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Multivariate regression analysis of core-collapse simulations

Project Description
ccsnmultivar companion code

This `Python <>`_ module aids the analysis of
core-collapse supernova gravitational waves. It is the companion code
for `this paper <>`_.

- **Multivariate Regression** of Fourier Transformed or Time Domain
- **Hypothesis testing** for measuring the influence of physical
- Optionally incorporate additional uncertainty due to detector noise
- Approximate waveforms from anywhere within the parameter space
- Includes the `Abdikamalov et. al. <>`_
catalog for example use


- A simplified formula language (like in R, or patsy) specific to this
- `Documentation <>`_


Make sure that the python packages numpy, scipy, pandas, and patsy are
already installed. pip installer will install patsy, pandas and tabular
if they aren't installed already.


cd /path/to/ccsnmultivar

1. Download github zip file here
2. Unzip

\`\`\`python # cd /CCSNMultivar-master

python install \`\`\` or


pip install ccsnmultivar

Its a good idea to update often because the package is being changed
often. To update, type


pip install -U ccsnmultivar

Basic Walkthrough

Using the code happens in five steps:

1. Instantiate a Catalog object
2. Instantiate a Basis object.
3. Instantiate a DesignMatrix object.
4. Wrapping them in a Multivar object.
5. Analysis using the Multivar object's methods.


# import code
import ccsnmultivar as cc

# load waveforms
path_to_waveforms = "/path/to/Abdika13_waveforms.csv"
# the Abdikamalov waveform file is called "Abdika13_waveforms.csv"

# we want to analyze the waveforms in the time domain, so instantiate
# a Catalog object with the transform_type arguement specified
Y = cc.Catalog(path_to_waveforms,transform_type='time')

Note that Abdikamalov et al's 2013 waveform catalog and parameter file
are included in the Example\_Waveforms directory of the GitHub repo as
an example of how to format the raw files for input. To access these for
the walkthrough, look at the right side of the GitHub page, there is a
toolbar with a Download button. Download, then unzip. The directory
Example\_Waveforms isn't included when the package is installed using

Now we need to make two objects, a Basis object and a DesignMatrix

First we instantiate a Basis object. Currently, there are two available
types of Basis objects, with more planned.

1. PCA - using the Singular Value Decompostion (SVD)
2. ICA - Independent Component Ananlysis. A wrapper for skearns
`FastICA <>`_

``python # use a PCA basis keeping the first 10 Principal Components pca = cc.PCA(num_components=10)``
Next we instantiate a DesignMatrix object.


# first, define a formula string describing how the physical parameters
# need to be translated to the design matrix. Say we only want to use
# encodings of the parameters A and B (A is discrete, B is continuous)

formula = "A + beta + A*beta | Dum(A,ref=2), Poly(beta,degree=4)"

The formula contains 5 peices of information that determine how the
design matrix is encoded. Reading the formula from left to right:

1. Include columns for the physical parameter named "A".
2. Include columns for the physical parameter named "beta".
3. Include columns for interaction terms between parameters "A" and
The "\|" character seperates instructions for *what* goes into the
design matrix from *how* it goes in.
4. Use a dummy variable encoding on parameter "A". One value of "A"
needs to be used as a reference in a dummy variable encoding, we
chose value "2".
5. Use a Chebyshev polynomial encoding on parameter "beta". Fit "beta"
with a 4th degree polynomial.

Now we instantiate the DesignMatrix object with two arguments: the
formula, and the path to the parameter file. \`\`\`python

note that the provided Abdikamalov+ parameterfile is called "Abdika13\_params.csv"

path\_to\_parameterfile = "/path/to/Abdika13\_params.csv"

note that we dont need to load the paramfile, just supply the path.

X = cc.DesignMatrix(path\_to\_parameterfile, formula) \`\`\`

Now with the waveforms in the Catalog object Y, the Basis object pca,
and DesignMatrix object X on hand, we instantiate a Multivar object with
these three arguements.


# instantiate Multivar object
M = cc.Multivar(Y,X, pca)

This makes it easy to create many different Catalog, DesignMatrix,
Basis, and Multivar objects to test different fits and parameter
influences very quickly.

\`\`\`python # now we do a fit to time domain waveforms (solve for B)

print summary of the hypothesis tests, metadata, and other

facts defined by the particular formula and basis used to make M.


Waveform Domain time Number of Waveforms 92 Catalog Mean Subtracted?
False Catalog Name Abdika13\_waveforms.csv Normalization Factor
2.45651978042e+20 Decomposition PCA num\_components 10 ================
================ =========== Comparison Hotellings T^2 p-value
================ ================ =========== Intercept 1129.44
1.11022e-16 A:[1 - 2] 87.9454 1.11022e-16 A:[3 - 2] 8.06119 5.49626e-08
A:[4 - 2] 1.8598 0.0700502 A:[5 - 2] 0.823121 0.607991 beta^1 257.711
1.11022e-16 beta^2 383.961 1.11022e-16 beta^3 93.1575 1.11022e-16 beta^4
18.3438 1.55431e-14 A:[1 - 2]*beta^1 77.7596 1.11022e-16 A:[1 -
2]*beta^2 14.0067 3.68272e-12 . . . . . . . . .

we can view the waveform reconstructions with the Multivar method .reconstruct()

Y\_reconstructed = M.reconstruct()

and pull out the original catalog waveforms for comparison

Y\_original = M.get\_waveforms()

plot the last waveform in the array with its reconstruction (requires matplotlib)

import matplotlib.pyplot as plt
plt.legend() \`\`\` Using the Abdikamalov catalog, this is what you
should see:

|alt tag| \`\`\`python # look at a summary of the overlaps between the
waveforms and their reconstructions M.overlap\_summary()

============ ============== Percentile Overlap ============
============== 5%: 0.64866522524 25%: 0.809185728124 50%: 0.879262580569
75%: 0.949587383571 95%: 0.97311500202

Min: 0.518678320514 Mean: 0.858585006085 Max: 0.98214781409 ============

\`\`\` One of the main goals of this method is to predict new waveforms,
given a set of physical parameters that wasn't originally used in the
catalog. For instance:


# make a dictionary of the new parameters
new_parameters = {}
# quickly generate two waveforms, one with A = 1, beta = .1, another with
# A = 3, beta = 0.05 (using the abdikamalov example)
new_parameters['A'] = [str(1), str(3)]
new_parameters['beta'] = [.1, .05]

# use the predict method of the multivar object
Y_new = M.predict(new_parameters)

# plot the two waveform predictions (requires matplotlib)
import matplotlib.pyplot as plt
plt.plot(Y_new[0,8000:9000],label='A = 1, beta = .1')
plt.plot(Y_new[1,8000:9000],label='A = 3, beta = .05')

With the Abdikamalov catalog, this is what you should see:

.. figure:: Example_Catalogs/example_prediction.png
:align: center
:alt: alt tag

alt tag
This allows one to rapidly interpolate the parameter space for
core-collapse waveforms


- numpy
- scipy
- scikits-learn
- tabulate


- Hotellings T2 with more than one GW detector
- Catalog objects
- amplitude/phase decomposition, spectrograms
- other PC basis methods
- sparse basis decompositions, kmeans, etc.
- other design matrix fitting methods
- splines, rbfs, etc.
- different types of crossvalidation methods
- Gaussian Process (or other interpolation/machine learning method)

.. |alt tag| image:: Example_Catalogs/example_reconstruction.png
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