Aghast: aggregated, histogram-like statistics, sharable as Flatbuffers.
Project description
aghast
Aghast is a histogramming library that does not fill histograms and does not plot them. Its role is behind the scenes, to provide better communication between histogramming libraries.
Specifically, it is a structured representation of aggregated, histogram-like statistics as sharable "ghasts." It has all of the "bells and whistles" often associated with plain histograms, such as number of entries, unbinned mean and standard deviation, bin errors, associated fit functions, profile plots, and even simple ntuples (needed for unbinned fits or machine learning applications). ROOT has all of these features; Numpy has none of them.
The purpose of aghast is to be an intermediate when converting ROOT histograms into Numpy, or vice-versa, or both of these into Boost.Histogram, Physt, Pandas, etc. Without an intermediate representation, converting between N libraries (to get the advantages of all) would equire N(N ‒ 1)/2 conversion routines; with an intermediate representation, we only need N, and the mapping of feature to feature can be made explicit in terms of a common language.
Furthermore, aghast is a Flatbuffers schema, so it can be deciphered in many languages, with lazy, random-access, and uses a small amount of memory. A collection of histograms, functions, and ntuples can be shared among processes as shared memory, used in remote procedure calls, processed incrementally in a memory-mapped file, or saved in files with future-proof schema evolution.
Installation from packages
Install aghast like any other Python package:
pip install aghast # maybe with sudo or --user, or in virtualenv
(Not on conda yet.)
Manual installation
After you git-clone this GitHub repository and ensure that numpy is installed, somehow:
pip install "flatbuffers>=1.8.0" # for the flatbuffers runtime (with Numpy)
cd python # only implementation so far is in Python
python setup.py install # to use it outside of this directory
Now you should be able to import aghast or from aghast import * in Python.
If you need to change flatbuffers/aghast.fbs, you'll need to additionally:
- Get
flatcto generate Python sources fromflatbuffers/aghast.fbs. I useconda install -c conda-forge flatbuffers. (Theflatcexecutable is not included in the pipflatbufferspackage, and the Python runtime is not included in the condaflatbufferspackage. They're disjoint.) - In the
pythondirectory, run./generate_flatbuffers.py(which callsflatcand does some post-processing).
Every time you change flatbuffers/aghast.fbs, re-run ./generate_flatbuffers.py.
Documentation
Full specification:
- Introduction
- Data types
- Collection
- Histogram
- Axis
- IntegerBinning
- RegularBinning
- RealInterval
- RealOverflow
- HexagonalBinning
- EdgesBinning
- IrregularBinning
- CategoryBinning
- SparseRegularBinning
- FractionBinning
- PredicateBinning
- VariationBinning
- Variation
- Assignment
- UnweightedCounts
- WeightedCounts
- InterpretedInlineBuffer
- InterpretedInlineInt64Buffer
- InterpretedInlineFloat64Buffer
- InterpretedExternalBuffer
- Profile
- Statistics
- Moments
- Quantiles
- Modes
- Extremes
- StatisticFilter
- Covariance
- ParameterizedFunction
- Parameter
- EvaluatedFunction
- BinnedEvaluatedFunction
- Ntuple
- Column
- NtupleInstance
- Chunk
- ColumnChunk
- Page
- RawInlineBuffer
- RawExternalBuffer
- Metadata
- Decoration
Tutorial examples
Conversions
The main purpose of aghast is to move aggregated, histogram-like statistics (called "ghasts") from one framework to the next. This requires a conversion of high-level domain concepts.
Consider the following example: in Numpy, a histogram is simply a 2-tuple of arrays with special meaning—bin contents, then bin edges.
import numpy
numpy_hist = numpy.histogram(numpy.random.normal(0, 1, int(10e6)), bins=80, range=(-5, 5))
numpy_hist
(array([ 2, 5, 9, 15, 29, 49, 80, 104,
237, 352, 555, 867, 1447, 2046, 3037, 4562,
6805, 9540, 13529, 18584, 25593, 35000, 46024, 59103,
76492, 96441, 119873, 146159, 177533, 210628, 246316, 283292,
321377, 359314, 393857, 426446, 453031, 474806, 489846, 496646,
497922, 490499, 473200, 453527, 425650, 393297, 358537, 321099,
282519, 246469, 211181, 177550, 147417, 120322, 96592, 76665,
59587, 45776, 34459, 25900, 18876, 13576, 9571, 6662,
4629, 3161, 2069, 1334, 878, 581, 332, 220,
135, 65, 39, 26, 19, 15, 4, 4]),
array([-5. , -4.875, -4.75 , -4.625, -4.5 , -4.375, -4.25 , -4.125,
-4. , -3.875, -3.75 , -3.625, -3.5 , -3.375, -3.25 , -3.125,
-3. , -2.875, -2.75 , -2.625, -2.5 , -2.375, -2.25 , -2.125,
-2. , -1.875, -1.75 , -1.625, -1.5 , -1.375, -1.25 , -1.125,
-1. , -0.875, -0.75 , -0.625, -0.5 , -0.375, -0.25 , -0.125,
0. , 0.125, 0.25 , 0.375, 0.5 , 0.625, 0.75 , 0.875,
1. , 1.125, 1.25 , 1.375, 1.5 , 1.625, 1.75 , 1.875,
2. , 2.125, 2.25 , 2.375, 2.5 , 2.625, 2.75 , 2.875,
3. , 3.125, 3.25 , 3.375, 3.5 , 3.625, 3.75 , 3.875,
4. , 4.125, 4.25 , 4.375, 4.5 , 4.625, 4.75 , 4.875,
5. ]))
We convert that into the aghast equivalent (a "ghast") with a connector (two functions: from_numpy and to_numpy).
import aghast
ghastly_hist = aghast.from_numpy(numpy_hist)
ghastly_hist
<Histogram at 0x7f0dc88a9b38>
This object is instantiated from a class structure built from simple pieces.
ghastly_hist.dump()
Histogram(
axis=[
Axis(binning=RegularBinning(num=80, interval=RealInterval(low=-5.0, high=5.0)))
],
counts=
UnweightedCounts(
counts=
InterpretedInlineInt64Buffer(
buffer=
[ 2 5 9 15 29 49 80 104 237 352
555 867 1447 2046 3037 4562 6805 9540 13529 18584
25593 35000 46024 59103 76492 96441 119873 146159 177533 210628
246316 283292 321377 359314 393857 426446 453031 474806 489846 496646
497922 490499 473200 453527 425650 393297 358537 321099 282519 246469
211181 177550 147417 120322 96592 76665 59587 45776 34459 25900
18876 13576 9571 6662 4629 3161 2069 1334 878 581
332 220 135 65 39 26 19 15 4 4])))
Now it can be converted to a ROOT histogram with another connector.
root_hist = aghast.to_root(ghastly_hist, "root_hist")
root_hist
<ROOT.TH1D object ("root_hist") at 0x55555e208ef0>
import ROOT
canvas = ROOT.TCanvas()
root_hist.Draw()
canvas.Draw()
And Pandas with yet another connector.
pandas_hist = aghast.to_pandas(ghastly_hist)
pandas_hist
| unweighted | |
|---|---|
| [-5.0, -4.875) | 2 |
| [-4.875, -4.75) | 5 |
| [-4.75, -4.625) | 9 |
| [-4.625, -4.5) | 15 |
| [-4.5, -4.375) | 29 |
| [-4.375, -4.25) | 49 |
| [-4.25, -4.125) | 80 |
| [-4.125, -4.0) | 104 |
| [-4.0, -3.875) | 237 |
| [-3.875, -3.75) | 352 |
| [-3.75, -3.625) | 555 |
| [-3.625, -3.5) | 867 |
| [-3.5, -3.375) | 1447 |
| [-3.375, -3.25) | 2046 |
| [-3.25, -3.125) | 3037 |
| [-3.125, -3.0) | 4562 |
| [-3.0, -2.875) | 6805 |
| [-2.875, -2.75) | 9540 |
| [-2.75, -2.625) | 13529 |
| [-2.625, -2.5) | 18584 |
| [-2.5, -2.375) | 25593 |
| [-2.375, -2.25) | 35000 |
| [-2.25, -2.125) | 46024 |
| [-2.125, -2.0) | 59103 |
| [-2.0, -1.875) | 76492 |
| [-1.875, -1.75) | 96441 |
| [-1.75, -1.625) | 119873 |
| [-1.625, -1.5) | 146159 |
| [-1.5, -1.375) | 177533 |
| [-1.375, -1.25) | 210628 |
| ... | ... |
| [1.25, 1.375) | 211181 |
| [1.375, 1.5) | 177550 |
| [1.5, 1.625) | 147417 |
| [1.625, 1.75) | 120322 |
| [1.75, 1.875) | 96592 |
| [1.875, 2.0) | 76665 |
| [2.0, 2.125) | 59587 |
| [2.125, 2.25) | 45776 |
| [2.25, 2.375) | 34459 |
| [2.375, 2.5) | 25900 |
| [2.5, 2.625) | 18876 |
| [2.625, 2.75) | 13576 |
| [2.75, 2.875) | 9571 |
| [2.875, 3.0) | 6662 |
| [3.0, 3.125) | 4629 |
| [3.125, 3.25) | 3161 |
| [3.25, 3.375) | 2069 |
| [3.375, 3.5) | 1334 |
| [3.5, 3.625) | 878 |
| [3.625, 3.75) | 581 |
| [3.75, 3.875) | 332 |
| [3.875, 4.0) | 220 |
| [4.0, 4.125) | 135 |
| [4.125, 4.25) | 65 |
| [4.25, 4.375) | 39 |
| [4.375, 4.5) | 26 |
| [4.5, 4.625) | 19 |
| [4.625, 4.75) | 15 |
| [4.75, 4.875) | 4 |
| [4.875, 5.0) | 4 |
80 rows × 1 columns
Serialization
A ghast is also a Flatbuffers object, which has a multi-lingual, random-access, small-footprint serialization:
ghastly_hist.tobuffer()
bytearray("\x04\x00\x00\x00\x90\xff\xff\xff\x10\x00\x00\x00\x00\x01\n\x00\x10\x00\x0c\x00\x0b\x00\x04
\x00\n\x00\x00\x00`\x00\x00\x00\x00\x00\x00\x01\x04\x00\x00\x00\x01\x00\x00\x00\x0c\x00\x00
\x00\x08\x00\x0c\x00\x0b\x00\x04\x00\x08\x00\x00\x00\x10\x00\x00\x00\x00\x00\x00\x02\x08\x00
(\x00\x1c\x00\x04\x00\x08\x00\x00\x00\x00\x00\x00\x00\x00\x00\x14\xc0\x00\x00\x00\x00\x00
\x00\x14@\x01\x00\x00\x00\x00\x00\x00\x00P\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x08
\x00\n\x00\t\x00\x04\x00\x08\x00\x00\x00\x0c\x00\x00\x00\x00\x02\x06\x00\x08\x00\x04\x00\x06
\x00\x00\x00\x04\x00\x00\x00\x80\x02\x00\x00\x02\x00\x00\x00\x00\x00\x00\x00\x05\x00\x00\x00
\x00\x00\x00\x00\t\x00\x00\x00\x00\x00\x00\x00\x0f\x00\x00\x00\x00\x00\x00\x00\x1d\x00\x00
\x00\x00\x00\x00\x001\x00\x00\x00\x00\x00\x00\x00P\x00\x00\x00\x00\x00\x00\x00h\x00\x00\x00
\x00\x00\x00\x00\xed\x00\x00\x00\x00\x00\x00\x00`\x01\x00\x00\x00\x00\x00\x00+\x02\x00\x00
\x00\x00\x00\x00c\x03\x00\x00\x00\x00\x00\x00\xa7\x05\x00\x00\x00\x00\x00\x00\xfe\x07\x00
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\x00\x00\x00\x00\xf9c\x00\x00\x00\x00\x00\x00\xb8\x88\x00\x00\x00\x00\x00\x00\xc8\xb3\x00\x00
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\x00\x00\xb2~\x06\x00\x00\x00\x00\x00Q\x00\x06\x00\x00\x00\x00\x00\x89x\x05\x00\x00\x00\x00
\x00K\xe6\x04\x00\x00\x00\x00\x00\x97O\x04\x00\x00\x00\x00\x00\xc5\xc2\x03\x00\x00\x00\x00
\x00\xed8\x03\x00\x00\x00\x00\x00\x8e\xb5\x02\x00\x00\x00\x00\x00\xd9?\x02\x00\x00\x00\x00
\x00\x02\xd6\x01\x00\x00\x00\x00\x00Py\x01\x00\x00\x00\x00\x00y+\x01\x00\x00\x00\x00\x00\xc3
\xe8\x00\x00\x00\x00\x00\x00\xd0\xb2\x00\x00\x00\x00\x00\x00\x9b\x86\x00\x00\x00\x00\x00\x00
,e\x00\x00\x00\x00\x00\x00\xbcI\x00\x00\x00\x00\x00\x00\x085\x00\x00\x00\x00\x00\x00c%\x00
\x00\x00\x00\x00\x00\x06\x1a\x00\x00\x00\x00\x00\x00\x15\x12\x00\x00\x00\x00\x00\x00Y\x0c
\x00\x00\x00\x00\x00\x00\x15\x08\x00\x00\x00\x00\x00\x006\x05\x00\x00\x00\x00\x00\x00n\x03
\x00\x00\x00\x00\x00\x00E\x02\x00\x00\x00\x00\x00\x00L\x01\x00\x00\x00\x00\x00\x00\xdc\x00
\x00\x00\x00\x00\x00\x00\x87\x00\x00\x00\x00\x00\x00\x00A\x00\x00\x00\x00\x00\x00\x00\'\x00
\x00\x00\x00\x00\x00\x00\x1a\x00\x00\x00\x00\x00\x00\x00\x13\x00\x00\x00\x00\x00\x00\x00\x0f
\x00\x00\x00\x00\x00\x00\x00\x04\x00\x00\x00\x00\x00\x00\x00\x04\x00\x00\x00\x00\x00\x00\x00")
print("Numpy size: ", numpy_hist[0].nbytes + numpy_hist[1].nbytes)
tmessage = ROOT.TMessage()
tmessage.WriteObject(root_hist)
print("ROOT size: ", tmessage.Length())
import pickle
print("Pandas size:", len(pickle.dumps(pandas_hist)))
print("Aghast size: ", len(ghastly_hist.tobuffer()))
Numpy size: 1288
ROOT size: 1962
Pandas size: 2984
Aghast size: 792
Aghast is generally forseen as a memory format, like Apache Arrow, but for statistical aggregations. Like Arrow, it reduces the need to implement $N(N - 1)/2$ conversion functions among $N$ statistical libraries to just $N$ conversion functions. (See the figure on Arrow's website.)
Translation of conventions
Aghast also intends to be as close to zero-copy as possible. This means that it must make graceful translations among conventions. Different histogramming libraries handle overflow bins in different ways:
fromroot = aghast.from_root(root_hist)
fromroot.axis[0].binning.dump()
print("Bin contents length:", len(fromroot.counts.array))
RegularBinning(
num=80,
interval=RealInterval(low=-5.0, high=5.0),
overflow=RealOverflow(loc_underflow=BinLocation.below1, loc_overflow=BinLocation.above1))
Bin contents length: 82
ghastly_hist.axis[0].binning.dump()
print("Bin contents length:", len(ghastly_hist.counts.array))
RegularBinning(num=80, interval=RealInterval(low=-5.0, high=5.0))
Bin contents length: 80
And yet we want to be able to manipulate them as though these differences did not exist.
sum_hist = fromroot + ghastly_hist
sum_hist.axis[0].binning.dump()
print("Bin contents length:", len(sum_hist.counts.array))
RegularBinning(
num=80,
interval=RealInterval(low=-5.0, high=5.0),
overflow=RealOverflow(loc_underflow=BinLocation.above1, loc_overflow=BinLocation.above2))
Bin contents length: 82
The binning structure keeps track of the existence of underflow/overflow bins and where they are located.
- ROOT's convention is to put underflow before the normal bins (
below1) and overflow after (above1), so that the normal bins are effectively 1-indexed. - Boost.Histogram's convention is to put overflow after the normal bins (
above1) and underflow after that (above2), so that underflow is accessed viamyhist[-1]in Numpy. - Numpy histograms don't have underflow/overflow bins.
- Pandas could have
Intervalsthat extend to infinity.
Aghast accepts all of these, so that it doesn't have to manipulate the bin contents buffer it receives, but knows how to deal with them if it has to combine histograms that follow different conventions.
Binning types
All the different axis types have an equivalent in aghast (and not all are single-dimensional).
aghast.IntegerBinning(5, 10).dump()
aghast.RegularBinning(100, aghast.RealInterval(-5, 5)).dump()
aghast.HexagonalBinning(0, 100, 0, 100, aghast.HexagonalBinning.cube_xy).dump()
aghast.EdgesBinning([0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100]).dump()
aghast.IrregularBinning([aghast.RealInterval(0, 5),
aghast.RealInterval(10, 100),
aghast.RealInterval(-10, 10)],
overlapping_fill=aghast.IrregularBinning.all).dump()
aghast.CategoryBinning(["one", "two", "three"]).dump()
aghast.SparseRegularBinning([5, 3, -2, 8, -100], 10).dump()
aghast.FractionBinning(error_method=aghast.FractionBinning.clopper_pearson).dump()
aghast.PredicateBinning(["signal region", "control region"]).dump()
aghast.VariationBinning([aghast.Variation([aghast.Assignment("x", "nominal")]),
aghast.Variation([aghast.Assignment("x", "nominal + sigma")]),
aghast.Variation([aghast.Assignment("x", "nominal - sigma")])]).dump()
IntegerBinning(min=5, max=10)
RegularBinning(num=100, interval=RealInterval(low=-5.0, high=5.0))
HexagonalBinning(qmin=0, qmax=100, rmin=0, rmax=100, coordinates=HexagonalBinning.cube_xy)
EdgesBinning(edges=[0.01 0.05 0.1 0.5 1 5 10 50 100])
IrregularBinning(
intervals=[
RealInterval(low=0.0, high=5.0),
RealInterval(low=10.0, high=100.0),
RealInterval(low=-10.0, high=10.0)
],
overlapping_fill=IrregularBinning.all)
CategoryBinning(categories=['one', 'two', 'three'])
SparseRegularBinning(bins=[5 3 -2 8 -100], bin_width=10.0)
FractionBinning(error_method=FractionBinning.clopper_pearson)
PredicateBinning(predicates=['signal region', 'control region'])
VariationBinning(
variations=[
Variation(assignments=[
Assignment(identifier='x', expression='nominal')
]),
Variation(
assignments=[
Assignment(identifier='x', expression='nominal + sigma')
]),
Variation(
assignments=[
Assignment(identifier='x', expression='nominal - sigma')
])
])
The meanings of these binning classes are given in the specification, but many of them can be converted into one another, and converting to CategoryBinning (strings) often makes the intent clear.
aghast.IntegerBinning(5, 10).toCategoryBinning().dump()
aghast.RegularBinning(10, aghast.RealInterval(-5, 5)).toCategoryBinning().dump()
aghast.EdgesBinning([0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100]).toCategoryBinning().dump()
aghast.IrregularBinning([aghast.RealInterval(0, 5),
aghast.RealInterval(10, 100),
aghast.RealInterval(-10, 10)],
overlapping_fill=aghast.IrregularBinning.all).toCategoryBinning().dump()
aghast.SparseRegularBinning([5, 3, -2, 8, -100], 10).toCategoryBinning().dump()
aghast.FractionBinning(error_method=aghast.FractionBinning.clopper_pearson).toCategoryBinning().dump()
aghast.PredicateBinning(["signal region", "control region"]).toCategoryBinning().dump()
aghast.VariationBinning([aghast.Variation([aghast.Assignment("x", "nominal")]),
aghast.Variation([aghast.Assignment("x", "nominal + sigma")]),
aghast.Variation([aghast.Assignment("x", "nominal - sigma")])]
).toCategoryBinning().dump()
CategoryBinning(categories=['5', '6', '7', '8', '9', '10'])
CategoryBinning(
categories=['[-5, -4)', '[-4, -3)', '[-3, -2)', '[-2, -1)', '[-1, 0)', '[0, 1)', '[1, 2)', '[2, 3)',
'[3, 4)', '[4, 5)'])
CategoryBinning(
categories=['[0.01, 0.05)', '[0.05, 0.1)', '[0.1, 0.5)', '[0.5, 1)', '[1, 5)', '[5, 10)', '[10, 50)',
'[50, 100)'])
CategoryBinning(categories=['[0, 5)', '[10, 100)', '[-10, 10)'])
CategoryBinning(categories=['[50, 60)', '[30, 40)', '[-20, -10)', '[80, 90)', '[-1000, -990)'])
CategoryBinning(categories=['pass', 'all'])
CategoryBinning(categories=['signal region', 'control region'])
CategoryBinning(categories=['x := nominal', 'x := nominal + sigma', 'x := nominal - sigma'])
This technique can also clear up confusion about overflow bins.
aghast.RegularBinning(5, aghast.RealInterval(-5, 5), aghast.RealOverflow(
loc_underflow=aghast.BinLocation.above2,
loc_overflow=aghast.BinLocation.above1,
loc_nanflow=aghast.BinLocation.below1
)).toCategoryBinning().dump()
CategoryBinning(
categories=['{nan}', '[-5, -3)', '[-3, -1)', '[-1, 1)', '[1, 3)', '[3, 5)', '[5, +inf]',
'[-inf, -5)'])
Fancy binning types
You might also be wondering about FractionBinning, PredicateBinning, and VariationBinning.
FractionBinning is an axis of two bins: #passing and #total, #failing and #total, or #passing and #failing. Adding it to another axis effectively makes an "efficiency plot."
h = aghast.Histogram([aghast.Axis(aghast.FractionBinning()),
aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5)))],
aghast.UnweightedCounts(
aghast.InterpretedInlineBuffer.fromarray(
numpy.array([[ 9, 25, 29, 35, 54, 67, 60, 84, 80, 94],
[ 99, 119, 109, 109, 95, 104, 102, 106, 112, 122]]))))
df = aghast.to_pandas(h)
df
| unweighted | ||
|---|---|---|
| pass | [-5.0, -4.0) | 9 |
| [-4.0, -3.0) | 25 | |
| [-3.0, -2.0) | 29 | |
| [-2.0, -1.0) | 35 | |
| [-1.0, 0.0) | 54 | |
| [0.0, 1.0) | 67 | |
| [1.0, 2.0) | 60 | |
| [2.0, 3.0) | 84 | |
| [3.0, 4.0) | 80 | |
| [4.0, 5.0) | 94 | |
| all | [-5.0, -4.0) | 99 |
| [-4.0, -3.0) | 119 | |
| [-3.0, -2.0) | 109 | |
| [-2.0, -1.0) | 109 | |
| [-1.0, 0.0) | 95 | |
| [0.0, 1.0) | 104 | |
| [1.0, 2.0) | 102 | |
| [2.0, 3.0) | 106 | |
| [3.0, 4.0) | 112 | |
| [4.0, 5.0) | 122 |
df = df.unstack(level=0)
df
| unweighted | ||
|---|---|---|
| all | pass | |
| [-5.0, -4.0) | 99 | 9 |
| [-4.0, -3.0) | 119 | 25 |
| [-3.0, -2.0) | 109 | 29 |
| [-2.0, -1.0) | 109 | 35 |
| [-1.0, 0.0) | 95 | 54 |
| [0.0, 1.0) | 104 | 67 |
| [1.0, 2.0) | 102 | 60 |
| [2.0, 3.0) | 106 | 84 |
| [3.0, 4.0) | 112 | 80 |
| [4.0, 5.0) | 122 | 94 |
df["unweighted", "pass"] / df["unweighted", "all"]
[-5.0, -4.0) 0.090909
[-4.0, -3.0) 0.210084
[-3.0, -2.0) 0.266055
[-2.0, -1.0) 0.321101
[-1.0, 0.0) 0.568421
[0.0, 1.0) 0.644231
[1.0, 2.0) 0.588235
[2.0, 3.0) 0.792453
[3.0, 4.0) 0.714286
[4.0, 5.0) 0.770492
dtype: float64
PredicateBinning means that each bin represents a predicate (if-then rule) in the filling procedure. Aghast doesn't have a filling procedure, but filling-libraries can use this to encode relationships among histograms that a fitting-library can take advantage of, for combined signal-control region fits, for instance. It's possible for those regions to overlap: an input datum might satisfy more than one predicate, and overlapping_fill determines which bin(s) were chosen: first, last, or all.
VariationBinning means that each bin represents a variation of one of the paramters used to calculate the fill-variables. This is used to determine sensitivity to systematic effects, by varying them and re-filling. In this kind of binning, the same input datum enters every bin.
xdata = numpy.random.normal(0, 1, int(1e6))
sigma = numpy.random.uniform(-0.1, 0.8, int(1e6))
h = aghast.Histogram([aghast.Axis(aghast.VariationBinning([
aghast.Variation([aghast.Assignment("x", "nominal")]),
aghast.Variation([aghast.Assignment("x", "nominal + sigma")])])),
aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5)))],
aghast.UnweightedCounts(
aghast.InterpretedInlineBuffer.fromarray(
numpy.concatenate([
numpy.histogram(xdata, bins=10, range=(-5, 5))[0],
numpy.histogram(xdata + sigma, bins=10, range=(-5, 5))[0]]))))
df = aghast.to_pandas(h)
df
| unweighted | ||
|---|---|---|
| x := nominal | [-5.0, -4.0) | 31 |
| [-4.0, -3.0) | 1309 | |
| [-3.0, -2.0) | 21624 | |
| [-2.0, -1.0) | 135279 | |
| [-1.0, 0.0) | 341683 | |
| [0.0, 1.0) | 341761 | |
| [1.0, 2.0) | 135675 | |
| [2.0, 3.0) | 21334 | |
| [3.0, 4.0) | 1273 | |
| [4.0, 5.0) | 31 | |
| x := nominal + sigma | [-5.0, -4.0) | 14 |
| [-4.0, -3.0) | 559 | |
| [-3.0, -2.0) | 10814 | |
| [-2.0, -1.0) | 84176 | |
| [-1.0, 0.0) | 271999 | |
| [0.0, 1.0) | 367950 | |
| [1.0, 2.0) | 209479 | |
| [2.0, 3.0) | 49997 | |
| [3.0, 4.0) | 4815 | |
| [4.0, 5.0) | 193 |
df.unstack(level=0)
| unweighted | ||
|---|---|---|
| x := nominal | x := nominal + sigma | |
| [-5.0, -4.0) | 31 | 14 |
| [-4.0, -3.0) | 1309 | 559 |
| [-3.0, -2.0) | 21624 | 10814 |
| [-2.0, -1.0) | 135279 | 84176 |
| [-1.0, 0.0) | 341683 | 271999 |
| [0.0, 1.0) | 341761 | 367950 |
| [1.0, 2.0) | 135675 | 209479 |
| [2.0, 3.0) | 21334 | 49997 |
| [3.0, 4.0) | 1273 | 4815 |
| [4.0, 5.0) | 31 | 193 |
Collections
You can gather many objects (histograms, functions, ntuples) into a Collection, partly for convenience of encapsulating all of them in one object.
aghast.Collection({"one": fromroot, "two": ghastly_hist}).dump()
Collection(
objects={
'one': Histogram(
axis=[
Axis(
binning=
RegularBinning(
num=80,
interval=RealInterval(low=-5.0, high=5.0),
overflow=RealOverflow(loc_underflow=BinLocation.below1, loc_overflow=BinLocation.above1)),
statistics=[
Statistics(
moments=[
Moments(sumwxn=InterpretedInlineInt64Buffer(buffer=[1e+07]), n=0),
Moments(sumwxn=InterpretedInlineFloat64Buffer(buffer=[1e+07]), n=0, weightpower=1),
Moments(sumwxn=InterpretedInlineFloat64Buffer(buffer=[1e+07]), n=0, weightpower=2),
Moments(sumwxn=InterpretedInlineFloat64Buffer(buffer=[2468.31]), n=1, weightpower=1),
Moments(
sumwxn=InterpretedInlineFloat64Buffer(buffer=[1.00118e+07]),
n=2,
weightpower=1)
])
])
],
counts=
UnweightedCounts(
counts=
InterpretedInlineFloat64Buffer(
buffer=
[0.00000e+00 2.00000e+00 5.00000e+00 9.00000e+00 1.50000e+01 2.90000e+01
4.90000e+01 8.00000e+01 1.04000e+02 2.37000e+02 3.52000e+02 5.55000e+02
8.67000e+02 1.44700e+03 2.04600e+03 3.03700e+03 4.56200e+03 6.80500e+03
9.54000e+03 1.35290e+04 1.85840e+04 2.55930e+04 3.50000e+04 4.60240e+04
5.91030e+04 7.64920e+04 9.64410e+04 1.19873e+05 1.46159e+05 1.77533e+05
2.10628e+05 2.46316e+05 2.83292e+05 3.21377e+05 3.59314e+05 3.93857e+05
4.26446e+05 4.53031e+05 4.74806e+05 4.89846e+05 4.96646e+05 4.97922e+05
4.90499e+05 4.73200e+05 4.53527e+05 4.25650e+05 3.93297e+05 3.58537e+05
3.21099e+05 2.82519e+05 2.46469e+05 2.11181e+05 1.77550e+05 1.47417e+05
1.20322e+05 9.65920e+04 7.66650e+04 5.95870e+04 4.57760e+04 3.44590e+04
2.59000e+04 1.88760e+04 1.35760e+04 9.57100e+03 6.66200e+03 4.62900e+03
3.16100e+03 2.06900e+03 1.33400e+03 8.78000e+02 5.81000e+02 3.32000e+02
2.20000e+02 1.35000e+02 6.50000e+01 3.90000e+01 2.60000e+01 1.90000e+01
1.50000e+01 4.00000e+00 4.00000e+00 0.00000e+00]))),
'two': Histogram(
axis=[
Axis(binning=RegularBinning(num=80, interval=RealInterval(low=-5.0, high=5.0)))
],
counts=
UnweightedCounts(
counts=
InterpretedInlineInt64Buffer(
buffer=
[ 2 5 9 15 29 49 80 104 237 352
555 867 1447 2046 3037 4562 6805 9540 13529 18584
25593 35000 46024 59103 76492 96441 119873 146159 177533 210628
246316 283292 321377 359314 393857 426446 453031 474806 489846 496646
497922 490499 473200 453527 425650 393297 358537 321099 282519 246469
211181 177550 147417 120322 96592 76665 59587 45776 34459 25900
18876 13576 9571 6662 4629 3161 2069 1334 878 581
332 220 135 65 39 26 19 15 4 4])))
})
Not only for convenience: you can also define an Axis in the Collection to subdivide all contents by that Axis. For instance, you can make a collection of qualitatively different histograms all have a signal and control region with PredicateBinning, or all have systematic variations with VariationBinning.
It is not necessary to rely on naming conventions to communicate this information from filler to fitter.
Histogram → histogram conversions
I said in the introduction that aghast does not fill histograms and does not plot histograms—the two things data analysts are expecting to do. These would be done by user-facing libraries.
Aghast does, however, transform histograms into other histograms, and not just among formats. You can combine histograms with +. In addition to adding histogram counts, it combines auxiliary statistics appropriately (if possible).
h1 = aghast.Histogram([
aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5)),
statistics=[aghast.Statistics(
moments=[
aghast.Moments(aghast.InterpretedInlineBuffer.fromarray(numpy.array([10])), n=1),
aghast.Moments(aghast.InterpretedInlineBuffer.fromarray(numpy.array([20])), n=2)],
quantiles=[
aghast.Quantiles(aghast.InterpretedInlineBuffer.fromarray(numpy.array([30])), p=0.5)],
mode=aghast.Modes(aghast.InterpretedInlineBuffer.fromarray(numpy.array([40]))),
min=aghast.Extremes(aghast.InterpretedInlineBuffer.fromarray(numpy.array([50]))),
max=aghast.Extremes(aghast.InterpretedInlineBuffer.fromarray(numpy.array([60]))))])],
aghast.UnweightedCounts(aghast.InterpretedInlineBuffer.fromarray(numpy.arange(10))))
h2 = aghast.Histogram([
aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5)),
statistics=[aghast.Statistics(
moments=[
aghast.Moments(aghast.InterpretedInlineBuffer.fromarray(numpy.array([100])), n=1),
aghast.Moments(aghast.InterpretedInlineBuffer.fromarray(numpy.array([200])), n=2)],
quantiles=[
aghast.Quantiles(aghast.InterpretedInlineBuffer.fromarray(numpy.array([300])), p=0.5)],
mode=aghast.Modes(aghast.InterpretedInlineBuffer.fromarray(numpy.array([400]))),
min=aghast.Extremes(aghast.InterpretedInlineBuffer.fromarray(numpy.array([500]))),
max=aghast.Extremes(aghast.InterpretedInlineBuffer.fromarray(numpy.array([600]))))])],
aghast.UnweightedCounts(aghast.InterpretedInlineBuffer.fromarray(numpy.arange(100, 200, 10))))
(h1 + h2).dump()
Histogram(
axis=[
Axis(
binning=RegularBinning(num=10, interval=RealInterval(low=-5.0, high=5.0)),
statistics=[
Statistics(
moments=[
Moments(sumwxn=InterpretedInlineInt64Buffer(buffer=[110]), n=1),
Moments(sumwxn=InterpretedInlineInt64Buffer(buffer=[220]), n=2)
],
min=Extremes(values=InterpretedInlineInt64Buffer(buffer=[50])),
max=Extremes(values=InterpretedInlineInt64Buffer(buffer=[600])))
])
],
counts=
UnweightedCounts(
counts=InterpretedInlineInt64Buffer(buffer=[100 111 122 133 144 155 166 177 188 199])))
The corresponding moments of h1 and h2 were matched and added, quantiles and modes were dropped (no way to combine them), and the correct minimum and maximum were picked; the histogram contents were added as well.
Another important histogram → histogram conversion is axis-reduction, which can take three forms:
- slicing an axis, either dropping the eliminated bins or adding them to underflow/overflow (if possible, depends on binning type);
- rebinning by combining neighboring bins;
- projecting out an axis, removing it entirely, summing over all existing bins.
All of these operations use a Pandas-inspired loc/iloc syntax.
h = aghast.Histogram(
[aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5)))],
aghast.UnweightedCounts(
aghast.InterpretedInlineBuffer.fromarray(numpy.array([0, 10, 20, 30, 40, 50, 60, 70, 80, 90]))))
loc slices in the data's coordinate system. 1.5 rounds up to bin index 6. The first five bins get combined into an overflow bin: 150 = 10 + 20 + 30 + 40 + 50.
h.loc[1.5:].dump()
Histogram(
axis=[
Axis(
binning=
RegularBinning(
num=4,
interval=RealInterval(low=1.0, high=5.0),
overflow=
RealOverflow(
loc_underflow=BinLocation.above1,
minf_mapping=RealOverflow.missing,
pinf_mapping=RealOverflow.missing,
nan_mapping=RealOverflow.missing)))
],
counts=UnweightedCounts(counts=InterpretedInlineInt64Buffer(buffer=[60 70 80 90 150])))
iloc slices by bin index number.
h.iloc[6:].dump()
Histogram(
axis=[
Axis(
binning=
RegularBinning(
num=4,
interval=RealInterval(low=1.0, high=5.0),
overflow=
RealOverflow(
loc_underflow=BinLocation.above1,
minf_mapping=RealOverflow.missing,
pinf_mapping=RealOverflow.missing,
nan_mapping=RealOverflow.missing)))
],
counts=UnweightedCounts(counts=InterpretedInlineInt64Buffer(buffer=[60 70 80 90 150])))
Slices have a start, stop, and step (start:stop:step). The step parameter rebins:
h.iloc[::2].dump()
Histogram(
axis=[
Axis(binning=RegularBinning(num=5, interval=RealInterval(low=-5.0, high=5.0)))
],
counts=UnweightedCounts(counts=InterpretedInlineInt64Buffer(buffer=[10 50 90 130 170])))
Thus, you can slice and rebin as part of the same operation.
Projecting uses the same mechanism, except that None passed as an axis's slice projects it.
h2 = aghast.Histogram(
[aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5))),
aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5)))],
aghast.UnweightedCounts(
aghast.InterpretedInlineBuffer.fromarray(numpy.arange(100))))
h2.iloc[:, None].dump()
Histogram(
axis=[
Axis(binning=RegularBinning(num=10, interval=RealInterval(low=-5.0, high=5.0)))
],
counts=
UnweightedCounts(
counts=InterpretedInlineInt64Buffer(buffer=[45 145 245 345 445 545 645 745 845 945])))
Thus, all three axis reduction operations can be performed in a single syntax.
In general, an n-dimensional ghastly histogram can be sliced like an n-dimensional Numpy array. This includes integer and boolean indexing (though that necessarily changes the binning to IrregularBinning).
h.iloc[[4, 3, 6, 7, 1]].dump()
Histogram(
axis=[
Axis(
binning=
IrregularBinning(
intervals=[
RealInterval(low=-1.0, high=0.0),
RealInterval(low=-2.0, high=-1.0),
RealInterval(low=1.0, high=2.0),
RealInterval(low=2.0, high=3.0),
RealInterval(low=-4.0, high=-3.0)
]))
],
counts=UnweightedCounts(counts=InterpretedInlineInt64Buffer(buffer=[40 30 60 70 10])))
h.iloc[[True, False, True, False, True, False, True, False, True, False]].dump()
Histogram(
axis=[
Axis(
binning=
IrregularBinning(
intervals=[
RealInterval(low=-5.0, high=-4.0),
RealInterval(low=-3.0, high=-2.0),
RealInterval(low=-1.0, high=0.0),
RealInterval(low=1.0, high=2.0),
RealInterval(low=3.0, high=4.0)
]))
],
counts=UnweightedCounts(counts=InterpretedInlineInt64Buffer(buffer=[0 20 40 60 80])))
loc for numerical binnings accepts
- a real number
- a real-valued slice
Nonefor projection- ellipsis (
...)
loc for categorical binnings accepts
- a string
- an iterable of strings
- an empty slice
Nonefor projection- ellipsis (
...)
iloc accepts
- an integer
- an integer-valued slice
Nonefor projection- integer-valued array-like
- boolean-valued array-like
- ellipsis (
...)
Bin counts → Numpy
Frequently, one wants to extract bin counts from a histogram. The loc/iloc syntax above creates histograms from histograms, not bin counts.
A histogram's counts property has a slice syntax.
allcounts = numpy.arange(12) * numpy.arange(12)[:, None] # multiplication table
allcounts[10, :] = -999 # underflows
allcounts[11, :] = 999 # overflows
allcounts[:, 0] = -999 # underflows
allcounts[:, 1] = 999 # overflows
print(allcounts)
[[-999 999 0 0 0 0 0 0 0 0 0 0]
[-999 999 2 3 4 5 6 7 8 9 10 11]
[-999 999 4 6 8 10 12 14 16 18 20 22]
[-999 999 6 9 12 15 18 21 24 27 30 33]
[-999 999 8 12 16 20 24 28 32 36 40 44]
[-999 999 10 15 20 25 30 35 40 45 50 55]
[-999 999 12 18 24 30 36 42 48 54 60 66]
[-999 999 14 21 28 35 42 49 56 63 70 77]
[-999 999 16 24 32 40 48 56 64 72 80 88]
[-999 999 18 27 36 45 54 63 72 81 90 99]
[-999 999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999]
[-999 999 999 999 999 999 999 999 999 999 999 999]]
h2 = aghast.Histogram(
[aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5),
aghast.RealOverflow(loc_underflow=aghast.RealOverflow.above1,
loc_overflow=aghast.RealOverflow.above2))),
aghast.Axis(aghast.RegularBinning(10, aghast.RealInterval(-5, 5),
aghast.RealOverflow(loc_underflow=aghast.RealOverflow.below2,
loc_overflow=aghast.RealOverflow.below1)))],
aghast.UnweightedCounts(
aghast.InterpretedInlineBuffer.fromarray(allcounts)))
print(h2.counts[:, :])
[[ 0 0 0 0 0 0 0 0 0 0]
[ 2 3 4 5 6 7 8 9 10 11]
[ 4 6 8 10 12 14 16 18 20 22]
[ 6 9 12 15 18 21 24 27 30 33]
[ 8 12 16 20 24 28 32 36 40 44]
[10 15 20 25 30 35 40 45 50 55]
[12 18 24 30 36 42 48 54 60 66]
[14 21 28 35 42 49 56 63 70 77]
[16 24 32 40 48 56 64 72 80 88]
[18 27 36 45 54 63 72 81 90 99]]
To get the underflows and overflows, set the slice extremes to -inf and +inf.
print(h2.counts[-numpy.inf:numpy.inf, :])
[[-999 -999 -999 -999 -999 -999 -999 -999 -999 -999]
[ 0 0 0 0 0 0 0 0 0 0]
[ 2 3 4 5 6 7 8 9 10 11]
[ 4 6 8 10 12 14 16 18 20 22]
[ 6 9 12 15 18 21 24 27 30 33]
[ 8 12 16 20 24 28 32 36 40 44]
[ 10 15 20 25 30 35 40 45 50 55]
[ 12 18 24 30 36 42 48 54 60 66]
[ 14 21 28 35 42 49 56 63 70 77]
[ 16 24 32 40 48 56 64 72 80 88]
[ 18 27 36 45 54 63 72 81 90 99]
[ 999 999 999 999 999 999 999 999 999 999]]
print(h2.counts[:, -numpy.inf:numpy.inf])
[[-999 0 0 0 0 0 0 0 0 0 0 999]
[-999 2 3 4 5 6 7 8 9 10 11 999]
[-999 4 6 8 10 12 14 16 18 20 22 999]
[-999 6 9 12 15 18 21 24 27 30 33 999]
[-999 8 12 16 20 24 28 32 36 40 44 999]
[-999 10 15 20 25 30 35 40 45 50 55 999]
[-999 12 18 24 30 36 42 48 54 60 66 999]
[-999 14 21 28 35 42 49 56 63 70 77 999]
[-999 16 24 32 40 48 56 64 72 80 88 999]
[-999 18 27 36 45 54 63 72 81 90 99 999]]
Also note that the underflows are now all below the normal bins and overflows are now all above the normal bins, regardless of how they were arranged in the ghast. This allows analysis code to be independent of histogram source.
Other types
Aghast can attach fit functions to histograms, can store standalone functions, such as lookup tables, and can store ntuples for unweighted fits or machine learning.
Acknowledgements
Support for this work was provided by NSF cooperative agreement OAC-1836650 (IRIS-HEP), grant OAC-1450377 (DIANA/HEP) and PHY-1520942 (US-CMS LHC Ops).
Thanks especially to the gracious help of aghast contributors!
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