histogram/doc/rationale.qbk

[section:rationale Rationale]

[section:guidelines Guidelines]

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].

Design goals of the library:

* Provide a simple and convenient default behavior for the casual user, yet allow a maximum of customization for the power user. Follow the "Don't pay for what you don't use" principle. Features that you don't use should not affect your performance negatively.

* 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.

* Hide the details of how the bin counters work. This design allows for interesting implementations, such as the default storage that provides a no-overflow-guarantee, which no other library offers.

* STL and Boost compatibility. The library should be compatible with STL algorithms and other Boost libraries, especially [@boost:/libs/accumulators/index.html Boost.Accumulators] and [@boost:/libs/range/index.html Boost.Range]. This potentiates what one can do with the library. Features provided by compatible libraries do not need to be explicitly implemented in this library.

[endsect]

[section:structure Structure]

The library consists of three orthogonal components:

* [link histogram.overview.rationale.structure.host histogram host class]: The histogram host class defines the public user interface and holds axis objects (one for each dimension) and a storage object. The user can chose whether axis objects are stored in a static tuple, a vector, or a vector of variants.

* [link histogram.overview.rationale.structure.axis axis types]: Defines how input values are mapped to bins. Several axis types are provided which implement different specializations. Users can make their own axis types following the axis concept and use them with the library. A variant type is provided, which can hold one of several concrete axis types.

* [link histogram.overview.rationale.structure.storage storage types]: Manages a collection of bin counters. The requirements for a storage differ from those of an STL container, it needs to follow the storage concept. Two implementations are provided.

[section:host Histogram host class]

Histograms store axis objects and a storage object. A one-dimensional histogram has one axis, a multi-dimensional histogram has several. When you pass an input tuple, say (v1, v2, v3), then the first axis will map v1 onto index i1, the second axis v2 onto i2, and so on, to generate the index tuple (i1, i2, i3). The histogram host class then converts these indices into a linear global index that is used to address bin counter in the storage object.

[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 approximate 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.overview.rationale.structure.axis axis types]. Users can furthermore chose between several ways of storing axis types in the histogram.

When the number and types of the axes are known at compile-time, the histogram host class stores axis types in a `std::tuple`. We call this a /static histogram/. To access a particular axis, one should use a compile-time number as index (a run-time index also works with some limitations). A static histogram is extremely fast (see [link histogram.benchmarks benchmark]), because there is no overhead and the compiler can inline code, unroll loops, and more. Also nice: many user errors are can be caught at compile-time rather than run-time.

Static histograms are the best kind, but cannot be used when histograms are to be created with an axis configuration that is only known at run-time. This is the case, for example, when histograms are created at run-time from Python.

There are two levels of dynamism. Firstly, the histogram can hold instances of a single axis type in a `std::vector`. Now the number of axis instances per histogram can vary at run-time, but the axis type must be the same for all instances. We call this /semi-dynamic histogram/.

If also the axis type should vary, one can use the `boost::histogram::axis::variant` type, which can hold one of a set of different concrete axis types and can be placed in a `std::vector`. When the histogram is configured to store axis types like this, we obtain a /dynamic histogram/. The dynamic histogram is a single type that can store arbitrary sequences of different axes types, which may be generated at run-time. The polymorphic behavior of the generic `boost::histogram::axis::variant` type has a run-time cost, however. Typically, the performance is reduced by a factor of two compared to a static histogram.

[note
The design decision to store axis types in the variant-like type `boost::histogram::axis::variant` has several advantages over forms of run-time polymorphism. Firstly, it guarantees that axis objects which belong to the same histogram are stored locally together in memory, which reduces cache misses when the histogram iterates over axis objects in a tight loop, which it often does. Secondly, each axis can accept a different value type in this scheme. Classic polymorphism with vtables requires that all overloads provided by derived classes share the same method call signature, but that was found to be too restrictive for this library. In this library, the first axis of a histogram may convert numbers to indices, the second strings. The method signatures of different axis types are allowed to differ. Classic run-time polymorphism does not work, but variants do.
]

[endsect]

[section:axis Axis types]

An axis defines an injective mapping of (a range of) input values to a 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 four builtin types, which implement different specializations.

* [classref boost::histogram::axis::regular] sorts real numbers into bins with equal width. The regular axis also supports monotonic transforms, which are applied when the input values are passed to the axis. This can be used to make a fast logarithmic axis, where the bins have equal width in the logarithm of the variable.
* [classref boost::histogram::axis::variable] sorts real numbers into bins with varying width.
* [classref boost::histogram::axis::integer] is a specialization of a regular axis for a range of integers with unit bin width. It is much faster than a regular axis.
* [classref boost::histogram::axis::category] is a bijective mapping of unique values onto bin indices and vice versa. This can be used with discrete categorical data, like "red", "green", "blue", for example.

Each builtin axis type has a few compile-time options, which change its behavior.

* All axis types can have an optional overflow bin. When the overflow bin is enabled and an input value is above the range covered by the axis, it is not discarded but counted in the overflow bin.
* All axis types except the category axis can have an optional underflow bin. When the underflow bin is enabled and an input value is below the range covered by the axis, it is not discarded but counted in the underflow bin.
* All axis types except the category axis can be circular, meaning that the axis range is periodic. This is useful for periodic data like polar angles.
* All axis types can optionally grow. When an input value is outside of the range of an axis which is configured to grow, the range of the axis is extended until the value is in range. This option is incompatible with the circular option, only either can be active.

[endsect]

[section:storage Storage types]

A storage type stores the actual cell values. It uses a one-dimensional index for cell lookup, computed by the histogram host from the indices generated from all its axes. The storage 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]. Standard containers are automatically adapted by the library to storage types.

Most storage types are optimized for fast look-up of the random-access variety. They use dense (aka contiguous) storage of cell values in memory. Cell lookup is often happening in a tight loop. A normal `std::vector` can be adapted to serve as a storage using the [classref boost::histogram::storage_adaptor]. Sometimes this is the best solution, but there are some caveats to this approach. The user has to decide which type should represents the cell counts and it is not an obvious choice. An integer type needs to be large enough to avoid counter overflow, but only a fraction of the bits are used if it is too large. Using an integral type that is too large is a waste of memory. When memory is wasted, more cache misses occur and performance is noticeably degraded. The performance of modern CPUs depends a lot on effective utilization of the CPU cache, which is small. Using floating point numbers instead of integers is also dangerous. They don't overflow, but cap the bin count when the bits in the mantissa are used up.

The default storage used in the library is [classref boost::histogram::unlimited_storage]. It solves these issues with a dynamic counter type 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. This means a small counter is often sufficient for high-dimensional histograms.

The default storage therefore starts with a minimum amount of memory per cell, it uses an 1 byte integer type to hold a count. If the count in any cell is about to overflow, all cells switch to the next larger integer type. The capacity per cell is doubled whenever this happens, until 8 byte per bin are reached. The following images illustrate this progression for a storage of 3 bin counters. A new memory block is always allocated for all counters, when the first 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, the default storage switches 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 provides the strong guarantee that bin counters cannot overflow.

[note
The no-overflow-guarantee only applies when the histogram is not using weighted fills or if all weights are integral numbers. When floating point weights are used, the default storage switches to a double counter per cell to store the sum of such weights. A double cannot provide the no-overflow-guarantee.
]

The best part: this approach is even faster for a histogram with sufficient size despite the run-time overheads of handling the counter type 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 occasional allocation costs. Doubling the size of the bin counters each time helps, too, because the allocations happen only O(logN) times for N increments.

[note
In a sense, [classref boost::histogram::unlimited_storage] is the complement 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]

[endsect]

[section:no_lambdas No lambdas as axis types]

It was considered whether histograms should accept lambdas as axis types, which simply implemented the mapping from the argument, the value type, to an integral index. The problem is that this wouldn't be enough be enough. The histogram frequently needs to query the size of axis without calling the lambda function. Because of this, the idea to allow lambdas as axis types was rejected.

This does not limit users in any way, because lambdas can easily be replaced by locally defined structs. Structs can be locally defined inside functions, if they are not templated and do not have templated methods. In the local context where the histogram is created, all relevant types should be known so that locally defined structs can simply use these concrete types within the context. There is no need for templates.

[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 have them. The normal bins, excluding under- and overflow, are called *inner bins*.

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 the histogram was filled. Even NaN values are counted, they are put in the overflow-bin by convention.

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

* Ability to reduce histograms: 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 some axes by summing all counts along that dimension, this would lead to distortions of the histogram along the remaining axes. When under- and overflow bins are present, the `reduce` operation always produces a sub-histogram identical to one obtained, if it was filled with the original data.

The presence of the extra bins does not interfere with normal indexing. 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. This choice is optimized for users who are unaware of the existence of these extra bins. They would find the other indexing scheme surprising, where you start with `0` at the underflow bin and the first normal bin is at `1`. Also, the chosen scheme allows one to turn off the extra bins in the code where the histogram is created, without changing any code downstream that addresses inner bins with indices.

[endsect]

[section:index_type Size method of axis returns signed integer]

The standard library returns a container size as an unsigned integer, because a container size cannot be negative. The `size()` method of the histogram class follows this rule, but the `size()` methods of axis types return a signed integral type. Why?

As explained in the [link histogram.overview.rationale.uoflow section about under- and overflow], a histogram axis may have an optional underflow bin, which is addressed by the index `-1`. It follows that the index type must be signed integer for all axis types.

The `size()` method of any axis returns the same signed integer type. The size of an axis cannot be negative, but this choice has two advantages. Firstly, the value returned by `size()` itself is guaranteed to be a valid index, which is good since it may address the overflow bin. Secondly, comparisons between an index and the value returned by `size()` are frequent. If `size()` returned an unsigned integral type, compilers would produce a warning for each comparisons, and rightly so. [@https://www.youtube.com/watch?v=wvtFGa6XJDU Something awful happens] on most machines when you compare `-1` with an unsigned integer, `-1 < 1u == false`, which causes a serious bug in the following innocent-looking loop:
```
auto my_axis = /* ... */;
// naive loop to iterate over all bins, including underflow and overflow
for (int i = -1; i <= my_axis.size(); ++i) {
  // body is never executed if return value of my_axis.size() is an unsigned integral type
}
```
The advantages clearly override the disadvantages of this choice.

[endsect]

[section:real_index_type Continuous axis maps real-valued indices to values]

An axis has a method which converts an index into the equivalent value at that index. If the axis is continuous, there are many possible values in the interval between two adjacent integer indices. User often want to access the center of such an interval. The easiest and most efficient way to compute the center value is to accept real-valued indices in the conversion method. Then, the center of the first bin between index `i` and `i+1` is simply obtained by passing `i+0.5` to the method which computes the value.

This scheme is computationally efficient and intuitive. It is so useful that each continuous axis is required to accept a real-valued index. Library code relies on this to detect whether an axis is continuous or discrete. It introspects the argument type of the conversion function and checks whether it is a floating point or integral type, respectively.

[endsect]

[section:variance Variance estimates]

Once a histogram is filled, the bin counter can be accessed with the `at(...)` method. The standard counter type has a `value()` method to return the count ['k]. It also offers a `variance()` method, which returns an estimate ['v] of the [@https://en.wikipedia.org/wiki/Variance variance] 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 questionnaire [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 is a number that 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 visualization 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 questionnaire). 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 that variances can be trivially added. [@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 more efficient for the same reason. 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-parametric 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 number returned by the `variance()` method is just the same as the number return by `value()` method, why bother with adding a `variance()` method? There is a reason, which becomes apparent if the histograms are filled with weights, which is discussed next.

[endsect]

[section:weights Support of weighted fills]

A histogram sorts input values into bins and increments a bin counter if an input value falls into the range covered by that bin. The [classref boost::histogram::unlimited_storage standard storage] uses integer types to store these counts, see the [link histogram.overview.rationale.structure.storage 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 weight value ['w].
[note
There are several use-cases 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::unlimited_storage unlimited_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 `operator()(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, which is the variance estimate in this case. The former is accessed with the `value()` method of the bin counter, and the latter with the `variance()` method.
[note
Why the sum of weights squared is the 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 a fixed 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 must be designed to support Python bindings, which are developed separately. The histogram is usable 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 a histogram in Python, let it be filled on the C++ side, and then get it back for further data analysis or plotting.

[endsect]

[section Support of Boost.Accumulators]

Boost.Histogram can be configured to use arbitrary accumulators as cells, in particular the accumulators from [@boost:/libs/accumulators/index.html Boost.Accumulators]. Sample values can be passed to the cell accumulator, which it may use to compute the mean, median, variance or other statistics of the samples sorted into each cell.

[endsect]

[section Support of Boost.Range]

The histogram class is a valid range and can be used with the [@boost:/libs/range/index.html Boost.Range] library. This library provides a custom adaptor generator, `indexed`, analog to the corresponding adaptor generator in Boost.Range, but with a potentially multi-dimensional index.

[endsect]

[section Serialization]

Serialization is implemented using [@boost:/libs/serialization/index.html Boost.Serialization]. It would be great to have a portable binary archive with support for floating point data to store and retrieve histograms efficiently, which is currently not available. The library has to be open for other serialization libraries.

[endsect]

[section Comparison to Boost.Accumulators]

Boost.Histogram has a minor overlap with [@boost:/libs/accumulators/index.html Boost.Accumulators], but the scopes are rather different. The statistical accumulators `density` and `weighted_density` in Boost.Accumulators generate one-dimensional histograms. The axis range and the bin widths are determined automatically from a cached sample of initial values. They cannot be used for multi-dimensional data. Boost.Histogram focuses on multi-dimensional data and gives the user full control of how the binning should be done for each dimension.

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]

[section Why is Boost.Histogram not built on top of Boost.MultiArray?]

Boost.MultiArray implements a multi-dimensional array, it also converts an index tuple into a global index that is used to access an element in the array. Boost.Histogram and Boost.MultiArray share this functionality, but Boost.Histogram cannot use Boost.MultiArray as a back-end. Boost.MultiArray makes the rank of the array a compile-time property, while this library needs the rank to be dynamic.

Boost.MultiArray also does not allow to change the element type dynamically. This is needed to implement the adaptive storage mentioned further up. Using a variant type as the element type of a Boost.MultiArray would not work, because it creates this wasteful layout:

`[type-index 1][value 1][type-index 2][value 2]...`

A type index is stored for each cell. Moreover, the variant is always as large as the largest type in the union, so there is no way to safe memory by using a smaller type when the bin count is low, as it is done by the adaptive storage. The adaptive storage uses only one type-index for the whole array and allocates a homogeneous array of values of the same type that exactly matches their sizes, creating the following layout:

`[type-index][value 1][value 2][value 3]...`

There is only one type index and the number of allocated bytes for the array can adapted dynamically to the size of the value type.

[endsect]

[endsect]