Question about the fit and pdf methods for discrete distributions
Dear all, In the context of my research on a way to find the distribution (discrete or continuous) the most adapted to a set of data (as well as the associated parameters), I use scipy a lot. Concerning continuous distributions, the present methods are very useful for my work (especially fit, pdf and ppf), but I have more difficulties with discrete distributions. I saw that some topics have been opened concerning a fit method for discrete distributions. Is this work still going on? Indeed, I would like to be able to estimate the parameters of a sample of data with a discrete distribution (like Poisson, Bernoulli, ...) and thus be able to use a PDF and fit method. On another note, I would like to know if the PERT distribution will be integrated in scipy? Don't hesitate to come back to me for more information. Best regards, Julian CHAMBRIER
On Tue, Sep 27, 2022 at 8:24 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
On another note, I would like to know if the PERT distribution will be integrated in scipy?
The PERT distribution is just a mildly-constrained, shifted and scaled parameterization of the `beta` distribution. I don't think we'll add it as its own distribution object, but showing how you would construct the `beta` parameters from the PERT parameters would make a very good example in the `beta` docstring. -- Robert Kern
`scipy.stats.fit` fits continuous and discrete distributions to data using a global optimizer, differential evolution. It is available in the latest releases of SciPy (1.9.0/1.9.1). https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fit.html#sc... I am not aware of plans to add a PERT distribution, but the PDF/CDF seem manageable, so it looks relatively straightforward to implement. If the mailing list agrees it is a good fit for SciPy, I'd suggest talking a look at: https://github.com/scipy/scipy/pull/12694 as an example of adding a new distribution. On Tue, Sep 27, 2022, 5:23 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Dear all,
In the context of my research on a way to find the distribution (discrete or continuous) the most adapted to a set of data (as well as the associated parameters), I use scipy a lot.
Concerning continuous distributions, the present methods are very useful for my work (especially fit, pdf and ppf), but I have more difficulties with discrete distributions.
I saw that some topics have been opened concerning a fit method for discrete distributions. Is this work still going on?
Indeed, I would like to be able to estimate the parameters of a sample of data with a discrete distribution (like Poisson, Bernoulli, ...) and thus be able to use a PDF and fit method.
On another note, I would like to know if the PERT distribution will be integrated in scipy?
Don't hesitate to come back to me for more information.
Best regards,
Julian CHAMBRIER _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
Oops, I didn't see Robert's reply about PERT before sending. I agree that a docstring example rather than separate distribution is the way to go. (I knew those functions looked familiar...) On Tue, Sep 27, 2022, 7:03 AM Matt Haberland <mhaberla@calpoly.edu> wrote:
`scipy.stats.fit` fits continuous and discrete distributions to data using a global optimizer, differential evolution. It is available in the latest releases of SciPy (1.9.0/1.9.1).
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fit.html#sc...
I am not aware of plans to add a PERT distribution, but the PDF/CDF seem manageable, so it looks relatively straightforward to implement. If the mailing list agrees it is a good fit for SciPy, I'd suggest talking a look at:
https://github.com/scipy/scipy/pull/12694
as an example of adding a new distribution.
On Tue, Sep 27, 2022, 5:23 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Dear all,
In the context of my research on a way to find the distribution (discrete or continuous) the most adapted to a set of data (as well as the associated parameters), I use scipy a lot.
Concerning continuous distributions, the present methods are very useful for my work (especially fit, pdf and ppf), but I have more difficulties with discrete distributions.
I saw that some topics have been opened concerning a fit method for discrete distributions. Is this work still going on?
Indeed, I would like to be able to estimate the parameters of a sample of data with a discrete distribution (like Poisson, Bernoulli, ...) and thus be able to use a PDF and fit method.
On another note, I would like to know if the PERT distribution will be integrated in scipy?
Don't hesitate to come back to me for more information.
Best regards,
Julian CHAMBRIER _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
Hello Matt, Thank you very much for your answer, I have one last question about stats.fit: In scipy.stats.rv_continuous, it was possible to retrieve the PDF and calculate, with it, an estimation error to then compare several probability distributions between them on a dataset, (in case we are looking for the best fitting distribution for a sample), like this: ... for distribution in DISTRIBUTION: # Calculate fitted PDF and error with fit in distribution pdf = distribution.pdf(x, loc=loc, scale=scale, *arg) sse = np.sum(np.power(y - pdf, 2.0)) best_distributions.append((distribution, params, sse)) ... Is something similar possible with stats.fit ? Thank you in advance, Sincerely, Julian Le mar. 27 sept. 2022 à 16:05, Matt Haberland <mhaberla@calpoly.edu> a écrit :
`scipy.stats.fit` fits continuous and discrete distributions to data using a global optimizer, differential evolution. It is available in the latest releases of SciPy (1.9.0/1.9.1).
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fit.html#sc...
I am not aware of plans to add a PERT distribution, but the PDF/CDF seem manageable, so it looks relatively straightforward to implement. If the mailing list agrees it is a good fit for SciPy, I'd suggest talking a look at:
https://github.com/scipy/scipy/pull/12694
as an example of adding a new distribution.
On Tue, Sep 27, 2022, 5:23 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Dear all,
In the context of my research on a way to find the distribution (discrete or continuous) the most adapted to a set of data (as well as the associated parameters), I use scipy a lot.
Concerning continuous distributions, the present methods are very useful for my work (especially fit, pdf and ppf), but I have more difficulties with discrete distributions.
I saw that some topics have been opened concerning a fit method for discrete distributions. Is this work still going on?
Indeed, I would like to be able to estimate the parameters of a sample of data with a discrete distribution (like Poisson, Bernoulli, ...) and thus be able to use a PDF and fit method.
On another note, I would like to know if the PERT distribution will be integrated in scipy?
Don't hesitate to come back to me for more information.
Best regards,
Julian CHAMBRIER _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: chambrierjulian@gmail.com
Yes. Like the `fit` method of distributions, `scipy.stats.fit` returns the optimal (MLE) parameters - among other things. Please see the documentation for the attributes you need. On Thu, Sep 29, 2022, 7:08 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Hello Matt,
Thank you very much for your answer,
I have one last question about stats.fit:
In scipy.stats.rv_continuous, it was possible to retrieve the PDF and calculate, with it, an estimation error to then compare several probability distributions between them on a dataset, (in case we are looking for the best fitting distribution for a sample), like this:
... for distribution in DISTRIBUTION: # Calculate fitted PDF and error with fit in distribution pdf = distribution.pdf(x, loc=loc, scale=scale, *arg) sse = np.sum(np.power(y - pdf, 2.0))
best_distributions.append((distribution, params, sse)) ...
Is something similar possible with stats.fit ?
Thank you in advance,
Sincerely,
Julian
Le mar. 27 sept. 2022 à 16:05, Matt Haberland <mhaberla@calpoly.edu> a écrit :
`scipy.stats.fit` fits continuous and discrete distributions to data using a global optimizer, differential evolution. It is available in the latest releases of SciPy (1.9.0/1.9.1).
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fit.html#sc...
I am not aware of plans to add a PERT distribution, but the PDF/CDF seem manageable, so it looks relatively straightforward to implement. If the mailing list agrees it is a good fit for SciPy, I'd suggest talking a look at:
https://github.com/scipy/scipy/pull/12694
as an example of adding a new distribution.
On Tue, Sep 27, 2022, 5:23 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Dear all,
In the context of my research on a way to find the distribution (discrete or continuous) the most adapted to a set of data (as well as the associated parameters), I use scipy a lot.
Concerning continuous distributions, the present methods are very useful for my work (especially fit, pdf and ppf), but I have more difficulties with discrete distributions.
I saw that some topics have been opened concerning a fit method for discrete distributions. Is this work still going on?
Indeed, I would like to be able to estimate the parameters of a sample of data with a discrete distribution (like Poisson, Bernoulli, ...) and thus be able to use a PDF and fit method.
On another note, I would like to know if the PERT distribution will be integrated in scipy?
Don't hesitate to come back to me for more information.
Best regards,
Julian CHAMBRIER _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: chambrierjulian@gmail.com
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: mhaberla@calpoly.edu
Matt, Yes, stats.fit is able to estimate the best parameters for a given data set and distribution. For example, it will tell us that on the data set X, the best estimate of the parameters of the Poisson distribution is mu=Y. But for example on two given distributions (e.g. one Poisson and one Beta), stats.fit will give us the best for the Poisson distribution and the best for the Beta distribution, independently. But how do we know if the Poisson or Beta distribution is adapted to our dataset? Can we compare which distribution is the most adapted to our dataset (for example by studying a minimal loss). Thanks in advance, Le jeu. 29 sept. 2022 à 16:52, Matt Haberland <mhaberla@calpoly.edu> a écrit :
Yes. Like the `fit` method of distributions, `scipy.stats.fit` returns the optimal (MLE) parameters - among other things. Please see the documentation for the attributes you need.
On Thu, Sep 29, 2022, 7:08 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Hello Matt,
Thank you very much for your answer,
I have one last question about stats.fit:
In scipy.stats.rv_continuous, it was possible to retrieve the PDF and calculate, with it, an estimation error to then compare several probability distributions between them on a dataset, (in case we are looking for the best fitting distribution for a sample), like this:
... for distribution in DISTRIBUTION: # Calculate fitted PDF and error with fit in distribution pdf = distribution.pdf(x, loc=loc, scale=scale, *arg) sse = np.sum(np.power(y - pdf, 2.0))
best_distributions.append((distribution, params, sse)) ...
Is something similar possible with stats.fit ?
Thank you in advance,
Sincerely,
Julian
Le mar. 27 sept. 2022 à 16:05, Matt Haberland <mhaberla@calpoly.edu> a écrit :
`scipy.stats.fit` fits continuous and discrete distributions to data using a global optimizer, differential evolution. It is available in the latest releases of SciPy (1.9.0/1.9.1).
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fit.html#sc...
I am not aware of plans to add a PERT distribution, but the PDF/CDF seem manageable, so it looks relatively straightforward to implement. If the mailing list agrees it is a good fit for SciPy, I'd suggest talking a look at:
https://github.com/scipy/scipy/pull/12694
as an example of adding a new distribution.
On Tue, Sep 27, 2022, 5:23 AM Julian CHAMBRIER < chambrierjulian@gmail.com> wrote:
Dear all,
In the context of my research on a way to find the distribution (discrete or continuous) the most adapted to a set of data (as well as the associated parameters), I use scipy a lot.
Concerning continuous distributions, the present methods are very useful for my work (especially fit, pdf and ppf), but I have more difficulties with discrete distributions.
I saw that some topics have been opened concerning a fit method for discrete distributions. Is this work still going on?
Indeed, I would like to be able to estimate the parameters of a sample of data with a discrete distribution (like Poisson, Bernoulli, ...) and thus be able to use a PDF and fit method.
On another note, I would like to know if the PERT distribution will be integrated in scipy?
Don't hesitate to come back to me for more information.
Best regards,
Julian CHAMBRIER _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: chambrierjulian@gmail.com
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: mhaberla@calpoly.edu
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: chambrierjulian@gmail.com
Looking at the documentation, I would add that nllf() doesn't seem to do this job (unless I use it wrong) Le jeu. 29 sept. 2022 à 17:03, Julian CHAMBRIER <chambrierjulian@gmail.com> a écrit :
Matt,
Yes, stats.fit is able to estimate the best parameters for a given data set and distribution. For example, it will tell us that on the data set X, the best estimate of the parameters of the Poisson distribution is mu=Y.
But for example on two given distributions (e.g. one Poisson and one Beta), stats.fit will give us the best for the Poisson distribution and the best for the Beta distribution, independently. But how do we know if the Poisson or Beta distribution is adapted to our dataset?
Can we compare which distribution is the most adapted to our dataset (for example by studying a minimal loss).
Thanks in advance,
Le jeu. 29 sept. 2022 à 16:52, Matt Haberland <mhaberla@calpoly.edu> a écrit :
Yes. Like the `fit` method of distributions, `scipy.stats.fit` returns the optimal (MLE) parameters - among other things. Please see the documentation for the attributes you need.
On Thu, Sep 29, 2022, 7:08 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Hello Matt,
Thank you very much for your answer,
I have one last question about stats.fit:
In scipy.stats.rv_continuous, it was possible to retrieve the PDF and calculate, with it, an estimation error to then compare several probability distributions between them on a dataset, (in case we are looking for the best fitting distribution for a sample), like this:
... for distribution in DISTRIBUTION: # Calculate fitted PDF and error with fit in distribution pdf = distribution.pdf(x, loc=loc, scale=scale, *arg) sse = np.sum(np.power(y - pdf, 2.0))
best_distributions.append((distribution, params, sse)) ...
Is something similar possible with stats.fit ?
Thank you in advance,
Sincerely,
Julian
Le mar. 27 sept. 2022 à 16:05, Matt Haberland <mhaberla@calpoly.edu> a écrit :
`scipy.stats.fit` fits continuous and discrete distributions to data using a global optimizer, differential evolution. It is available in the latest releases of SciPy (1.9.0/1.9.1).
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fit.html#sc...
I am not aware of plans to add a PERT distribution, but the PDF/CDF seem manageable, so it looks relatively straightforward to implement. If the mailing list agrees it is a good fit for SciPy, I'd suggest talking a look at:
https://github.com/scipy/scipy/pull/12694
as an example of adding a new distribution.
On Tue, Sep 27, 2022, 5:23 AM Julian CHAMBRIER < chambrierjulian@gmail.com> wrote:
Dear all,
In the context of my research on a way to find the distribution (discrete or continuous) the most adapted to a set of data (as well as the associated parameters), I use scipy a lot.
Concerning continuous distributions, the present methods are very useful for my work (especially fit, pdf and ppf), but I have more difficulties with discrete distributions.
I saw that some topics have been opened concerning a fit method for discrete distributions. Is this work still going on?
Indeed, I would like to be able to estimate the parameters of a sample of data with a discrete distribution (like Poisson, Bernoulli, ...) and thus be able to use a PDF and fit method.
On another note, I would like to know if the PERT distribution will be integrated in scipy?
Don't hesitate to come back to me for more information.
Best regards,
Julian CHAMBRIER _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: chambrierjulian@gmail.com
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: mhaberla@calpoly.edu
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: chambrierjulian@gmail.com
On Thu, Sep 29, 2022 at 11:05 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
Matt,
Yes, stats.fit is able to estimate the best parameters for a given data set and distribution. For example, it will tell us that on the data set X, the best estimate of the parameters of the Poisson distribution is mu=Y.
But for example on two given distributions (e.g. one Poisson and one Beta), stats.fit will give us the best for the Poisson distribution and the best for the Beta distribution, independently. But how do we know if the Poisson or Beta distribution is adapted to our dataset?
Can we compare which distribution is the most adapted to our dataset (for example by studying a minimal loss).
The term you are looking for is "goodness-of-fit test". It's a broad and subtle topic, so there is not one single way to do it. Matt does have a PR open for consolidating the variety of goodness-of-fit tests that we have in scipy.stats into a single interface. The PR description points to the existing tests. https://github.com/scipy/scipy/pull/16967 -- Robert Kern
It sounds like Julian also wants to compare multiple distributions.
But for example on two given distributions (e.g. one Poisson and one Beta), stats.fit will give us the best for the Poisson distribution and the best for the Beta distribution, independently This is the same as for `rv_continuous.fit`. Once you retrieve the optimal parameters from the result object of `scipy.stats.fit` - which is all that `rv_continuous.fit` returns - you can freeze a distribution object and calculate whatever distance metrics on the PDF that you like, as you showed in the example code. Neither automatically tries different distributions for you. You will still need to do it manually as you showed.
Looking at the documentation, I would add that nllf() doesn't seem to do this job (unless I use it wrong) It does, but you may need to remember to include the parameters of the distribution in a single tuple. `nllf((loc, scale), data)`
On Thu, Sep 29, 2022, 8:16 AM Robert Kern <robert.kern@gmail.com> wrote:
On Thu, Sep 29, 2022 at 11:05 AM Julian CHAMBRIER < chambrierjulian@gmail.com> wrote:
Matt,
Yes, stats.fit is able to estimate the best parameters for a given data set and distribution. For example, it will tell us that on the data set X, the best estimate of the parameters of the Poisson distribution is mu=Y.
But for example on two given distributions (e.g. one Poisson and one Beta), stats.fit will give us the best for the Poisson distribution and the best for the Beta distribution, independently. But how do we know if the Poisson or Beta distribution is adapted to our dataset?
Can we compare which distribution is the most adapted to our dataset (for example by studying a minimal loss).
The term you are looking for is "goodness-of-fit test". It's a broad and subtle topic, so there is not one single way to do it. Matt does have a PR open for consolidating the variety of goodness-of-fit tests that we have in scipy.stats into a single interface. The PR description points to the existing tests.
https://github.com/scipy/scipy/pull/16967
-- Robert Kern _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
This is the same as for `rv_continuous.fit`. Once you retrieve the optimal parameters from the result object of `scipy.stats.fit` - which is all that `rv_continuous.fit` returns - you can freeze a distribution object and calculate whatever distance metrics on the PDF that you like, as you showed in the example code. Neither automatically tries different distributions for you. You will still need to do it manually as you showed.
yes I agree for continuous distributions but discrete distributions don't have PDF method in scipy. Le jeu. 29 sept. 2022 à 17:32, Matt Haberland <mhaberla@calpoly.edu> a écrit :
It sounds like Julian also wants to compare multiple distributions.
But for example on two given distributions (e.g. one Poisson and one Beta), stats.fit will give us the best for the Poisson distribution and the best for the Beta distribution, independently This is the same as for `rv_continuous.fit`. Once you retrieve the optimal parameters from the result object of `scipy.stats.fit` - which is all that `rv_continuous.fit` returns - you can freeze a distribution object and calculate whatever distance metrics on the PDF that you like, as you showed in the example code. Neither automatically tries different distributions for you. You will still need to do it manually as you showed.
Looking at the documentation, I would add that nllf() doesn't seem to do this job (unless I use it wrong) It does, but you may need to remember to include the parameters of the distribution in a single tuple. `nllf((loc, scale), data)`
On Thu, Sep 29, 2022, 8:16 AM Robert Kern <robert.kern@gmail.com> wrote:
On Thu, Sep 29, 2022 at 11:05 AM Julian CHAMBRIER < chambrierjulian@gmail.com> wrote:
Matt,
Yes, stats.fit is able to estimate the best parameters for a given data set and distribution. For example, it will tell us that on the data set X, the best estimate of the parameters of the Poisson distribution is mu=Y.
But for example on two given distributions (e.g. one Poisson and one Beta), stats.fit will give us the best for the Poisson distribution and the best for the Beta distribution, independently. But how do we know if the Poisson or Beta distribution is adapted to our dataset?
Can we compare which distribution is the most adapted to our dataset (for example by studying a minimal loss).
The term you are looking for is "goodness-of-fit test". It's a broad and subtle topic, so there is not one single way to do it. Matt does have a PR open for consolidating the variety of goodness-of-fit tests that we have in scipy.stats into a single interface. The PR description points to the existing tests.
https://github.com/scipy/scipy/pull/16967
-- Robert Kern _______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: haberland@ucla.edu
_______________________________________________ SciPy-Dev mailing list -- scipy-dev@python.org To unsubscribe send an email to scipy-dev-leave@python.org https://mail.python.org/mailman3/lists/scipy-dev.python.org/ Member address: chambrierjulian@gmail.com
On Thu, Sep 29, 2022 at 11:47 AM Julian CHAMBRIER <chambrierjulian@gmail.com> wrote:
This is the same as for `rv_continuous.fit`. Once you retrieve the optimal parameters from the result object of `scipy.stats.fit` - which is all that `rv_continuous.fit` returns - you can freeze a distribution object and calculate whatever distance metrics on the PDF that you like, as you showed in the example code. Neither automatically tries different distributions for you. You will still need to do it manually as you showed.
yes I agree for continuous distributions but discrete distributions don't have PDF method in scipy.
The equivalent is `.pmf()` (because it is a probability mass function rather than a probability density function in the discrete case). -- Robert Kern
participants (3)
-
Julian CHAMBRIER -
Matt Haberland -
Robert Kern