Identifying changes in new data using previously trained regression model
$begingroup$
I would like some ideas on the following problem. I have a data $x$:
$$x(y,z) = [x_1(y,z), x_2(y,z), x_3(y,z)]$$
such that they are function of $(y,z)$, but neither of $(y,z)$ is available. Therefore I set up a regression model, such that:
$$tilde{x}_1 = ax_2 + bx_3$$
$$tilde{x}_2 = ax_1 + bx_3 tag{1}$$
$$tilde{x}_3 = ax_1 + bx_2$$
Such model, let's simplify the notation to $tilde{x}_i = ABx_{notin{i}}$, can reflect when there is a change of $x$, other than due to $(y,z)$. This is because the relationship, other than relationship to $(y,z)$ is broken and the model will output $tilde{x}$, which is different from the given $x$.
Now, I also know, that $x_{new}$, that is the new incoming data, can differ from $x$, only in some particular way, lets say:
$$x = n^{(1)}x_{new} + n^{(2)} tag{2}$$
where $n^{(1)}$ and $n^{(2)}$ are vectors, the same dimension as $x$. This gives $2*dim(x)$ unknowns.
QUESTION:
Is it possible to recover the vectors $n^{(1)}$, $n^{(2)}$, given sufficient data $x_{new}$ ?
The whole problem again in simple words. The regression model is trained on $x_{train}$, such that $(1)$ is obtained. Then there is a change in the input $x$, such that $x_{new}$ in $(1)$ produces error. Can I find a function $(2)$, such that $x_{new}$ in $(2)$, will give $x$ using $(1)$ ?
Is there any material/technique that could help me to analyze this situation / solve the problem ?
P.S.
I have simply plugged $(2)$ into $(1)$, and using at least 2 $x_{new}$ to obtain $n^{(1)}$, $n^{(2)}$. Why 2 $x_{new}$ ? Because there are $2*dim(x)$ unknowns. This probably should work, but it is non-convex and relatively difficult to optimize.
P.S.2
I can use some stochastic search algorithm, but I would rather try some more mathematically reasoned techniques.
optimization regression machine-learning
asked Jan 21 at 13:14
Martin GMartin G
548
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add a comment |
$begingroup$
I would like some ideas on the following problem. I have a data $x$:
$$x(y,z) = [x_1(y,z), x_2(y,z), x_3(y,z)]$$
such that they are function of $(y,z)$, but neither of $(y,z)$ is available. Therefore I set up a regression model, such that:
$$tilde{x}_1 = ax_2 + bx_3$$
$$tilde{x}_2 = ax_1 + bx_3 tag{1}$$
$$tilde{x}_3 = ax_1 + bx_2$$
Such model, let's simplify the notation to $tilde{x}_i = ABx_{notin{i}}$, can reflect when there is a change of $x$, other than due to $(y,z)$. This is because the relationship, other than relationship to $(y,z)$ is broken and the model will output $tilde{x}$, which is different from the given $x$.
Now, I also know, that $x_{new}$, that is the new incoming data, can differ from $x$, only in some particular way, lets say:
$$x = n^{(1)}x_{new} + n^{(2)} tag{2}$$
where $n^{(1)}$ and $n^{(2)}$ are vectors, the same dimension as $x$. This gives $2*dim(x)$ unknowns.
QUESTION:
Is it possible to recover the vectors $n^{(1)}$, $n^{(2)}$, given sufficient data $x_{new}$ ?
The whole problem again in simple words. The regression model is trained on $x_{train}$, such that $(1)$ is obtained. Then there is a change in the input $x$, such that $x_{new}$ in $(1)$ produces error. Can I find a function $(2)$, such that $x_{new}$ in $(2)$, will give $x$ using $(1)$ ?
Is there any material/technique that could help me to analyze this situation / solve the problem ?
P.S.
I have simply plugged $(2)$ into $(1)$, and using at least 2 $x_{new}$ to obtain $n^{(1)}$, $n^{(2)}$. Why 2 $x_{new}$ ? Because there are $2*dim(x)$ unknowns. This probably should work, but it is non-convex and relatively difficult to optimize.
P.S.2
I can use some stochastic search algorithm, but I would rather try some more mathematically reasoned techniques.
optimization regression machine-learning
asked Jan 21 at 13:14
Martin GMartin G
548
$endgroup$
add a comment |
$begingroup$
I would like some ideas on the following problem. I have a data $x$:
$$x(y,z) = [x_1(y,z), x_2(y,z), x_3(y,z)]$$
such that they are function of $(y,z)$, but neither of $(y,z)$ is available. Therefore I set up a regression model, such that:
$$tilde{x}_1 = ax_2 + bx_3$$
$$tilde{x}_2 = ax_1 + bx_3 tag{1}$$
$$tilde{x}_3 = ax_1 + bx_2$$
Such model, let's simplify the notation to $tilde{x}_i = ABx_{notin{i}}$, can reflect when there is a change of $x$, other than due to $(y,z)$. This is because the relationship, other than relationship to $(y,z)$ is broken and the model will output $tilde{x}$, which is different from the given $x$.
Now, I also know, that $x_{new}$, that is the new incoming data, can differ from $x$, only in some particular way, lets say:
$$x = n^{(1)}x_{new} + n^{(2)} tag{2}$$
where $n^{(1)}$ and $n^{(2)}$ are vectors, the same dimension as $x$. This gives $2*dim(x)$ unknowns.
QUESTION:
Is it possible to recover the vectors $n^{(1)}$, $n^{(2)}$, given sufficient data $x_{new}$ ?
The whole problem again in simple words. The regression model is trained on $x_{train}$, such that $(1)$ is obtained. Then there is a change in the input $x$, such that $x_{new}$ in $(1)$ produces error. Can I find a function $(2)$, such that $x_{new}$ in $(2)$, will give $x$ using $(1)$ ?
Is there any material/technique that could help me to analyze this situation / solve the problem ?
P.S.
I have simply plugged $(2)$ into $(1)$, and using at least 2 $x_{new}$ to obtain $n^{(1)}$, $n^{(2)}$. Why 2 $x_{new}$ ? Because there are $2*dim(x)$ unknowns. This probably should work, but it is non-convex and relatively difficult to optimize.
P.S.2
I can use some stochastic search algorithm, but I would rather try some more mathematically reasoned techniques.
optimization regression machine-learning
asked Jan 21 at 13:14
Martin GMartin G
548
$endgroup$
I would like some ideas on the following problem. I have a data $x$:
$$x(y,z) = [x_1(y,z), x_2(y,z), x_3(y,z)]$$
such that they are function of $(y,z)$, but neither of $(y,z)$ is available. Therefore I set up a regression model, such that:
$$tilde{x}_1 = ax_2 + bx_3$$
$$tilde{x}_2 = ax_1 + bx_3 tag{1}$$
$$tilde{x}_3 = ax_1 + bx_2$$
Such model, let's simplify the notation to $tilde{x}_i = ABx_{notin{i}}$, can reflect when there is a change of $x$, other than due to $(y,z)$. This is because the relationship, other than relationship to $(y,z)$ is broken and the model will output $tilde{x}$, which is different from the given $x$.
Now, I also know, that $x_{new}$, that is the new incoming data, can differ from $x$, only in some particular way, lets say:
$$x = n^{(1)}x_{new} + n^{(2)} tag{2}$$
where $n^{(1)}$ and $n^{(2)}$ are vectors, the same dimension as $x$. This gives $2*dim(x)$ unknowns.
QUESTION:
Is it possible to recover the vectors $n^{(1)}$, $n^{(2)}$, given sufficient data $x_{new}$ ?
The whole problem again in simple words. The regression model is trained on $x_{train}$, such that $(1)$ is obtained. Then there is a change in the input $x$, such that $x_{new}$ in $(1)$ produces error. Can I find a function $(2)$, such that $x_{new}$ in $(2)$, will give $x$ using $(1)$ ?
Is there any material/technique that could help me to analyze this situation / solve the problem ?
P.S.
I have simply plugged $(2)$ into $(1)$, and using at least 2 $x_{new}$ to obtain $n^{(1)}$, $n^{(2)}$. Why 2 $x_{new}$ ? Because there are $2*dim(x)$ unknowns. This probably should work, but it is non-convex and relatively difficult to optimize.
P.S.2
I can use some stochastic search algorithm, but I would rather try some more mathematically reasoned techniques.
optimization regression machine-learning
optimization regression machine-learning
asked Jan 21 at 13:14
Martin GMartin G
548
asked Jan 21 at 13:14
Martin GMartin G
548
asked Jan 21 at 13:14
Martin GMartin G
548
asked Jan 21 at 13:14
Martin GMartin G
548
asked Jan 21 at 13:14
Martin GMartin G
548
548
add a comment |
add a comment |
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