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"source": [
"<h1><center>Regresja jądrowa</center></h1>\n",
"\n",
"#### <center>Karolina Oparczyk, Tomasz Grzybowski, Jan Nowak</center>\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",
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"Można ją porównać do rysowania krzywej na wykresie punktowym tak, aby była jak najlepiej do nich dopasowana."
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]
},
{
"cell_type": "markdown",
"id": "ce35888e",
"metadata": {
"slideshow": {
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"source": [
"Właściwości regresji jądrowej:\n",
"* symetryczna - wartość maksymalna leży pośrodku krzywej\n",
"<img src=\"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"
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},
"source": [
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"Do implementacji regresji jądrowej można użyć wielu różnych jąder. Przykłady użyte w projekcie:\n",
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"* jądro Gaussa\n",
"\\begin{equation}\n",
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"K(x) = \\frac1{h\\sqrt{2\\pi}}e^{-\\frac12(\\frac{x - x_i}h)^2}\n",
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"\\end{equation}\n",
"<img src=\"files/gauss.png\">"
]
},
{
"cell_type": "markdown",
"id": "f46e92e5",
"metadata": {},
"source": [
"* jądro Epanechnikova\n",
"\\begin{equation}\n",
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"K(x) = (\\frac34)(1-(\\frac{x - x_i}h)^2) \\text{ dla } {|x|\\leq1}\n",
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"\\end{equation}\n",
"<img src=\"files/epanechnikov.png\">"
]
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},
{
"cell_type": "markdown",
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"id": "6d60bbc1",
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"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",
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" 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."
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]
},
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"execution_count": 10,
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"id": "4ae1bce9",
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"metadata": {},
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"outputs": [],
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"source": [
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"import ipywidgets as widgets\n",
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"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",
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"YYY = np.array(fires_thefts.y)"
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]
},
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{
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"execution_count": 11,
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"id": "loved-clinton",
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"\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",
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" 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",
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" 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()"
]
},
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"cell_type": "code",
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"execution_count": 12,
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"id": "injured-english",
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},
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"execution_count": 12,
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"metadata": {},
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},
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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",
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"Y_pred_gauss = KernelRegression.ker_reg(XXX, YYY, bw_manual, 'gauss')\n",
"Y_pred_epanechnikov = KernelRegression.ker_reg(XXX, YYY, bw_manual, 'epanechnikov')\n",
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"\n",
"fig = px.scatter(x=XXX,y=YYY)\n",
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"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",
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"fig.add_trace(go.Scatter(x=XXX, y=np.array(Y_pred_linear.flatten().tolist()), name='Linear', mode='lines'))"
]
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}
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}
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