# Accumulators

## Contents

# Accumulators¶

## Common properties¶

All accumulators can be filled like a histogram. You just call `.fill`

with
values, and this looks and behaves like filling a single-bin or “scalar”
histogram. Like histograms, the fill is inplace.

All accumulators have a `.value`

property as well, which gives the primary
value being accumulated.

## Types¶

There are several accumulators.

### Sum¶

This is the simplest accumulator, and is never returned from a histogram. This is internally used by the Double and Unlimited storages to perform sums when needed. It uses a highly accurate Neumaier sum to compute the floating point sum with a correction term. Since this accumulator is never returned by a histogram, it is not available in a view form, but only as a single accumulator for comparison and access to the algorithm. Usage example in Python 3.8, showing how non-accurate sums fail to produce the obvious answer, 2.0:

```
import math
import numpy as np
import hist
values = [1.0, 1e100, 1.0, -1e100]
print(f"{sum(values) = } (simple)")
print(f"{math.fsum(values) = }")
print(f"{np.sum(values) = } (pairwise)")
print(f"{hist.accumulators.Sum().fill(values) = }")
```

```
sum(values) = 0.0 (simple)
math.fsum(values) = 2.0
np.sum(values) = 0.0 (pairwise)
hist.accumulators.Sum().fill(values) = Sum(0 + 2)
```

Note that this is still intended for performance and does not guarantee
correctness as `math.fsum`

does. In general, you must not have more than two
orders of values:

```
values = [1., 1e100, 1e50, 1., -1e50, -1e100]
print(f"{math.fsum(values) = }")
print(f"{hist.accumulators.Sum().fill(values) = }")
```

```
math.fsum(values) = 2.0
hist.accumulators.Sum().fill(values) = Sum(0 + 0)
```

You should note that this is a highly contrived example and the Sum accumulator should still outperform simple and pairwise summation methods for a minimal performance cost. Most notably, you have to have large cancellations with negative values, which histograms generally do not have.

You can use `+=`

with a float value or a Sum to fill as well.

### WeightedSum¶

This accumulator is contained in the Weight storage, and supports Views. It
provides two values; `.value`

, and `.variance`

. The value is the sum of the
weights, and the variance is the sum of the squared weights.

For example, you could sum the following values:

```
import hist
values = [10]*10
smooth = hist.accumulators.WeightedSum().fill(values)
print(f"{smooth = }")
values = [1]*9 + [91]
rough = hist.accumulators.WeightedSum().fill(values)
print(f"{rough = }")
```

```
smooth = WeightedSum(value=100, variance=1000)
rough = WeightedSum(value=100, variance=8290)
```

When filling, you can optionally provide a `variance=`

keyword, with either a
single value or a matching length array of values.

You can also fill with `+=`

on a value or another WeighedSum.

### Mean¶

This accumulator is contained in the Mean storage, and supports Views. It
provides three values; `.count`

, `.value`

, and `.variance`

. Internally,
the variance is stored as `_sum_of_deltas_squared`

, which is used to compute
`variance`

.

For example, you could compute the mean of the following values:

```
import hist
values = [10]*10
smooth = hist.accumulators.Mean().fill(values)
print(f"{smooth = }")
values = [1]*9 + [91]
rough = hist.accumulators.Mean().fill(values)
print(f"{rough = }")
```

```
smooth = Mean(count=10, value=10, variance=0)
rough = Mean(count=10, value=10, variance=810)
```

You can add a `weight=`

keyword when filling, with either a single value
or a matching length array of values.

You can call a Mean with a value or with another Mean to fill inplace, as well.

### WeightedMean¶

This accumulator is contained in the WeightedMean storage, and supports Views.
It provides four values; `.sum_of_weights`

, `sum_of_weights_squared`

,
`.value`

, and `.variance`

. Internally, the variance is stored as
`_sum_of_weighted_deltas_squared`

, which is used to compute `variance`

.

For example, you could compute the mean of the following values:

```
import hist
values = [1]*9 + [91]
wm = hist.accumulators.WeightedMean().fill(values, weight=2)
print(f"{wm = }")
```

```
wm = WeightedMean(sum_of_weights=20, sum_of_weights_squared=40, value=10, variance=810)
```

You can add a `weight=`

keyword when filling, with either a single value or a
matching length array of values.

You can call a WeightedMean with a value or with another WeightedMean to fill inplace, as well.

## Views¶

Most of the accumulators (except Sum) support a View. This is what is returned from
a histogram when `.view()`

is requested. This is a structured Numpy ndarray, with a few small
additions to make them easier to work with. Like a Numpy recarray, you can access the fields with
attributes; you can even access (but not set) computed attributes like `.variance`

. A view will
also return an accumulator instance if you select a single item.