<!--overview-start-->
# Lmo - Trimmed L-moments and L-comoments
![Lmo - License][IMG-BSD]
[![Lmo - PyPI][IMG-PYPI]](https://pypi.org/project/Lmo/)
[![Lmo - Versions][IMG-VER]](https://github.com/jorenham/Lmo)
![Lmo - CI][IMG-CI]
[![Lmo - Pre-commit][IMG-PC]](https://github.com/pre-commit/pre-commit)
[![Lmo - Ruff][IMG-RUFF]](https://github.com/astral-sh/ruff)
[![Lmo - BassedPyright][IMG-BPR]](https://detachhead.github.io/basedpyright)
[IMG-CI]: https://img.shields.io/github/actions/workflow/status/jorenham/Lmo/ci.yml?branch=master
[IMG-BSD]: https://img.shields.io/github/license/jorenham/Lmo
[IMG-PYPI]: https://img.shields.io/pypi/v/Lmo
[IMG-VER]: https://img.shields.io/pypi/pyversions/Lmo
[IMG-pC]: https://img.shields.io/badge/pre--commit-enabled-orange?logo=pre-commit
[IMG-RUFF]: https://img.shields.io/endpoint?url=https://raw.githubusercontent.com/astral-sh/ruff/main/assets/badge/v2.json
[IMG-BPR]: https://img.shields.io/badge/basedpyright-checked-42b983
Unlike the legacy
[product-moments](https://wikipedia.org/wiki/Moment_(mathematics)), the
[*L-moments*](https://wikipedia.org/wiki/L-moment) **uniquely describe** a
probability distribution, and are more robust and efficient.
The "L" stands for Linear; it is a linear combination of order statistics.
So Lmo is as fast as sorting your samples (in terms of time-complexity).
## Key Features
- Calculates trimmed L-moments and L-*co*moments, from samples or any
`scipy.stats` distribution.
- Full support for trimmed L-moment (TL-moments), e.g.
`lmo.l_moment(..., trim=(1/137, 3.1416))`.
- Generalized Method of L-moments: robust distribution fitting that beats MLE.
- Fast estimation of L-*co*moment matrices from your multidimensional data
or multivariate distribution.
- Goodness-of-fit test, using L-moment or L-moment ratio's.
- Exact (co)variance structure of the sample- and population L-moments.
- Theoretical & empirical influence functions of L-moments & L-ratio's.
- Complete [docs](https://jorenham.github.io/lmo/), including detailed API
reference with usage examples and with mathematical $\TeX$ definitions.
- Clean Pythonic syntax for ease of use.
- Vectorized functions for very fast fitting.
- Fully typed, tested, and tickled.
- Optional Pandas integration.
## Quick example
Even if your data is pathological like
[Cauchy](https://wikipedia.org/wiki/Cauchy_distribution), and the L-moments
are not defined, the trimmed L-moments (TL-moments) can be used instead.
Let's calculate the TL-location and TL-scale of a small amount of samples:
```pycon
>>> import numpy as np
>>> import lmo
>>> rng = np.random.default_rng(1980)
>>> x = rng.standard_cauchy(96) # pickle me, Lmo
>>> lmo.l_moment(x, [1, 2], trim=(1, 1)).
array([-0.17937038, 0.68287665])
```
Now compare with the theoretical standard Cauchy TL-moments:
```pycon
>>> from scipy.stats import cauchy
>>> cauchy.l_moment([1, 2], trim=(1, 1))
array([0. , 0.69782723])
```
---
See the [documentation](https://jorenham.github.io/lmo/) for more examples and
the API reference.
## Roadmap
- Automatic trim-length selection.
- Plotting utilities (deps optional), e.g. for L-moment ratio diagrams.
## Installation
Lmo is on [PyPI](https://pypi.org/project/lmo/), so you can do something like:
```shell
pip install lmo
```
### Dependencies
These are automatically installed by your package manager when installing Lmo.
| Package | Supported versions |
| ------------ | ------------------ |
| [Python][PY] | `>=3.10` |
| [NumPy][NP] | `>=1.23` |
| [SciPy][SP] | `>=1.9` |
Additionally, Lmo supports the following optional packages:
| Package | Supported versions | Installation |
| ------------ | ------------------ | ------------------------- |
| [Pandas][PD] | `>=1.5` | `pip install Lmo[pandas]` |
See [SPEC 0][SPEC0] for more information.
[PY]: https://github.com/python/cpython
[NP]: https://github.com/numpy/numpy
[SP]: https://github.com/scipy/scipy
[PD]: https://github.com/pandas-dev/pandas
[SPEC0]: https://scientific-python.org/specs/spec-0000/
## Foundational Literature
- [*J.R.M. Hosking* (1990) – L-moments: Analysis and Estimation of
Distributions using Linear Combinations of Order Statistics
](https://doi.org/10.1111/j.2517-6161.1990.tb01775.x)
- [*E.A.H. Elamir & A.H. Seheult* (2003) – Trimmed L-moments
](https://doi.org/10.1016/S0167-9473(02)00250-5)
- [*J.R.M. Hosking* (2007) – Some theory and practical uses of trimmed
L-moments](https://doi.org/10.1016/j.jspi.2006.12.002)
- [*R. Serfling & P. Xiao* (2007) – A contribution to multivariate
L-moments: L-comoment matrices](https://doi.org/10.1016/j.jmva.2007.01.008)
<!--overview-end-->
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"description": "<!--overview-start-->\n\n# Lmo - Trimmed L-moments and L-comoments\n\n![Lmo - License][IMG-BSD]\n[![Lmo - PyPI][IMG-PYPI]](https://pypi.org/project/Lmo/)\n[![Lmo - Versions][IMG-VER]](https://github.com/jorenham/Lmo)\n![Lmo - CI][IMG-CI]\n[![Lmo - Pre-commit][IMG-PC]](https://github.com/pre-commit/pre-commit)\n[![Lmo - Ruff][IMG-RUFF]](https://github.com/astral-sh/ruff)\n[![Lmo - BassedPyright][IMG-BPR]](https://detachhead.github.io/basedpyright)\n\n[IMG-CI]: https://img.shields.io/github/actions/workflow/status/jorenham/Lmo/ci.yml?branch=master\n[IMG-BSD]: https://img.shields.io/github/license/jorenham/Lmo\n[IMG-PYPI]: https://img.shields.io/pypi/v/Lmo\n[IMG-VER]: https://img.shields.io/pypi/pyversions/Lmo\n[IMG-pC]: https://img.shields.io/badge/pre--commit-enabled-orange?logo=pre-commit\n[IMG-RUFF]: https://img.shields.io/endpoint?url=https://raw.githubusercontent.com/astral-sh/ruff/main/assets/badge/v2.json\n[IMG-BPR]: https://img.shields.io/badge/basedpyright-checked-42b983\n\nUnlike the legacy\n[product-moments](https://wikipedia.org/wiki/Moment_(mathematics)), the\n[*L-moments*](https://wikipedia.org/wiki/L-moment) **uniquely describe** a\nprobability distribution, and are more robust and efficient.\n\nThe \"L\" stands for Linear; it is a linear combination of order statistics.\nSo Lmo is as fast as sorting your samples (in terms of time-complexity).\n\n## Key Features\n\n- Calculates trimmed L-moments and L-*co*moments, from samples or any\n `scipy.stats` distribution.\n- Full support for trimmed L-moment (TL-moments), e.g.\n `lmo.l_moment(..., trim=(1/137, 3.1416))`.\n- Generalized Method of L-moments: robust distribution fitting that beats MLE.\n- Fast estimation of L-*co*moment matrices from your multidimensional data\n or multivariate distribution.\n- Goodness-of-fit test, using L-moment or L-moment ratio's.\n- Exact (co)variance structure of the sample- and population L-moments.\n- Theoretical & empirical influence functions of L-moments & L-ratio's.\n- Complete [docs](https://jorenham.github.io/lmo/), including detailed API\nreference with usage examples and with mathematical $\\TeX$ definitions.\n- Clean Pythonic syntax for ease of use.\n- Vectorized functions for very fast fitting.\n- Fully typed, tested, and tickled.\n- Optional Pandas integration.\n\n## Quick example\n\nEven if your data is pathological like\n[Cauchy](https://wikipedia.org/wiki/Cauchy_distribution), and the L-moments\nare not defined, the trimmed L-moments (TL-moments) can be used instead.\nLet's calculate the TL-location and TL-scale of a small amount of samples:\n\n```pycon\n>>> import numpy as np\n>>> import lmo\n>>> rng = np.random.default_rng(1980)\n>>> x = rng.standard_cauchy(96) # pickle me, Lmo\n>>> lmo.l_moment(x, [1, 2], trim=(1, 1)).\narray([-0.17937038, 0.68287665])\n```\n\nNow compare with the theoretical standard Cauchy TL-moments:\n\n```pycon\n>>> from scipy.stats import cauchy\n>>> cauchy.l_moment([1, 2], trim=(1, 1))\narray([0. , 0.69782723])\n```\n\n---\n\nSee the [documentation](https://jorenham.github.io/lmo/) for more examples and\nthe API reference.\n\n## Roadmap\n\n- Automatic trim-length selection.\n- Plotting utilities (deps optional), e.g. for L-moment ratio diagrams.\n\n## Installation\n\nLmo is on [PyPI](https://pypi.org/project/lmo/), so you can do something like:\n\n```shell\npip install lmo\n```\n\n### Dependencies\n\nThese are automatically installed by your package manager when installing Lmo.\n\n| Package | Supported versions |\n| ------------ | ------------------ |\n| [Python][PY] | `>=3.10` |\n| [NumPy][NP] | `>=1.23` |\n| [SciPy][SP] | `>=1.9` |\n\nAdditionally, Lmo supports the following optional packages:\n\n| Package | Supported versions | Installation |\n| ------------ | ------------------ | ------------------------- |\n| [Pandas][PD] | `>=1.5` | `pip install Lmo[pandas]` |\n\nSee [SPEC 0][SPEC0] for more information.\n\n[PY]: https://github.com/python/cpython\n[NP]: https://github.com/numpy/numpy\n[SP]: https://github.com/scipy/scipy\n[PD]: https://github.com/pandas-dev/pandas\n[SPEC0]: https://scientific-python.org/specs/spec-0000/\n\n## Foundational Literature\n\n- [*J.R.M. Hosking* (1990) – L-moments: Analysis and Estimation of\n Distributions using Linear Combinations of Order Statistics\n ](https://doi.org/10.1111/j.2517-6161.1990.tb01775.x)\n- [*E.A.H. Elamir & A.H. Seheult* (2003) – Trimmed L-moments\n ](https://doi.org/10.1016/S0167-9473(02)00250-5)\n- [*J.R.M. Hosking* (2007) – Some theory and practical uses of trimmed\n L-moments](https://doi.org/10.1016/j.jspi.2006.12.002)\n- [*R. Ser\ufb02ing & P. Xiao* (2007) – A contribution to multivariate\n L-moments: L-comoment matrices](https://doi.org/10.1016/j.jmva.2007.01.008)\n\n<!--overview-end-->\n",
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