diff --git a/kernel_regression.ipynb b/kernel_regression.ipynb
index a860370..54f4fa0 100644
--- a/kernel_regression.ipynb
+++ b/kernel_regression.ipynb
@@ -25,7 +25,7 @@
"source": [
"Regresja jądrowa używana jest jako funkcja wagi do opracowania modelu regresji nieparametrycznej. Nadaje ona niektórym elementom zbioru większą \"wagę\", która ma wpływ na ostateczny wynik. \n",
"\n",
- "Można ją porównać do rysowania krzywej na wykresie punktowym tak, aby była jak najlepiej do nich dopasowana. Jest mniej wrażliwa na wartości odstające, niż na przykład regresja liniowa."
+ "Można ją porównać do rysowania krzywej na wykresie punktowym tak, aby była jak najlepiej do nich dopasowana."
]
},
{
@@ -53,10 +53,10 @@
}
},
"source": [
- "Przykłady:\n",
+ "Do implementacji regresji jądrowej można użyć wielu różnych jąder. Przykłady użyte w projekcie:\n",
"* jądro Gaussa\n",
"\\begin{equation}\n",
- "K(x) = (2\\pi)^{-\\frac12} \\text{ exp } (-\\frac{{x}^{2}}2) \\text{ lg}(x)\n",
+ "K(x) = \\frac1{h\\sqrt{2\\pi}}e^{-\\frac12(\\frac{x - x_i}h)^2}\n",
"\\end{equation}\n",
"
"
]
@@ -68,10 +68,106528 @@
"source": [
"* jądro Epanechnikova\n",
"\\begin{equation}\n",
- "K(x) = (\\frac34)(1-x^2)l_{|x|\\leq1}(x)\n",
+ "K(x) = (\\frac34)(1-(\\frac{x - x_i}h)^2) \\text{ dla } {|x|\\leq1}\n",
"\\end{equation}\n",
"
"
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+ "Istotne znaczenie ma nie tylko dobór jądra, ale również parametru wygładzania, czyli szerokości okna. W zależności od niego, punkty są grupowane i dla każdej grupy wyliczana jest wartość funkcji. Jeśli okno będzie zbyt szerokie, funkcja będzie bardziej przypominała prostą (under-fitting). Natomiast jeśli będzie zbyt wąskie, funkcja będzie za bardzo \"skakać\" (over-fitting).\n",
+ "\n",
+ "Wyliczenie wartości funkcji polega na wzięciu średniej ważonej z $y_{i}$\n",
+ " dla takich $x_{i}$, które znajdują się blisko x, dla którego wyznaczamy wartość. Wagi przy $y_{i}$ dla x sumują się do 1 i są wyższe, kiedy $x_{i}$ jest bliżej x oraz niższe w przeciwnym przypadku.\n",
+ "\n",
+ "Poniższe przykłady używają jądra Gaussa do obliczania wag dla $y_{i}$, korzystając ze wzoru \\begin{equation}\n",
+ "K(x) = \\frac1{h\\sqrt{2\\pi}}e^{-\\frac12(\\frac{x - x_i}h)^2}\n",
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+ "import numpy as np\n",
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+ "import pandas as pd \n",
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+ "# kernel regression\n",
+ "kernel_x = np.arange(-20,20, 0.1)\n",
+ "\n",
+ "## Plotting gaussian for all input x points \n",
+ "kernel_fns = {'kernel_x': kernel_x}\n",
+ "for input_x in XXX: \n",
+ " input_string= 'x_value_{}'.format(np.round(input_x,2)) \n",
+ " kernel_fns[input_string] = KernelRegression.kernel_function(bw_manual, kernel_x, input_x)\n",
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+ "metadata": {},
+ "output_type": "display_data"
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+ "source": [
+ "from sklearn.linear_model import LinearRegression\n",
+ "# linear regression\n",
+ "reg = LinearRegression().fit(XXX.reshape(-1, 1), YYY.reshape(-1, 1))\n",
+ "Y_pred_linear = reg.predict(XXX.reshape(-1, 1))\n",
+ "\n",
+ "# kernel regression\n",
+ "kernel_x = np.arange(-20,20, 0.1)\n",
+ "\n",
+ "## Plotting gaussian for all input x points \n",
+ "kernel_fns = {'kernel_x': kernel_x}\n",
+ "for input_x in XXX: \n",
+ " input_string= 'x_value_{}'.format(np.round(input_x,2)) \n",
+ " kernel_fns[input_string] = KernelRegression.kernel_function(bw_manual, kernel_x, input_x)\n",
+ "\n",
+ "kernels_df = pd.DataFrame(data=kernel_fns)\n",
+ "y_all = kernels_df.drop(columns='kernel_x')\n",
+ "px.line(kernels_df, x='kernel_x', y=y_all.columns, title='Gaussian for all input points', range_x=[-5,100])\n",
+ "\n",
+ "\n",
+ "Y_pred = []\n",
+ "for input_x in XXX: \n",
+ " w = []\n",
+ " Y_single = KernelRegression.single_y_pred(bw_manual, input_x, XXX, YYY)\n",
+ " Y_pred.append(Y_single)\n",
+ " \n",
+ "fig = px.scatter(x=XXX,y=YYY)\n",
+ "fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred), name='Manual KR', mode='lines'))\n",
+ "fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred_linear.flatten().tolist()), name='Linear', mode='lines'))"
+ ]
}
],
"metadata": {
@@ -91,7 +106609,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.8.8"
+ "version": "3.9.2"
}
},
"nbformat": 4,