Numerical transformations and descriptive statistics

Centering of a matrix

mat.centered(res, data, population, opt)

Returns the centered version of the matrix

>>> 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

mat.standardized(res, data, population, opt)

Returns the standardized version of the matrix

>>> 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 [244].

Trace

mat.trace()

Returns the trace of the matrix. See also: Wikipedia [1513].

>>> 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

mat.squaredNorm(partialmode=full)

Returns the squared norm of the matrix.

>>> 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

mat.Norm(partialmode=full)

Returns the norm of the matrix.

>>> 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

mat.covariance_matrix(use_crossproduct=False)

Returns the covariance matrix of the matrix. See also Wikipedia [1509], Wikipedia [1487].

>>> 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

mat.correlation(use_crossproduct=False)

Returns the correlation matrix of matrix ?matA. See also Wikipedia [1485].

>>> 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

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

>>> 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", "")

Vector norm of a matrix

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 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')

Matrix norm

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

\[\frac{1}{\sqrt{\mathrm{rank}(A)}} \|A\|_F \le \|A\|_2 \le \|A\|_F\]

The Frobenius norm lacks some mathematical properties that might be expected of a norm.

For general elementwise \(p\)-norms, use 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')