histogram/doc/rationale.qbk

[section Rationale]

[section Philosophy and principles]

This library was written based on a decade of experience collected in working with big data, more precisely in the field of particle physics and astroparticle physics. The design is guided by advice from people like Bjarne Stroustrup, Scott Meyers, Herb Sutter, and Andrei Alexandrescu, and Chandler Carruth. I also like the [@https://www.python.org/dev/peps/pep-0020 Zen of Python] (also applies to other languages). I also borrowed ideas from the [@https://eigen.tuxfamily.org/ Eigen library].

Two design goals of the library:

* Provide the same interface for one-dimensional and multi-dimensional histograms. This makes the interface easier to learn, and makes it easier to move a project from one-dimensional to multi-dimensional analysis. This sounds obvious, but other libraries don't do that.

* Hide the details of how the bin counters work. Other implementations, notably those in the [@https://root.cern.ch ROOT framework] expose this, which forces the user to make a choice which is potentially dangerous. At best, the choice is merely inefficient. In the worst case, it can lead to information loss in form of overflowing or capped counters.

[endsect]

[section Structure]

The library consists of three orthogonal components:

* [link histogram.rationale.histogram_types histogram types]: Host classes which defines the user interface and responsible for holding axis objects. The two variants have the same user interface, but differ internally.

* [link histogram.rationale.axis_types axis types]: Defines how input values are mapped to bins. Several axis types are provided which implement different specializations. The list is user-extensible.

* [link histogram.rationale.storage_types storage types]: Manages memory to hold bin counters. The requirements for a storage differ from those of an STL container. Two implementations are provided.

[endsect]

[section:histogram_types Histograms types]

Histograms store a number of axes. A one-dimensional histogram has one axis, a multi-dimensional histogram has several. Each axis maps a value from an input tuple onto a bin in its range.

[note
To understand the need for multi-dimensional histograms, think of point coordinates. If all points that you consider lie on a line, you need only one value to describe the point. If all points lie in a plane, you need two values to describe the position. Three values are needed for a point in space. A histogram puts a discrete grid over the line, the plane or the space, and counts how many points lie in each cell of the grid. To reflect a point distribution on a line, a 1d-histogram is sufficient. To do the same in 3d-space, one needs a 3d-histogram.
]

This library supports different axis types, so that the user can customize how the mapping is done exactly, see [link histogram.rationale.axis_types axis types]. The number and concrete types of the axes objects held by the histogram may be known at compile time or only at runtime, depending on how the library is used.

Users can chose between two histogram variants, which have the same user interface, see [classref boost::histogram::static_histogram] and [classref boost::histogram::dynamic_histogram]. The static variant is faster (see [link histogram.benchmarks benchmark]), because it can access the different axis types without any indirections or dynamic type casting. This also means that user errors are caught at compile-time rather than run-time.

The static variant cannot be used when the axis configuration is only known at run-time, for example, if a histogram is created from Python. The dynamic variant addresses this and allows one to set the number of axes and their types at runtime. The interface of the dynamic variant is a strict superset of the static variant.

[endsect]

[section:axis_types Axis types]

An axis defines which range of input values is mapped to which bin. The logic is encapsulated in an axis type. Users can create their own axis classes and use them with the library, by providing a class compatible with the [link histogram.concepts.axis axis concept]. The library comes with five builtin types, which implement different specializations.

* [classref boost::histogram::axis::regular] sorts real numbers into bins with equal width.
* [classref boost::histogram::axis::variable] sorts real numbers into bins with varying width.
* [classref boost::histogram::axis::circular] is a specialization of a regular axis for angles and similar values defined on a closed circle.
* [classref boost::histogram::axis::integer] is a specialization of a regular axis for a range of integers with unit bin width.
* [classref boost::histogram::axis::category] is a one-on-one mapping of a set of unique values onto bins. This can be used to count labels, like "red", "green", "blue".

[endsect]

[section:storage_types Storage types]

A storage type stores the actual bin counters. It uses a one-dimensional index for lookup, computed by the histogram host from the multi-dimensional index generated by evaluating all its axes. The storage therefore needs to know nothing about axes. Users can integrate their own storage classes with the library, by providing a class compatible with the [link histogram.concepts.storage storage concept].

Dense (aka contiguous) storage in memory is needed for fast bin lookup, which is of the random-access variety, and may be happening in a tight loop. The builtin storage types therefore implement dense storage of bin counters. [classref boost::histogram::array_storage] implements a simple storage based on a heap-allocated array. That could be the end of story, but there are some issues with this approach. It is not convenient, because the user has to decide what (integral) type to use to hold the bin counts and it is not an obvious choice. The integer needs to be large enough to avoid counter overflow, but only a fraction of the bits are used if it is too large. Then it is a waste of memory. This is still a concern since the performance of modern CPUs requires frequent use of the CPU cache, which is small. Using floating point numbers is similarly dangerous. They don't overflow, but cap the bin count when the bits in the mantissa are used up.

The standard storage used in the library is [classref boost::histogram::adaptive_storage], which solves these issues with dynamic memory management, based on the following insight. The [@https://en.wikipedia.org/wiki/Curse_of_dimensionality curse of dimensionality] makes the total number of bins grow very fast as the dimension of the histogram grows. However, having many bins also reduces the number of counts per bin, since the input values are spread over many more bins now.

We therefore start with a minimum amount of memory per bin counter by using the smallest integer type to hold a count. If the bin counter is about to overflow, we switch to the next larger integer type. We start with 1 byte per bin counter and then double the size as needed, until 8 byte per bin are reached. The following images illustrate this progression for a storage of 3 bin counters. Note, that always a new memory block is allocated for all counters, when one of them hits its capacity limit.

[$../storage_3_uint8.svg]

[$../storage_3_uint16.svg]

[$../storage_3_uint32.svg]

[$../storage_3_uint64.svg]

When even that is not enough, we switch to the [@boost:/libs/multiprecision/index.html Boost.Multiprecision] type `cpp_int`, whose capacity is limited only by available memory. The following image is not to scale:

[$../storage_3_cpp_int.svg]

This approach is not only memory conserving, but also ultimately safe, because bin counters cannot overflow.

And now comes the best part: this approach is even faster in the multi-dimensional case despite the overheads involved in handling the storage dynamically. The benchmarks show, that the gains from better cache usage outweigh the run-time overheads of dynamic dispatching to the right bin counter type and the additional allocation costs. Doubling the size of the bin counters each time helps, because then allocations happen only O(logN) times for N bin increments.

In a sense, [classref boost::histogram::adaptive_storage adaptive_storage] is the opposite of a `std::vector`, which keeps the size of the stored type constant, but grows to hold a larger number of elements. Here, the number of elements remains the same, but the storage grows to hold a uniform collection of larger and larger elements.

[endsect]

[section:uoflow Under- and overflow bins]

Axis instances by default add extra bins that count values which fall below or above the range covered by the axis (for those types where that makes sense). These extra bins are called under- and overflow bins, respectively. The extra bins can be turned off individually for each axis to conserve memory, but it is generally recommended to keep them. The extra bins do not interfere with normal bin counting. On an axis with `n` bins, the first bin has the index `0`, the last bin `n-1`, while the under- and overflow bins are accessible at the indices `-1` and `n`, respectively.

Under- and overflow bins are useful in one-dimensional histograms, and nearly essential in multi-dimensional histograms. Here are the advantages:

* No loss: The total sum over all bin counts is strictly equal to the number of times `fill(...)` was called. Even NaN values are counted, they end up in the underflow bin by convention.

* Diagnosis: Unexpected extreme values show up in the extra bins, which otherwise may be overlooked.

* Reducibility: In multi-dimensional histograms, an out-of-range value along one axis may be paired with an in-range value along another axis. If under- and overflow bins are missing, such a value pair is lost completely. If you apply a `reduce` operation on a histogram, which removes somes axes by resummation of the bin counts, this would lead to distortions of the histogram along the remaining axes. When under- and overflow bins are present, the `reduce` operation always produces the same sub-histogram that would have been obtained if it was filled from scratch with the original data.

[endsect]

[section:variance Variance estimates]

Once a histogram is filled, the count ['k] in a bin can be queried with the `value(...)` method. The histogram also offers a `variance(...)` method, which returns an estimate of the [@https://en.wikipedia.org/wiki/Variance variance] ['v] of that count.

If the input values for the histogram come from a [@https://en.wikipedia.org/wiki/Stochastic_process stochastic process], the variance provides useful additional information. Examples for a stochastic process are a physics experiment or a random person filling out a questionaire[footnote The choices of the person are most likely not random, but if we pick a random person from a group, we randomly sample from a pool of opinions]. The variance ['v] is the square of the [@https://en.wikipedia.org/wiki/Standard_deviation standard deviation]. The standard deviation in a bin tells us how much we can expect the observed value to fluctuate if we or someone else would repeat our experiment with new random input.

Variance estimates are useful in many ways:

* Error bars: Drawing an [@https://en.wikipedia.org/wiki/Error_bar error bar] over the interval ['(k - sqrt(v), k + sqrt(v))] is a simple visualisation of the expected random scatter of the bin value ['k], if the histogram was cleared and filled again with another independent sample of the same size (e.g. by repeating the physics experiment or asking more people to fill a questionaire). If you compare the result with a fitted model (see next item), about 2/3 of the error bars should overlap with the model, if the model is correct.

* Least-squares fitting: Often you have a model of the expected number of counts ['lambda] per bin, which is a function of parameters with unknown values. A simple method to find good (sometimes the best) estimates for those parameter values is to vary them until the sum of squared residuals ['(k - lambda)^2/v] is minimized. This is the [@https://en.wikipedia.org/wiki/Least_squares method of least squares], in which both the bin values ['k] and variance estimates ['v] enter.

* Pull distributions: If you have two histograms filled with the same number of samples and you want to know whether they are in agreement, you can compare the so-called pull distribution. It is formed by subtracting the counts and dividing by the square root of their variances ['(k1 - k2)/sqrt(v1 + v2)]. If the histograms are identical, the pull distribution randomly scatters around zero, and about 2/3 of the values are in the interval ['[ -1, 1]].

Why return the variance ['v] and not the standard deviation ['s = sqrt(v)]? The reason is the additivity of variances. [@https://en.wikipedia.org/wiki/Variance#Properties Variances of independent samples can be added] like normal numbers ['v3 = v1 + v2]. This is not true for standard deviations, where the addition law is more complex ['s3 = sqrt(s1^2 + s2^2)]. In that sense, the variance is more straight-forward to use during data processing. It is also obviously more efficient. The user can take the square-root at the end of the processing obtain the standard deviation as needed.

How is the variance estimate ['v] computed? If we know the expected number of counts ['lambda] per bin, we could compute the variance as ['v = lambda], because counts in a histogram follow the [@https://en.wikipedia.org/wiki/Poisson_distribution Poisson distribution]
[footnote
The Poisson distribution is correct as far as the counts ['k] themselves are of interest. If the fractions per bin ['p = k / N] are of interest, where ['N] is the total number of counts, then the correct distribution to describe the fractions is the [@https://en.wikipedia.org/wiki/Multinomial_distribution multinomial distribution].
]. After filling a histogram, we do not know the expected number of counts ['lambda] for any particular bin, but we know the observed count ['k], which is not too far from ['lambda]. We therefore might be tempted to just replace ['lambda] with ['k] in the formula ['v = lambda = k]. This is in fact the so-called non-parameteric estimate for the variance based on the [@https://en.wikipedia.org/wiki/Plug-in_principle plug-in principle]. It is the best (and only) estimate for the variance, if we know nothing more about the underlying stochastic process which generated the inputs (or want to feign ignorance about it).

Now, if the value returned by the method `variance(...)` is just the same as the value return by `value(...)`, why bother with adding a `variance(...)` method, except perhaps for convenience? There is another reason, which becomes apparent if the histograms are filled with weighted counts, which is discussed next.

[endsect]

[section:weights Support of weighted increments]

A histogram categorizes input values and increments a bin counter if an input value falls into the value range covered by that bin. The [classref boost::histogram::adaptive_storage standard storage] uses integer types to store these counts, see the [link histogram.rationale.storage_types storage section] how integer overflow is avoided. However, sometimes histograms need to be filled with values that have a weight ['w] attached to them. In this case, the corresponding bin counter is not increased by one, but by the passed weight ['w].
[note
There are several uses for weighted increments. The main use in particle physics is to adapt simulated data of an experiment to real data. Simulations are needed to determine various corrections and efficiencies, but a simulated experiment is almost never a perfect replica of the real experiment. In addition, simulations are expensive to do. So, when deviations in a simulated distribution of a variable are found, one typically does not rerun the simulations, but assigns weights to match the simulated distribution to the real one.
]
When the [classref boost::histogram::adaptive_storage adaptive_storage] is used, histograms may also be filled with weighted value tuples. The choice of using weighted fills can be made at run-time. If the call `fill(..., weight(x))` is used, two doubles per bin are stored (previous integer counts are automatically converted). The first double keeps track of the sum of weights. The second double keeps track of the sum of weights squared. The latter is the variance estimate in this case and returned by a call to `variance(...)`.
[note
This variance estimate can be derived from the [@https://en.wikipedia.org/wiki/Variance#Properties mathematical properties of the variance]. Let us say a bin is filled ['k1] times with weight ['w1]. The sum of weights is then ['w1 k1]. It then follows from the variance properties that ['Var(w1 k1) = w1^2 Var(k1)]. Using the reasoning from before, the estimated variance of ['k1] is ['k1], so that ['Var(w1 k1) = w1^2 Var(k1) = w1^2 k1]. Variances of independent samples are additive. If the bin is further filled ['k2] times with weight ['w2], the sum of weights is ['w1 k1 + w2 k2], with variance ['w1^2 k1 + w2^2 k2]. This also holds for ['k1 = k2 = 1]. Therefore, the sum of weights ['w[i]] has variance sum of ['w[i]^2]. In other words, to incrementally keep track of the variance of the sum of weights, we need to keep track of the sum of weights squared.
]

[endsect]

[section Python support]

Python is a popular scripting language in the data science community. Thus, the library provides Python bindings. The histogram may be used as an interface between a complex simulation or data-storage system written in C++ and data-analysis/plotting in Python. Users are able to define the histogram in Python, let it be filled on the C++ side (using a few lines of Boost.Python code to define the interface), and then get it back for further data analysis or plotting.

Data analysis in Python is Numpy-based, so Numpy is fully supported. Histograms can be filled with chunks of data in a form of Numpy arrays, which is efficient, and the bin counts can be retrieved as a Numpy array without copying data.
[note
If number of dimensions is larger than one, this implementation is faster than the equivalent Numpy functions (while being more flexible), see [link histogram.benchmarks benchmark].
]

The Python and C++ interface were designed to be as consistent as possible, while following established style for the respective C++ or Python community. This leads to the following stylistic changes on the Python side.

Properties: Getter/setter-like functions on the C++ side are wrapped in Python as properties. Examples: `histogram.dim`, `axis.regular.uoflow`. In general, a C++ function that takes no argument but returns a value is using the property syntax on the Python side. An exception is made for the function `size()`, see next item.

`len(x)` versus `x.size()`: An axis instance behaves like a container of bins in C++ and like a sequence of bins in Python. To get the length of a sequence in Python one uses the `len(...)` function, while in C++ one uses the `size()` method.

Keyword-based parameters: the member function call `fill(..., weight(x))` in C++ is translated into a Python member function call `fill(..., weight=x)`.

[endsect]

[section Serialization]

Serialization is implemented using [@boost:/libs/serialization/index.html Boost.Serialization]. Pickling in Python is implemented based on the C++ serialization code. In the current implementation, the pickled stream is *not* portable between different hardware platforms, since it uses `boost::archive::binary_archive`. It would be great to switch to a portable binary representation in the future, when that becomes available. In practice, most computing clusters and most consumer hardware are x86 compatible nowadays. The binary_archive is portable between such hardware platforms. Naturally, it is portable between operating systems, like OSX, Windows, and Linux.

[endsect]

[section Comparison to Boost.Accumulators]

Boost.Histogram has a weak overlap with [@boost:/libs/accumulators/index.html Boost.Accumulators]. In particular, the statistical accumulators `density` and `weighted_density` also generate one-dimensional histograms. The axis range and the bin widths are determined automatically from a cached sample of initial values. In contrast, Boost.Histogram puts the responsibility to choose range and bin widths on the user.

Automatic binning is not an option for Boost.Histogram, because it does not scale well to many dimensions. Because of the Curse of Dimensionality, a prohibitive number of samples would need to be collected.

[note
There is no scientific consensus on how do automatic binning in an optimal way, mostly because there is no consensus over the cost function (there are many articles with different solutions in the literature). The problem is not solved for one-dimensional data, and even less so for multi-dimensional data.
]

Recommendation:

* Boost.Accumulators
  * You have one-dimensional data of which you know nothing about, and you want a histogram quickly without worrying about binning details.
* Boost.Histogram
  * You have multi-dimensional data or you suspect you will switch to multi-dimensional data later.
  * You want to customize the binning by hand, for example, to make bin edges coincide with special values or to handle special properties of your values, like angles defined on a circle.

[endsect]

[endsect]