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License: GPL3
ubuntu2004

Kernel: Python [conda env:blog.soerp]

In this post we discuss modeling covariances beyond the Gaussian approximation.

In [9]:

import mmf_setup;mmf_setup.nbinit(hgroot=False)

Out[9]:

<IPython.core.display.Javascript object>

In [2]:

%pylab inline --no-import-all
from scipy import stats

x = np.linspace
f = stats.lognorm.pdf()

Out[2]:

Populating the interactive namespace from numpy and matplotlib

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-2-3b06b66c0832> in <module>
      2 from scipy import stats
      3 
----> 4 f = stats.lognorm()

/data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages/scipy/stats/_distn_infrastructure.py in __call__(self, *args, **kwds)
    770 
    771     def __call__(self, *args, **kwds):
--> 772         return self.freeze(*args, **kwds)
    773     __call__.__doc__ = freeze.__doc__
    774 
/data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages/scipy/stats/_distn_infrastructure.py in freeze(self, *args, **kwds)
    767 
    768         """
--> 769         return rv_frozen(self, *args, **kwds)
    770 
    771     def __call__(self, *args, **kwds):
/data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages/scipy/stats/_distn_infrastructure.py in __init__(self, dist, *args, **kwds)
    435 
    436         # a, b may be set in _argcheck, depending on *args, **kwds. Ouch.
--> 437         shapes, _, _ = self.dist._parse_args(*args, **kwds)
    438         self.dist._argcheck(*shapes)
    439         self.a, self.b = self.dist.a, self.dist.b
TypeError: _parse_args() missing 1 required positional argument: 's'

Characterization

Consider a random variable $X$ with probability distribution $f(x)$ .

Addition of Stochastic Variables

Let $X_n$ be $N$ independent stochastic variables with joint probability density ParseError: KaTeX parse error: Undefined control sequence: \vect at position 31: …dots, x_N) = P(\̲v̲e̲c̲t̲{x}). Then, the probability distribution for the sum $Y = \sum_n X_n$ is:

ParseError: KaTeX parse error: Undefined control sequence: \idotsint at position 10: P_Y(y) = \̲i̲d̲o̲t̲s̲i̲n̲t̲ ̲\delta\Bigl(\su…

If the variables $X_n$ are independent, then ParseError: KaTeX parse error: Undefined control sequence: \vect at position 5: P_X(\̲v̲e̲c̲t̲{x}) = \prod_{n…, and this simplifies:

ParseError: KaTeX parse error: Undefined control sequence: \idotsint at position 10: P_Y(y) = \̲i̲d̲o̲t̲s̲i̲n̲t̲ ̲\delta\left(\su…

Environment

Execute the following two cells to define a conda environment for running this notebook.

In [7]:

%%file environment.soerp.yml
name: blog.soerp
channels:
  - defaults
  - conda-forge
dependencies:
  - ipykernel
  - notebook
  - uncertainties
  - scipy
  - matplotlib
  - pip:
    - soerp
    - mmf_setup

Out[7]:

Overwriting environment.soerp.yml

In [8]:

!conda env update -f environment.soerp.yml

Out[8]:

Solving environment: done
Preparing transaction: done
Verifying transaction: done
Executing transaction: done
Looking in links: https://bitbucket.org/mforbes/mypi/
Requirement already satisfied: soerp in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from -r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (0.9.6)
Requirement already satisfied: mmf_setup in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from -r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (0.1.11)
Requirement already satisfied: matplotlib in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (3.0.0)
Requirement already satisfied: numpy in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.15.2)
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Requirement already satisfied: scipy in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.1.0)
Requirement already satisfied: python-hglib in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (2.6.1)
Requirement already satisfied: nbstripout>=0.2.0 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (0.3.3)
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Requirement already satisfied: kiwisolver>=1.0.1 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.0.1)
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Requirement already satisfied: nbformat in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (4.4.0)
Requirement already satisfied: six in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from cycler>=0.10->matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.11.0)
Requirement already satisfied: setuptools in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from kiwisolver>=1.0.1->matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (40.4.3)
Requirement already satisfied: ipython-genutils in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (0.2.0)
Requirement already satisfied: jsonschema!=2.5.0,>=2.4 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (2.6.0)
Requirement already satisfied: traitlets>=4.1 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (4.3.2)
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Requirement already satisfied: decorator in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from traitlets>=4.1->nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (4.3.0)
mkl-random 1.0.1 requires cython, which is not installed.
#
# To activate this environment, use
#
#     $ conda activate blog.soerp
#
# To deactivate an active environment, use
#
#     $ conda deactivate

In [ ]:

Characterization

Addition of Stochastic Variables

Environment

Product

Resources

Company