scipy.stats.qqplot and scipy.stats.probplot axis labeling
The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values versus actual data values for visualization of fit to a distribution. First a one-D array of expected percentiles is generated for a sample of size N; then that is passed to dist.ppf, the per cent point function for the chosen distribution, to return an array of expected values. The visualized data points are pairs of expected and actual values, and a linear regression is done on these to produce the line data points in this distribution should lie on. Where x is the input data array and dist the chosen distribution we have:
osr = np.sort(x) osm_uniform = _calc_uniform_order_statistic_medians(len(x)) osm = dist.ppf(osm_uniform) slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and xticklabels produced produced by qqplot and problplot do not seem correct for the their intended interpretations. First the numbers on the x-axis do not represent quantiles; the intervals between them do not in general contain equal numbers of points. For a normal distribution with sigma=1, they represent standard deviations. Changing the label on the x-axis does not seem like a very good solution, because the interpretation of the values on the x-axis will be different for different distributions. Rather the right solution seems to be to actually show quantiles on the x-axis. The numbers on the x-axis can stay as they are, representing quantile indexes, but they need to be spaced so as to show the actual division points that carve the population up into groups of the same size. This can be done in something like the following way.
import numpy as np xt = np.arange(-3,3,dtype=int)
# Find the 5 quantiles to divide the data into sixths percentiles = [x*.167 + .502 for x in xt] percentiles = np.array(percentiles + [.999]) vals = dist.ppf(percentiles) ax.set_xticks(vals) xt = np.array(list(xt)+[3]) ax.set_xticklabels(xt) ax.set_xlabel('Quantile') plt.show()
I’ve attached two images to show the difference between the current visualization and the suggested one. Mark Gawron
Hi Mark, Note that the scipy-dev or scipy-user mailing list would have been more appropriate for this question. On Fri, Jun 10, 2016 at 9:06 AM, Mark Gawron <gawron@mail.sdsu.edu> wrote:
The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values versus actual data values for visualization of fit to a distribution. First a one-D array of expected percentiles is generated for a sample of size N; then that is passed to dist.ppf, the per cent point function for the chosen distribution, to return an array of expected values. The visualized data points are pairs of expected and actual values, and a linear regression is done on these to produce the line data points in this distribution should lie on.
Where x is the input data array and dist the chosen distribution we have:
osr = np.sort(x) osm_uniform = _calc_uniform_order_statistic_medians(len(x)) osm = dist.ppf(osm_uniform) slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and xticklabels produced produced by qqplot and problplot do not seem correct for the their intended interpretations. First the numbers on the x-axis do not represent quantiles; the intervals between them do not in general contain equal numbers of points. For a normal distribution with sigma=1, they represent standard deviations. Changing the label on the x-axis does not seem like a very good solution, because the interpretation of the values on the x-axis will be different for different distributions. Rather the right solution seems to be to actually show quantiles on the x-axis. The numbers on the x-axis can stay as they are, representing quantile indexes, but they need to be spaced so as to show the actual division points that carve the population up into groups of the same size. This can be done in something like the following way.
The ticks are correct I think, but they're theoretical quantiles and not sample quantiles. This was discussed in [1] and is consistent with R [2] and statsmodels [3]. I see that we just forgot to add "theoretical" to the x-axis label (mea culpa). Does adding that resolve your concern? [1] https://github.com/scipy/scipy/issues/1821 [2] http://data.library.virginia.edu/understanding-q-q-plots/ [3] http://statsmodels.sourceforge.net/devel/generated/statsmodels.graphics.gofp... Ralf
On Sat, Jun 11, 2016 at 8:53 AM, Ralf Gommers <ralf.gommers@gmail.com> wrote:
Hi Mark,
Note that the scipy-dev or scipy-user mailing list would have been more appropriate for this question.
On Fri, Jun 10, 2016 at 9:06 AM, Mark Gawron <gawron@mail.sdsu.edu> wrote:
The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values versus actual data values for visualization of fit to a distribution. First a one-D array of expected percentiles is generated for a sample of size N; then that is passed to dist.ppf, the per cent point function for the chosen distribution, to return an array of expected values. The visualized data points are pairs of expected and actual values, and a linear regression is done on these to produce the line data points in this distribution should lie on.
Where x is the input data array and dist the chosen distribution we have:
osr = np.sort(x) osm_uniform = _calc_uniform_order_statistic_medians(len(x)) osm = dist.ppf(osm_uniform) slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and xticklabels produced produced by qqplot and problplot do not seem correct for the their intended interpretations. First the numbers on the x-axis do not represent quantiles; the intervals between them do not in general contain equal numbers of points. For a normal distribution with sigma=1, they represent standard deviations. Changing the label on the x-axis does not seem like a very good solution, because the interpretation of the values on the x-axis will be different for different distributions. Rather the right solution seems to be to actually show quantiles on the x-axis. The numbers on the x-axis can stay as they are, representing quantile indexes, but they need to be spaced so as to show the actual division points that carve the population up into groups of the same size. This can be done in something like the following way.
The ticks are correct I think, but they're theoretical quantiles and not sample quantiles. This was discussed in [1] and is consistent with R [2] and statsmodels [3]. I see that we just forgot to add "theoretical" to the x-axis label (mea culpa). Does adding that resolve your concern?
[1] https://github.com/scipy/scipy/issues/1821 [2] http://data.library.virginia.edu/understanding-q-q-plots/ [3] http://statsmodels.sourceforge.net/devel/generated/statsmodels.graphics.gofp...
Ralf
as related link http://phobson.github.io/mpl-probscale/tutorial/closer_look_at_viz.html Paul Hobson has done a lot of work for getting different probabitlity scales attached to pp-plots or generalized versions of probability plots. I think qqplots are less ambiguous because they are on the original or standardized scale. I haven't worked my way through the various interpretation of probability axis yet because I find it "not obvious". It might be easier for fields that have a tradition of using probability papers. It's planned to be added to the statsmodels probability plots so that there will be a large choice of axis labels and scales. Josef
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
Thanks, Jozef. This is very helpful. And I will direct this to one of the other mailing lists, once I read the previous posts. Regarding your remark: Maybe Im having a terminology problem. It seems to me once you do
osm = dist.ppf(osm_uniform)
you’re back in the value space for the particular distribution. So this gives you known probability intervals, but not UNIFORM probability intervals (the interval between 0 and 1 STD covers a bigger prob interval than the the interval between 1 and 2). And the idea of a quantile is that it’s a division point in a UNIFORM division of the probability axis. Mark On Jun 11, 2016, at 10:03 AM, josef.pktd@gmail.com wrote:
On Sat, Jun 11, 2016 at 8:53 AM, Ralf Gommers <ralf.gommers@gmail.com> wrote: Hi Mark,
Note that the scipy-dev or scipy-user mailing list would have been more appropriate for this question.
On Fri, Jun 10, 2016 at 9:06 AM, Mark Gawron <gawron@mail.sdsu.edu> wrote:
The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values versus actual data values for visualization of fit to a distribution. First a one-D array of expected percentiles is generated for a sample of size N; then that is passed to dist.ppf, the per cent point function for the chosen distribution, to return an array of expected values. The visualized data points are pairs of expected and actual values, and a linear regression is done on these to produce the line data points in this distribution should lie on.
Where x is the input data array and dist the chosen distribution we have:
osr = np.sort(x) osm_uniform = _calc_uniform_order_statistic_medians(len(x)) osm = dist.ppf(osm_uniform) slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and xticklabels produced produced by qqplot and problplot do not seem correct for the their intended interpretations. First the numbers on the x-axis do not represent quantiles; the intervals between them do not in general contain equal numbers of points. For a normal distribution with sigma=1, they represent standard deviations. Changing the label on the x-axis does not seem like a very good solution, because the interpretation of the values on the x-axis will be different for different distributions. Rather the right solution seems to be to actually show quantiles on the x-axis. The numbers on the x-axis can stay as they are, representing quantile indexes, but they need to be spaced so as to show the actual division points that carve the population up into groups of the same size. This can be done in something like the following way.
The ticks are correct I think, but they're theoretical quantiles and not sample quantiles. This was discussed in [1] and is consistent with R [2] and statsmodels [3]. I see that we just forgot to add "theoretical" to the x-axis label (mea culpa). Does adding that resolve your concern?
[1] https://github.com/scipy/scipy/issues/1821 [2] http://data.library.virginia.edu/understanding-q-q-plots/ [3] http://statsmodels.sourceforge.net/devel/generated/statsmodels.graphics.gofp...
Ralf
as related link http://phobson.github.io/mpl-probscale/tutorial/closer_look_at_viz.html
Paul Hobson has done a lot of work for getting different probabitlity scales attached to pp-plots or generalized versions of probability plots. I think qqplots are less ambiguous because they are on the original or standardized scale.
I haven't worked my way through the various interpretation of probability axis yet because I find it "not obvious". It might be easier for fields that have a tradition of using probability papers.
It's planned to be added to the statsmodels probability plots so that there will be a large choice of axis labels and scales.
Josef
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
On Sat, Jun 11, 2016 at 2:49 PM, Mark Gawron <gawron@mail.sdsu.edu> wrote:
Thanks, Jozef. This is very helpful. And I will direct this to one of the other mailing lists, once I read the previous posts.
Regarding your remark: Maybe Im having a terminology problem. It seems to me once you do
you’re back in the value space for the particular distribution. So this gives you known probability intervals, but not UNIFORM probability intervals (the interval between 0 and 1 STD covers a bigger prob interval
osm = dist.ppf(osm_uniform) than the the interval between 1 and 2). And the idea of a quantile is that it’s a division point in a UNIFORM division of the probability axis.
Yes and No, quantile, i.e. what you get from ppf, are units of the random variable. So it is on the scale of the random variable not on a probability scale. The axis labels are in units of the random variable. pp-plots have probabilities on the axis and are uniform scaled in probabilities but non-uniform in the values of the random variable. The difficult part to follow is if the plot is done uniform in one scale, but the axis are labeled non-uniform in the other scale. That's what Paul's probscale does and what you have in mind, AFAIU. Josef
Mark
On Jun 11, 2016, at 10:03 AM, josef.pktd@gmail.com wrote:
On Sat, Jun 11, 2016 at 8:53 AM, Ralf Gommers <ralf.gommers@gmail.com> wrote:
Hi Mark,
Note that the scipy-dev or scipy-user mailing list would have been more appropriate for this question.
On Fri, Jun 10, 2016 at 9:06 AM, Mark Gawron <gawron@mail.sdsu.edu> wrote:
The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values versus actual data values for visualization of fit to a distribution. First a one-D array of expected percentiles is generated for a sample of size N; then that is passed to dist.ppf, the per cent point function for the chosen distribution, to return an array of expected values. The visualized data points are pairs of expected and actual values, and a linear regression is done on these to produce the line data points in this distribution should lie on.
Where x is the input data array and dist the chosen distribution we have:
osr = np.sort(x) osm_uniform = _calc_uniform_order_statistic_medians(len(x)) osm = dist.ppf(osm_uniform) slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and xticklabels produced produced by qqplot and problplot do not seem correct for the their intended interpretations. First the numbers on the x-axis do not represent quantiles; the intervals between them do not in general contain equal numbers of points. For a normal distribution with sigma=1, they represent standard deviations. Changing the label on the x-axis does not seem like a very good solution, because the interpretation of the values on the x-axis will be different for different distributions. Rather the right solution seems to be to actually show quantiles on the x-axis. The numbers on the x-axis can stay as they are, representing quantile indexes, but they need to be spaced so as to show the actual division points that carve the population up into groups of the same size. This can be done in something like the following way.
The ticks are correct I think, but they're theoretical quantiles and not sample quantiles. This was discussed in [1] and is consistent with R [2] and statsmodels [3]. I see that we just forgot to add "theoretical" to the x-axis label (mea culpa). Does adding that resolve your concern?
[1] https://github.com/scipy/scipy/issues/1821 [2] http://data.library.virginia.edu/understanding-q-q-plots/ [3] http://statsmodels.sourceforge.net/devel/generated/statsmodels.graphics.gofp...
Ralf
as related link http://phobson.github.io/mpl-probscale/tutorial/closer_look_at_viz.html
Paul Hobson has done a lot of work for getting different probabitlity scales attached to pp-plots or generalized versions of probability plots. I think qqplots are less ambiguous because they are on the original or standardized scale.
I haven't worked my way through the various interpretation of probability axis yet because I find it "not obvious". It might be easier for fields that have a tradition of using probability papers.
It's planned to be added to the statsmodels probability plots so that there will be a large choice of axis labels and scales.
Josef
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
Ok, Our messages crossed. I understand now. Thanks. Mark On Jun 11, 2016, at 12:24 PM, josef.pktd@gmail.com wrote:
On Sat, Jun 11, 2016 at 2:49 PM, Mark Gawron <gawron@mail.sdsu.edu> wrote: Thanks, Jozef. This is very helpful. And I will direct this to one of the other mailing lists, once I read the previous posts.
Regarding your remark: Maybe Im having a terminology problem. It seems to me once you do
osm = dist.ppf(osm_uniform)
you’re back in the value space for the particular distribution. So this gives you known probability intervals, but not UNIFORM probability intervals (the interval between 0 and 1 STD covers a bigger prob interval than the the interval between 1 and 2). And the idea of a quantile is that it’s a division point in a UNIFORM division of the probability axis.
Yes and No, quantile, i.e. what you get from ppf, are units of the random variable. So it is on the scale of the random variable not on a probability scale. The axis labels are in units of the random variable.
pp-plots have probabilities on the axis and are uniform scaled in probabilities but non-uniform in the values of the random variable.
The difficult part to follow is if the plot is done uniform in one scale, but the axis are labeled non-uniform in the other scale. That's what Paul's probscale does and what you have in mind, AFAIU.
Josef
Mark
On Jun 11, 2016, at 10:03 AM, josef.pktd@gmail.com wrote:
On Sat, Jun 11, 2016 at 8:53 AM, Ralf Gommers <ralf.gommers@gmail.com> wrote: Hi Mark,
Note that the scipy-dev or scipy-user mailing list would have been more appropriate for this question.
On Fri, Jun 10, 2016 at 9:06 AM, Mark Gawron <gawron@mail.sdsu.edu> wrote:
The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values versus actual data values for visualization of fit to a distribution. First a one-D array of expected percentiles is generated for a sample of size N; then that is passed to dist.ppf, the per cent point function for the chosen distribution, to return an array of expected values. The visualized data points are pairs of expected and actual values, and a linear regression is done on these to produce the line data points in this distribution should lie on.
Where x is the input data array and dist the chosen distribution we have:
osr = np.sort(x) osm_uniform = _calc_uniform_order_statistic_medians(len(x)) osm = dist.ppf(osm_uniform) slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and xticklabels produced produced by qqplot and problplot do not seem correct for the their intended interpretations. First the numbers on the x-axis do not represent quantiles; the intervals between them do not in general contain equal numbers of points. For a normal distribution with sigma=1, they represent standard deviations. Changing the label on the x-axis does not seem like a very good solution, because the interpretation of the values on the x-axis will be different for different distributions. Rather the right solution seems to be to actually show quantiles on the x-axis. The numbers on the x-axis can stay as they are, representing quantile indexes, but they need to be spaced so as to show the actual division points that carve the population up into groups of the same size. This can be done in something like the following way.
The ticks are correct I think, but they're theoretical quantiles and not sample quantiles. This was discussed in [1] and is consistent with R [2] and statsmodels [3]. I see that we just forgot to add "theoretical" to the x-axis label (mea culpa). Does adding that resolve your concern?
[1] https://github.com/scipy/scipy/issues/1821 [2] http://data.library.virginia.edu/understanding-q-q-plots/ [3] http://statsmodels.sourceforge.net/devel/generated/statsmodels.graphics.gofp...
Ralf
as related link http://phobson.github.io/mpl-probscale/tutorial/closer_look_at_viz.html
Paul Hobson has done a lot of work for getting different probabitlity scales attached to pp-plots or generalized versions of probability plots. I think qqplots are less ambiguous because they are on the original or standardized scale.
I haven't worked my way through the various interpretation of probability axis yet because I find it "not obvious". It might be easier for fields that have a tradition of using probability papers.
It's planned to be added to the statsmodels probability plots so that there will be a large choice of axis labels and scales.
Josef
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org https://mail.scipy.org/mailman/listinfo/numpy-discussion
On Sat, Jun 11, 2016 at 2:53 PM, Ralf Gommers <ralf.gommers@gmail.com> wrote:
Hi Mark,
Note that the scipy-dev or scipy-user mailing list would have been more appropriate for this question.
On Fri, Jun 10, 2016 at 9:06 AM, Mark Gawron <gawron@mail.sdsu.edu> wrote:
The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values versus actual data values for visualization of fit to a distribution. First a one-D array of expected percentiles is generated for a sample of size N; then that is passed to dist.ppf, the per cent point function for the chosen distribution, to return an array of expected values. The visualized data points are pairs of expected and actual values, and a linear regression is done on these to produce the line data points in this distribution should lie on.
Where x is the input data array and dist the chosen distribution we have:
osr = np.sort(x) osm_uniform = _calc_uniform_order_statistic_medians(len(x)) osm = dist.ppf(osm_uniform) slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and xticklabels produced produced by qqplot and problplot do not seem correct for the their intended interpretations. First the numbers on the x-axis do not represent quantiles; the intervals between them do not in general contain equal numbers of points. For a normal distribution with sigma=1, they represent standard deviations. Changing the label on the x-axis does not seem like a very good solution, because the interpretation of the values on the x-axis will be different for different distributions. Rather the right solution seems to be to actually show quantiles on the x-axis. The numbers on the x-axis can stay as they are, representing quantile indexes, but they need to be spaced so as to show the actual division points that carve the population up into groups of the same size. This can be done in something like the following way.
The ticks are correct I think, but they're theoretical quantiles and not sample quantiles. This was discussed in [1] and is consistent with R [2] and statsmodels [3]. I see that we just forgot to add "theoretical" to the x-axis label (mea culpa). Does adding that resolve your concern?
Sent a PR for this: https://github.com/scipy/scipy/pull/6249 Ralf
[1] https://github.com/scipy/scipy/issues/1821 [2] http://data.library.virginia.edu/understanding-q-q-plots/ [3] http://statsmodels.sourceforge.net/devel/generated/statsmodels.graphics.gofp...
Ralf
participants (3)
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josef.pktd＠gmail.com -
Mark Gawron -
Ralf Gommers