Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

noise-share

One function in the tls library, called 5 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 17 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws which of the two least-squares methods is more accurate, as a fixed amount of noise moves from b into a. One curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.28 when all the noise is in b — where ordinary least squares is the more accurate — to 2.47 when all of it is in A. The line at one is where the two methods are equally right.

noise-share is one function in lib/figures/tls.js — errors in the matrix too — where the residual orders two methods backwards. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.

Which of the two least-squares methods is more accurate, as a fixed amount of noise moves from b into AOne curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.28 when all the noise is in b — where ordinary least squares is the more accurate — to 2.47 when all of it is in A. The line at one is where the two methods are equally right.00.250.50.75110⁻¹110¹share of the noise placed in the matrixleast-squares error ÷ total least-squares errorequally accuratetotal leastsquares aheadordinary leastsquares aheadthe model, not the methodadvantage, all noise in b0.28advantage, all noise in A2.5seeds at each share40the same total noise at every pointand only where it sits changes

One curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.28 when all the noise is in b — where ordinary least squares is the more accurate — to 2.47 when all of it is in A. The line at one is where the two methods are equally right.

total: 0.2

The arguments are the ones The projection and the right angle passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Which of the two least-squares methods is more accurate, as a fixed amount of noise moves from b into AOne curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.28 when all the noise is in b — where ordinary least squares is the more accurate — to 2.47 when all of it is in A. The line at one is where the two methods are equally right.00.250.50.75110⁻¹110¹share of the noise placed in the matrixleast-squares error ÷ total least-squares errorequally accuratetotal leastsquares aheadordinary leastsquares aheadthe model, not the methodadvantage, all noise in b0.28advantage, all noise in A2.5seeds at each share40the same total noise at every pointand only where it sits changes

One curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.28 when all the noise is in b — where ordinary least squares is the more accurate — to 2.47 when all of it is in A. The line at one is where the two methods are equally right.

total: 0.3

The arguments are the ones When the matrix is wrong too passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Which of the two least-squares methods is more accurate, as a fixed amount of noise moves from b into AOne curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.17 when all the noise is in b — where ordinary least squares is the more accurate — to 3.14 when all of it is in A. The line at one is where the two methods are equally right.00.250.50.75110⁻²10⁻¹110¹share of the noise placed in the matrixleast-squares error ÷ total least-squares errorequally accuratetotal leastsquares aheadordinary leastsquares aheadthe model, not the methodadvantage, all noise in b0.17advantage, all noise in A3.1seeds at each share40the same total noise at every pointand only where it sits changes

One curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.17 when all the noise is in b — where ordinary least squares is the more accurate — to 3.14 when all of it is in A. The line at one is where the two methods are equally right.

total: 0.05

The arguments are the ones When the matrix is wrong too passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Which of the two least-squares methods is more accurate, as a fixed amount of noise moves from b into AOne curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.99 when all the noise is in b — where ordinary least squares is the more accurate — to 0.91 when all of it is in A. The line at one is where the two methods are equally right.00.250.50.75110⁻¹110¹share of the noise placed in the matrixleast-squares error ÷ total least-squares errorequally accuratetotal leastsquares aheadordinary leastsquares aheadthe model, not the methodadvantage, all noise in b0.99advantage, all noise in A0.91seeds at each share40the same total noise at every pointand only where it sits changes

One curve: the ordinary least-squares error over the total least-squares error, at each share of the noise placed in the matrix, as the median over 40 seeds. It runs from 0.99 when all the noise is in b — where ordinary least squares is the more accurate — to 0.91 when all of it is in A. The line at one is where the two methods are equally right.

What it checked while drawing

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

17 distinct claims across 4 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

every problem at share 0 has a total least-squares solution — asserted 5 times

and the advantage moves the same way at every share, at 0.3 — asserted 4 times

a noise level the fit survives

and total least squares when all of it is

at this noise level the two methods are indistinguishable at every share

enough seeds for a median to mean something

least squares is the more accurate when none of the noise is in A

over a range wider than the seeds can account for

with a crossing inside the sweep

with neither ahead by more than a few per cent anywhere

Against the rule

It draws a decomposition and prints its residual. It calls overSeeds, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 90 of 174 generators — 75 print a residual and 15 are exempt with a published reason; 84 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Least squares, and the road not to take

The projection and the right angle

The least-squares solution is the one whose residual is perpendicular to everything the columns can reach. That is not a mnemonic — it is an equation, Aᵀr = 0, and the computed answer satisfies it to 10⁻¹⁶.

 Least squares, and the road not to take

When the matrix is wrong too

Every least-squares problem on this site has assumed A is exact and b is not, and moved b onto the column space of A. Where both were measured, the smallest correction that makes the system consistent moves the matrix as well — and on the problems where that answer is more accurate, it has the larger residual, by construction rather than by luck.

The whole library · All essays · What must fail