Stats module¶

pygeode.correlate(X, Y, axes=None, output='r,p', pbar=None)[source]¶

Computes Pearson correlation coefficient between variables X and Y.

Parameters

X, YVar: Variables to correlate. Must have at least one axis in common.
axeslist, optional: Axes over which to compute correlation; if nothing is specified, the correlation is computed over all axes common to shared by X and Y.
outputstring, optional: A string determining which parameters are returned; see list of possible outputs in the Returns section. The specifications must be separated by a comma. Defaults to ‘r,p’.
pbarprogress bar, optional: A progress bar object. If nothing is provided, a progress bar will be displayed if the calculation takes sufficiently long.

Returns

resultsDataset

The names of the variables match the output request string (i.e. if ds is the returned dataset, the correlation coefficient can be obtained through ds.r2).

‘r’: The Pearson correlation coefficient \(\rho_{XY}\)
‘r2’: The coefficient of determination \(\rho^2_{XY}\)
‘p’: The p-value; see notes.

Notes

The coefficient \(\rho_{XY}\) is computed following von Storch and Zwiers 1999, section 8.2.2. The p-value is the probability of finding a correlation coeefficient of equal or greater magnitude (two-sided) to the given result under the hypothesis that the true correlation coefficient between X and Y is zero. It is computed from the t-statistic given in eq (8.7), in section 8.2.3, and assumes normally distributed quantities.

pygeode.regress(X, Y, axes=None, N_fac=None, output='m,b,p', pbar=None)[source]¶

Computes least-squares linear regression of Y against X.

Parameters

X, YVar: Variables to regress. Must have at least one axis in common.
axeslist, optional: Axes over which to compute correlation; if nothing is specified, the correlation is computed over all axes common to X and Y.
N_facinteger: A factor by which to rescale the estimated number of degrees of freedom; the effective number will be given by the number estimated from the dataset divided by N_fac.
outputstring, optional: A string determining which parameters are returned; see list of possible outputs in the Returns section. The specifications must be separated by a comma. Defaults to ‘m,b,p’.
pbarprogress bar, optional: A progress bar object. If nothing is provided, a progress bar will be displayed if the calculation takes sufficiently long.

Returns

resultsDataset

The returned variables are specified by the output argument. The names of the variables match the output request string (i.e. if ds is the returned dataset, the linear coefficient of the regression can be obtained by ds.m).

A fit of the form \(Y = m X + b + \epsilon\) is assumed, and the following parameters can be returned:

‘m’: Linear coefficient of the regression
‘b’: Constant coefficient of the regression
‘r2’: Fraction of the variance in Y explained by X (\(R^2\))
‘p’: Probability of this fit under null hypothesis that true linear coefficient is zero
‘sm’: Standard deviation of linear coefficient estimate (\(\hat{\sigma}_E/\sqrt{S_{XX}}\))
‘se’: Standard deviation of residuals (\(\hat{\sigma}_E\))

Notes

The statistics described are computed following von Storch and Zwiers 1999, section 8.3. The p-value ‘p’ is computed using the t-statistic given in section 8.3.8, and confidence intervals for the slope and intercept can be computed from ‘sm’ or ‘se’. The data is assumed to be normally distributed.

pygeode.multiple_regress(Xs, Y, axes=None, N_fac=None, output='B,p', pbar=None)[source]¶

Computes least-squares multiple regression of Y against variables Xs.

Parameters

Xslist of Var instances: Variables to treat as independent regressors. Must have at least one axis in common with each other and with Y.
YVar: The dependent variable. Must have at least one axis in common with the Xs.
axeslist, optional: Axes over which to compute correlation; if nothing is specified, the correlation is computed over all axes common to the Xs and Y.
N_facinteger: A factor by which to rescale the estimated number of degrees of freedom; the effective number will be given by the number estimated from the dataset divided by N_fac.
outputstring, optional: A string determining which parameters are returned; see list of possible outputs in the Returns section. The specifications must be separated by a comma. Defaults to ‘B,p’.
pbarprogress bar, optional: A progress bar object. If nothing is provided, a progress bar will be displayed if the calculation takes sufficiently long.

Returns

resultstuple of floats or Var instances.

The return values are specified by the output argument. The names of the variables match the output request string (i.e. if ds is the returned dataset, the linear coefficient of the regression can be obtained by ds.m).

A fit of the form \(Y = \sum_i \beta_i X_i + \epsilon\) is assumed. Note that a constant term is not included by default. The following parameters can be returned:

‘B’: Linear coefficients \(\beta_i\) of each regressor
‘r2’: Fraction of the variance in Y explained by all Xs (\(R^2\))
‘p’: p-value of regession; see notes.
‘sb’: Standard deviation of each linear coefficient
‘covb’: Covariance matrix of the linear coefficients
‘se’: Standard deviation of residuals

The outputs ‘B’, ‘p’, and ‘sb’ will produce as many outputs as there are regressors.

Notes

The statistics described are computed following von Storch and Zwiers 1999, section 8.4. The p-value ‘p’ is computed using the t-statistic appropriate for the multi-variate normal estimator \(\hat{\vec{a}}\) given in section 8.4.2; it corresponds to the probability of obtaining the regression coefficient under the null hypothesis that there is no linear relationship. Note this may not be the best way to determine if a given parameter is contributing a significant fraction to the explained variance of Y. The variances ‘se’ and ‘sb’ are \(\hat{\sigma}_E\) and the square root of the diagonal elements of \(\hat{\sigma}^2_E (\chi^T\chi)\) in von Storch and Zwiers, respectively. The data is assumed to be normally distributed.

pygeode.difference(X, Y, axes=None, alpha=0.05, Nx_fac=None, Ny_fac=None, output='d,p,ci', pbar=None)[source]¶

Computes the mean value and statistics of X - Y.

Parameters

X, YVar: Variables to difference. Must have at least one axis in common.
axeslist, optional, defaults to None: Axes over which to compute means; if othing is specified, the mean is computed over all axes common to X and Y.
alphafloat, optional; defaults to 0.05: Confidence level for which to compute confidence interval.
Nx_facinteger, optional: defaults to None: A factor by which to rescale the estimated number of degrees of freedom of X; the effective number will be given by the number estimated from the dataset divided by Nx_fac.
Ny_facinteger, optional: defaults to None: A factor by which to rescale the estimated number of degrees of freedom of Y; the effective number will be given by the number estimated from the dataset divided by Ny_fac.
outputstring, optional: A string determining which parameters are returned; see list of possible outputs in the Returns section. The specifications must be separated by a comma. Defaults to ‘d,p,ci’.
pbarprogress bar, optional: A progress bar object. If nothing is provided, a progress bar will be displayed if the calculation takes sufficiently long.

Returns

resultsDataset

The returned variables are specified by the output argument. The names of the variables match the output request string (i.e. if ds is the returned dataset, the average of the difference can be obtained by ds.d). The following four quantities can be computed:

‘d’: The difference in the means, X - Y
‘df’: The effective number of degrees of freedom, \(df\)
‘p’: The p-value; see notes.
‘ci’: The confidence interval of the difference at the level specified by alpha