10.3 The physics of springs  (Page 8/11)

Example

Reading the number shown on a die is a random variable. The number shown is in R .

Calculating the displacements of the nodes is a random variable. The actual displacements are in R ¹⁴

Definition 3 Probability Distribution

A function on a set $B\subset {\mathbb{R}}^n$ describing the probability that $X\left(\omega \right)\in B$ .

Example

The distribution, the log-normal distribution, and the Poisson distribution are all well-known distributions.

Definition 4 Expectation, Variance, Covariance

The expectation of a distribution is the “center of mass”:

The empirical expectation of a sample $\left\{x_1,,,x_2,,,\cdots ,,,x_N\right\}$ is the mean:

The variance of a distribution is the expectation of the squared difference from the expectation of the distribution, $var\left(X\right)={\sigma }^2=E\left\{{\left(X,-,\overline{x}\right)}^2\right\}={\int }_{\mathbb{R}}{\left(X,-,\overline{x}\right)}^2\pi \left(x\right)dx.$ This is only for a one-dimensional distribution.

The covariance of two single-variable distributions is the expectation of the deviation from the expectation of one, times the expectation of the deviation from the expectation of the other, $cov(X,Y)=E\left\{(X-\overline{x}),(Y-\overline{y})\right\}=E\left\{X,Y\right\}-E\left\{X\right\}E\left\{Y\right\}$ .

The covariance matrix of a distribution has, for its $(i,j)$ entry the covariance of the i and j components of the distribution. The empirical covariance matrix of a sample $\left\{x_1,,,x_2,,,\cdots ,,,x_N\right\}$ is the mean of deviation from the mean:

Definition 5 Normal (Gaussian) Distribution

In one dimension, the normal distribution has probability distribution function

In n dimensions, the normal distribution has probability distribution function

Definition 6 Maximum Likelihood Estimate

Given a sample from a known distribution depending on unknown parameter θ , the value of θ that maximizes the probability that the distribution returns the given sample.

Definition 7 Equiprobability Curve

Given an n -dimensional distribution, the locus of points in n -dimensions with equivalent probability.

Example

The equiprobability curves for a normal n -dimensional distribution are n -dimensional ellipses.

Normality

Because we wish to use statitsical inference to solve this problem, we begin by looking at the probability distribution of the displacements.

Calvetti and Somersalo [link] present a method for studying the normality of a 2-dimensional sample S , by using the equiprobability curves, which they also call credibility curves. Let π , corresponding to a random variable X , be a probability density. Let

that is, the set of all points whose probability is greater than or equal to some positive number α . This corresponds to either the empty set (for $\alpha >1$ ) or the interior of an equiprobability curve; if π is normally distributed and $\alpha \le 1$ , then B _α is the interior of an ellipse. We calculate the probability that X is in B _α , the credibility:

We would like to find an explicit expression relating α and p , $\alpha =\alpha \left(p\right).$ If S is normally distributed, then the number of points inside $B_{\alpha \left(p\right)}$ is about $pn.$

The immediate goal, then is to calculate the number of points in S that lie within B _α for some α . We will evaluate the integral $p={\int }_{B_{\alpha }}\pi \left(x\right)dx$ , but first we will need a change of variable. Let $U{D}^{-1}{U}^T$ be the eigenvalue of decomposition of ${\widehat{\Gamma }}^{-1}$ (the inverse of the empirical covariance matrix), where