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|newpage| Numerical transformations and descriptive statistics =============================================================================== Centering of a matrix ------------------------------------------------------------------------------- .. method:: mat.centered(res, data, population, opt) Returns the centered version of the matrix .. code-block:: pycon >>> from arbeigenlab import mp14 >>> ctx = mp14.drf(); mp14.setdps(15) >>> A = ctx.read_from_sqlite(mp14.dbpath(), "DecTableA6x6", ""); A.show("A") A: 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, >>> B = A.centered(); B.show("B") B: -25, -25, -25, -25, -25, -25, -15, -15, -15, -15, -15, -15, -5, -5, -5, -5, -5, -5, 5, 5, 5, 5, 5, 5, 15, 15, 15, 15, 15, 15, 25, 25, 25, 25, 25, 25, >>> B = A.variance(); B.show("B") B: 350, 350, 350, 350, 350, 350, >>> B = A.stdev(); B.show("B") Standardization of a matrix ------------------------------------------------------------------------------- .. method:: mat.standardized(res, data, population, opt) Returns the standardized version of the matrix .. code-block:: pycon >>> from arbeigenlab import mp14 >>> ctx = mp14.drf(); mp14.setdps(5) >>> A = ctx.read_from_sqlite(mp14.dbpath(), "DecTableA6x6", ""); A.show("A") A: 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, >>> B = A.standardized(); B.show("B") B: -1.336, -1.336, -1.336, -1.336, -1.336, -1.336, -0.8017, -0.8017, -0.8017, -0.8017, -0.8017, -0.8017, -0.2672, -0.2672, -0.2672, -0.2672, -0.2672, -0.2672, 0.2672, 0.2672, 0.2672, 0.2672, 0.2672, 0.2672, 0.8017, 0.8017, 0.8017, 0.8017, 0.8017, 0.8017, 1.336, 1.336, 1.336, 1.336, 1.336, 1.336, >>> C = B.variance(); C.show("C") C: 0.9996, 0.9996, 0.9996, 0.9996, 0.9996, 0.9996, See also: https://en.m.wikipedia.org/wiki/Quantile Hyndman, 1996. The quantile functions are not working: https://numpy.org/doc/stable/reference/generated/numpy.ptp.html#numpy.ptp https://numpy.org/doc/stable/reference/generated/numpy.percentile.html#numpy.percentile https://numpy.org/doc/stable/reference/generated/numpy.nanpercentile.html#numpy.nanpercentile https://numpy.org/doc/stable/reference/generated/numpy.quantile.html#numpy.quantile https://numpy.org/doc/stable/reference/generated/numpy.nanquantile.html#numpy.nanquantile The correlation functions are not working: https://numpy.org/doc/stable/reference/generated/numpy.corrcoef.html#numpy.corrcoef https://numpy.org/doc/stable/reference/generated/numpy.correlate.html#numpy.correlate https://numpy.org/doc/stable/reference/generated/numpy.cov.html#numpy.cov See also Eigen :cite:p:`EigenMat101`. Trace ------------------------------------------------------------------------------- .. method:: mat.trace() Returns the trace of the matrix. See also: Wikipedia :cite:p:`WikipediaMat10`. .. code-block:: pycon >>> from arbeigenlab import mp14 >>> ctx = mp14.drf() >>> A = ctx.read_from_sqlite(mp14.dbpath(), "DecTableA6x6", ""); A.show("A") A: 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, >>> Res = A.trace(); Res.show("Res") !!! MISSING !!! Squared Norm ------------------------------------------------------------------------------- .. method:: mat.squaredNorm(partialmode=full) Returns the squared norm of the matrix. .. code-block:: pycon >>> from arbeigenlab import mp14 >>> ctx = mp14.drf() >>> A = ctx.read_from_sqlite(mp14.dbpath(), "DecTableA6x6", ""); A.show("A") A: 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, >>> Res = A.squaredNorm(); Res.show("Res") Res: 9526, 9964, 10414, 10876, 11350, 11836, Euclidian Norm ------------------------------------------------------------------------------- .. method:: mat.Norm(partialmode=full) Returns the norm of the matrix. .. code-block:: pycon >>> from arbeigenlab import mp14 >>> ctx = mp14.drf(); mp14.setdps(6) >>> A = ctx.read_from_sqlite(mp14.dbpath(), "DecTableA6x6", ""); A.show("A") A: 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, >>> Res = A.Norm(); Res.show("Res") Res: 97.601, 99.820, 102.05, 104.29, 106.54, 108.79, Covariance matrix ------------------------------------------------------------------------------- .. method:: mat.covariance_matrix(use_crossproduct=False) Returns the covariance matrix of the matrix. See also Wikipedia :cite:p:`WikipediaMat11`, Wikipedia :cite:p:`WikipediaMat12`. .. code-block:: pycon >>> from arbeigenlab import mp14 >>> ctx = mp14.drf(); mp14.setdps(15) >>> A = ctx.read_from_sqlite(mp14.dbpath(), "DecTableA6x6", ""); A.show("A") A: 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, >>> B = A.centered(); B.show("B") B: -25, -25, -25, -25, -25, -25, -15, -15, -15, -15, -15, -15, -5, -5, -5, -5, -5, -5, 5, 5, 5, 5, 5, 5, 15, 15, 15, 15, 15, 15, 25, 25, 25, 25, 25, 25, >>> C = (B.T * B)/(B.cols-1); C.show("C") C: 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, >>> D = A.covariance(); D.show("D") D: 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, Correlation matrix ------------------------------------------------------------------------------- .. method:: mat.correlation(use_crossproduct=False) Returns the correlation matrix of matrix ?matA. See also Wikipedia :cite:p:`WikipediaMat13`. .. code-block:: pycon >>> from arbeigenlab import mp14 >>> ctx = mp14.drf(); mp14.setdps(6) >>> A = ctx.read_from_sqlite(mp14.dbpath(), "DecTableA6x6", ""); A.show("A") A: 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, >>> B = A.standardized(); B.show("B") B: -1.3363, -1.3363, -1.3363, -1.3363, -1.3363, -1.3363, -0.80180, -0.80180, -0.80180, -0.80180, -0.80180, -0.80180, -0.26727, -0.26727, -0.26727, -0.26727, -0.26727, -0.26727, 0.26727, 0.26727, 0.26727, 0.26727, 0.26727, 0.26727, 0.80180, 0.80180, 0.80180, 0.80180, 0.80180, 0.80180, 1.3363, 1.3363, 1.3363, 1.3363, 1.3363, 1.3363, >>> C = (B.T * B)/(B.cols-1); C.show("C") C: 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, >>> D = A.correlation(); D.show("D") D: 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, Summary statistic ------------------------------------------------------------------------------- .. method:: mat.summary(summary, data, population, opt) Sets *summary* to a matrix containing summary statistics of *data*. ======================================== ================================================================= Constant Meaning ======================================== ================================================================= ARB_STAT_COUNT 1 ARB_STAT_SUM 4 = `\sum_0^10 7^e+11^f` ARB_STAT_MEAN 4 ARB_STAT_MIN 13 ARB_STAT_MEDIAN 1 ARB_STAT_MAX 11 ARB_STAT_AVERAGE_DEVIATION 2 ARB_STAT_SUM_OF_SQUARES_OF_DEV 14 ARB_STAT_SUM_OF_SQUARES 8 ARB_STAT_VARIANCE 11 ARB_STAT_STANDARD_DEVIATION 6 ARB_STAT_SKEWNESS 6 ARB_STAT_KURTOSIS 8 ARB_STAT_TRIMMED_MEAN 8 ARB_STAT_HARMONIC_MEAN 9 ARB_STAT_GEOMETRIC_MEAN 9 ======================================== ================================================================= .. code-block:: pycon >>> from arbeigenlab import mp14 >>> mpm.dps = 40; >>> A = mp14.xrf().read_from_sqlite(mp14.dbpath(), "MpfrTableA4x4", "") >>> B = mp14.xrf().read_from_sqlite(mp14.dbpath(), "MpfrTableB4x4", "") .. _rst_mpm_norm: Vector norm of a matrix ------------------------------------------------------------------------------- .. method:: ctx.norm(x, p=2) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Gives the entrywise `p`-norm of an iterable *x*, i.e. the vector norm `\left(\sum_k |x_k|^p\right)^{1/p}`, for any given `1 \le p \le \infty`. Special cases: If *x* is not iterable, this just returns ``absmax(x)``. ``p=1`` gives the sum of absolute values. ``p=2`` is the standard Euclidean vector norm. ``p=inf`` gives the magnitude of the largest element. For *x* a matrix, ``p=2`` is the Frobenius norm. For operator matrix norms, use :func:`~mpmath.mnorm` instead. You can use the string 'inf' as well as float('inf') or mpf('inf') to specify the infinity norm. **Examples** >>> from mpmath import * >>> mp.dps = 15; mp.pretty = False >>> x = matrix([-10, 2, 100]) >>> norm(x, 1) mpf('112.0') >>> norm(x, 2) mpf('100.5186549850325') >>> norm(x, inf) mpf('100.0') .. _rst_mpm_mnorm: Matrix norm --------------------------------------------------------------------------------- .. method:: ctx.mnorm(A, p=1) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Gives the matrix (operator) `p`-norm of A. Currently ``p=1`` and ``p=inf`` are supported: ``p=1`` gives the 1-norm (maximal column sum) ``p=inf`` gives the `\infty`-norm (maximal row sum). You can use the string 'inf' as well as float('inf') or mpf('inf') ``p=2`` (not implemented) for a square matrix is the usual spectral matrix norm, i.e. the largest singular value. ``p='f'`` (or 'F', 'fro', 'Frobenius, 'frobenius') gives the Frobenius norm, which is the elementwise 2-norm. The Frobenius norm is an approximation of the spectral norm and satisfies .. math :: \frac{1}{\sqrt{\mathrm{rank}(A)}} \|A\|_F \le \|A\|_2 \le \|A\|_F The Frobenius norm lacks some mathematical properties that might be expected of a norm. For general elementwise `p`-norms, use :func:`~mpmath.norm` instead. **Examples** >>> from mpmath import * >>> mp.dps = 15; mp.pretty = False >>> A = matrix([[1, -1000], [100, 50]]) >>> mnorm(A, 1) mpf('1050.0') >>> mnorm(A, inf) mpf('1001.0') >>> mnorm(A, 'F') mpf('1006.2310867787777')