Optimised tools for group-indexing operations: aggregated sum and more
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README.md

numpy-groupies

This package consists of a small library of optimised tools for doing things that can roughly be considered "group-indexing operations". The most prominent tool is aggregate, which is descibed in detail further down the page.

Installation

If you have pip, then simply:

pip install numpy_groupies

Note that numpy_groupies doesn't have any compulsory dependencies (even numpy is optional) so you should be able to install it fairly easily even without a package manager. If you just want one particular implementation of aggregate (e.g. aggregate_numpy.py), you can download that one file, and copy-paste the contents of utils.py into the top of that file (replacing the from .utils import (...) line).

aggregate

aggregate_diagram

import numpy as np
import numpy_groupies as npg
group_idx = np.array([   3,   0,   0,   1,   0,   3,   5,   5,   0,   4])
a =         np.array([13.2, 3.5, 3.5,-8.2, 3.0,13.4,99.2,-7.1, 0.0,53.7])
npg.aggregate(group_idx, a, func='sum', fill_value=0)
# >>>          array([10.0, -8.2, 0.0, 26.6, 53.7, 92.1])

aggregate takes an array of values, and an array giving the group number for each of those values. It then returns the sum (or mean, or std, or any, ...etc.) of the values in each group. You have probably come across this idea before - see Matlab's accumarray function, or pandas groupby concept, or MapReduce paradigm, or simply the basic histogram.

A couple of implemented functions do not reduce the data, instead it calculates values cumulatively while iterating over the data or permutates them. The output size matches the input size.

group_idx = np.array([4, 3, 3, 4, 4, 1, 1, 1, 7, 8, 7, 4, 3, 3, 1, 1])
a =         np.array([3, 4, 1, 3, 9, 9, 6, 7, 7, 0, 8, 2, 1, 8, 9, 8])
npg.aggregate(group_idx, a, func='cumsum')
# >>>          array([3, 4, 5, 6,15, 9,15,22, 7, 0,15,17, 6,14,31,39])

Inputs

The function accepts various different combinations of inputs, producing various different shapes of output. We give a brief description of the general meaning of the inputs and then go over the different combinations in more detail:

  • group_idx - array of non-negative integers to be used as the "labels" with which to group the values in a.
  • a - array of values to be aggregated.
  • func='sum' - the function to use for aggregation. See the section below for nore details.
  • size=None - the shape of the output array. If None, the maximum value in group_idx will set the size of the output.
  • fill_value=0 - value to use for output groups that do not appear anywhere in the group_idx input array.
  • order='C' - for multidimensional output, this controls the layout in memory, can be 'F' for fortran-style.
  • dtype=None - thedtype of the output. None means choose a sensible type for the given a, func, and fill_value.
  • axis=None - explained below.
  • ddof=0 - passed through into calculations of variance and standard deviation (see section on functions).

aggregate_dims_diagram

  • Form 1 is the simplest, taking group_idx and a of matching 1D lengths, and producing a 1D output.
  • Form 2 is similar to Form 1, but takes a scalar a, which is broadcast out to the length of group_idx. Note that this is generally not that useful.
  • Form 3 is more complicated. group_idx is the same length as the a.shape[axis]. The groups are broadcast out along the other axis/axes of a, thus the output is of shape n_groups x a.shape[0] x ... x a.shape[axis-1] x a.shape[axis+1] x ... a.shape[-1], i.e. the output has two or more dimensions.
  • Form 4 also produces output with two or more dimensions, but for very different reasons to Form 3. Here a is 1D and group_idx is exactly 2D, whereas in Form 3 a is ND, group_idx is 1D, and we provide a value for axis. The length of a must match group_idx.shape[1], the value of group_idx.shape[0] determines the number of dimensions in the ouput, i.e. group_idx[:,99] gives the (x,y,z) group indices for the a[99].
  • Form 5 is the same as Form 4 but with scalar a. As with Form 2, this is rarely that helpful.

Note on performance. The order of the output is unlikely to affect performance of aggregate (although it may affect your downstream usage of that output), however the order of multidimensional a or group_idx can affect performance: in Form 4 it is best if columns are contiguous in memory within group_idx, i.e. group_idx[:, 99] corresponds to a contiguous chunk of memory; in Form 3 it's best if all the data in a for group_idx[i] is contiguous, e.g. if axis=1 then we want a[:, 55] to be contiguous.

Available functions

By default, aggregate assumes you want to sum the values within each group, however you can specify another function using the func kwarg. This func can be any custom callable, however you will likely want one of the following optimized functions. Note that not all functions might be provided by all implementations.

  • 'sum' - sum of items within each group (see example above).
  • 'prod' - product of items within each group
  • 'mean' - mean of items within each group
  • 'var'- variance of items within each group. Use ddof kwarg for degrees of freedom. The divisor used in calculations is N - ddof, where N represents the number of elements. By default ddof is zero.
  • 'std' - standard deviation of items within each group. Use ddof kwarg for degrees of freedom (see var above).
  • 'min' - minimum value of items within each group.
  • 'max' - maximum value of items within each group.
  • 'first' - first item in a from each group.
  • 'last' - last item in a from each group.
  • 'argmax' - the index in a of the maximum value in each group.
  • 'argmin' - the index in a of the minimum value in each group.

The above functions also have a nan-form, which skip the nan values instead of propagating them to the result of the calculation:

  • 'nansum', 'nanprod', 'nanmean', 'nanvar', 'nanstd', 'nanmin', 'nanmax', 'nanfirst', 'nanlast', nanargmax, nanargmin

The following functions are slightly different in that they always return boolean values. Their treatment of nans is also different from above:

  • 'all' - True if all items within a group are truethy. Note that np.all(nan) is True, i.e. nan is actually truethy.
  • 'any' - True if any items within a group are truethy.
  • 'allnan' - True if all items within a group are nan.
  • 'anynan' - True if any items within a gorup are nan.

The following functions don't reduce the data, but instead produce an output matching the size of the input:

  • cumsum - cumulative sum of items within each group.
  • cumprod - cumulative product of items within each group. (numba only)
  • cummin - cumulative minimum of items within each group. (numba only)
  • cummax - cumulative maximum of items within each group. (numba only)
  • 'sort' - sort the items within each group in ascending order, use reverse=True to invert the order.

Finally, there are three functions which don't reduce each group to a single value, instead they return the full set of items within the group:

  • 'array' - simply returns the grouped items, using the same order as appeared in a. (numpy only)

Examples

Compute sums of consecutive integers, and then compute products of those consecutive integers.

group_idx = np.arange(5).repeat(3)
# group_idx: array([0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4])
a = np.arange(group_idx.size)
# a: array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14])
x = aggregate(group_idx, a) # sum is default
# x: array([ 3, 12, 21, 30, 39])
x = aggregate(group_idx, a, 'prod')
# x: array([ 0, 60, 336, 990, 2184])

Get variance ignoring nans, setting all-nan groups to nan.

x = aggregate(group_idx, a, func='nanvar', fill_value=nan)

Count the number of elements in each group. Note that this is equivalent to doing np.bincount(group_idx), indeed that is how the numpy implementation does it.

x = aggregate(group_idx, 1)

Sum 1000 values into a three-dimensional cube of size 15x15x15. Note that in this example all three dimensions have the same size, but that doesn't have to be the case.

group_idx = np.random.randint(0, 15, size=(3, 1000))
a = np.random.random(group_idx.shape[1])
x = aggregate(group_idx, a, func="sum", size=(15,15,15), order="F")
# x.shape: (15, 15, 15)
# np.isfortran(x): True

Use a custom function to generate some strings.

group_idx = array([1, 0,  1,  4,  1])
a = array([12.0, 3.2, -15, 88, 12.9])
x = aggregate(group_idx, a,
              func=lambda g: ' or maybe '.join(str(gg) for gg in g), fill_value='')
# x: ['3.2', '12.0 or maybe -15.0 or maybe 12.9', '', '', '88.0']

Use the axis arg in order to do a sum-aggregation on three rows simultaneously.

a = array([[99, 2,  11, 14,  20],
	   	   [33, 76, 12, 100, 71],
		   [67, 10, -8, 1,   9]])
group_idx = array([[3, 3, 7, 0, 0]])
x = aggregate(group_idx, a, axis=1)
# x : [[ 34, 0, 0, 101, 0, 0, 0, 11],
#      [171, 0, 0, 109, 0, 0, 0, 12],
#      [ 10, 0, 0,  77, 0, 0, 0, -8]]

Multiple implementations

There are multiple implementations of aggregate provided. If you use from numpy_groupies import aggregate, the best available implementation will automatically be selected. Otherwise you can pick a specific version directly like from numpy_groupies import aggregate_nb as aggregate or by importing aggregate from the implementing module from numpy_groupies.aggregate_weave import aggregate.

Currently the following implementations exist:

  • numpy - This is the default implementation. It uses plain numpy, mainly relying on np.bincount and basic indexing magic. It comes without other dependencies except numpy and shows reasonable performance for the occasional usage.
  • numba - This is the most performant implementation in average, based on jit compilation provided by numba and LLVM.
  • weave - weave compiles C-code on demand at runtime, producing binaries that get executed from within python. The performance of this implementation is comparable to the numba implementation.
  • pure python - This implementation has no dependencies and uses only the standard library. It's horribly slow and should only be used, if there is no numpy available.
  • numpy ufunc - Only for benchmarking. This impelmentation uses the .at method of numpy's ufuncs (e.g. add.at), which would appear to be designed for perfoming excactly the same calculation that aggregate executes, however the numpy implementation is rather incomplete and slow (as of v1.14.0). A numpy issue has been created to address this issue.
  • pandas - Only for reference. The pandas' groupby concept is the same as the task performed by aggregate. However, pandas is not actually faster than the default numpy implementation. Also, note that there may be room for improvement in the way that pandas is utilized here. Most notably, when computing multiple aggregations of the same data (e.g. 'min' and 'max') pandas could potentially be used more efficiently.

All implementations have the same calling syntax and produce the same outputs, to within some floating-point error. However some implementations only support a subset of the valid inputs and will sometimes throw NotImplementedError.

Benchmarks

Scripts for testing and benchmarking are included in this repository. For benchmarking, run python -m numpy_groupies.benchmarks.generic from the root of this repository.

Below we are using 500,000 indices uniformly picked from [0, 1000). The values of a are uniformly picked from the interval [0,1), with anything less than 0.2 then set to 0 (in order to serve as falsy values in boolean operations). For nan- operations another 20% of the values are set to nan, leaving the remainder on the interval [0.2,0.8).

The benchmarking results are given in ms for an i7-7560U running at 2.40GHz:

function         ufunc         numpy         numba         weave
-----------------------------------------------------------------
sum              29.738         1.454         0.901         1.176
prod             29.552        29.575         0.905         1.162
amin             29.944        29.850         0.979         1.169
amax             29.943        29.956         1.009         1.201
len              28.865         1.250         0.748         1.053
all              34.013         3.962         0.995         1.195
any              32.913         6.690         1.023         1.189
anynan           28.956         2.551         0.961         1.210
allnan           29.966         5.943         0.924         1.204
mean               ----         2.302         0.843         1.030
std                ----         5.587         1.015         1.210
var                ----         5.635         1.004         1.205
first              ----         1.586         0.977         0.994
last               ----         1.335         0.665         0.887
argmax             ----        29.734         0.903          ----
argmin             ----        33.152         0.868          ----
nansum             ----         4.576         1.633         1.176
nanprod            ----        26.832         1.618         1.186
nanmin             ----        27.198         1.852         1.211
nanmax             ----        27.194         1.786         1.252
nanlen             ----         4.602         1.611         1.948
nanall             ----         8.512         1.706         1.237
nanany             ----         9.492         1.719         1.245
nanmean            ----         5.089         1.669         1.282
nanvar             ----         8.783         1.815         1.485
nanstd             ----         8.771         1.805         1.482
nanfirst           ----         4.899         1.851         1.095
nanlast            ----         4.834         1.625         1.051
cumsum             ----       132.572         1.161          ----
cumprod            ----          ----         1.179          ----
cummax             ----          ----         1.146          ----
cummin             ----          ----         1.114          ----
arbitrary          ----       217.585        72.019          ----
sort               ----       237.604          ----          ----
Linux(x86_64), Python 2.7.12, Numpy 1.15.2, Numba 0.40.1

Development

This project was started by @ml31415 and the numba and weave implementations are by him. The pure python and numpy implementations were written by @d1manson.

The authors hope that numpy's ufunc.at methods will eventually be fast enough that hand-optimisation of individual functions will become unneccessary. However even if that does happen, there may still be a role for the aggregate function as a light-weight wrapper around those functions.

Maybe at some point a version of aggregate will make its way into numpy itself (or at least scipy).