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notations for estimated versus non estimated params #1

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notations for estimated versus non estimated params #1

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32 changes: 26 additions & 6 deletions glm_intro.ipynb
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Original file line number Diff line number Diff line change
@@ -1,7 +1,7 @@
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Expand Down Expand Up @@ -285,7 +285,7 @@
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"$\\evec$ is the vector of errors $e_1 ... e_{12}$:"
"$\\bf{e}$ is the vector of errors $e_1 ... e_{12}$:"
]
},
{
Expand Down Expand Up @@ -318,7 +318,7 @@
"Bear with with us for a little trick. If $\\mathbf{o}$ is a vector of ones, then we can rewrite the formula as:\n",
"\n",
"$$\n",
"\\yvec = c\\mathbf{o} + b\\xvec + \\evec\n",
"\\yvec = c\\mathbf{o} + b\\xvec + \\bf{e}\n",
"$$\n",
"\n",
"because $o_i = 1$ and so $co_i = c$."
Expand Down Expand Up @@ -380,15 +380,27 @@
"In symbols:\n",
"\n",
"$$\n",
"\\yvec = \\Xmat \\bvec + \\evec\n",
"\\yvec = \\Xmat \\bf{B} + \\bf{e}\n",
"$$"
]
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"We still haven't found our best fitting line. But before we go further, it might be obvious that we can easily add a new predictor here."
"We use the symbols $\\bf{B}$ and $\\bf{e}$ when we have given them some values (here, we estimated $\\bf{B}$ - albeit visually- and therefore $\\bf{e}$ because we gave B the values (10, .9). We still haven't found our best fitting line. But before we go further, it might be obvious that we can easily add a new predictor here."
]
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"If we have not estimated the parameters $\\bf{B}$, then we use the greek notations for the unknowns:\n",
"\n",
"\n",
"$$\n",
"\\yvec = \\Xmat \\bf{\\bvec} + \\bf{\\evec}\n",
"$$"
]
},
{
Expand Down Expand Up @@ -459,7 +471,7 @@
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"$\\newcommand{\\bhat}{\\hat{\\bvec}} \\newcommand{\\yhat}{\\hat{\\yvec}}$\n",
"Our students and their psychopathy scores are a *sample* from the population of all students' psychopathy scores. The parameters $\\bvec$ are the parameters that fit the design $\\Xmat$ to the *population* scores. We only have a sample from this population, so we cannot get the true population $\\bvec$ vector, we can only *estimate* $\\bvec$ from our sample. We will write this sample estimate as $\\bhat$ to distinguish it from the true population parameters $\\bvec$."
"Our students and their psychopathy scores are a *sample* from the population of all students' psychopathy scores. The parameters $\\bvec$ are the parameters that fit the design $\\Xmat$ to the *population* scores. We only have a sample from this population, so we cannot get the true population $\\bvec$ vector, we can only *estimate* $\\bvec$ from our sample. We will write this sample estimate as $\\bhat$ or $\\bf{B}$ to distinguish it from the true population parameters $\\bvec$."
]
},
{
Expand Down Expand Up @@ -1055,6 +1067,14 @@
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