103
votes
Accepted
Impractical question: is it possible to find the regression line using a ruler and compass?
Loosely speaking, it's apparently possible to compute any quantity which can be expressed "using only the integers 0 and 1 and the operations for addition, subtraction, multiplication, division, ...
- 26.5k
14
votes
Ellipse formula from points
A straightforward way, especially when you expect the points to fall exactly on an ellipse (yet which works even when they don't), is to observe that an ellipse is the set of zeros of a second order ...
- 334k
13
votes
Accepted
What is the geometric relationship between the covariance matrix and the inverse of the covariance matrix?
Before I answer your questions, allow me to share how I think about covariance and precision matrices.
Covariance matrices have a special structure: they are positive semi-definite (PSD), which means ...
- 605
11
votes
Appropriate measure to find smallest covariance matrix
The ordering of matrices you refer to is known as the Loewner order and is a partial order much used in the study of positive definite matrices. A book-length treatment of the geometry on the manifold ...
- 82.8k
11
votes
Accepted
How does a quadratic kernel look like?
There are (at least) two ways to think about this.
One is as you mentioned: imagine the points being lifted into the shape of a quadratic function, and then being cut by a plane, producing an ellipse....
- 25.2k
10
votes
What type of data are dates?
This is a tricky question, and personally I feel this question is more about semantics and conventions.
Let's go to basics. What is Date? It's just a name we give to 86,400 seconds period. Date by ...
- 1,055
9
votes
What type of data are dates?
It is correct that dates do not fit nicely into the Stevens' typology of different levels of measurement. Dates are certainly ordered, so we could say that dates are ordinal type, but they are ...
- 82.8k
8
votes
Accepted
Average absolute value of a coordinate of a random unit vector?
This problem has a very nice geometrical interpretation if we can assume that the distribution $f(\vec{x})$ is constant over the surface of the $n-1$-unit-sphere in $n$-dimensional space. Then $f(\vec{...
- 86.6k
7
votes
Accepted
How is the spherical elevation angle distributed when $(x,y,z)$ are uniformly and normally chosen?
In my discussion here I am assuming your $\theta$ is effectively a longitude and $\phi$ is effectively a latitude. Perhaps more typical spherical co-ordinates use an angle down from the north pole ...
- 291k
7
votes
How does a quadratic kernel look like?
Suppose we have two features $(x_1, x_2)$, and we expand it into five features $(x_1^2, x_2^2, x_1, x_2, x_1x_2)$
The decision boundary is
$$
\beta_0+\beta_1x_1^2+\beta_2x_2^2+\beta_3x_1+\beta_4x_2+\...
- 37.3k
7
votes
Expected triangle area from normal distribution
Rather than an answer I want to extend your speculation: The distribution of the area with $\sigma=1$ has a Gamma distribution with parameters 2 and $\sqrt{3}/2$.
Why? First, a histogram of random ...
- 4,505
7
votes
Accepted
Bivariate normal covering circles and ellipses
What you want is the radius $r$ such that the cdf $P(\sqrt{X^2+Y^2}\leq r)=\alpha$ for some alpha. It is clear that $P(\sqrt{X^2+Y^2} \leq r)=P(X^2+Y^2 \leq r^2)$, and it is easier to work with the ...
- 1,415
7
votes
Accepted
Regression coefficient on a triangle using geometry
Because $(X,Y)$ has a uniform distribution over the triangle shown, the expectation of $Y$ conditional on $X$ evidently splits the lower and upper boundaries of the triangle, shown as the dotted line $...
- 334k
7
votes
Accepted
Origin of the term "spherical" in relation to covariance matrices?
This form of covariance matrix is actually more "elliptical" than "spherical"
I have not heard of this form of matrix being called a "spherical" covariance matrix, and ...
- 133k
6
votes
How is the spherical elevation angle distributed when $(x,y,z)$ are uniformly and normally chosen?
The complementary cumulative distribution for the spherical latitude $\phi$ gives the chance that a random point in the cube $[-1,1]^3$ will lie above the cone that graphs the function $z = \cot(\phi)\...
- 334k
6
votes
How is the kurtosis of a distribution related to the geometry of the density function?
A different kind of answer: We can illustrate kurtosis geometrically, using ideas from http://www.quantdec.com/envstats/notes/class_06/properties.htm: graphical moments.
Start with the definition of ...
- 82.8k
6
votes
Accepted
what means to be outside unit circle?
The roots in this case are roots of a polynomial, and they can be (and often are) complex numbers. That means they have coordinates, in this case called the real part and the imaginary part. As an ...
- 82.8k
6
votes
Accepted
Expected triangle area from normal distribution
This problem can be solved through a series of simplifications and then looking things up.
First, $\sigma$ merely establishes a unit of measurement: in a system where $\sigma$ is one unit, the ...
- 334k
5
votes
How is the kurtosis of a distribution related to the geometry of the density function?
Kurtosis is not related to the geometry of the distribution at all, at least not in the central portion of the distribution. In the central portion of the distribution (within the $\mu \pm \sigma$ ...
5
votes
Appropriate measure to find smallest covariance matrix
@kjetil b halvorsen gives a nice discussion of the geometric intuition behind positive semi-definiteness as a partial ordering. I'll give a more grubby-handed take on that same intuition. One which ...
- 8,004
5
votes
Accepted
Bounding data by two parallel lines with minimum distance between them
It is immediate (from the definitions) that the lines must pass through extremal points of the point set. Because at least one of them must contain at least two of the points (provided there is more ...
- 334k
4
votes
Is the expectation of the sufficient statistics $S(X)$ transverse the whole space in an exponential family?
Short answer to OP
Not necessarily, it depends on whether the extended convex cone spanned by varying the $\lambda_i$ outside the range of your parameter space can cover the whole mean parameter space....
- 2,490
4
votes
Average absolute value of a coordinate of a random unit vector?
The answer is $1/2$
This paper has the probability density $f_n(x_i)$ of $x_i$ for the vector inside an n-dimensional hypersphere. You're interested in the vector from origin to the random point on a ...
- 62.3k
4
votes
Geometric interpretation of mathematical expectation of a random variable
The mathematical expectation is the x-coordinate of the centre of gravity.
The picture above is borrowed from Wikipedia.
- 69.5k
4
votes
Why do we need to triangulate a convex polygon in order to sample uniformly from it?
The tldr answer is that in the square case, there are multiple ways to express a "deep" interior point as a convex combination of the vertices, but only one way for points that are nearer to ...
- 2,124
4
votes
Accepted
Does the distribution $f(x) \propto (1-x^2)^{n/2}$ have a name?
It is known as a Power semi-circle distribution with pdf $f(x)$:
$$f(x) = \frac{1}{\sqrt{\pi }}\frac{\Gamma (\theta +2) }{ \Gamma \left(\theta +\frac{3}{2}\right)} \sqrt{1-x^2}^{2 \theta +1} \quad \...
- 8,001
4
votes
Accepted
How is the set of probability distributions on $m$ values an $m-1$-dimensional simplex?
This comes from the definition of the standard simplex:
$$\left\{x \in \mathbb{R}^{k} : x_0 + \dots + x_{k-1} = 1, x_i \ge 0 \text{ for } i = 0, \dots, k-1 \right\} $$
It’s the set of $k$-dimensional ...
- 9,001
4
votes
Accepted
Derivation for the probability density of the squared sum of k coordinates of a uniform distributed vector/point on a unit n-sphere
The following closely follows whuber's exposition of deriving the t-distribution here.
It comes down to parameterizing the unit sphere in $\mathbb{R}^{d_1 + d_2}$ by using a point on the unit sphere ...
- 670
3
votes
Accepted
What is the physical intuition behind the equality $\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar x) x_i$?
This is the geometry of (orthogonal) projections and Pythagoras' theorem. Let $J$ be the all 1's $n\times n$ matrix, note that $J^2=nJ$. Now the centering operation $x_i \to x_i-\bar{x}$ is the ...
- 82.8k
3
votes
Accepted
Distribution of quadratic equation roots where coefficients are generated uniformly
There are two steps in any question of this nature: (1) find a useful way to characterize the event and (2) compute the probability of this event (in general, by integrating a probability density over ...
- 334k
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