10.3 The physics of springs  (Page 9/11)

So we can set up a change of variable:

Notice that

This change of variable transforms the equiprobability ellipses of x into circles centered at the origin:

We are now ready to evaluate the integral. We will first perform a change of variable from x to w , then change to polar coordinates.

Given this, we can solve for δ :

This gives us a straightforward way to check whether or not a point in the sample, x _j , lies in the credibility region. If

holds, then the point is in the credibility ellipse.

Recall that this expression is for two dimensions. If we repeat the integral in different dimensions, we get a different expression relating δ and p . We are also interested in distributions in R , R ¹⁴ , and R ¹⁶ , so we repeat the calculations for those spaces. In R ¹⁴ and R ¹⁶ , we do not get a nice expression, so we approximate the values of δ for $p=\{0.1,0.2,\cdots ,0.9\}$ using a TI-89 calculator (MATLAB was unable to carry out the calculation symbolically). In R , using $w={\sigma }^{-1}$ , we find that

We can then plot the number of points inside a credibility ellipse against the credibility. A perfectly normal distribution will be a straight line from the origin to $(1,1)$ . If a sample significantly deviates from this line, then this criterion suggests that the sample is not normal.

Distributions

Armed with these criteria for looking at the normality of a sample, we study the displacements of the nodes. First we look at the nodes individually. We look at how a single node moves under a particular load. We expect the sample to be normally distributed: under a fixed load, most of the displacements should be about the same, with a few outliers. Data Set A has 104 samples split evenly between two loads, so we have 14 samples in R ² with 52 points each. [link] plots the number of points inside the credibility ellipse against the credibility for Node 4, Pattern 2. The red line is from the sample, the blue line is a randomly generated normally distributed sample with 52 points, and the black line is a reference line. The blue allows us to have an idea for how much we can expect a sample this size to deviate from the straight line. [link] is a histogram of the positions of the node. This plot suggests that the distribution of this node is normally distributed. All of the nodes, under both loads, give similar results (See [link] ). Because of this, we believe that the displacements are normally distributed, as we expected.

Next we look at the nodes collectively, giving us two samples in R ¹⁴ with 52 points each. Because we believe that each node is normally distributed, we expect the combined displacements to be normally distributed. [link] is the plot from Load 1; [link] is the plot from Load 2. [link] suggests that the displacements of the nodes are indeed normally distributed.