What is an affine transformation? Which families of distribution are closed under affine transformation?
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6$\begingroup$ Could you please be more specific in what context you need to know that, whether this is a homework question and what is wrong with en.wikipedia.org/wiki/Affine_transformation ? $\endgroup$– MomoCommented Jun 26, 2014 at 13:48
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3$\begingroup$ I was asked in an interview.... the question was "give an example of statistical distribution, other than normal distribution, which is closed under affine transformation". $\endgroup$– arnabCommented Jun 26, 2014 at 15:03
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6$\begingroup$ All location-scale families are, by definition, closed under affine transformations, because in one dimension an affine transformation is just a shift in location together with a rescaling. Any family that is not already a location-scale family can be made into one by including all such transformations of its members. $\endgroup$– whuber ♦Commented Jun 26, 2014 at 15:20
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3$\begingroup$ To add to @whuber's excellent comment that shows that the answer to the interview questions was "any": one can make any univariate distribution into a location-scale family by replacing the density $f(x)$ with $f^*(x)=\frac{f(\frac{x-mean}{sd})}{sd}$. $\endgroup$– MomoCommented Jun 26, 2014 at 15:32
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$\begingroup$ en.wikipedia.org/wiki/Affine_transformation $\endgroup$– kjetil b halvorsen ♦Commented Oct 31, 2019 at 10:39
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4 Answers
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An affine transformation has the form $f(x) = Ax + b$ where $A$ is a matrix and $b$ is a vector (of proper dimensions, obviously).
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Affine transformation(left multiply a matrix), also called linear transformation(for more intuition please refer to this blog: A Geometrical Understanding of Matrices), is parallel preserving, and it only stretches, reflects, rotates(for example diagonal matrix or orthogonal matrix) or shears(matrix with off-diagonal elements) a vector(the same applies to many vectors/a matrix), and the "non-affine"(also a type of projective transformation as explained in the comment by @whuber) transformation may be like the first example in the following diagram:
More generally speaking affine transformation has the following three properties:
straight lines preserved
parallel lines preserved
ratios of lengths along lines preserved (midpoints preserved)
answered Dec 18, 2019 at 0:09
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3$\begingroup$ It is difficult to see in what sense a projective transformation is any kind of "opposite" to an affine transformation. Indeed, affine transformations are special kinds of projective transformations. $\endgroup$– whuber ♦Commented Dec 18, 2019 at 14:50
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$\begingroup$ @whuber Glad to know that, then I wonder what the other projective transformations are except affine transformation? I mean the formal name of the "non-affine" transformation? Thanks for helping me figure out the "bug". $\endgroup$ Commented Dec 18, 2019 at 14:57
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2$\begingroup$ The projective transformations are the rational functions of order 1. On the line, for instance, they are given by four numbers $a,b,c,d$ as $$x\to \frac{ax+b}{cx+d}.$$ Usually they are limited to numbers where $ad-bc\ne 0$ for which the transformation is invertible. The transformations associated with $(a,b,c,d)$ and $(a\lambda,b\lambda,c\lambda,d\lambda)$ are the same when $\lambda\ne 0,$ making this a three-dimensional family of transformations. The affine transformations are those for which $c=0$ and $d\ne 0.$ $\endgroup$– whuber ♦Commented Dec 18, 2019 at 16:58
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$\begingroup$ FWIW, what makes a transformation "affine" instead of just "linear" is that in addition to multiplication by a (noninvertible) matrix, one is allowed to add a constant vector to the result, thereby shifting it away from the origin. You give only the definition of "linear transformation" here. $\endgroup$– whuber ♦Commented May 12, 2023 at 19:10
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In your comment the interview question you were asked was "give an example of statistical distribution, other than normal distribution, which is closed under affine transformation".
The example to which the question refers is the fact that if you have a normally distributed random variable $X$, say $X\sim \text{N}(\mu,\sigma^2)$, then an affine transformation is also normally distributed $aX+b \sim \text{N}(a\mu+b,a^2\sigma^2)$.
The terminology in Statistics for distributions which are 'closed under affine transformation' is $\textbf{location-scale family}$.
One example which would answer the question is the continuous uniform distribution. If $X\sim U[\alpha,\beta]$, and $Y= aX+b$ then $$Y\sim U[a\alpha+b, a\beta+b].$$
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2$\begingroup$ +1. Some care is needed with the notation when $a\lt 0,$ though. $\endgroup$– whuber ♦Commented May 12, 2023 at 15:45
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So I look here: http://mathworld.wolfram.com/AffineTransformation.html
It is a rotation. All points on a line, stay on the same line.
Per @Luca: It can have scaling, shear, translation as well. No bending. Straight lines are always straight.
answered Jun 26, 2014 at 14:09
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10$\begingroup$ Affine is not just a rotation. It could be translation, shearing, scaling, rotation (and others) or any combinations of it AFAIK. I know normal distribution is closed under Affine transformations and it may be the case for the whole exponential family, but I am not sure about that. $\endgroup$– LucaCommented Jun 26, 2014 at 14:13
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$\begingroup$ @Luca, I was thinking rotation in the homogenous sense, where a translation was a rotation through another dimension. Thank you for the clarification. The center of the rotation is in $\mathbb{R}^n$ then projected back to the original dimensionality of $\mathbb{R}^{m \leqslant n}$ $\endgroup$ Commented Jun 26, 2014 at 14:19
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$\begingroup$ a common type of affine transformation is $\frac{aX-b}{c}$ btw $\endgroup$– emcorCommented Jun 26, 2014 at 14:19
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6$\begingroup$ @Luca All affine transformations of $\mathbb{R}^n$ can be derived from the linear transformations of $\mathbb{R}^{n+1}=\{x_1,x_2,\ldots,x_{n+1}\}$ that preserve the hyperplane $x_{n+1}=1$. Few of those linear transformations will be "rotations," though: by definition, the matrix of a rotation is orthogonal. $\endgroup$– whuber ♦Commented Jun 26, 2014 at 15:24
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1$\begingroup$ It would have to be some extraordinarily flexible "augmentation." After all, suppose a transformation increases the length (Euclidean norm) of a vector: there does not appear to be any meaningful way to make that transformation somehow a component of an orthogonal transformation, which preserves all lengths. Part of the issue might be that transformations which preserve lines, as you say in your second sentence, are projective transformations, which includes much more than rotations; yet not all projective transformations are affine. $\endgroup$– whuber ♦Commented May 12, 2023 at 20:02