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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "kernel_x = np.arange(-2,2,0.1)\n", + "col = KernelRegression.epanechnikov_list(1, kernel_x, 0)\n", + "px.line(x=kernel_x, y=col, title='Funkcja jądrowa Epanechnikova')" ] }, { @@ -81,22 +2216,24 @@ "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." + " 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", + "
\\begin{equation}w_i= \\frac{K_i}{\\sum\\limits_{i=1}^n K_i} \\end{equation}\n", + "

\n", + "\\begin{equation} Y=w_1*x_1+w_2*x_2+ \\text{...} +w_n*x_n \\end{equation}
\n", + "" ] }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 3, "id": "4ae1bce9", "metadata": {}, "outputs": [], "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", @@ -105,7 +2242,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 4, "id": "loved-clinton", "metadata": {}, "outputs": [], @@ -113,31 +2250,42 @@ "\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", + "def interactive_kernel(bw_manual): \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()\n", + " \n", + " # kernel regression\n", + " kernel_x = np.arange(min(XXX)-5,max(XXX)+5, 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", + " fig = px.line(kernels_df, x='kernel_x', y=y_all.columns, title='Gaussian for all input points', range_x=[min(XXX)-5,max(XXX)+5]) \n", " fig.show()" ] }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 5, "id": "injured-english", - "metadata": {}, + "metadata": { + "scrolled": false + }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { - "model_id": "a38d1ceed30b42d19a56951cf88e16e6", + "model_id": "ad45339f5a46492393ca1158653290e3", "version_major": 2, "version_minor": 0 }, @@ -154,7 +2302,7 @@ "" ] }, - "execution_count": 12, + "execution_count": 5, "metadata": {}, "output_type": "execute_result" } @@ -165,7 +2313,452 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 43, + "id": "collect-imperial", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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82.0K(5.0)0.080656908231
92.0K(5.4)0.069964098725
102.0K(5.6)0.064728685034
112.0K(5.7)0.062156096214
122.0K(6.2)0.049909155211
132.0K(6.9)0.035033879111
142.0K(7.2)0.029606153722
152.0K(7.3)0.027928534316
162.0K(7.7)0.021871938327
172.0K(8.6)0.01182486439
182.0K(9.0)0.008740629729
192.0K(9.5)0.005842766830
202.0K(10.5)0.002402033340
212.0K(10.5)0.00240203333254.121103
222.0K(10.7)0.001984177541
232.0K(10.8)0.0018003521147
242.0K(11.0)0.001477282822
252.0K(11.3)0.001088939729
262.0K(11.9)0.000574189646
272.0K(12.2)0.000410739723
282.0K(15.1)0.00000962224
292.0K(15.1)0.000009622231
302.0K(16.5)0.000001124639
312.0K(17.4)0.000000252215
322.0K(18.4)0.000000043132
332.0K(18.5)0.000000035927
342.0K(21.6)0.000000000132
352.0K(21.8)0.000000000034
362.0K(23.3)0.000000000017
372.0K(28.6)0.000000000046
382.0K(29.1)0.000000000042
392.0K(34.1)0.000000000043
402.0K(36.2)0.000000000034
412.0K(39.7)0.000000000019
\n", + "
" + ], + "text/plain": [ + " Input_x kernel_x weigth Y Y=w0*x0+w1*x1+...+w41*x41\n", + "0 2.0 K(2.0) 0.1329807601 29 \n", + "1 2.0 K(2.2) 0.1326855754 44 \n", + "2 2.0 K(2.2) 0.1326855754 36 \n", + "3 2.0 K(2.5) 0.1311465720 37 \n", + "4 2.0 K(3.4) 0.1192611431 53 \n", + "5 2.0 K(3.6) 0.1153512977 68 \n", + "6 2.0 K(4.0) 0.1064826685 75 \n", + "7 2.0 K(4.8) 0.0860259420 18 \n", + "8 2.0 K(5.0) 0.0806569082 31 \n", + "9 2.0 K(5.4) 0.0699640987 25 \n", + "10 2.0 K(5.6) 0.0647286850 34 \n", + "11 2.0 K(5.7) 0.0621560962 14 \n", + "12 2.0 K(6.2) 0.0499091552 11 \n", + "13 2.0 K(6.9) 0.0350338791 11 \n", + "14 2.0 K(7.2) 0.0296061537 22 \n", + "15 2.0 K(7.3) 0.0279285343 16 \n", + "16 2.0 K(7.7) 0.0218719383 27 \n", + "17 2.0 K(8.6) 0.0118248643 9 \n", + "18 2.0 K(9.0) 0.0087406297 29 \n", + "19 2.0 K(9.5) 0.0058427668 30 \n", + "20 2.0 K(10.5) 0.0024020333 40 \n", + "21 2.0 K(10.5) 0.0024020333 32 54.121103\n", + "22 2.0 K(10.7) 0.0019841775 41 \n", + "23 2.0 K(10.8) 0.0018003521 147 \n", + "24 2.0 K(11.0) 0.0014772828 22 \n", + "25 2.0 K(11.3) 0.0010889397 29 \n", + "26 2.0 K(11.9) 0.0005741896 46 \n", + "27 2.0 K(12.2) 0.0004107397 23 \n", + "28 2.0 K(15.1) 0.0000096222 4 \n", + "29 2.0 K(15.1) 0.0000096222 31 \n", + "30 2.0 K(16.5) 0.0000011246 39 \n", + "31 2.0 K(17.4) 0.0000002522 15 \n", + "32 2.0 K(18.4) 0.0000000431 32 \n", + "33 2.0 K(18.5) 0.0000000359 27 \n", + "34 2.0 K(21.6) 0.0000000001 32 \n", + "35 2.0 K(21.8) 0.0000000000 34 \n", + "36 2.0 K(23.3) 0.0000000000 17 \n", + "37 2.0 K(28.6) 0.0000000000 46 \n", + "38 2.0 K(29.1) 0.0000000000 42 \n", + "39 2.0 K(34.1) 0.0000000000 43 \n", + "40 2.0 K(36.2) 0.0000000000 34 \n", + "41 2.0 K(39.7) 0.0000000000 19 " + ] + }, + "execution_count": 43, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "bw_manual = 3\n", + "\n", + "kernel_x = XXX\n", + "col3 = KernelRegression.kernel_function(bw_manual, kernel_x, XXX[0])\n", + "# Dataframe for a single observation point x_i. In the code x_i comes from new_x\n", + "data = {'Input_x': [XXX[0] for x in kernel_x],\n", + " 'kernel_x': kernel_x,\n", + " 'weigth': col3,\n", + " 'Y': YYY,\n", + " 'Y=w0*x0+w1*x1+...+w41*x41': ''\n", + " }\n", + "single_pt_KE = pd.DataFrame(data=data)\n", + "single_pt_KE.at[len(YYY)//2, 'Y=w0*x0+w1*x1+...+w41*x41'] = sum([col3[index]*YYY[index] for index, x in enumerate(kernel_x)])\n", + "single_pt_KE['weigth'] = single_pt_KE['weigth'].map('{:,.10f}'.format)\n", + "single_pt_KE['kernel_x'] = single_pt_KE['kernel_x'].map('K({})'.format)\n", + "single_pt_KE" + ] + }, + { + "cell_type": "code", + "execution_count": 7, "id": "0406c2c4", "metadata": {}, "outputs": [ @@ -530,14 +3123,14 @@ 34.3922802405316, 34.354702225608875, 34.342176220634634, - 34.29207220073767, + 34.29207220073768, 34.17933815596951, 34.12923413607255, 34.06660411120134, 33.941344061458935, 33.941344061458935, 33.91629205151045, - 33.90376604653621, + 33.90376604653622, 33.87871403658774, 33.841136021665015, 33.76597999181957, @@ -550,12 +3143,12 @@ 32.93926366351969, 32.55095750931823, 32.52590549936975, - 32.338015424756136, + 32.33801542475614, 31.674137161121386, 31.611507136250186, 30.985206887538155, 30.7221607830791, - 30.28375060898068 + 30.283750608980682 ] } ], @@ -1395,9 +3988,9 @@ } }, "text/html": [ - "