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    Prediction interval using a function that combines regression parameters




    I have a question about the prediction interval on a function that combines parameters obtained via linear regression. I was wondering if someone could point me in the right direction or recommend a text that addresses the issue. Or maybe just tell me what terms I should be looking for in a websearch.

    I have a collection of samples giving elevations at closely spaced but irregular locations near a point of interest. I've adjusted the X/Y coordinates of the samples so that the point of interest is essentially the origin. I am modeling the surface using 1st, 2nd, and 3rd order polynomials such that elevation is given by

    z = p (x, y) = \beta_0 + \beta_1 x + \beta_2 y + \beta_3 x^2 + \beta_4 y^2 ...

    I can derive a prediction interval for the elevation at the point-of-interest (origin) easily enough. But I am also interested in deriving the slope and getting a prediction interval for that value.

    The slope is just

    \sqrt{ ( \frac {\partial p}{\partial x})^2 + ( \frac {\partial p}{\partial y})^2 }


    Because my point of interest is at the origin, all but the first-order terms of the polynomial either drop out when I take the derivatives or get zeroed out when the polynomial is evaluated at (0,0). So the slope is


    slope = \sqrt{ (\beta_1^2 + \beta_2^2 }

    I am fairly sure that the uncertainty of the slope result will have to combine the uncertainty for both parameters. I'm just not quite sure how to get there.

    I am currently working both in R and also using some code I've written in Java (using the Apache Commons Math library).

    Thanks for you help.

    Gary
    Last edited by Dason; 12-16-2014 at 10:21 AM. Reason: trying to get LaTex tags to work

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    Re: Prediction interval using a function that combines regression parameters

    Just so we're on the same page...

    When you say "slope" do you mean the slope in the direction of (1,1)? And what do you mean by "prediction interval". I ask because typically it doesn't make sense to talk about a prediction interval for a "slope"
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    Re: Prediction interval using a function that combines regression parameters

    For this application, I am thinking of slope not a as a mathematical concept, but in ordinary human terms... You stand on the ground and notice that it is not horizontal. The form I used in the earlier post is simply the magnitude of the slope without regard to the direction. It's the tangent of the angle between the horizontal plane and the plane tangent to the surface at the point of interest. It's also the tangent of the angle between the surface normal and the vertical axis. To get the expression I gave in my post, I simply found the unit surface normal, dotted it with (0, 0, 1) to get the cosine of the angle then used a bit of trig to get the tangent.

    Later on, I do think I am going to be interested in directional derivatives. But however I approach the problem, what I'm really interested is both an estimated value for slope, but also some indication of the general accuracy of that estimate.

    I am fairly new to regression/stats and think I kinda understand the ideas of a confidence interval versus prediction interval, but I could be foolishly optimistic here. Should I be asking a different question?

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    Re: Prediction interval using a function that combines regression parameters


    Also, thanks for fixing my math tags. I was baffled as to why the LaTex stuff wasn't working... I can't believe I used the wrong slash in the close tags.

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