )
    ]

    fig.update_layout(
        autosize=True,
        sliders=sliders,
        title=f"LLR: {kernel_name} with Bandwidth λ = {round(initial_λ, 2)}",
        xaxis_title="X",
        yaxis_title="Y",
    )

    plots.append(fig)

# Show and save the plots with themed backgrounds
themes = [
    {
        "name": "light",
        "template": "plotly_white",
        "font_color": "#141413",
        "background": "#f0efea",
        "axis_color": "#141413",
        "gridcolor": "rgba(20, 20, 19, 0.2)",
    },
    {
        "name": "dark",
        "template": "plotly_dark",
        "font_color": "#f0efea",
        "background": "#141413",
        "axis_color": "#f0efea",
        "gridcolor": "rgba(240, 239, 234, 0.2)",
    },
]

output_dir = Path(__file__).resolve().parents[3] / "static"
output_dir.mkdir(parents=True, exist_ok=True)

for kernel_name, fig in zip(kernels.keys(), plots):
    fig.show()
    for theme in themes:
        themed_fig = go.Figure(fig)
        themed_fig.update_layout(
            template=theme["template"],
            font=dict(color=theme["font_color"]),
            paper_bgcolor=theme["background"],
            plot_bgcolor=theme["background"],
        )
        themed_fig.update_xaxes(
            showline=True,
            linecolor=theme["axis_color"],
            tickcolor=theme["axis_color"],
            tickfont=dict(color=theme["axis_color"]),
            title_font=dict(color=theme["axis_color"]),
            gridcolor=theme["gridcolor"],
            zeroline=False,
        )
        themed_fig.update_yaxes(
            showline=True,
            linecolor=theme["axis_color"],
            tickcolor=theme["axis_color"],
            tickfont=dict(color=theme["axis_color"]),
            title_font=dict(color=theme["axis_color"]),
            gridcolor=theme["gridcolor"],
            zeroline=False,
        )

        filename = (
            output_dir
            / f"llr_{kernel_name.lower().replace(' ', '_')}_{theme['name']}.html"
        )
        themed_fig.write_html(filename)
        print(f"Saved interactive plot for {kernel_name} to {filename}")

我觉得结果看起来平滑多了！

参考文献

统计学习基础 - Hastie, Tibshirani, 和 Friedman (2009). 一本关于数据挖掘、推断和预测的全面指南。了解更多.