Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On 2 April 2014 16:06, Sturla Molden sturla.mol...@gmail.com wrote: josef.p...@gmail.com wrote: pandas came later and thought ddof=1 is worth more than consistency. Pandas is a data analysis package. NumPy is a numerical array package. I think ddof=1 is justified for Pandas, for consistency with statistical software (SPSS et al.) For NumPy, there are many computational tasks where the Bessel correction is not wanted, so providing a uncorrected result is the correct thing to do. NumPy should be a low-level array library that does very little magic. All this discussion reminds me of the book Numerical Recipes: if the difference between N and N - 1 ever matters to you, then you are probably up to no good anyway -- e.g., trying to substantiate a questionable hypothesis with marginal data. For any reasonably sized data set, it is a correction in the second significant figure. ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Fri, Apr 4, 2014 at 8:50 AM, Daπid davidmen...@gmail.com wrote: On 2 April 2014 16:06, Sturla Molden sturla.mol...@gmail.com wrote: josef.p...@gmail.com wrote: pandas came later and thought ddof=1 is worth more than consistency. Pandas is a data analysis package. NumPy is a numerical array package. I think ddof=1 is justified for Pandas, for consistency with statistical software (SPSS et al.) For NumPy, there are many computational tasks where the Bessel correction is not wanted, so providing a uncorrected result is the correct thing to do. NumPy should be a low-level array library that does very little magic. All this discussion reminds me of the book Numerical Recipes: if the difference between N and N − 1 ever matters to you, then you are probably up to no good anyway — e.g., trying to substantiate a questionable hypothesis with marginal data. For any reasonably sized data set, it is a correction in the second significant figure. I fully agree, but sometimes you don't have much choice. `big data` == `statistics with negative degrees of freedom` ? or maybe `machine learning` == `statistics with negative degrees of freedom` ? Josef ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
alex argri...@ncsu.edu wrote: I don't have any opinion about this debate, but I love the justification in that thread Any surprise that is created by the different default should be mitigated by the fact that it's an opportunity to learn something about what you are doing. That is so true. Sturla ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
josef.p...@gmail.com wrote: pandas came later and thought ddof=1 is worth more than consistency. Pandas is a data analysis package. NumPy is a numerical array package. I think ddof=1 is justified for Pandas, for consistency with statistical software (SPSS et al.) For NumPy, there are many computational tasks where the Bessel correction is not wanted, so providing a uncorrected result is the correct thing to do. NumPy should be a low-level array library that does very little magic. Those who need the Bessel correction can multiply with sqrt(n/float(n-1)) or specify ddof. Bu that belongs in the docs. Sturla P.S. Personally I am not convinced unbiased is ever a valid argument, as the biased estimator has smaller error. This is from experience in marksmanship: I'd rather shoot a tight series with small systematic error than scatter my bullets wildly but unbiased on the target. It is the total error that counts. The series with smallest total error gets the best score. It is better to shoot two series and calibrate the sight in between than use a calibration-free sight that don't allow us to aim. That's why I think classical statistics got this one wrong. Unbiased is never a virtue, but the smallest error is. Thus, if we are to repeat an experiment, we should calibrate our estimator just like a marksman calibrates his sight. But the aim should always be calibrated to give the smallest error, not an unbiased scatter. Noone in their right mind would claim a shotgun is more precise than a rifle because it has smaller bias. But that is what applying the Bessel correction implies. ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Wed, Apr 2, 2014 at 10:06 AM, Sturla Molden sturla.mol...@gmail.com wrote: josef.p...@gmail.com wrote: pandas came later and thought ddof=1 is worth more than consistency. Pandas is a data analysis package. NumPy is a numerical array package. I think ddof=1 is justified for Pandas, for consistency with statistical software (SPSS et al.) For NumPy, there are many computational tasks where the Bessel correction is not wanted, so providing a uncorrected result is the correct thing to do. NumPy should be a low-level array library that does very little magic. Those who need the Bessel correction can multiply with sqrt(n/float(n-1)) or specify ddof. Bu that belongs in the docs. Sturla P.S. Personally I am not convinced unbiased is ever a valid argument, as the biased estimator has smaller error. This is from experience in marksmanship: I'd rather shoot a tight series with small systematic error than scatter my bullets wildly but unbiased on the target. It is the total error that counts. The series with smallest total error gets the best score. It is better to shoot two series and calibrate the sight in between than use a calibration-free sight that don't allow us to aim. calibration == bias correction ? That's why I think classical statistics got this one wrong. Unbiased is never a virtue, but the smallest error is. Thus, if we are to repeat an experiment, we should calibrate our estimator just like a marksman calibrates his sight. But the aim should always be calibrated to give the smallest error, not an unbiased scatter. Noone in their right mind would claim a shotgun is more precise than a rifle because it has smaller bias. But that is what applying the Bessel correction implies. https://www.youtube.com/watch?v=i4xcEZZDW_I I spent several days trying to figure out what Stata is doing for small sample corrections to reduce the bias of the rejection interval with uncorrected variance estimates. Josef ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
Sturla P.S. Personally I am not convinced unbiased is ever a valid argument, as the biased estimator has smaller error. This is from experience in marksmanship: I'd rather shoot a tight series with small systematic error than scatter my bullets wildly but unbiased on the target. It is the total error that counts. The series with smallest total error gets the best score. It is better to shoot two series and calibrate the sight in between than use a calibration-free sight that don't allow us to aim. That's why I think classical statistics got this one wrong. Unbiased is never a virtue, but the smallest error is. Thus, if we are to repeat an experiment, we should calibrate our estimator just like a marksman calibrates his sight. But the aim should always be calibrated to give the smallest error, not an unbiased scatter. Noone in their right mind would claim a shotgun is more precise than a rifle because it has smaller bias. But that is what applying the Bessel correction implies. I agree with the point, and what makes it even worse is that ddof=1 does not even produce an unbiased standard deviation estimate. I produces an unbiased variance estimate but the sqrt of this variance estimate is a biased standard deviation estimate, http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation. Bago ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Thu, Apr 3, 2014 at 2:21 PM, Bago mrb...@gmail.com wrote: Sturla P.S. Personally I am not convinced unbiased is ever a valid argument, as the biased estimator has smaller error. This is from experience in marksmanship: I'd rather shoot a tight series with small systematic error than scatter my bullets wildly but unbiased on the target. It is the total error that counts. The series with smallest total error gets the best score. It is better to shoot two series and calibrate the sight in between than use a calibration-free sight that don't allow us to aim. That's why I think classical statistics got this one wrong. Unbiased is never a virtue, but the smallest error is. Thus, if we are to repeat an experiment, we should calibrate our estimator just like a marksman calibrates his sight. But the aim should always be calibrated to give the smallest error, not an unbiased scatter. Noone in their right mind would claim a shotgun is more precise than a rifle because it has smaller bias. But that is what applying the Bessel correction implies. I agree with the point, and what makes it even worse is that ddof=1 does not even produce an unbiased standard deviation estimate. I produces an unbiased variance estimate but the sqrt of this variance estimate is a biased standard deviation estimate, http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation. But ddof=1 still produces a smaller bias than ddof=0 I think the main point in stats is that without ddof, the variance will be too small and t-test or similar will be liberal in small samples, or confidence intervals will be too short. (for statisticians that prefer to have tests that maintain their level and prefer to err on the conservative side.) Josef Bago ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
[Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
While most other Python applications (scipy, pandas) use for the calculation of the standard deviation the default ddof=1 (i.e. they calculate the sample standard deviation), the Numpy implementation uses the default ddof=0. Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? Thomas --- Prof. (FH) PD Dr. Thomas Haslwanter School of Applied Health and Social Sciences University of Applied Sciences Upper Austria FH OÖ Studienbetriebs GmbH Garnisonstraße 21 4020 Linz/Austria Tel.: +43 (0)5 0804 -52170 Fax: +43 (0)5 0804 -52171 E-Mail: thomas.haslwan...@fh-linz.atmailto:thomas.haslwan...@fh-linz.at Web: me-research.fh-linz.athttp://work.thaslwanter.at or work.thaslwanter.at ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
Because np.mean() is ddof=0? (I mean effectively, not that it actually has a parameter for that) There is consistency within the library, and I certainly wouldn't want to have NaN all of the sudden coming from my calls to mean() that I apply to an arbitrary non-empty array of values that happened to have only one value. So, if we can't change the default for mean, then it only makes sense to keep np.std() consistent with np.mean(). My 2 cents... Ben Root On Tue, Apr 1, 2014 at 2:27 PM, Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: While most other Python applications (scipy, pandas) use for the calculation of the standard deviation the default ddof=1 (i.e. they calculate the sample standard deviation), the Numpy implementation uses the default ddof=0. Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? Thomas --- Prof. (FH) PD Dr. Thomas Haslwanter School of Applied Health and Social Sciences *University of Applied Sciences* *Upper Austria* *FH OÖ Studienbetriebs GmbH* Garnisonstraße 21 4020 Linz/Austria Tel.: +43 (0)5 0804 -52170 Fax: +43 (0)5 0804 -52171 E-Mail: thomas.haslwan...@fh-linz.at Web: me-research.fh-linz.at http://work.thaslwanter.at or work.thaslwanter.at ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. If you are not eatimating from a sample, but rather calculating for the whole population, you always want ddof=0. What does Matlab do by default? (Yes, it is a retorical question.) I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? It is a bad idea to suddenly break everyone's code. Sturla ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
I agree; breaking code over this would be ridiculous. Also, I prefer the zero default, despite the mean/std combo probably being more common. On Tue, Apr 1, 2014 at 10:02 PM, Sturla Molden sturla.mol...@gmail.comwrote: Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. If you are not eatimating from a sample, but rather calculating for the whole population, you always want ddof=0. What does Matlab do by default? (Yes, it is a retorical question.) I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? It is a bad idea to suddenly break everyone's code. Sturla ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Tue, Apr 1, 2014 at 9:02 PM, Sturla Molden sturla.mol...@gmail.com wrote: Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. It's true, but the counter-arguments are also strong. And regardless of whether ddof=1 or ddof=0 is better, surely the same one is better for both numpy and scipy. If you are not eatimating from a sample, but rather calculating for the whole population, you always want ddof=0. What does Matlab do by default? (Yes, it is a retorical question.) R (which is probably a more relevant comparison) does do ddof=1 by default. I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? It is a bad idea to suddenly break everyone's code. It would be a disruptive transition, but OTOH having inconsistencies like this guarantees the ongoing creation of new broken code. -n -- Nathaniel J. Smith Postdoctoral researcher - Informatics - University of Edinburgh http://vorpus.org ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Tue, Apr 1, 2014 at 10:08 PM, Nathaniel Smith n...@pobox.com wrote: On Tue, Apr 1, 2014 at 9:02 PM, Sturla Molden sturla.mol...@gmail.com wrote: Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. It's true, but the counter-arguments are also strong. And regardless of whether ddof=1 or ddof=0 is better, surely the same one is better for both numpy and scipy. If we could still choose here without any costs, obviously that's true. This particular ship sailed a long time ago though. By the way, there isn't even a `scipy.stats.std`, so we're comparing with differently named functions (nanstd for example). If you are not eatimating from a sample, but rather calculating for the whole population, you always want ddof=0. What does Matlab do by default? (Yes, it is a retorical question.) R (which is probably a more relevant comparison) does do ddof=1 by default. I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? It is a bad idea to suddenly break everyone's code. It would be a disruptive transition, but OTOH having inconsistencies like this guarantees the ongoing creation of new broken code. Not much of an argument to change return values for a so heavily used function. Ralf -n -- Nathaniel J. Smith Postdoctoral researcher - Informatics - University of Edinburgh http://vorpus.org ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Tue, Apr 1, 2014 at 2:08 PM, Nathaniel Smith n...@pobox.com wrote: On Tue, Apr 1, 2014 at 9:02 PM, Sturla Molden sturla.mol...@gmail.com wrote: Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. It's true, but the counter-arguments are also strong. And regardless of whether ddof=1 or ddof=0 is better, surely the same one is better for both numpy and scipy. If you are not eatimating from a sample, but rather calculating for the whole population, you always want ddof=0. What does Matlab do by default? (Yes, it is a retorical question.) R (which is probably a more relevant comparison) does do ddof=1 by default. I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? It is a bad idea to suddenly break everyone's code. It would be a disruptive transition, but OTOH having inconsistencies like this guarantees the ongoing creation of new broken code. This topic comes up regularly. The original choice was made for numpy 1.0b1 by Travis, see this later thread.http://thread.gmane.org/gmane.comp.python.numeric.general/25720/focus=25721At this point it is probably best to leave it alone. Chuck ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Tue, Apr 1, 2014 at 9:51 PM, Ralf Gommers ralf.gomm...@gmail.com wrote: On Tue, Apr 1, 2014 at 10:08 PM, Nathaniel Smith n...@pobox.com wrote: On Tue, Apr 1, 2014 at 9:02 PM, Sturla Molden sturla.mol...@gmail.com wrote: Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. It's true, but the counter-arguments are also strong. And regardless of whether ddof=1 or ddof=0 is better, surely the same one is better for both numpy and scipy. If we could still choose here without any costs, obviously that's true. This particular ship sailed a long time ago though. By the way, there isn't even a `scipy.stats.std`, so we're comparing with differently named functions (nanstd for example). Presumably nanstd is a lot less heavily used than std, and presumably people expect 'nanstd' to be a 'nan' version of 'std' -- what do you think of changing nanstd to ddof=0 to match numpy? (With appropriate FutureWarning transition, etc.) -- Nathaniel J. Smith Postdoctoral researcher - Informatics - University of Edinburgh http://vorpus.org ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Tue, Apr 1, 2014 at 4:54 PM, Charles R Harris charlesr.har...@gmail.com wrote: On Tue, Apr 1, 2014 at 2:08 PM, Nathaniel Smith n...@pobox.com wrote: On Tue, Apr 1, 2014 at 9:02 PM, Sturla Molden sturla.mol...@gmail.com wrote: Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. It's true, but the counter-arguments are also strong. And regardless of whether ddof=1 or ddof=0 is better, surely the same one is better for both numpy and scipy. If you are not eatimating from a sample, but rather calculating for the whole population, you always want ddof=0. What does Matlab do by default? (Yes, it is a retorical question.) R (which is probably a more relevant comparison) does do ddof=1 by default. I am wondering if there is a good reason to stick to ddof=0 as the default for std, or if others would agree with my suggestion to change the default to ddof=1? It is a bad idea to suddenly break everyone's code. It would be a disruptive transition, but OTOH having inconsistencies like this guarantees the ongoing creation of new broken code. This topic comes up regularly. The original choice was made for numpy 1.0b1 by Travis, see this later thread. At this point it is probably best to leave it alone. I don't have any opinion about this debate, but I love the justification in that thread Any surprise that is created by the different default should be mitigated by the fact that it's an opportunity to learn something about what you are doing. This masterpiece of rhetoric will surely help me win many internet arguments in the future! ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Standard Deviation (std): Suggested change for ddof default value
On Tue, Apr 1, 2014 at 5:11 PM, Nathaniel Smith n...@pobox.com wrote: On Tue, Apr 1, 2014 at 9:51 PM, Ralf Gommers ralf.gomm...@gmail.com wrote: On Tue, Apr 1, 2014 at 10:08 PM, Nathaniel Smith n...@pobox.com wrote: On Tue, Apr 1, 2014 at 9:02 PM, Sturla Molden sturla.mol...@gmail.com wrote: Haslwanter Thomas thomas.haslwan...@fh-linz.at wrote: Personally I cannot think of many applications where it would be desired to calculate the standard deviation with ddof=0. In addition, I feel that there should be consistency between standard modules such as numpy, scipy, and pandas. ddof=0 is the maxiumum likelihood estimate. It is also needed in Bayesian estimation. It's true, but the counter-arguments are also strong. And regardless of whether ddof=1 or ddof=0 is better, surely the same one is better for both numpy and scipy. If we could still choose here without any costs, obviously that's true. This particular ship sailed a long time ago though. By the way, there isn't even a `scipy.stats.std`, so we're comparing with differently named functions (nanstd for example). Presumably nanstd is a lot less heavily used than std, and presumably people expect 'nanstd' to be a 'nan' version of 'std' -- what do you think of changing nanstd to ddof=0 to match numpy? (With appropriate FutureWarning transition, etc.) numpy is numpy, a numerical library scipy.stats is stats and behaves differently. (axis=0) nanstd in scipy.stats will hopefully also go away soon, so I don't think it's worth changing there either. pandas came later and thought ddof=1 is worth more than consistency. I don't think ddof defaults's are worth jumping through deprecation hoops. (bias in cov, corrcoef is non-standard ddof) Josef -- Nathaniel J. Smith Postdoctoral researcher - Informatics - University of Edinburgh http://vorpus.org ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion ___ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
