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", "" ] + }, + { + "cell_type": "markdown", + "id": "8096322f", + "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.\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", + "\\end{equation}" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "1f48cc94", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + " \n", + " " + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "application/vnd.plotly.v1+json": { + "config": { + "plotlyServerURL": "https://plot.ly" + }, + "data": [ + { + "hovertemplate": "x=%{x}
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" + ] + }, + "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", + "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,