{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"name":"AdvPythonVisualVariance_2_ind_1_dep.ipynb","provenance":[],"collapsed_sections":[],"authorship_tag":"ABX9TyNx46tHLxcL0I39nCY0l3rU"},"kernelspec":{"name":"python3","display_name":"Python 3"}},"cells":[{"cell_type":"code","metadata":{"id":"CB3DFanzE_GC","colab_type":"code","colab":{}},"source":["import copy\n","import plotly\n","import plotly.express as px\n","import plotly.graph_objects as go\n","import numpy as np\n","import sklearn\n","from sklearn import linear_model\n"],"execution_count":null,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"ERixrjIYI8g9","colab_type":"text"},"source":["# **Regression revisited: 2 independent, 1 dependent**\n","\n","Last time we did linear regression with 1 independent variable, x, and 1 dependent, y. Now lets use 2 independent variables, x and y, to predict one dependent variable, z.\n","\n","### We'll reuse this function from last time, to plot point clouds and their means.\n","\n"]},{"cell_type":"code","metadata":{"id":"z2bC73DAH-nO","colab_type":"code","colab":{}},"source":["def plot_cloud_with_mean( A, \n"," point_cloud_name = '',\n"," plot_title = '', \n"," axis_limits = None, \n"," return_plotting_data = False,\n"," show_plot = True):\n"," # RECALCULATE THE MEAN X AND Y COORDINATES\n"," mean_x, mean_y, mean_z = A.mean(axis=0)\n"," # GENERATE PLOTTING DATA FOR THE POINT CLOUD \n"," trace_cloud = go.Scatter3d(x=A[:,0], y=A[:,1], z=A[:,2], mode='markers', marker=dict(size=10,color='blue'), showlegend=True, name = \"point cloud\"+point_cloud_name)\n"," # GENERATE PLOTTING DATA FOR THE MEAN \n"," trace_mean = go.Scatter3d(x=[mean_x], y=[mean_y], z=[mean_z], mode='markers', showlegend=True, marker=dict(size=10,color='red'), name = f'mean(point cloud{point_cloud_name})')\n"," # STORE ALL THE TRACES IN A LIST\n"," data = [trace_cloud, trace_mean]\n"," # GENERATE A \"LAYOUT\" OBJECT\n"," if axis_limits is None:\n"," layout = go.Layout(title=plot_title)\n"," else:\n"," layout = go.Layout(title=plot_title, scene = dict( xaxis=dict(range=axis_limits), yaxis=dict(range=axis_limits), zaxis=dict(range=axis_limits), aspectmode = 'cube'))\n"," # COMBINE THE PLOTTING DATA WITH THE LAYOUT OBJECT, AND PLOT \n"," if show_plot:\n"," fig = go.Figure(data = data,layout = layout)\n"," fig.show()\n"," if return_plotting_data:\n"," return data,layout\n"],"execution_count":null,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"kl1fCSvU2oKj","colab_type":"text"},"source":["## Step 1: Player 1 chooses a point cloud\n","\n","For convenience let's take the one we've already been working with."]},{"cell_type":"code","metadata":{"id":"oHDrDRzwfQ8p","colab_type":"code","colab":{"base_uri":"https://localhost:8080/","height":542},"executionInfo":{"status":"ok","timestamp":1594331796624,"user_tz":240,"elapsed":593,"user":{"displayName":"Gregory Henselman-Petrusek","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14GhckxaTMcRtkWS6WTTfLWaTxbu5kyRx4KZCGhjo=s64","userId":"16161081938846221309"}},"outputId":"e61b6591-5d8d-4356-be9d-3d7a285d0107"},"source":["# DEFINE THE ARRAY\n","A = 2 + np.array( [ [3, 3, 1 ],\n"," [2, -1, 4],\n"," [-5, -4, -2],\n"," [-6, 1, -4]])\n","\n","# AS BEFORE, CONVERT TO FLOAT TYPE\n","A = A.astype(float)\n","\n","# PLOT THE CLOUD\n","plot_cloud_with_mean(A, axis_limits = [-8,8])\n"],"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/html":["\n","\n","\n","

\n"," \n"," \n"," \n","

\n"," \n","

\n","\n",""]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"WHAa3bIu22aX","colab_type":"text"},"source":["## Step 2: Player 2 draws a plane parallel to the x-y plane, passing through the center point."]},{"cell_type":"code","metadata":{"id":"Pq7hvjFf20JI","colab_type":"code","colab":{"base_uri":"https://localhost:8080/","height":542},"executionInfo":{"status":"ok","timestamp":1594331802974,"user_tz":240,"elapsed":636,"user":{"displayName":"Gregory Henselman-Petrusek","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14GhckxaTMcRtkWS6WTTfLWaTxbu5kyRx4KZCGhjo=s64","userId":"16161081938846221309"}},"outputId":"fedeccef-d0cb-411f-ba61-b9738c9bac05"},"source":["\n","# GET THE PLOTTING DATA AND LAYOUT FOR THE ORIGINAL PLOT\n","data, layout = plot_cloud_with_mean(A, axis_limits = [-8,8], return_plotting_data=True, show_plot=False)\n","\n","# CREATE HORIZONTAL DASHED LINE FOR HEIGHT\n","mean_x, mean_y, mean_z = np.mean(A,axis=0)\n","axis_limits = [-8,8]\n","z_coords = mean_z * np.ones((2,2))\n","\n","trace_horizontal_line = go.Surface( \n"," x=axis_limits, \n"," y=axis_limits,\n"," z=z_coords,\n"," opacity = 0.5,\n"," name='average height')\n","\n","# COMBINE + PLOT\n","data_player1 = data + [trace_horizontal_line]\n","fig = go.Figure(data=data_player1, layout=layout)\n","fig.show()"],"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/html":["\n","\n","\n","

\n"," \n"," \n"," \n","

\n"," \n","

\n","\n",""]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"uU2F4MtP42QJ","colab_type":"text"},"source":["## Step 3: Player 1 draws a line from each point in the cloud to a point on the dotted line (specifically, the point with the same x-coordinate)\n","\n","The sume of squares of these lines is the \"starting\" quantity that Player 2 wants to reduce. In this case it's\n","\n","$$\\text{SumOfSquares} = 3^2 + 0^2 + 3^2 = 18$$"]},{"cell_type":"code","metadata":{"id":"O0UlxFlV4eeX","colab_type":"code","colab":{"base_uri":"https://localhost:8080/","height":542},"executionInfo":{"status":"ok","timestamp":1594331807987,"user_tz":240,"elapsed":599,"user":{"displayName":"Gregory Henselman-Petrusek","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14GhckxaTMcRtkWS6WTTfLWaTxbu5kyRx4KZCGhjo=s64","userId":"16161081938846221309"}},"outputId":"98b200e1-70ae-40c5-e662-8fc6c413020c"},"source":["# CREATE A NEW POINT CLOUD BY REPLACING EACH Y-COORDINATE WITH mean_y, AND EACH\n","# Z-COORDINATE WITH mean_z\n","B = copy.deepcopy(A)\n","B[:,2] = np.mean(A[:,2])\n","\n","# LET'S MAKE A COPY OF THIS FOR LATER USE, CALLED B_baseline\n","B_baseline = copy.deepcopy(B)\n","\n","# WRITE A FUNCTION TO GENERATE TRACES FOR THE JOINING LINES\n","def residual_traces(cloud0, cloud1):\n"," \"\"\"\n"," :param cloud0: an m x n matrix whose rows represent points in a point cloud\n"," :param cloud1: an m x n matrix whose rows represent points in a point cloud \n"," :return: a list of traces connecting the mth point in cloud0 to the mth \n"," point in cloud1\n"," \"\"\"\n"," data_residual = [ go.Scatter3d( x=[cloud0[q,0], cloud1[q,0]], \n"," y=[cloud0[q,1], cloud1[q,1]], \n"," z=[cloud0[q,2], cloud1[q,2]], \n"," # text=['', f'offset= {np.linalg.norm(cloud0[q]-cloud1[q])}', ''],\n"," textposition='top left',\n"," mode='lines', \n"," line=dict(color='red'),\n"," name = f'vertical offset= {np.linalg.norm(cloud0[q]-cloud1[q])}') \\\n"," for q in range(cloud0.shape[0])] \n"," return data_residual\n","\n","# COMBINE THE PLOT WITH THE VISUAL LINES\n","data_residual = residual_traces(A,B)\n","data_player1_vlines = data_player1 + data_residual\n","fig = go.Figure(data = data_player1_vlines, layout=layout)\n","\n","# CALCULATE DISTANCE FROM EACH POINT TO ITS \"GUESSED\" POINT\n","differences = [np.linalg.norm(A[row,[1,2]]-B[row,[1,2]]) for row in range(A.shape[0])]\n","differences_sq = np.array(differences)**2\n","differences_sq = np.round(differences_sq,4)\n","\n","# MODIFY TITLE\n","fig.update_layout(title=f'SumOfSquaredLengths = {differences_sq[0]}+{differences_sq[1]}+{differences_sq[2]} = {differences_sq.sum()}')\n","\n","# DISPLAY\n","fig.show()"],"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/html":["\n","\n","\n","

\n"," \n"," \n"," \n","

\n"," \n","

\n","\n",""]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"x9_mnyvYGT5P","colab_type":"text"},"source":["# Step 4: Payer 2 draws a new plane\n","\n","The plane is given by equation\n","\\begin{align*}\n","z &= m_1 x + m_2 y + b \\\\\n","\\end{align*}\n","or equivalently by the matrix expression\n","\n","\n","**If the data lay perfectly along this plane** then we could think of $m_1$ as the rate at which $z$ increases with $x$ and $m_2$ as the rate at which $z$ increases with $y$. We think of $b$ as the value of $z$ when $x = y = 0$.\n","\n","Let's eyeball the values for $m_1, m_2$ and $b$. Looking at the plot, $z$ seems to increase as $x$ increases, and decrease as $y$ increases. So let's try $m_1 = 1$ and and $m_2 = -1$. We'll try $b=0$ cause it looks about right and we don't feel like thinking about it too hard.\n","\n","We can rephrase these formulae in terms of matrix multiplication:\n","\n","\\begin{align*}\n","\\left[\\begin{array}{c}z \\end{array}\\right]\n","=\n","\\left[\\begin{array}{cc} x & y \\end{array}\\right]\n","* \n","\\left[\\begin{array}{c} m_1 \\\\ m_2 \\end{array}\\right]\n","+\n","\\left[\\begin{array}{c} b\\end{array}\\right]\n","\\end{align*}\n","\n","If we have a whole sequence of x-values, $(x_1, ..., x_n)$, we can write the seqeunce of y-values in almost exactly the same way **by using matrix multiplication**\n","\n","\\begin{align*}\n","\\left[\\begin{array}{c}z_0 \\\\ \\vdots \\\\ z_n\\end{array}\\right]\n","=\n","\\left[\\begin{array}{cc}x_0 & y_0 \\\\ \\vdots & \\vdots \\\\ x_n & y_n\\end{array}\\right]\n","* \n","\\left[\\begin{array}{c} m_1 \\\\ m_2\\end{array}\\right]\n","+\n","\\left[\\begin{array}{c} b \\\\ \\vdots \\\\ b \\end{array}\\right]\n","\\end{align*}\n","\n","or more specifically, in our case,\n","\n","\\begin{align*}\n","\\left[\\begin{array}{c}z_0 \\\\ \\vdots \\\\ z_n\\end{array}\\right]\n","=\n","\\left[\\begin{array}{cc}x_0 & y_0 \\\\ \\vdots & \\vdots \\\\ x_n & y_n\\end{array}\\right]\n","* \n","\\left[\\begin{array}{c} 1 \\\\ -1\\end{array}\\right]\n","+\n","\\left[\\begin{array}{c} 0 \\\\ \\vdots \\\\ 0 \\end{array}\\right]\n","\\end{align*}\n","\n","### Let's write a function to do this so we don't have to\n","Let's write a function that will take a sequence of x-values, and complete it to a sequence of x,y,z values (recording the output as the rows of a matrix)"]},{"cell_type":"code","metadata":{"id":"kEX-UZsYCpFu","colab_type":"code","colab":{}},"source":["\n","\n","# WRITE A FUNCTION TO \"ADD\" Y AND Z COORDINATES TO EACH X\n","\n","def extend_xy_to_xyz(xy,slopes=[1,1],intercept=0):\n"," \"\"\"\n"," Input: a list or vector of x values\n"," Output: an array B such that \n"," (i ) the first column of B equals x, \n"," (ii) each row of B is a point on the line that we have guessed\n"," \"\"\"\n"," xy = np.array(xy).astype(float).reshape(-1,2) # convert x to a width-2 array, if it isn't one already\n"," slopes = np.array(slopes).reshape(-1,1) # convert the slope list into a column vector\n"," z = np.matmul(xy, slopes) + intercept # APPLY THE MATRIX FORMULA\n"," xyz = np.concatenate((xy,z),axis=1) # concatenate the x and xy matrices\n"," return xyz"],"execution_count":null,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"678yDAcrHTJ9","colab_type":"text"},"source":["These equations determine a line. For each point in our cloud, let's find the point on this line that shares the same x-coordinate. How do we do this? Well, pick a point $(x_i, y_i, z_i)$. We already know that the x-coordinate should be *and* we have formulas for y and z!\n","\n"]},{"cell_type":"code","metadata":{"id":"PWOSU89sDWiz","colab_type":"code","colab":{"base_uri":"https://localhost:8080/","height":187},"executionInfo":{"status":"ok","timestamp":1594331828323,"user_tz":240,"elapsed":444,"user":{"displayName":"Gregory Henselman-Petrusek","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14GhckxaTMcRtkWS6WTTfLWaTxbu5kyRx4KZCGhjo=s64","userId":"16161081938846221309"}},"outputId":"f3a9bf2c-88a3-40cb-d20a-8a320f8fea1e"},"source":["xy_coords = A[:,[0,1]]\n","\n","xyz_coords_guesscloud = extend_xy_to_xyz( xy_coords, \n"," slopes=[1,1], \n"," intercept = 0)\n","\n","print(\"x coordinates\")\n","display(xy_coords)\n","\n","print(\"xyz coordinates\")\n","display(xyz_coords_guesscloud)"],"execution_count":null,"outputs":[{"output_type":"stream","text":["x coordinates\n"],"name":"stdout"},{"output_type":"display_data","data":{"text/plain":["array([[ 5., 5.],\n"," [ 4., 1.],\n"," [-3., -2.],\n"," [-4., 3.]])"]},"metadata":{"tags":[]}},{"output_type":"stream","text":["xyz coordinates\n"],"name":"stdout"},{"output_type":"display_data","data":{"text/plain":["array([[ 5., 5., 10.],\n"," [ 4., 1., 5.],\n"," [-3., -2., -5.],\n"," [-4., 3., -1.]])"]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"byF7oZ6-U1Hd","colab_type":"text"},"source":["### Let's plot.\n","\n"]},{"cell_type":"code","metadata":{"id":"MxNnjY31DXmp","colab_type":"code","colab":{"base_uri":"https://localhost:8080/","height":542},"executionInfo":{"status":"ok","timestamp":1594332107626,"user_tz":240,"elapsed":674,"user":{"displayName":"Gregory Henselman-Petrusek","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14GhckxaTMcRtkWS6WTTfLWaTxbu5kyRx4KZCGhjo=s64","userId":"16161081938846221309"}},"outputId":"03c0d3dd-fffb-443b-8c1e-1f990c0278ad"},"source":["# PLOT\n","\n","# RECALCULATE THE POINTS ON THE GUESSED LINE\n","xy_coords = A[:,[0,1]]\n","xyz_coords_guesscloud = extend_xy_to_xyz( xy_coords, \n"," slopes=[1,-1], \n"," intercept=0)\n","\n","# GET THE PLOTTING DATA AND LAYOUT FOR THE ORIGINAL POINT CLOUD\n","data, layout = plot_cloud_with_mean(A, axis_limits = [-8,8], return_plotting_data=True, show_plot=False)\n","\n","# CREATE A NEW PLANE (WE'LL DO THIS BY JOINING FOUR CORNER POINTS)\n","\n","# WRITE A FUNCTION TO CALCULATE Z-COORDINATES\n","def axis_limits_to_z_coordinates(axis_limits, slopes = [1,-1], intercept=0):\n"," z_coords = np.zeros((2,2)) # we'll put the z-coordinate for corner (i,j) into entry (i,j) of this matrix\n"," for row in range(2):\n"," for col in range(2):\n"," # notice that we are reversing row and column below; this has to do with \n"," # an unfortunate indexing convention in plotly\n"," z_coords[col,row] = extend_xy_to_xyz( [axis_limits[row],axis_limits[col]], \n"," slopes=slopes, \n"," intercept=intercept)[0,2] # <-- since this function returns a row vector, we have to extract out the last entry of the row\n"," return z_coords\n","\n","# CALCULATE THE Z-COORDINATES \n","axis_limits = [-8,8] # the (x,y coordinates of each point will be -8 or 8)\n","z_coords = axis_limits_to_z_coordinates(axis_limits)\n","\n","# MAKE THE SURFACE TRACE\n","trace_surface_guess = go.Surface( \n"," x=axis_limits, \n"," y=axis_limits,\n"," z=z_coords,\n"," opacity = 0.5,\n"," name='player 2 guess')\n","\n","# ADD THE TRACE TO THE LIST OF TRACES\n","data_player2 = data + [trace_surface_guess]\n","\n","vline_traces_player2 = residual_traces(A, xyz_coords_guesscloud)\n","data_player2 = data_player2 + vline_traces_player2\n","\n","# CREATE FIGURE\n","fig = go.Figure(data = data_player2, layout=layout)\n","\n","# CALCULATE DISTANCE FROM EACH POINT TO ITS \"GUESSED\" POINT\n","differences = A[:,2] - xyz_coords_guesscloud[:,2]\n","differences_sq = np.array(differences)**2\n","differences_sq = np.round(differences_sq,4)\n","\n","# MODIFY TITLE\n","fig.update_layout(title=f'SumOfSquaredLengths = {differences_sq[0]}+{differences_sq[1]}+{differences_sq[2]} = {differences_sq.sum()}')\n","\n","# DISPLAY\n","fig.show()"],"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/html":["\n","\n","\n","

\n"," \n"," \n"," \n","

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\n","\n",""]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"mVvFeqA_gRgS","colab_type":"text"},"source":["This is better - but still not best!"]},{"cell_type":"markdown","metadata":{"id":"2ybaKfu3aDFu","colab_type":"text"},"source":["# Step 5: Linear regression paracheuts into the game, and wins!"]},{"cell_type":"code","metadata":{"id":"FtC18dIoDdI9","colab_type":"code","colab":{"base_uri":"https://localhost:8080/","height":610},"executionInfo":{"status":"ok","timestamp":1594332161219,"user_tz":240,"elapsed":691,"user":{"displayName":"Gregory Henselman-Petrusek","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14GhckxaTMcRtkWS6WTTfLWaTxbu5kyRx4KZCGhjo=s64","userId":"16161081938846221309"}},"outputId":"ac4c127d-a9ff-47b8-b875-604de73ed4ce"},"source":["# FORMAT THE DATA\n","xy_matrix = A[:,[0,1]] # why the extra brackets? because the package we use to run linear regression wants *column vectors*\n","z_matrix = A[:,[2]] # why the extra brackets? because the package we use to run linear regression wants *column vectors*\n","\n","# FORMAT THE REGRESSOR\n","reg = sklearn.linear_model.LinearRegression()\n","reg.fit(xy_matrix, z_matrix)\n","\n","# EXTRACT SLOPE AND INTERCEPT\n","slopes_linreg = reg.coef_\n","intercepts_linreg = reg.intercept_\n","\n","print(\"slope coefficients\")\n","print(slopes_linreg)\n","print(\"intercept\")\n","print(intercepts_linreg)\n","\n","# GET THE PROJECTED POINT CLOUD\n","xy_coords = A[:,[0,1]]\n","xyz_coords_guesscloud = extend_xy_to_xyz( xy_coords, \n"," slopes=slopes_linreg, \n"," intercept=intercepts_linreg)\n","\n","# MAKE A COPY FOR FUTURE USE\n","B_linreg = copy.deepcopy(xyz_coords_guesscloud)\n","\n","# GET THE PLOTTING DATA AND LAYOUT FOR THE ORIGINAL POINT CLOUD\n","data, layout = plot_cloud_with_mean(A, axis_limits = [-8,8], return_plotting_data=True, show_plot=False)\n","\n","# CREATE A NEW PLANE (WE'LL DO THIS BY JOINING FOUR CORNER POINTS)\n","\n","# CALCULATE THE Z-COORDINATES \n","axis_limits = [-8,8] # the (x,y coordinates of each point will be -8 or 8)\n","z_coords = axis_limits_to_z_coordinates(axis_limits,slopes=slopes_linreg, intercept=intercepts_linreg)\n","\n","# MAKE THE SURFACE TRACE\n","trace_surface_guess = go.Surface( \n"," x=axis_limits, \n"," y=axis_limits,\n"," z=z_coords,\n"," opacity = 0.5,\n"," name='player 2 guess')\n","\n","# ADD THE TRACE TO THE LIST OF TRACES\n","data_player2 = data + [trace_surface_guess]\n","\n","# COMBINE THE PLOT WITH THE VISUAL LINES\n","vline_traces_player2 = residual_traces(A, xyz_coords_guesscloud)\n","data_player2 = data_player2 + vline_traces_player2\n","\n","# CREATE FIGURE\n","fig = go.Figure(data = data_player2, layout=layout)\n","\n","# CALCULATE DISTANCE FROM EACH POINT TO ITS \"GUESSED\" POINT\n","differences = A[:,2] - xyz_coords_guesscloud[:,2]\n","differences_sq = np.array(differences)**2\n","differences_sq = np.round(differences_sq,4)\n","\n","# MODIFY TITLE\n","fig.update_layout(title=f'SumOfSquaredLengths = {differences_sq[0]}+{differences_sq[1]}+{differences_sq[2]} = {differences_sq.sum()}')\n","\n","# DISPLAY\n","fig.show()"],"execution_count":null,"outputs":[{"output_type":"stream","text":["slope coefficients\n","[[ 0.81707766 -0.49282297]]\n","intercept\n","[2.20390136]\n"],"name":"stdout"},{"output_type":"display_data","data":{"text/html":["\n","\n","\n","

\n"," \n"," \n"," \n","

\n"," \n","

\n","\n",""]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"LTvzJA5y4-7L","colab_type":"text"},"source":["### Notice: the best fit plane passes through the mean\n","\n","As we can see, in this example. Interestingly, it *always* passes through the mean (a fact which takes some real work to prove)."]},{"cell_type":"markdown","metadata":{"id":"WlUVrlDNgVvU","colab_type":"text"},"source":["# Lessons from this example\n","\n","* Mathematically, this is similar to the preceding examples\n"," * we use formulas similar to $y = mx + b$\n"," * we connect \"original\" points to points on a plane (instead of line)\n"," * in particular, we send each point in the original cloud to the point on the plane that shares the same x and y coordinates"]},{"cell_type":"code","metadata":{"id":"u3F274hkGwbA","colab_type":"code","colab":{}},"source":[""],"execution_count":null,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"Me4r8CF5OmJp","colab_type":"text"},"source":[""]}]}

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