Fixed assignment to F. dt=.1, but used dt=1.

06-Multivariate-Kalman-Filters.ipynb

@@ -425,7 +425,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x410f5c0>"
+       "<matplotlib.figure.Figure at 0x407b908>"
       ]
      },
      "metadata": {},
@@ -706,7 +706,7 @@
     "\n",
     "$$\\begin{bmatrix}2& 3 \\\\ 3&-1\\end{bmatrix} \\begin{bmatrix}x\\\\y\\end{bmatrix} = \\begin{bmatrix}8\\\\1\\end{bmatrix}$$\n",
     "\n",
-    "If you perform the matrix multiplication in this equation the result will be the two equations above. In linear algebra we would write this as $\\mathbf{Ax}=\\mathbf{b}$, where\n",
+    "If you perform the [matrix multiplication](https://en.wikipedia.org/wiki/Matrix_multiplication#General_definition_of_the_matrix_product) in this equation the result will be the two equations above. In linear algebra we would write this as $\\mathbf{Ax}=\\mathbf{b}$, where\n",
     "\n",
     "$$\\mathbf{A} = \\begin{bmatrix}2& 3 \\\\ 3&-1\\end{bmatrix},\\, \\mathbf{x} = \\begin{bmatrix}x\\\\y\\end{bmatrix}, \\mathbf{b}=\\begin{bmatrix}8\\\\1\\end{bmatrix}$$\n",
     "\n",
@@ -888,7 +888,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x3dc2f60>"
+       "<matplotlib.figure.Figure at 0x77e0dd8>"
       ]
      },
      "metadata": {},
@@ -1227,13 +1227,15 @@
    },
    "outputs": [],
    "source": [
-    "dog_filter.x = np.array([0., 0.])                         # state (location and velocity)\n",
-    "dog_filter.F = np.array([[1., 1.], \n",
-    "                         [0., 1.]])                       # state transition function\n",
-    "dog_filter.H = np.array([[1., 0.]])                       # Measurement function\n",
-    "dog_filter.R =  5                                         # measurement noise\n",
-    "dog_filter.Q = Q_discrete_white_noise(2, dt=0.1, var=0.1) # process noise\n",
-    "dog_filter.P = np.diag([500., 400.])                      # covariance matrix "
+    "dt = .1\n",
+    "dog_filter.x = np.array([0., 0.])           # state (location and velocity)\n",
+    "dog_filter.F = np.array([[1., dt], \n",
+    "                         [0., 1.]])         # state transition function\n",
+    "dog_filter.H = np.array([[1., 0.]])         # Measurement function\n",
+    "dog_filter.R =  5                           # measurement noise\n",
+    "dog_filter.Q = Q_discrete_white_noise(\n",
+    "                 2, dt=dt, var=0.1)         # process noise\n",
+    "dog_filter.P = np.diag([500., 400.])        # covariance matrix "
    ]
   },
   {
@@ -1244,7 +1246,7 @@
     "\n",
     "**1**: We assign the initial value for our state. Here we initialize both the position and velocity to zero.\n",
     "\n",
-    "**2**: We set $\\textbf{F}=\\begin{bmatrix}1&1\\\\0&1\\end{bmatrix}$, as in design step 2 above. \n",
+    "**2**: We set $\\textbf{F}=\\begin{bmatrix}1&\\Delta t\\\\0&1\\end{bmatrix}$, as in design step 2 above. \n",
     "\n",
     "**3**: We set $\\textbf{H}=\\begin{bmatrix}1&0\\end{bmatrix}$, as in design step 4 above.\n",
     "\n",
@@ -1265,16 +1267,19 @@
    },
    "outputs": [],
    "source": [
-    "def pos_vel_filter(x, P, R, Q=0.):\n",
+    "def pos_vel_filter(x, P, R, Q=0., dt=1.0):\n",
     "    dog_filter = KalmanFilter(dim_x=2, dim_z=1)\n",
     "    dog_filter.x = np.array([x[0], x[1]]) # (location and velocity)\n",
-    "    dog_filter.F = np.array([[1, 1],\n",
-    "                             [0, 1]])     # state transition matrix\n",
+    "    dog_filter.F = np.array([[1, dt],\n",
+    "                             [0,  1]])    # state transition matrix\n",
     "    dog_filter.H = np.array([[1, 0]])     # Measurement function\n",
     "    dog_filter.R *= R                     # measurement uncertainty\n",
-    "    dog_filter.P *= P                     # covariance matrix \n",
+    "    if np.isscalar(P):\n",
+    "        dog_filter.P *= P                 # covariance matrix \n",
+    "    else:\n",
+    "        dog_filter.P[:] = P\n",
     "    if np.isscalar(Q):\n",
-    "        dog_filter.Q = Q_discrete_white_noise(dim=2, dt=0.1, var=Q)\n",
+    "        dog_filter.Q = Q_discrete_white_noise(dim=2, dt=dt, var=Q)\n",
     "    else:\n",
     "        dog_filter.Q = Q\n",
     "    return dog_filter"
@@ -1295,7 +1300,7 @@
    },
    "outputs": [],
    "source": [
-    "def filter_dog(data, R=0, Q=0, P=500., initial_x=(0., 0.)):\n",
+    "def filter_dog(data, R=0, Q=0, P=500, initial_x=(0., 0.)):\n",
     "    \"\"\" Filters 'data' using the filter provided by \n",
     "    dog_tracking_filter().\n",
     "    \n",
@@ -1370,8 +1375,8 @@
    },
    "outputs": [],
    "source": [
-    "def filter_dog(data, R=0, Q=0, P=500., initial_x=(0., 0.)):\n",
-    "    dog_filter = pos_vel_filter(initial_x, R=R, Q=Q, P=P)  \n",
+    "def filter_dog(data, R=0, Q=0, P=500, dt=1.0, initial_x=(0., 0.)):\n",
+    "    dog_filter = pos_vel_filter(initial_x, R=R, Q=Q, P=P, dt=dt)  \n",
     "    pos, cov, _, _ = dog_filter.batch_filter(data)\n",
     "    return (pos, cov)"
    ]
@@ -1387,14 +1392,16 @@
     "from DogSimulation import DogSimulation\n",
     "from mkf_internal import plot_track\n",
     "\n",
-    "def run_filter(Q, R, P, initial_x=(0.,0.), data=None, count=0, std_scale=1, **kwargs):\n",
+    "def run_filter(Q, R, P, dt=1.0, initial_x=(0.,0.), data=None, \n",
+    "               count=0, std_scale=1, do_plot=True, **kwargs):\n",
     "    if data is None:\n",
     "        dog = DogSimulation(velocity=1, measurement_variance=R)\n",
-    "        zs = [dog.move_and_sense() for t in range(count)]\n",
+    "        zs = [dog.move_and_sense(dt) for t in range(count)]\n",
     "    else:\n",
     "        zs = data\n",
-    "    ps, cov = filter_dog(data=zs, R=R, Q=Q, P=P, initial_x=initial_x)\n",
-    "    plot_track(ps[:, 0], zs, cov, std_scale, **kwargs)\n",
+    "    ps, cov = filter_dog(data=zs, R=R, Q=Q, P=P, dt=dt, initial_x=initial_x)\n",
+    "    if do_plot:\n",
+    "        plot_track(ps[:, 0], zs, cov, std_scale, dt=dt, **kwargs)\n",
     "    return ps, cov"
    ]
   },
@@ -1414,9 +1421,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7755fd0>"
+       "<matplotlib.figure.Figure at 0x796ce48>"
       ]
      },
      "metadata": {},
@@ -1424,9 +1431,9 @@
     },
     {
      "data": {
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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtoAAAEpCAYAAACtAIUZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e96dd8>"
+       "<matplotlib.figure.Figure at 0x79968d0>"
       ]
      },
      "metadata": {},
@@ -1435,7 +1442,7 @@
    ],
    "source": [
     "P = np.diag([500., 400.])\n",
-    "Ms, Ps = run_filter(count=50, R=10, Q=0.01, P=P, y_lim=(-20, 50))"
+    "Ms, Ps = run_filter(count=50, R=10, Q=0.01, P=P, dt=1., y_lim=(-20, 50))"
    ]
   },
   {
@@ -1460,7 +1467,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "I want you to get a better feel for how the Gaussians change over time, so here is a 3D plot showing the Gaussians from the plot above. I generated this by computing the probability distribution for each over a large area, and then summing them together so I can show them all in the same plot. Note that it makes no mathematical sense to add these together, it is merely a way to display them all at once. The scale of the y axis is much smaller than the x axis, so the Gaussians look very stretched horizontally, but they are not. This scaling makes it easier to see everything and minimizes the amount of computation needed. You should still be able to tell that our certainty in both the position and velocity improve over time. We know this because the Gaussian gets narrow in both axis as time increases, and it simultaneously gets taller."
+    "I want you to get a better feel for how the Gaussians change over time, so here is a 3D plot showing the Gaussians. The scale of the y axis is much smaller than the x axis, so the Gaussians look very stretched horizontally. This scaling makes it easier to see everything. You should still be able to tell that our certainty in both the position and velocity improve over time. We know this because the Gaussian gets narrow in both axis as time increases, and it simultaneously gets taller. I plot every 7th epoch (using the Python index `::7`) so the Gaussians are separated from each other. I also started with a small value for `P`; `[500 400]` yields a probability density that is indistinguisable from a flat plane at this scale."
    ]
   },
   {
@@ -1473,9 +1480,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAr8AAAE5CAYAAACZGEi2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e9df28>"
+       "<matplotlib.figure.Figure at 0x78f5a90>"
       ]
      },
      "metadata": {},
@@ -1485,8 +1492,11 @@
    "source": [
     "from book_format import set_figsize, figsize\n",
     "from nonlinear_plots import plot_multiple_gaussians\n",
+    "P = np.diag([3., 1.])\n",
+    "np.random.seed(3)\n",
+    "Ms, Ps = run_filter(count=25, R=10, Q=0.01, P=P, y_lim=(-20, 50), do_plot=False)\n",
     "with figsize(y=5):\n",
-    "    plot_multiple_gaussians(Ms[::7], Ps[::7], (0,52), (-1, 2), 75)"
+    "    plot_multiple_gaussians(Ms[::7], Ps[::7], (-5,25), (-5, 5), 75)"
    ]
   },
   {
@@ -1561,7 +1571,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvEAAAEoCAYAAADR4a6bAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7daed68>"
+       "<matplotlib.figure.Figure at 0x796bf98>"
       ]
      },
      "metadata": {},
@@ -1610,7 +1620,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f08518>"
+       "<matplotlib.figure.Figure at 0x787d240>"
       ]
      },
      "metadata": {},
@@ -1673,7 +1683,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d9ed30>"
+       "<matplotlib.figure.Figure at 0x79010b8>"
       ]
      },
      "metadata": {},
@@ -1814,7 +1824,7 @@
     "\n",
     "This notation allows a mathematician to express himself exactly, and when it comes to formal publications presenting new results this precision is necessary. As a programmer I find all of that fairly unreadable; I am used to thinking about variables changing state as a program runs, and do not use a different variable name for each new computation. There is no agreed upon format, so each author makes different choices. I find it challenging to switch quickly between books an papers, and so have adopted my admittedly less precise notation. Mathematicians will write scathing emails to me, but I hope the programmers and students will rejoice.\n",
     "\n",
-    "Here are some examples for how other authors write the prior: $X^*_{n+1,n}$, $\\underline{\\hat{x}}_k(-)$ (really!), $\\hat{\\textbf{x}}^-_{k+1}$, $\\hat{x}_{k}$. If you are lucky an author defines the notation; more often you have to read the equations in context to recognize what the author is doing. Of course, people write within a tradition; papers on Kalman filters in finance are likely to use one set of notations while papers on radar tracking is likely to use a different set. Over time you will start to become familiar with trends, and also instantly recognize when somebody just copied equations wholesale from another work. For example - the equations I gave above were copied from the  Wikipedia [Kalman Filter](https://en.wikipedia.org/wiki/Kalman_filter#Details) [[1]](#[wiki_article]) article.\n",
+    "Here are some examples for how other authors write the prior: $X^*_{n+1,n}$, $\\underline{\\hat{x}}_k(-)$ (really!), $\\hat{\\textbf{x}}^-_{k+1}$, $\\hat{x}_{k}$. If you are lucky an author defines the notation; more often you have to read the equations in context to recognize what the author is doing. Of course, people write within a tradition; papers on Kalman filters in finance are likely to use one set of notations while papers on radar tracking are likely to use a different set. Over time you will start to become familiar with trends, and also instantly recognize when somebody just copied equations wholesale from another work. For example - the equations I gave above were copied from the  Wikipedia [Kalman Filter](https://en.wikipedia.org/wiki/Kalman_filter#Details) [[1]](#[wiki_article]) article.\n",
     "\n",
     "The *Symbology* Chapter lists the notation used by various authors. This brings up another difficulty. Different authors use different variable names. $\\mathbf{x}$ is fairly universal, but after that it is anybody's guess. Again, you need to read carefully, and hope that the author defines their variables (they often do not).\n",
     "\n",
@@ -1861,9 +1871,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9163780>"
+       "<matplotlib.figure.Figure at 0x92aa278>"
       ]
      },
      "metadata": {},
@@ -1970,9 +1980,9 @@
     },
     {
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x917fa90>"
+       "<matplotlib.figure.Figure at 0x9209898>"
       ]
      },
      "metadata": {},
@@ -1980,9 +1990,9 @@
     },
     {
      "data": {
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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9197b38>"
+       "<matplotlib.figure.Figure at 0x8f78fd0>"
       ]
      },
      "metadata": {},
@@ -2039,9 +2049,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f3fc88>"
+       "<matplotlib.figure.Figure at 0x9201550>"
       ]
      },
      "metadata": {},
@@ -2069,9 +2079,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x778e6a0>"
+       "<matplotlib.figure.Figure at 0x794e4e0>"
       ]
      },
      "metadata": {},
@@ -2100,9 +2110,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d9c828>"
+       "<matplotlib.figure.Figure at 0x92fdef0>"
       ]
      },
      "metadata": {},
@@ -2173,7 +2183,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x77c9828>"
+       "<matplotlib.figure.Figure at 0x7932358>"
       ]
      },
      "metadata": {},
@@ -2207,7 +2217,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x91a9940>"
+       "<matplotlib.figure.Figure at 0x9284a90>"
       ]
      },
      "metadata": {},
@@ -2268,9 +2278,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x92e2358>"
+       "<matplotlib.figure.Figure at 0x91d3780>"
       ]
      },
      "metadata": {},
@@ -2278,9 +2288,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8f6ca58>"
+       "<matplotlib.figure.Figure at 0x91e6cc0>"
       ]
      },
      "metadata": {},
@@ -2322,9 +2332,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f39dd8>"
+       "<matplotlib.figure.Figure at 0x7939d30>"
       ]
      },
      "metadata": {},
@@ -2332,9 +2342,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x779e8d0>"
+       "<matplotlib.figure.Figure at 0x78d1668>"
       ]
      },
      "metadata": {},
@@ -2366,9 +2376,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9303dd8>"
+       "<matplotlib.figure.Figure at 0x8fe1860>"
       ]
      },
      "metadata": {},
@@ -2400,9 +2410,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x92b97b8>"
+       "<matplotlib.figure.Figure at 0x92f1be0>"
       ]
      },
      "metadata": {},
@@ -2439,9 +2449,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f08940>"
+       "<matplotlib.figure.Figure at 0x7681400>"
       ]
      },
      "metadata": {},
@@ -2449,9 +2459,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8f6f2e8>"
+       "<matplotlib.figure.Figure at 0x7938b38>"
       ]
      },
      "metadata": {},
@@ -2559,9 +2569,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e2a7b8>"
+       "<matplotlib.figure.Figure at 0x7962860>"
       ]
      },
      "metadata": {},
@@ -2694,9 +2704,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d9ca90>"
+       "<matplotlib.figure.Figure at 0x9122518>"
       ]
      },
      "metadata": {},
@@ -2745,9 +2755,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e98978>"
+       "<matplotlib.figure.Figure at 0x8f784e0>"
       ]
      },
      "metadata": {},
@@ -2786,9 +2796,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x6b18a20>"
+       "<matplotlib.figure.Figure at 0x77b2ba8>"
       ]
      },
      "metadata": {},
@@ -2852,9 +2862,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f511d0>"
+       "<matplotlib.figure.Figure at 0x8f935c0>"
       ]
      },
      "metadata": {},
@@ -2863,8 +2873,9 @@
    ],
    "source": [
     "dx = np.diff(xs[:, 0], axis=0)\n",
-    "plt.plot(range(1,count), dx, c='r',ls='--', lw=2, label='Raw velocity')\n",
-    "plt.plot(xs[:, 1], label='Filter')\n",
+    "plt.scatter(range(1,count), dx, facecolor='none', edgecolor='k',\n",
+    "            lw=2, label='Raw velocity')\n",
+    "plt.plot(xs[:, 1], ls='--', label='Filter')\n",
     "plt.plot(Ms[:, 1], label='RTS')\n",
     "plt.legend(loc='best');"
    ]

code/book_plots.py

@@ -59,7 +59,8 @@ def set_limits(x, y):
     plt.gca().set_ylim(y)
 
 
-def plot_measurements(xs, ys=None, color='r', lw=2, label='Measurements', **kwargs):
+def plot_measurements(xs, ys=None, color='k', lw=2, label='Measurements',
+                      lines=False, **kwargs):
     """ Helper function to give a consistant way to display
     measurements in the book.
     """
@@ -76,11 +77,16 @@ def plot_measurements(xs, ys=None, color='r', lw=2, label='Measurements', **kwar
         if connect:
            plt.plot(range(len(xs)), xs, lw=1, c=c, alpha=alpha)'''
 
-    if ys is not None:
-        plt.plot(xs, ys, color=color, lw=lw, ls='--', label=label, **kwargs)
+    if lines:
+        if ys is not None:
+            plt.plot(xs, ys, color=color, lw=lw, ls='--', label=label, **kwargs)
+        else:
+            plt.plot(xs, color=color, lw=lw, ls='--', label=label, **kwargs)
     else:
-        plt.plot(xs, color=color, lw=lw, ls='--', label=label, **kwargs)
-
+        if ys is not None:
+            plt.scatter(xs, ys, edgecolor=color, facecolor='none', lw=2, label=label, **kwargs)
+        else:
+            plt.scatter(range(len(xs)), xs, edgecolor=color, facecolor='none', lw=2, label=label, **kwargs)
 
 
 def plot_residual_limits(Ps, stds=1.):

code/mkf_internal.py

@@ -356,11 +356,12 @@ def plot_correlation_covariance():
 
 import book_plots as bp
 def plot_track(ps, zs, cov, std_scale=1,
-               plot_P=True, y_lim=None,
+               plot_P=True, y_lim=None, dt=1.,
+               xlabel='time', ylabel='position',
                title='Kalman Filter'):
 
     count = len(zs)
-    actual = np.linspace(0, count - 1, count)
+    actual = np.linspace(0, count - 1, count) * dt
     cov = np.asarray(cov)
     std = std_scale*np.sqrt(cov[:,0,0])
     std_top = np.minimum(actual+std, [count + 10])
@@ -378,6 +379,8 @@ def plot_track(ps, zs, cov, std_scale=1,
     plt.fill_between(range(len(std_top)), std_top, std_btm,
                      facecolor='yellow', alpha=0.2, interpolate=True)
     plt.legend(loc=4)
+    plt.xlabel(xlabel)
+    plt.ylabel(ylabel)
     if y_lim is not None:
         plt.ylim(y_lim)
     else:
@@ -389,10 +392,10 @@ def plot_track(ps, zs, cov, std_scale=1,
 
     if plot_P:
         ax = plt.subplot(121)
-        ax.set_title("$\sigma^2_x$")
+        ax.set_title("$\sigma^2_x$ (pos variance)")
         plot_covariance(cov, (0, 0))
         ax = plt.subplot(122)
-        ax.set_title("$\sigma^2_y$")
+        ax.set_title("$\sigma^2_y$ (vel variance)")
         plot_covariance(cov, (1, 1))
         plt.show()
 

code/nonlinear_plots.py

@@ -270,7 +270,6 @@ def plot_multiple_gaussians(xs, ps, x_range, y_range, N):
     """ given a list of 2d states (x,y) and 2x2 covariance matrices, produce
     a surface plot showing all of the gaussians"""
 
-
     xs = np.asarray(xs)
     x = np.linspace (x_range[0], x_range[1], N)
     y = np.linspace (y_range[0], y_range[1], N)
@@ -285,11 +284,11 @@ def plot_multiple_gaussians(xs, ps, x_range, y_range, N):
     ax = plt.figure().add_subplot(111, projection='3d')
     ax.plot_surface(xx, yy, zv, rstride=1, cstride=1, lw=.5, edgecolors='#191919',
                     antialiased=True, shade=True, cmap=cm.autumn)
-    ax.view_init(elev=40., azim=250)
+    ax.view_init(elev=40., azim=230)
 
 
 if __name__ == "__main__":
-    plot_cov_ellipse_colormap(cov=[[2, 1.2], [1.2, 2]])
+    #plot_cov_ellipse_colormap(cov=[[2, 1.2], [1.2, 2]])
     '''
     from numpy.random import normal
     import numpy as np
@@ -308,4 +307,26 @@ if __name__ == "__main__":
     #plot_transfer_func (data, g, lims=(-3,3), num_bins=100)
     plot_nonlinear_func (data, g, gaussian=x0,
                         num_bins=100)
-    '''
+    '''
+
+    Ps = np.array([[[ 2.85841814,  0.71772898],
+        [ 0.71772898,  0.93786824]],
+
+       [[ 3.28939458,  0.52634978],
+        [ 0.52634978,  0.13435503]],
+
+       [[ 2.40532661,  0.29692055],
+        [ 0.29692055,  0.07671416]],
+
+       [[ 2.23084082,  0.27823192],
+        [ 0.27823192,  0.07488681]]])
+
+    Ms = np.array([[  0.68040795,   0.17084572],
+       [  8.46201389,   1.15070342],
+       [ 13.7992229 ,   0.96022707],
+       [ 19.95838208,   0.87524265]])
+
+    plot_multiple_gaussians(Ms, Ps, (-5,25), (-5, 5), 75)
+
+
+