update on book
This commit is contained in:
Morten Hjorth-Jensen
@@ -239,7 +239,7 @@
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"\n",
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"What follows is a simple Python code where we have defined a function\n",
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"$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n",
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"The numbers in the vector $\\hat{x}$ are given\n",
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"The numbers in the vector $\\boldsymbol{x}$ are given\n",
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"by random numbers generated with a uniform distribution with entries\n",
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"$x_i \\in [0,1]$ (more about probability distribution functions\n",
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"later). These values are then used to define a function $y(x)$\n",
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@@ -275,7 +275,7 @@
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"distribution. From **Scikit-Learn** we import then the\n",
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"**LinearRegression** functionality and make a prediction $\\tilde{y} =\n",
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"\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n",
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"data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n",
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"data $(\\boldsymbol{x},\\boldsymbol{y})$ for our training data. The Python package\n",
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"**scikit-learn** has also a functionality which extracts the above\n",
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"fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n",
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"distinguish between training data and test data.\n",
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@@ -407,7 +407,7 @@
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"metadata": {},
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"source": [
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"$$\n",
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"\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n",
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"\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\boldsymbol{y} -\\boldsymbol{\\tilde{y}}\\vert}{\\vert \\boldsymbol{y}\\vert}.\n",
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"$$"
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]
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},
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@@ -521,7 +521,7 @@
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"metadata": {},
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"source": [
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"$$\n",
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"MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n",
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"MSE(\\boldsymbol{y},\\boldsymbol{\\tilde{y}}) = \\frac{1}{n}\n",
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"\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n",
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"$$"
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]
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@@ -538,10 +538,10 @@
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"determination. It provides a measure of how well future samples are\n",
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"likely to be predicted by the model. Best possible score is 1.0 and it\n",
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"can be negative (because the model can be arbitrarily worse). A\n",
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"constant model that always predicts the expected value of $\\hat{y}$,\n",
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"constant model that always predicts the expected value of $\\boldsymbol{y}$,\n",
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"disregarding the input features, would get a $R^2$ score of $0.0$.\n",
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"\n",
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"If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as"
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"If $\\tilde{\\boldsymbol{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as"
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]
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},
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{
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@@ -549,7 +549,7 @@
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"metadata": {},
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"source": [
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"$$\n",
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"R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
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"R^2(\\boldsymbol{y}, \\tilde{\\boldsymbol{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
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"$$"
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]
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},
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@@ -557,7 +557,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"where we have defined the mean value of $\\hat{y}$ as"
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"where we have defined the mean value of $\\boldsymbol{y}$ as"
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]
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},
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{
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@@ -583,7 +583,7 @@
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"metadata": {},
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"source": [
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"$$\n",
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"\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n",
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"\\text{MAE}(\\boldsymbol{y}, \\boldsymbol{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n",
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"$$"
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]
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},
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@@ -600,7 +600,7 @@
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"metadata": {},
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"source": [
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"$$\n",
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"\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n",
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"\\text{MSLE}(\\boldsymbol{y}, \\boldsymbol{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n",
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"$$"
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]
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},
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@@ -3314,7 +3314,7 @@
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"metadata": {},
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"source": [
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"$$\n",
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"MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n",
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"MSE(\\boldsymbol{y},\\boldsymbol{\\tilde{y}}) = \\frac{1}{n}\n",
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"\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n",
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"$$"
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]
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@@ -3324,7 +3324,7 @@
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"metadata": {},
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"source": [
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"and the $R^2$ score function.\n",
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"If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as"
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"If $\\tilde{\\boldsymbol{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as"
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]
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},
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{
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@@ -3332,7 +3332,7 @@
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"metadata": {},
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"source": [
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"$$\n",
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"R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
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"R^2(\\boldsymbol{y}, \\tilde{\\boldsymbol{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
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"$$"
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]
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},
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@@ -3340,7 +3340,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"where we have defined the mean value of $\\hat{y}$ as"
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"where we have defined the mean value of $\\boldsymbol{y}$ as"
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]
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},
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{
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