= SciPy 1.16.2 documentation
{c}
{scope}
Publisher: SciPy project.
The scipy.stats.multivariate_normal Notes describe singular covariance using a pseudo-determinant, pseudoinverse and covariance rank when singular evaluation is allowed. The pinned implementation separately checks the covariance support: _PSD._support_mask tests the null-space residual against its numerical tolerance, and multivariate_normal_gen.pdf/logpdf return zero density or negative-infinite log density off support. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.multivariate_normal.html[Notes, singular covariance]{ref} https://github.com/scipy/scipy/blob/v1.16.2/scipy/stats/_multivariate.py[_PSD._support_mask; multivariate_normal_gen.pdf and logpdf]{ref}
Covariance.whiten describes transforming observations to whitened coordinates. Its example constructs Covariance.from_precision from a positive-definite covariance, whitens a Gaussian sample and displays a near-identity sample covariance. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.Covariance.whiten.html[Description and Examples, from_precision and np.cov]{ref}
The scipy.stats.norm method list identifies cdf as the cumulative distribution function, sf as the survival function, and ppf as the inverse cdf. The loc and scale parameters specify the mean and standard deviation. The documentation notes that sf can be more accurate than subtracting cdf from one when evaluating a tail. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.norm.html[Parameters and Methods: cdf, sf, ppf]{ref}
The scipy.stats.uniform description specifies the standard uniform distribution on the unit interval and the loc/scale transformation to the interval from loc to loc plus scale. Its Methods list includes rvs for random variates, alongside cdf and ppf. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.uniform.html[Description and Methods, standard support, loc/scale and rvs]{ref}
The scipy.signal.group_delay definition differentiates continuous unwrapped phase with respect to digital angular frequency and reports delay in samples; the fs parameter sets the frequency units. An analog derivative with respect to angular frequency in radians per second instead yields seconds. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.signal.group_delay.html[Definition, Parameters: fs, and Returns: gd]{ref}
https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.foldnorm.html[Notes]: standardized folded-normal density, nonnegative shape parameter and loc/scale transformation. Taking the absolute value of a normal variable with mean $m$ and standard deviation $s$ corresponds to shape $|m|/s$, `loc=0` and `scale=s`.
The scipy.stats.multivariate_normal cdf method evaluates joint lower-orthant probabilities using the specified mean vector and covariance matrix. Its maxpts, abseps, and releps arguments control numerical integration limits and tolerances; lower_limit specifies the lower integration bounds. This supports evaluating a correlated maximum at a vector of equal upper thresholds without multiplying marginal probabilities. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.multivariate_normal.html[Parameters: mean, cov; Methods: cdf signature]{ref}
The scipy.fft.fft norm parameter documents an unscaled forward transform for the default backward convention, a 1/n-scaled forward transform for forward, and symmetric 1/sqrt(n) scaling for ortho. Its Notes specify the negative-exponent forward sum, ordering of positive and negative frequencies, and conjugate symmetry for real inputs. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.fft.fft.html[Parameters: norm; Notes: forward sum, bin ordering, and real-input symmetry]{ref}
The scipy.stats.binom Notes define the binomial mass function, including its positive probability at zero for a finite nondegenerate calibration sample; pmf evaluates each count probability. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.binom.html[Notes: probability mass function; Methods: pmf]{ref}
The signal-processing guide Spectral Analysis section derives the sinc leakage of a finite-duration sinusoid and discusses the tradeoff between window main-lobe width and sidelobe height. https://docs.scipy.org/doc/scipy-1.16.2/tutorial/signal.html#spectral-analysis[Spectral Analysis, equations 1–3 and window comparison]{ref}
The scipy.signal.periodogram scaling parameter distinguishes squared spectrum units from power spectral density units. Its Notes give the window-dependent spectrum/density conversion and one-sided power folding; its flattop example estimates RMS amplitude from the square root of a spectrum peak. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.signal.periodogram.html[Parameters: scaling; Notes and Examples]{ref}
BinomTestResult.proportion_ci documents its confidence_level parameter, the exact binomial proportion-interval method, and the low/high attributes of the returned bounds. These are parameter-estimation bounds; scipy.stats.binom.interval instead describes count-distribution intervals for supplied parameters. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats._result_classes.BinomTestResult.proportion_ci.html[Parameters: confidence_level and method; Returns: low and high]{ref} https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.binom.html[Methods: interval]{ref}
https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.signal.windows.boxcar.html[scipy.signal.windows.boxcar, description and Returns]: the rectangular window uses all-ones coefficients. https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.signal.windows.hann.html[scipy.signal.windows.hann, sym parameter, Returns and Notes]: the symmetric endpoint convention differs from the periodic spectral convention; the documented even-length symmetric array need not attain coefficient one.
https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.ndimage.zoom.html[scipy.ndimage.zoom, description and order parameter]: array zoom uses spline interpolation of the requested order, with orders zero through five; resizing is a numerical reconstruction operation rather than an additional physical measurement.
https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.signal.freqz.html[scipy.signal.freqz, defining frequency-response sum; Parameters: fs; Returns: w]: evaluates a finite numerator/denominator frequency response; the default frequency unit is radians per sample, with frequencies expressed in the same units as fs when supplied.
https://raw.githubusercontent.com/scipy/scipy/v1.16.2/scipy/linalg/_decomp_svd.py[null_space, lines 333–403]: the pinned function constructs a null-space basis from right singular vectors after SVD; rcond controls the relative cutoff for effective numerical rank. The returned basis `Q` uses `vh[num:,:].T.conj()`.
https://docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.stats.chi2.html[scipy.stats.chi2, Notes]: the chi-squared density and its gamma-distribution representation supply the density used in the explicitly derived reciprocal-moment calculation.
Back to article page
