trying to make sense of expected test error
$begingroup$
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 Jan 4 at 17:07
yjntyjnt
113
$endgroup$
add a comment |
$begingroup$
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 Jan 4 at 17:07
yjntyjnt
113
$endgroup$
add a comment |
$begingroup$
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 Jan 4 at 17:07
yjntyjnt
113
$endgroup$
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 Jan 4 at 17:07
yjntyjnt
113
asked Jan 4 at 17:07
yjntyjnt
113
asked Jan 4 at 17:07
yjntyjnt
113
asked Jan 4 at 17:07
yjntyjnt
113
asked Jan 4 at 17:07
yjntyjnt
113
113
add a comment |
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