The thread: A model is not a corpus
The matrix a corpus wrote
A substitution matrix is not a property of an alphabet. Fit one to four hundred pairs of sequences that rarely change and the dearest substitution costs five; fit the same model to four hundred pairs that often change and it costs two. Two hundred test pairs aligned under each matrix give different alignments in 115 cases — and a matrix fitted to eight pairs of the first kind moves 79 of them, from sampling alone.
What the libraries doWhat the generated collection was right about
Five strands of conclusions, drawn on collections made by one line with one dial, checked against a corpus nobody made. Most hold. One headline was a property of the generator's alphabet, and one crossing that was guessed at turns out to be met — but only with the structure the strand on range minima built.
The data that is not a numberA corpus that was not generated
Every collection in five strands has been copies of a generated text with a fraction of its characters replaced — three numbers, one dial. Here is one that was not — twelve essays, eight source modules, ten revisions of one file — measured beside the model of it.
The index that replaces the textDocuments that are not the same length
A separator per document costs a whole bit per character — on a generated collection whose alphabet is 21 symbols, where adding a few crosses 32. Real prose has 87 symbols and sits 41 short of the next power of two, so the same separators cost 0.04%.
The data that is not a numberA million characters of the same thing
Every measurement this collection has published about real text was taken on twenty-four thousand characters, because the phrase count was quadratic. It is linear now, so here is the same corpus at forty times the size — and what forty times does to its own numbers.
When the algorithm is a tableThe counts only short words carry
A spelling checker capped at two edits still gives a false suggestion to many words its vocabulary lacks, and the one evidence about a writer's intention it already holds is how often each vocabulary word is used. Scored by its nearest words' counts, weighed a tenth for each edit, an answer was predicted to separate a slip from an unfamiliar word on short queries. On the streams the checker was built against, whose typos are of words drawn evenly from the list, the false suggestions go to commoner words than the corrections do, and the test gives up sixteen corrections for each false suggestion it removes. Drawn as a writer mistypes, in proportion to use, the counts separate the two at last — and still trade worse than a plain cap of one edit, except on words of four letters or fewer, where a threshold removes a ninth of the false suggestions for one correction in ninety-six.
What is taught wronglyThe half of a fall that is the logarithm
Phrases per character on a real collection of essays fall by a factor of 2.34 as it grows. A shuffle of the same characters falls by 1.70. Nearly three quarters of the movement is arithmetic, and no definition of the measure says so.
The floorsWhere a context stops naming the letter before
A sort of rotations by their first k characters matters only where those k characters fail to decide the character before them, and that share of positions can be counted in one pass with no sorting. It was proposed as a replacement for sweeping k. Counted as proposed, it misses by a factor of four: on a stream of independent words 37% of positions are still ambiguous at four characters, where the sweep has already gone flat. Counted only where the context sits inside a word that goes on past it — the one place more context can still help — it lands within a step of the sweep on four sources of five.
The index that replaces the textTwo thousand documents of two hundred characters
Every collection this field has measured has been a dozen documents of ten thousand characters. A real collection is usually the other shape, and the other shape moves every term in a document index — one of them by a factor of seventy.
What the libraries doThe crossing that never arrives
Output-sensitive document listing exists because a pattern can occur four thousand times in eight documents. On a real collection of two thousand short documents it occurs 1.04 times per document, and the whole apparatus buys nothing at all.
The floorsThe repeats a context cannot see past
Sorting rotations by their first k characters stops losing to the full Burrows–Wheeler transform at four or five characters on four sources and at about ten on a corpus of technical writing. The count proposed to explain the exception — how much a context spanning a word boundary says about the letter before it — cannot be read off its own estimate, which falls on words drawn independently where there is nothing to find. Measured against each text's own shuffled twin, the information the word order carries is real, but it has all arrived by four or five characters, and at eight, twelve and sixteen the two corpora are indistinguishable. What does separate them is repetition. Stream by stream, the share of positions inside a repeated sixteen-character string predicts the technical corpus's loss with a correlation of 0.98.
The data that is not a numberThe shape of a real history's depth
Four essays here are about capping how far an extraction follows a chain of copies, and every number in them came from a generated collection. Here is the depth histogram of a real version history, which is a bell, and of twelve unrelated essays, which is nearly the same bell.
StructuresOne revision, one level
The cap ladder assumed depth is generations of copying. On a real file it is exactly that — one more revision, one more level, nine times running — plus eight levels the first revision already had before any history existed.
The data that is not a numberThe cell nobody filled
Every structure here was measured either on one long repetitive text or on prose cut into short documents. The collection that is both is a two-by-two with one empty corner, and what is in it is not the product of its margins.
The data that is not a numberWhat repetition is worth once the logarithm is gone
On prose, nearly three quarters of the fall in phrases per character with size is arithmetic that any text pays. On a collection built of copies it is a fifth, and what is left is a factor of three that is genuinely the arrangement.
What the libraries doThe cap that would ship
The published sweep put the knee at four to eight. On four real collections it is at nine to twelve, a cap of one costs forty-eight times the free parse rather than twenty-one, and the number a system should actually set is none of those.
What is taught wronglyThe dial that has no setting
The generator has one parameter. The real version history's run count asks it for 2.3% and its phrase count asks for 3.2% — and the reason is not that the dial is badly calibrated. Real edits average 7.3 characters a block and generated ones average 1.04.
What the libraries doA collection is a construction
The same characters, arranged as one copy per document or cut across the copies, give a different run count, a different boundary cost and a different answer about which structure to build. Which one a benchmark used is usually not recorded.
The index that replaces the textThe collections the saving was quoted for
A document array compressed to the entropy of its own symbols saves exactly what the collection's length distribution allows, and the measurements that showed it used four constructed collections. The estimate they ended on — a directory of source files is roughly log-normal, so its array falls about a bit short of the plain one, around 17% — can be checked on a real directory. A project's 222 source files fall short by 0.66 bits and save 10.5%, a fifth of which is only the rounding of ⌈log₂ d⌉. Across six real collections the skew of the lengths is worth between nothing and 0.76 bits, and the log-normal formula overstates it wherever short documents are common.
The index that replaces the textThe shape arrived early and the bytes late
A document array coded by its documents' lengths was predicted to survive a growing collection: fitted once at a quarter of a project's files, it would lose under a tenth of a bit a symbol by the end, because the spread of the sizes is set early. The spread was set early — 0.88 at 56 files, 0.90 at 222. The fitted code loses 0.50 to 2.08 bits all the same, because the files added after the fit hold 43% of the final bytes and every one of their symbols is named through an escape. Rebuilding only the escape's subtree brings it to 0.18.
The index that replaces the textOne doubling late
A document array coded by its documents' lengths and refitted each time a growing directory's files double needs an escape for the files added in between, and the escape's right weight is the share of the characters still to come. Weighting it by the share that arrived since the previous refit was predicted to cut the mean loss below a quarter of a bit a symbol. It cuts nothing: 0.518 bits a symbol, against 0.389 for an escape that always takes half the coding space. The growth seen is the growth to come one doubling late — 71%, 50% and 27% where the doublings that followed brought 50%, 27% and 8% — because this directory's growth slowed at every doubling, and before the first refit there is no growth to read at all. A Huffman code also rounds every weight to a whole number of bits, so being right within a factor of two buys nothing. Started from half, the growth seen averages 0.356, the best a store can know, and the weight nobody can know reaches 0.183.