Whitening — where it appears
Named by 31 essays across 9 fields — each of them below, with the objects they name alongside it.
A break that was looked for
A two-regime whitening finds its change point by maximising a profile, and then reads a criterion that counts parameters. Under no break there is no parameter to count, because every position describes the same model.
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 family before a fit
A regression's coefficients and one correlation can be maximised together. Replace the correlation with an estimated covariance and there is nothing left for "jointly" to mean — until a set of covariances is named, and the set turns out not to contain the truth.
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.
Two searches, one sample
A searched break in a regression manufactures 34.7 of likelihood ratio where a count of coefficients says 11.1. A searched window manufactures 84.0. The two together manufacture 99.4, not 118.7.
The fit that takes the memory out
A candidate's residuals report less dependence than its errors do, and how much less is arithmetic rather than noise. The rule used for a good reason reads the series that has lost the most.
The plug-in and the maximum
A tapered covariance estimate sits five and a half log-likelihood units below the maximum of the likelihood it is substituted into. Four fifths of that is what the optimiser would have found if nothing were missing.
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.
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 charge that depends on the rule
The break search's charge is 34.7 on its own and 15.4 once a window has been chosen from the same sample. Most of what a break search finds under correlated errors is the correlation, and a whitening has taken it already.
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.
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.
Three quarters of the way to one search
The pair that started this reads 0.762 on a scale whose one is containment. And the pair that shares nothing but its response reads −0.306, so the sign the earlier field found does not transport at all.
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 fitting them together buys
Maximising over the coefficients and the covariance together beats the two-step under one of four dependences and ties under the other three. It is the one the band family contains, and the likelihood said so before any coefficient was compared.
Two values the criterion cannot tell apart
Fifteen candidates disagree about a whitening's order on two draws in five when the list holds all thirteen orders — and on more than one in five when it holds only one and two. Every adjacent pair of positive orders disagrees about as often as that pair does, and every pair holding an order the criterion never picks disagrees on none. A list's disagreement is set by the pairs of values on it that the criterion cannot tell apart, not by how many values it has, and one candidate's own doubt about its pick predicts all fifteen candidates' disagreement with a rank correlation of 0.96.
A break that is there
Put a real break in the regression at row 60 and the profile's two-unit set still misses it on most draws — it contains the true row on 12% of draws at a small break and 69% at a large one, without whitening; with the window searched as well, 23% to 75%. A set that would contain it on 95% of draws needs a threshold of about 19 units without whitening and about 8.5 with it, and either honest set holds most of the sample: fifty-seven of the seventy-three candidate rows at a break of one standard deviation. Searching the window does not move where the break is found.
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 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.
The interval the joint fit reports
Maximising a regression's coefficients and an eight-lag covariance band together gives a better estimate than plugging the band in. It also prints an interval that covers 71% to 85% of the time where 95% was promised — on the white predictor too, where every other rule is honest — because it reports almost the precision of the law's own covariance and delivers less than half of that advantage. Counting the eight lags it estimated buys one to two points.
Named alongside it
The objects these essays reach for when they reach for this one.
Model selectionInformation criterionRegretNuisance parameterMonte CarloSelection effectCovariance matrixDependenceLong memoryStructural breakGeneralised least squaresAutocorrelation