Explained row order access of cholesky #316

10-Unscented-Kalman-Filter.ipynb

@@ -80,7 +80,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": "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 truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 900x400 with 3 Axes>"
       ]
@@ -122,7 +122,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 2 Axes>"
       ]
@@ -218,12 +218,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Difference in mean x=-0.037, y=43.103\n"
+      "Difference in mean x=-0.157, y=43.481\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 2 Axes>"
       ]
@@ -350,7 +350,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -789,7 +789,7 @@
     "\\mu + \\left[ \\sqrt{(n+\\lambda)\\Sigma}\\right ]_{i}& \\text{for i=1 .. n} \\\\\n",
     "\\mu - \\left[ \\sqrt{(n+\\lambda)\\Sigma}\\right]_{i-n} &\\text{for i=(n+1) .. 2n}\\end{cases}\n",
     "$$\n",
-    "The $i$ subscript chooses the i$^{th}$ column vector of the matrix.\n",
+    "The $i$ subscript chooses the i$^{th}$ roth vector of the matrix.\n",
     "\n",
     "In other words, we scale the covariance matrix by a constant, take the square root of it, and ensure symmetry by both adding and subtracting it from the mean. We will discuss how you take the square root of a matrix later.\n",
     "\n",
@@ -2067,7 +2067,7 @@
     "\n",
     "The Python is not difficult once we understand the $\\left[\\sqrt{(n+\\lambda)\\Sigma}  \\right]_i$ term.\n",
     "\n",
-    "The term $\\sqrt{(n+\\lambda)\\Sigma}$ is a matrix because $\\Sigma$ is a matrix. The subscript $i$ in $[\\sqrt{(n+\\lambda)\\Sigma}]_i$ is choosing the column vector of the matrix. What is the square root of a matrix? There is no unique definition. One definition is that the square root of a matrix $\\Sigma$ is the matrix $S$ that, when multiplied by itself, yields $\\Sigma$: if  $\\Sigma = SS$ then $S = \\sqrt{\\Sigma}$.\n",
+    "The term $\\sqrt{(n+\\lambda)\\Sigma}$ is a matrix because $\\Sigma$ is a matrix. The subscript $i$ in $[\\sqrt{(n+\\lambda)\\Sigma}]_i$ is choosing the i-th row vector of the matrix. What is the square root of a matrix? There is no unique definition. One definition is that the square root of a matrix $\\Sigma$ is the matrix $S$ that, when multiplied by itself, yields $\\Sigma$: if  $\\Sigma = SS$ then $S = \\sqrt{\\Sigma}$.\n",
     "\n",
     "We will choose an alternative definition that has numerical properties which make it easier to compute. We can define the square root as the matrix S, which when multiplied by its transpose, returns $\\Sigma$:\n",
     "\n",
@@ -2088,7 +2088,7 @@
     "\n",
     "SciPy provides `cholesky()` method in `scipy.linalg`. If your language of choice is Fortran, C, or C++, libraries such as LAPACK provide this routine. Matlab provides `chol()`.\n",
     "\n",
-    "By default `scipy.linalg.cholesky()` returns a upper triangular matrix, so I elected to write the code to expect an upper triangular matrix. If you provide your own square root implementation you will need to take this into account."
+    "By default `scipy.linalg.cholesky()` returns a upper triangular matrix, so I elected to write the code to expect an upper triangular matrix. For this reason I access the result by row, so that the first sigma point, which is the center point, is affected by a full row of non-zero values.  If you provide your own square root implementation you will need to take this into account. You will find UKF algorithms in the literature that take the values column first. This is fine if the cholesky is lower-triangular, or if you are using a different algorithm that computes a symmetric matrix so row vs column order doesn't matter. "
    ]
   },
   {
@@ -2096,6 +2096,15 @@
    "execution_count": 41,
    "metadata": {},
    "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "cholesky:\n",
+      "[[1.414 0.071]\n",
+      " [0.    1.731]]\n"
+     ]
+    },
     {
      "data": {
       "text/plain": [
@@ -2112,6 +2121,8 @@
     "import scipy\n",
     "a = np.array([[2., .1], [.1, 3]])\n",
     "s = scipy.linalg.cholesky(a)\n",
+    "print(\"cholesky:\")\n",
+    "print(s)\n",
     "np.dot(s, s.T)"
    ]
   },