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"cell_type": "markdown",
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"source": [
"Regresja jądrowa
\n",
"\n",
"#### Karolina Oparczyk, Tomasz Grzybowski, Jan Nowak\n"
]
},
{
"cell_type": "markdown",
"id": "f792be04",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"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."
]
},
{
"cell_type": "markdown",
"id": "ce35888e",
"metadata": {
"slideshow": {
"slide_type": "slide"
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},
"source": [
"Właściwości regresji jądrowej:\n",
"* symetryczna - wartość maksymalna leży pośrodku krzywej\n",
"![](\"files/symmetric.PNG\")
\n",
"* powierzchnia pod krzywą funkcji wynosi 1\n",
"* wartość funkcji jądrowej nie jest ujemna"
]
},
{
"cell_type": "markdown",
"id": "dbe6165c",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Do implementacji regresji jądrowej można użyć wielu różnych jąder. Przykłady użyte w projekcie:\n",
"* jądro Gaussa\n",
"\\begin{equation}\n",
"K(x) = \\frac1{h\\sqrt{2\\pi}}e^{-\\frac12(\\frac{x - x_i}h)^2}\n",
"\\end{equation}\n",
"![](\"files/gauss.png\")
"
]
},
{
"cell_type": "markdown",
"id": "f46e92e5",
"metadata": {},
"source": [
"* jądro Epanechnikova\n",
"\\begin{equation}\n",
"K(x) = (\\frac34)(1-(\\frac{x - x_i}h)^2) \\text{ dla } {|x|\\leq1}\n",
"\\end{equation}\n",
"![](\"files/epanechnikov.png\")
"
]
},
{
"cell_type": "markdown",
"id": "6d60bbc1",
"metadata": {},
"source": [
"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."
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "4ae1bce9",
"metadata": {},
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"source": [
"import ipywidgets as widgets\n",
"import numpy as np\n",
"import plotly.express as px\n",
"import plotly.graph_objs as go\n",
"import pandas as pd \n",
"import KernelRegression\n",
"\n",
"fires_thefts = pd.read_csv('fires_thefts.csv', names=['x','y'])\n",
"XXX = np.sort(np.array(fires_thefts.x))\n",
"YYY = np.array(fires_thefts.y)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "loved-clinton",
"metadata": {},
"outputs": [],
"source": [
"\n",
"dropdown_bw = widgets.Dropdown(options=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10], value=1, description='Szerokość okna')\n",
"\n",
"def interactive_kernel(bw_manual):\n",
" fig = px.scatter(x=XXX,y=YYY)\n",
" for i in range(0, len(XXX), bw_manual):\n",
" fig.add_vline(x=i)\n",
" fig.show()\n",
" \n",
" Y_pred_gauss = KernelRegression.ker_reg(XXX, YYY, bw_manual, 'gauss')\n",
" Y_pred_epanechnikov = KernelRegression.ker_reg(XXX, YYY, bw_manual, 'epanechnikov')\n",
"\n",
" fig = px.scatter(x=XXX,y=YYY)\n",
" fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred_gauss), name='Gauss', mode='lines'))\n",
" fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred_epanechnikov), name='Epanechnikov', mode='lines'))\n",
" fig.show()"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "injured-english",
"metadata": {},
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"data": {
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""
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"execution_count": 12,
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},
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"fill": {
"color": "#C8D4E3"
},
"line": {
"color": "white"
}
},
"type": "table"
}
]
},
"layout": {
"annotationdefaults": {
"arrowcolor": "#2a3f5f",
"arrowhead": 0,
"arrowwidth": 1
},
"autotypenumbers": "strict",
"coloraxis": {
"colorbar": {
"outlinewidth": 0,
"ticks": ""
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},
"colorscale": {
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0,
"#8e0152"
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0.1,
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0.2,
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0.3,
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0.4,
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"font": {
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"geo": {
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"landcolor": "#E5ECF6",
"showlakes": true,
"showland": true,
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"hoverlabel": {
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"hovermode": "closest",
"mapbox": {
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"bgcolor": "#E5ECF6",
"radialaxis": {
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"scene": {
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},
"shapedefaults": {
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"bgcolor": "#E5ECF6",
"caxis": {
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},
"title": {
"x": 0.05
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"xaxis": {
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"ticks": "",
"title": {
"standoff": 15
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"yaxis": {
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"xaxis": {
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"title": {
"text": "x"
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"title": {
"text": "y"
}
}
}
},
"text/html": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"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",
"Y_pred_gauss = KernelRegression.ker_reg(XXX, YYY, bw_manual, 'gauss')\n",
"Y_pred_epanechnikov = KernelRegression.ker_reg(XXX, YYY, bw_manual, 'epanechnikov')\n",
"\n",
"fig = px.scatter(x=XXX,y=YYY)\n",
"fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred_gauss), name='Gauss', mode='lines'))\n",
"fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred_epanechnikov), name='Epanechnikov', mode='lines'))\n",
"fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred_linear.flatten().tolist()), name='Linear', mode='lines'))"
]
}
],
"metadata": {
"celltoolbar": "Slideshow",
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.5"
}
},
"nbformat": 4,
"nbformat_minor": 5
}