feat: replace linear scaling law with power law with offset

- Replace log-log linear regression with nonlinear power law fitting: y = a * x^b + c
- Use scipy.optimize.curve_fit for parameter estimation
- Replace R² metric with Raw RMSE and Log-RMSE for better fit quality assessment
- Update all three plot modes: Overall, By Dataset (average), By Dataset (separate)
- Display complete fitting formula in legends for consistency
- Add test_fitting.py to validate the fitting algorithm

The new approach fits the power law directly in original space, then displays
the fitted curve in log-log coordinates, resulting in a natural curve rather
than forcing a straight line.

app.py

@@ -6,6 +6,7 @@ from dotenv import load_dotenv
 from matplotlib.colors import LinearSegmentedColormap
 import plotly.graph_objects as go
 import numpy as np
+from scipy.optimize import curve_fit
 from huggingface_hub import HfApi
 from huggingface_hub.hf_api import HTTPError
 from huggingface_hub.utils import GatedRepoError
@@ -223,6 +224,82 @@ def submit_model(name):
         return "ERROR: Unexpected error. Please try again later."
 
 
+def power_law_with_offset(x, a, b, c):
+    """带偏置的幂律函数: y = a * x^b + c"""
+    return a * np.power(x, b) + c
+
+
+def fit_power_law_with_offset(x_values, y_values):
+    """
+    使用带偏置的幂律拟合原始数据
+    返回: (params, raw_rmse, log_rmse, fit_x, fit_y)
+    """
+    x_arr = np.array(x_values)
+    y_arr = np.array(y_values)
+
+    # 初始参数估计
+    # 使用简单的幂律拟合作为初始值
+    log_x = np.log10(x_arr)
+    log_y = np.log10(y_arr)
+    slope, intercept = np.polyfit(log_x, log_y, 1)
+
+    a_init = 10**intercept
+    b_init = slope
+    c_init = 0  # 偏置初始值设为0
+
+    try:
+        # 使用curve_fit进行非线性拟合
+        params, _ = curve_fit(
+            power_law_with_offset,
+            x_arr,
+            y_arr,
+            p0=[a_init, b_init, c_init],
+            maxfev=10000
+        )
+        a, b, c = params
+
+        # 计算预测值
+        y_pred = power_law_with_offset(x_arr, a, b, c)
+
+        # 计算原始空间 RMSE
+        raw_rmse = np.sqrt(np.mean((y_arr - y_pred) ** 2))
+
+        # 计算对数空间 RMSE
+        log_y_actual = np.log10(y_arr)
+        log_y_pred = np.log10(y_pred)
+        log_rmse = np.sqrt(np.mean((log_y_actual - log_y_pred) ** 2))
+
+        # 生成拟合曲线的点
+        x_min, x_max = min(x_values), max(x_values)
+        fit_x = np.linspace(x_min * 0.8, x_max * 1.2, 100)
+        fit_y = power_law_with_offset(fit_x, a, b, c)
+
+        return params, raw_rmse, log_rmse, fit_x, fit_y
+    except Exception as e:
+        print(f"Fitting failed: {e}")
+        # 如果拟合失败，返回简单幂律拟合结果
+        a = a_init
+        b = b_init
+        c = 0
+        params = (a, b, c)
+
+        y_pred = a * np.power(x_arr, b)
+
+        # 计算原始空间 RMSE
+        raw_rmse = np.sqrt(np.mean((y_arr - y_pred) ** 2))
+
+        # 计算对数空间 RMSE
+        log_y_actual = np.log10(y_arr)
+        log_y_pred = np.log10(y_pred)
+        log_rmse = np.sqrt(np.mean((log_y_actual - log_y_pred) ** 2))
+
+        x_min, x_max = min(x_values), max(x_values)
+        fit_x = np.linspace(x_min * 0.8, x_max * 1.2, 100)
+        fit_y = a * np.power(fit_x, b)
+
+        return params, raw_rmse, log_rmse, fit_x, fit_y
+
+
 def create_scaling_plot(data_manager: DataManager, period: str):
     new_df = data_manager.query(
         period=period,
@@ -256,25 +333,9 @@ def create_scaling_plot(data_manager: DataManager, period: str):
     x_dtick = (x_max - x_min) / 4
     y_dtick = (y_max - y_min) / 4
 
-    # 在对数空间中进行线性回归拟合
-    # log(y) = a * log(x) + b  =>  y = 10^b * x^a
-    log_x = np.log10(np.array(x_values))
-    log_y = np.log10(np.array(y_values))
-
-    # 线性拟合: log_y = slope * log_x + intercept
-    slope, intercept = np.polyfit(log_x, log_y, 1)
-
-    # 计算 R² 值
-    log_y_pred = slope * log_x + intercept
-    ss_res = np.sum((log_y - log_y_pred) ** 2)
-    ss_tot = np.sum((log_y - np.mean(log_y)) ** 2)
-    r_squared = 1 - (ss_res / ss_tot) if ss_tot > 0 else 0
-
-    # 生成拟合线的点（在对数空间中是直线）
-    fit_x_log = np.linspace(x_min - 0.1, x_max + 0.1, 100)
-    fit_y_log = slope * fit_x_log + intercept
-    fit_x = 10**fit_x_log
-    fit_y = 10**fit_y_log
+    # 使用带偏置的幂律拟合原始数据
+    params, raw_rmse, log_rmse, fit_x, fit_y = fit_power_law_with_offset(x_values, y_values)
+    a, b, c = params
 
     fig = go.Figure()
 
@@ -294,8 +355,11 @@ def create_scaling_plot(data_manager: DataManager, period: str):
         )
     )
 
-    # 添加拟合直线
-    fit_label = f"Fit: y = {10**intercept:.2f} × x^{slope:.3f} (R² = {r_squared:.3f})"
+    # 添加拟合曲线
+    if abs(c) < 0.01:
+        fit_label = f"Fit: y = {a:.2f} × x^{b:.3f}<br>Raw RMSE: {raw_rmse:.2f}, Log-RMSE: {log_rmse:.3f}"
+    else:
+        fit_label = f"Fit: y = {a:.2f} × x^{b:.3f} + {c:.2f}<br>Raw RMSE: {raw_rmse:.2f}, Log-RMSE: {log_rmse:.3f}"
     fig.add_trace(
         go.Scatter(
             x=fit_x.tolist(),
@@ -412,24 +476,9 @@ def create_category_scaling_plot(data_manager: DataManager, period: str, selecte
 
             color = "#39C5BB"
 
-            # 在对数空间中进行线性回归拟合
-            log_x = np.log10(np.array(x_vals))
-            log_y = np.log10(np.array(y_vals))
-
-            slope, intercept = np.polyfit(log_x, log_y, 1)
-
-            # 计算 R² 值
-            log_y_pred = slope * log_x + intercept
-            ss_res = np.sum((log_y - log_y_pred) ** 2)
-            ss_tot = np.sum((log_y - np.mean(log_y)) ** 2)
-            r_squared = 1 - (ss_res / ss_tot) if ss_tot > 0 else 0
-
-            # 生成拟合线的点
-            x_min_local, x_max_local = np.log10(min(x_vals)), np.log10(max(x_vals))
-            fit_x_log = np.linspace(x_min_local - 0.05, x_max_local + 0.05, 100)
-            fit_y_log = slope * fit_x_log + intercept
-            fit_x = 10**fit_x_log
-            fit_y = 10**fit_y_log
+            # 使用带偏置的幂律拟合原始数据
+            params, raw_rmse, log_rmse, fit_x, fit_y = fit_power_law_with_offset(x_vals, y_vals)
+            a, b, c = params
 
             # 构建数据集名称列表（用于hover显示）
             datasets_label = f"Average of {len(selected_datasets)} datasets"
@@ -450,8 +499,11 @@ def create_category_scaling_plot(data_manager: DataManager, period: str, selecte
                 )
             )
 
-            # 添加拟合直线
-            fit_label = f"Fit: y = {10**intercept:.2f} × x^{slope:.3f} (R² = {r_squared:.3f})"
+            # 添加拟合曲线
+            if abs(c) < 0.01:
+                fit_label = f"Fit: y = {a:.2f} × x^{b:.3f}<br>Raw RMSE: {raw_rmse:.2f}, Log-RMSE: {log_rmse:.3f}"
+            else:
+                fit_label = f"Fit: y = {a:.2f} × x^{b:.3f} + {c:.2f}<br>Raw RMSE: {raw_rmse:.2f}, Log-RMSE: {log_rmse:.3f}"
             fig.add_trace(
                 go.Scatter(
                     x=fit_x.tolist(),
@@ -487,24 +539,9 @@ def create_category_scaling_plot(data_manager: DataManager, period: str, selecte
 
             color = color_palette[idx % len(color_palette)]
 
-            # 在对数空间中进行线性回归拟合
-            log_x = np.log10(np.array(x_vals))
-            log_y = np.log10(np.array(y_vals))
-
-            slope, intercept = np.polyfit(log_x, log_y, 1)
-
-            # 计算 R² 值
-            log_y_pred = slope * log_x + intercept
-            ss_res = np.sum((log_y - log_y_pred) ** 2)
-            ss_tot = np.sum((log_y - np.mean(log_y)) ** 2)
-            r_squared = 1 - (ss_res / ss_tot) if ss_tot > 0 else 0
-
-            # 生成拟合线的点
-            x_min_local, x_max_local = np.log10(min(x_vals)), np.log10(max(x_vals))
-            fit_x_log = np.linspace(x_min_local - 0.05, x_max_local + 0.05, 100)
-            fit_y_log = slope * fit_x_log + intercept
-            fit_x = 10**fit_x_log
-            fit_y = 10**fit_y_log
+            # 使用带偏置的幂律拟合原始数据
+            params, raw_rmse, log_rmse, fit_x, fit_y = fit_power_law_with_offset(x_vals, y_vals)
+            a, b, c = params
 
             # 添加数据点
             fig.add_trace(
@@ -523,8 +560,11 @@ def create_category_scaling_plot(data_manager: DataManager, period: str, selecte
                 )
             )
 
-            # 添加拟合直线
-            fit_label = f"{dataset} fit (slope={slope:.3f}, R²={r_squared:.3f})"
+            # 添加拟合曲线
+            if abs(c) < 0.01:
+                fit_label = f"{dataset}: y = {a:.2f} × x^{b:.3f}<br>Raw RMSE: {raw_rmse:.2f}, Log-RMSE: {log_rmse:.3f}"
+            else:
+                fit_label = f"{dataset}: y = {a:.2f} × x^{b:.3f} + {c:.2f}<br>Raw RMSE: {raw_rmse:.2f}, Log-RMSE: {log_rmse:.3f}"
             fig.add_trace(
                 go.Scatter(
                     x=fit_x.tolist(),

test_fitting.py

@@ -0,0 +1,109 @@
+"""测试带偏置的幂律拟合功能"""
+import numpy as np
+from scipy.optimize import curve_fit
+
+def power_law_with_offset(x, a, b, c):
+    """带偏置的幂律函数: y = a * x^b + c"""
+    return a * np.power(x, b) + c
+
+
+def fit_power_law_with_offset(x_values, y_values):
+    """
+    使用带偏置的幂律拟合原始数据
+    返回: (params, raw_rmse, log_rmse, fit_x, fit_y)
+    """
+    x_arr = np.array(x_values)
+    y_arr = np.array(y_values)
+
+    # 初始参数估计
+    # 使用简单的幂律拟合作为初始值
+    log_x = np.log10(x_arr)
+    log_y = np.log10(y_arr)
+    slope, intercept = np.polyfit(log_x, log_y, 1)
+
+    a_init = 10**intercept
+    b_init = slope
+    c_init = 0  # 偏置初始值设为0
+
+    try:
+        # 使用curve_fit进行非线性拟合
+        params, _ = curve_fit(
+            power_law_with_offset,
+            x_arr,
+            y_arr,
+            p0=[a_init, b_init, c_init],
+            maxfev=10000
+        )
+        a, b, c = params
+
+        # 计算预测值
+        y_pred = power_law_with_offset(x_arr, a, b, c)
+
+        # 计算原始空间 RMSE
+        raw_rmse = np.sqrt(np.mean((y_arr - y_pred) ** 2))
+
+        # 计算对数空间 RMSE
+        log_y_actual = np.log10(y_arr)
+        log_y_pred = np.log10(y_pred)
+        log_rmse = np.sqrt(np.mean((log_y_actual - log_y_pred) ** 2))
+
+        # 生成拟合曲线的点
+        x_min, x_max = min(x_values), max(x_values)
+        fit_x = np.linspace(x_min * 0.8, x_max * 1.2, 100)
+        fit_y = power_law_with_offset(fit_x, a, b, c)
+
+        return params, raw_rmse, log_rmse, fit_x, fit_y
+    except Exception as e:
+        print(f"Fitting failed: {e}")
+        # 如果拟合失败，返回简单幂律拟合结果
+        a = a_init
+        b = b_init
+        c = 0
+        params = (a, b, c)
+
+        y_pred = a * np.power(x_arr, b)
+
+        # 计算原始空间 RMSE
+        raw_rmse = np.sqrt(np.mean((y_arr - y_pred) ** 2))
+
+        # 计算对数空间 RMSE
+        log_y_actual = np.log10(y_arr)
+        log_y_pred = np.log10(y_pred)
+        log_rmse = np.sqrt(np.mean((log_y_actual - log_y_pred) ** 2))
+
+        x_min, x_max = min(x_values), max(x_values)
+        fit_x = np.linspace(x_min * 0.8, x_max * 1.2, 100)
+        fit_y = a * np.power(fit_x, b)
+
+        return params, raw_rmse, log_rmse, fit_x, fit_y
+
+
+if __name__ == "__main__":
+    # 测试数据：模拟一些模型参数和压缩率的关系
+    # 假设真实关系为 y = 50 * x^(-0.1) + 10
+    x_test = np.array([1, 3, 7, 13, 20, 30])
+    y_true = 50 * np.power(x_test, -0.1) + 10
+    # 添加一些噪声
+    np.random.seed(42)
+    y_test = y_true + np.random.normal(0, 0.5, len(x_test))
+
+    print("测试数据:")
+    print(f"x: {x_test}")
+    print(f"y: {y_test}")
+    print()
+
+    # 进行拟合
+    params, raw_rmse, log_rmse, fit_x, fit_y = fit_power_law_with_offset(x_test.tolist(), y_test.tolist())
+    a, b, c = params
+
+    print("拟合结果:")
+    print(f"a = {a:.4f}")
+    print(f"b = {b:.4f}")
+    print(f"c = {c:.4f}")
+    print(f"Raw RMSE = {raw_rmse:.4f}")
+    print(f"Log-RMSE = {log_rmse:.4f}")
+    print()
+    print(f"拟合公式: y = {a:.2f} * x^{b:.3f} + {c:.2f}")
+    print()
+    print("真实参数: a=50, b=-0.1, c=10")
+    print("拟合成功!" if raw_rmse < 2.0 else "拟合可能需要调整")