Timeline for Spherical restriction of OLS
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16 events
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| Jun 7, 2017 at 15:13 | answer | added | user795305 | timeline score: 2 | |
| Apr 13, 2017 at 12:44 | history | edited | CommunityBot |
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| Jan 19, 2017 at 0:30 | history | edited | Antoni Parellada | CC BY-SA 3.0 |
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| Jan 18, 2017 at 21:15 | history | tweeted | twitter.com/StackStats/status/821828432258535425 | ||
| Jan 18, 2017 at 17:52 | history | edited | Antoni Parellada | CC BY-SA 3.0 |
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| Jan 18, 2017 at 17:15 | comment | added | Antoni Parellada | @whuber Very good. Thank you very much! You have solved my main hurdle in understanding this. | |
| Jan 18, 2017 at 17:13 | comment | added | whuber♦ | I did not intend to pre-empt any answers: after all, none of these comments have actually addressed your question, which appears to concern possible connections to spherical errors. I was only trying to clarify the meaning of "spherical restriction" (which does not appear to be a commonly used term in statistics). | |
| Jan 18, 2017 at 17:08 | comment | added | Antoni Parellada | @whuber Magisterial! As always! From your comments, I gather that, although you pre-empted the answer, it's still worthwhile to keep the OP open. Let me know if you disagree. | |
| Jan 18, 2017 at 17:07 | comment | added | whuber♦ | After Ridge Regression was invented, it was re-interpreted as the solution to OLS with a particular kind of Bayes prior on $\beta$. Within that interpretation, $\delta$ does have some meaning (related to how small you think the coefficients ought to be). | |
| Jan 18, 2017 at 17:05 | comment | added | Antoni Parellada | @whuber Oh, I see it now... but I thought that we were still within OLS... That ridge regression with the elliptical form was the next step, but that $\delta$ had already surfaced as a concept before moving from OLS to ridge. | |
| Jan 18, 2017 at 17:02 | comment | added | whuber♦ | In Ridge Regression, you don't know $\delta$, so you solve this optimization problem for a wide range of values of $\delta$ and you explore how the solutions vary with $\delta$. See stats.stackexchange.com/questions/154706 for an example. (Usually the columns of $X$ are first standardized. The coefficient estimates are graphed against $\delta$ in a "ridge trace" plot.) | |
| Jan 18, 2017 at 16:59 | comment | added | Antoni Parellada | @whuber I see the picture now - not on the surface of the sphere, but within. Where are the values of $\delta$ specified? | |
| Jan 18, 2017 at 16:58 | comment | added | whuber♦ | You seem to be confusing a ball with a line. The constraints require only that $\beta$ lie within the ball, not that it lie on the line $\beta_1=\beta_2=\cdots=\beta_k$! Note that both $\delta$ and $d$ are "given values": that is, they are not free to vary, but are specified as part of the problem. | |
| Jan 18, 2017 at 16:56 | comment | added | Antoni Parellada | @whuber Even centered, though, each $ \beta_i$ coefficient is different. Do you draw this hypersphere with a radius equal to the smallest coefficient? And is it just a statement implying that the coefficients are all less than infinity? I am still not seeing it... | |
| Jan 18, 2017 at 16:51 | comment | added | whuber♦ | For $\beta\in\mathbb{R}^k$, the equation $||\beta||^2\le\delta^2$ is satisfied by the points in the ball (aka "sphere") of radius $\delta$ centered at the origin of $\mathbb{R}^k$. I believe that's all that the phrase "spherical restriction" was intended to mean. | |
| Jan 18, 2017 at 16:02 | history | asked | Antoni Parellada | CC BY-SA 3.0 |