trying to make sense of expected test error
I'm trying to understand the definition of training error and test error given by Ryan Tibshirani in this handout where he states that:
in a model given by $y_i=f(x_i)+epsilon_i$, for $iin {1,...,n}$ where $epsilon_i$ is a sequence of i.i.d $sim mathcal{N}(0,sigma^2)$, and $hat{f}$ is an estimator such as $hat{y}_i=hat{f}(x_i)$. The expected training error and test error are given by:
begin{align}
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y_i-hat{y}_i)^2Big] \
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y'_i-hat{y}_i)^2Big]
end{align}
where $y'_i=f(x_i)+epsilon'_i$ and the $x_i$'s are fixed and are the same as in the expected training error. Now what i don't seem understand is why do we use the same $x_i$'s. For me the $x_i$'s are observations that are associated to their corresponding output $y_i$ and the test error is when you take, let's say $p$ new predictors measurements $x'_i$, that need not to be the same as the ones used in the training error, for which you already know the outcome $y_i$ and you calculate the average:
begin{equation}
mathbb{E}Big[frac{1}{p} sum_{i=1}^p big(y'_i - f(x'_i)big)^2Big]
end{equation}
where $y'_i=f(x'_i)+epsilon_i$.
It seems that i have a lot of misconceptions and i would be glad if someone could correct me. Thank you in advance
statistics machine-learning
asked yesterday
yjnt
63
add a comment |
I'm trying to understand the definition of training error and test error given by Ryan Tibshirani in this handout where he states that:
in a model given by $y_i=f(x_i)+epsilon_i$, for $iin {1,...,n}$ where $epsilon_i$ is a sequence of i.i.d $sim mathcal{N}(0,sigma^2)$, and $hat{f}$ is an estimator such as $hat{y}_i=hat{f}(x_i)$. The expected training error and test error are given by:
begin{align}
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y_i-hat{y}_i)^2Big] \
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y'_i-hat{y}_i)^2Big]
end{align}
where $y'_i=f(x_i)+epsilon'_i$ and the $x_i$'s are fixed and are the same as in the expected training error. Now what i don't seem understand is why do we use the same $x_i$'s. For me the $x_i$'s are observations that are associated to their corresponding output $y_i$ and the test error is when you take, let's say $p$ new predictors measurements $x'_i$, that need not to be the same as the ones used in the training error, for which you already know the outcome $y_i$ and you calculate the average:
begin{equation}
mathbb{E}Big[frac{1}{p} sum_{i=1}^p big(y'_i - f(x'_i)big)^2Big]
end{equation}
where $y'_i=f(x'_i)+epsilon_i$.
It seems that i have a lot of misconceptions and i would be glad if someone could correct me. Thank you in advance
statistics machine-learning
asked yesterday
yjnt
63
add a comment |
I'm trying to understand the definition of training error and test error given by Ryan Tibshirani in this handout where he states that:
in a model given by $y_i=f(x_i)+epsilon_i$, for $iin {1,...,n}$ where $epsilon_i$ is a sequence of i.i.d $sim mathcal{N}(0,sigma^2)$, and $hat{f}$ is an estimator such as $hat{y}_i=hat{f}(x_i)$. The expected training error and test error are given by:
begin{align}
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y_i-hat{y}_i)^2Big] \
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y'_i-hat{y}_i)^2Big]
end{align}
where $y'_i=f(x_i)+epsilon'_i$ and the $x_i$'s are fixed and are the same as in the expected training error. Now what i don't seem understand is why do we use the same $x_i$'s. For me the $x_i$'s are observations that are associated to their corresponding output $y_i$ and the test error is when you take, let's say $p$ new predictors measurements $x'_i$, that need not to be the same as the ones used in the training error, for which you already know the outcome $y_i$ and you calculate the average:
begin{equation}
mathbb{E}Big[frac{1}{p} sum_{i=1}^p big(y'_i - f(x'_i)big)^2Big]
end{equation}
where $y'_i=f(x'_i)+epsilon_i$.
It seems that i have a lot of misconceptions and i would be glad if someone could correct me. Thank you in advance
statistics machine-learning
asked yesterday
yjnt
63
I'm trying to understand the definition of training error and test error given by Ryan Tibshirani in this handout where he states that:
in a model given by $y_i=f(x_i)+epsilon_i$, for $iin {1,...,n}$ where $epsilon_i$ is a sequence of i.i.d $sim mathcal{N}(0,sigma^2)$, and $hat{f}$ is an estimator such as $hat{y}_i=hat{f}(x_i)$. The expected training error and test error are given by:
begin{align}
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y_i-hat{y}_i)^2Big] \
mathbb{E}Big[frac{1}{N} sum_{i=1}^N (y'_i-hat{y}_i)^2Big]
end{align}
where $y'_i=f(x_i)+epsilon'_i$ and the $x_i$'s are fixed and are the same as in the expected training error. Now what i don't seem understand is why do we use the same $x_i$'s. For me the $x_i$'s are observations that are associated to their corresponding output $y_i$ and the test error is when you take, let's say $p$ new predictors measurements $x'_i$, that need not to be the same as the ones used in the training error, for which you already know the outcome $y_i$ and you calculate the average:
begin{equation}
mathbb{E}Big[frac{1}{p} sum_{i=1}^p big(y'_i - f(x'_i)big)^2Big]
end{equation}
where $y'_i=f(x'_i)+epsilon_i$.
It seems that i have a lot of misconceptions and i would be glad if someone could correct me. Thank you in advance
statistics machine-learning
statistics machine-learning
asked yesterday
yjnt
63
asked yesterday
yjnt
63
asked yesterday
yjnt
63
asked yesterday
yjnt
63
asked yesterday
yjnt
63
63
add a comment |
add a comment |
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