Regret — where it appears
Named by 35 essays across 13 fields — each of them below, with the objects they name alongside it.
A dependence fitted with the line
Every whitening in this collection reads the dependence off a set of residuals, and residuals are not errors. Fitting the two together recovers most of what that costs, and changes almost nothing about the decision it feeds.
A dependence with a shape
Four ways for errors to repeat, all with the same first lag and nothing else in common. A rule told the errors are a first-order autoregression finds the same number in all four, and is right about one of them.
A rate times a size
A sweep reported what it costs to let every candidate choose its own tuning parameter and found it flat across the list. It was reporting a product, and the two things multiplied together do not behave the same way at all.
The comparison that was not made
Choosing a whitening's window separately for every candidate costs 0.00401 of regret. The same question about an order was named and left, because the two lists are different lengths. The order's answer is 0.00360, and matching the lists changes almost nothing.
The eighth that was not a constant
How often a per-candidate tuning list changes which candidate wins is reported flat at about an eighth across list length. Vary how far apart the candidates are instead and it runs from 17.6% to 1.5%.
The quarrel that changes the winner
A disagreement about the tuning parameter costs 0.031 when it changes which candidate the table selects and −0.0007 when it does not. The distance between the values disagreed about has nothing to do with it.
The volume a whitening moves
A sieve's whitening has a determinant and this collection's criterion for it never carried one. Shared across a table the term cancels exactly, which is why nothing ever noticed; used per candidate it is worth more than a parameter and the whole comparison turns on it.
Two factors pointing opposite ways
As the candidates on a table are pulled apart, they quarrel about the tuning parameter three times as often and the quarrel decides the winner thirty times less often. A sweep that reads the first factor has read the one pointing the wrong way.
Where the generality runs out
A covariance that changes half way through a sample is not one a window can estimate. One number, a window and an order are worth the same as each other on it — and letting the model change once, at a point nobody can locate, is worth as much again as all three.
A list is not a rule
How often fifteen candidates disagree about a tuning parameter runs from nothing at two values on the list to two draws in five at thirteen. What the disagreement costs does not move at all.
A table and a list
A nested ladder of candidates differing by one coefficient was predicted to turn over more often at every list length. It turns over less at every one, and its list changes the winner half as often.
A width that moves and an error that does not
Four charges give four widths a factor of four apart and four errors half a per cent apart. The derived charge wins, significantly, by a quarter of what was on offer — and none of the four is an estimate of anything.
Choosing whether to break
Charging what the search manufactures takes a rule from splitting a stationary sample on 99% of draws to 16%. It also costs regret, because the two mistakes a rule can make are not the same size.
How often it matters
The disagreement rate rises by half across the list and the share of disagreements that decide anything falls by nearly the same factor. Their product — how often the tuning list changes which candidate wins — sits at an eighth and does not move.
Iterating is not maximising
Re-reading a correlation from the generalised residuals and refitting converges in seven steps. What it converges to solves the first-order condition of a sum of squares, and the likelihood has one term more than that.
Nothing in the fit picks the width
A wider band is always a better fit, and it is better by about one unit of log-likelihood a lag — which is the order of what a criterion charges for a parameter. Three defensible rules choose widths a factor of three apart.
The window a whitening wants
Every law here is best whitened by a window several times longer than its own memory, including the one whose memory ends at the fourth lag. The three ways of choosing it from the sample all land in the same place, and it is the wrong one.
What choosing the length costs
The gap between two block windows at the best available length is 2.12 points. What the best rule a practitioner could run gives up against that same length is 7.26. The argument is a third of the size of the thing it is inside.
A window for every candidate
The window and the order a whitening needs are chosen once, from the fullest candidate, on an argument that was made about an estimated covariance. A tuning parameter is not a covariance, and the two cost different amounts.
The order the tail is drawn at
A fitted autoregression reproduces the sample exactly at the lags it was fitted on, so everything it says past them is extrapolation — and the order is the dial that decides how much of it there is.
What a better charge buys
Four charges derived from the same measurements pick band widths within six per cent of each other and deliver errors within two per cent of the gap any of them leaves. The scale a charge is levied on decides the width; the shape of the charge decides nothing.
A step that is not a ratio
Run the separation sweep on a tuning list of integers rather than a geometric ladder and the two factors still point opposite ways. The invariant does not survive: along a row of integers the probability moves by 2.163 where along the geometric ladder it moves by 1.208.
What a longer sample settles
At a hundred and twenty rows, letting each candidate choose its own whitening changes which of fifteen candidates wins on about an eighth of draws, whatever the list, the table or the law. At four hundred and eighty rows it changes the winner on 1.0% of draws for a window and 2.3% for an order, and at nine hundred and sixty on a quarter of one per cent. The sample settles it through one factor only: how often the candidates disagree about the tuning value hardly moves, and the share of their disagreements that change the winner collapses.
The residue was the weights
A regression whose errors change persistence at a known row was whitened as two autoregressions and still gave up 0.0246 against the law's own covariance. That residue was put down to estimating the two persistences from residuals, and the repair named was to fit them with the line. Fitting them with the line moves the persistent regime's estimate from 0.81 to 0.89 and recovers a fifth of the residue — exactly what being told the true persistences recovers. The rest was the transform: it never divided the two regimes by their own innovation scales, which differ sixfold. The law's own recursion, told the persistences, is the oracle to machine precision; fitted with the line, it estimates the persistent one at 0.945 and comes within 0.0006 of the oracle.
The same dial under a second law
On a first-order autoregression, separating the candidates moved the chance that a tuning list changes the winner 2.35 times as far on a whitening window as on a sieve order, and the difference was put down to the step: an integer step is worth a third of a geometric one. Rerun the whole sweep with errors from a five-period moving average and the difference is gone — the probability spans a factor of 5.53 on the window and 5.13 on the order, a ratio of 1.08 — and near the null both lists leave about five and a half candidates within reach, where the autoregression left 3.6 and 1.9. The two factors still point opposite ways under both laws. The step's size was a property of the law the lists were tuned on.
A penalty that counts independent rows
Akaike's penalty charges a fit two error variances a coefficient because each coefficient is fitted from n independent rows. When the design and the errors both persist, a row is worth less than a row, and the honest charge is twice the coefficient's variance inflation. On a hundred and twenty rows at persistence 0.7 that charge is 34.7 against an optimism counted at 34.5, where Akaike's charges 10. Charged it, the criterion's regret at 0.95 falls from 0.128 to 0.032 with the persistences given and to 0.065 when they are read off the sample — and at 0.5, where the charge is right and Akaike's is half of it, Akaike's still selects better.
The break a shared scale hid
The break searches in this field all maximised a likelihood with one innovation variance for the whole series, and on a break from 0.95 to 0.65 they found it badly: an average of 74.8 where the truth is 60, within twelve rows on 47% of draws, from a profile everyone called flat. Give each regime its own innovation variance — one more parameter, and the one the law actually has — and the same search lands at 61.8 with a spread of 5.8, within twelve rows on 94% of draws, reporting four times the likelihood. The break was never barely there; the likelihood was blind to the half of its evidence that is in the scale. Under no break the extra parameter costs about one and a half units of manufactured likelihood.
The table decides when the sample settles it
Letting each candidate regression choose its own whitening changes which candidate wins on about an eighth of draws at 120 rows, and a longer sample settles it — for fifteen nested-and-crossed subsets, at 0.25% by 960 rows. A nested sequence of four candidates gets there a sample size sooner, from half the rate. Six candidates that all hold two predictors never get there: at 960 rows their tuning list still changes the winner on 2.3% of draws, nine times the full table's, because a disagreement among candidates of one size goes on deciding the winner as often at 960 rows as at 240.
A fixed point with no likelihood under it
A tapered band of autocovariances has no likelihood to maximise, so the one rule that can fit its dependence together with the regression is to iterate: read the band off the residuals, refit, read it again. It converges on every draw, in seven to eleven fits at the median, gives back about half the memory the first fit took out of the residuals under a short-memory law, and lowers prediction error by two to four thousandths — five to eight paired standard errors. Under the law a first-order likelihood describes, its maximum still wins; under the two it does not, the band's fixed point beats it.
A quarrel's cost is its width
When fifteen candidate regressions disagree about a whitening's autoregressive order, the disagreement costs in proportion to how far apart the two orders are — 0.0027 of regret for every order of distance, a line that explains 96% of the cost across sixty-six pairs — and not at all by which orders they are: a quarrel between eleven and twelve costs what one between one and two costs, though one of those holds the truth. Fitted on pairs, the line predicts every longer list's cost. The window's quarrels follow no such line.
A covariance repaired
A truncated autocovariance sequence fails to be a covariance matrix on most draws of a persistent series, and this line of essays has always scored it on the draws where it survives. The textbook repair is to floor its eigenvalues. At a floor of 0.001 that moves the matrix by four per cent and gives up 1.56 of regret, seventeen times what least squares gives up, because a whitening reads an inverse and an inverse reads the smallest eigenvalues. At a floor of 0.2 it beats the taper at the shortest windows. So the repair is not a repair: it is a second dial.
A charge that reads the draw
Three charges built to read the sample track the best band width on their own draw at −0.012, −0.019 and −0.041, deliver more error than the fixed rule they are calibrated to, and pick a width half again as variable. The statistic moves; the answer does not.
A charge the sample can afford
The inflation charge for dependent rows runs to twice the counted optimism at persistence 0.95, and the reason offered was that a series of n rows cannot be optimistic about movements slower than n rows. Summing the inflation over n rows does almost nothing: 234 becomes 209, against 119 counted. Evaluating the same variance on the sample's own design gives 132, and moving the fresh rows away from the sample raises the count to 159. So the charge can be made nearly right, and when it is, the regret doubles from 0.032 to 0.061. A selection wants a much larger charge than the optimism, and past about fifty per predictor it is no longer selecting.
The surface under every law
A draw-reading charge for a covariance band failed on one law, and the failure was deferred to a law with a steeper error surface. Run under all four laws in the collection, the surface is flat from six lags to thirty under every one, no draw-reading charge tracks anything, and long memory — the law expected to be steepest — is the flattest end to end. What the law moves is the level of the charge, not whether a draw can be read.
A threshold reported with its interval
Resample a validation study of a hundred diseased and three hundred healthy people, refit the binormal model each time, and read off the threshold each resample recommends. Far from the prevalence at which the best threshold jumps to flagging everyone, the interval is tight — flagging 12% to 24% of healthy people at a prevalence of 5%. Near the jump, the median study's interval runs from flagging under half of the healthy to flagging everyone, and 91% of intervals cross the jump. The two ends differ by fifteen false alarms' worth of harm per hundred people, and no rule for choosing inside the interval beats the study's own choice at every prevalence.
Named alongside it
The objects these essays reach for when they reach for this one.
Model selectionInformation criterionWhiteningMonte CarloSelection effectBandwidth selectionTuning parameterDependenceAutocorrelationAutoregressionNuisance parameterCovariance matrix