Octave supports various helpful statistical functions. Many are useful as initial steps to prepare a data set for further analysis. Others provide different measures from those of the basic descriptive statistics.
y = center (x) ¶y = center (x, dim) ¶Center data by subtracting its mean.
If x is a vector, subtract its mean.
If x is a matrix, do the above for each column.
If the optional argument dim is given, operate along this dimension.
Programming Note: center has obvious application for normalizing
statistical data. It is also useful for improving the precision of general
numerical calculations. Whenever there is a large value that is common
to a batch of data, the mean can be subtracted off, the calculation
performed, and then the mean added back to obtain the final answer.
See also: zscore.
z = zscore (x) ¶z = zscore (x, opt) ¶z = zscore (x, opt, dim) ¶[z, mu, sigma] = zscore (…) ¶Compute the Z score of x.
If x is a vector, subtract its mean and divide by its standard deviation. If the standard deviation is zero, divide by 1 instead.
The optional parameter opt determines the normalization to use when
computing the standard deviation and has the same definition as the
corresponding parameter for std.
If x is a matrix, calculate along the first non-singleton dimension. If the third optional argument dim is given, operate along this dimension.
The optional outputs mu and sigma contain the mean and standard deviation.
z = normalize (x) ¶z = normalize (x, dim) ¶z = normalize (…, method) ¶z = normalize (…, method, option) ¶z = normalize (…, scale, scaleoption, center, centeroption) ¶[z, c, s] = normalize (…) ¶Return a normalization of the data in x using one of several available scaling and centering methods.
normalize by default will return the zscore of x,
defined as the number of standard deviations each element is from the mean
of x. This is equivalent to centering at the mean of the data and
scaling by the standard deviation.
The returned value z will have the same size as x. The optional return variables c and s are the centering and scaling factors used in the normalization such that:
z = (x - c) ./ s
If x is a vector, normalize will operate on the data in
x.
If x is a matrix, normalize will operate independently on
each column in x.
If x is an N-dimensional array, normalize will operate
independently on the first non-singleton dimension in x.
If the optional second argument dim is given, operate along this dimension.
normalize ignores NaN values is x similar to the behavior of
the omitnan option in std, mean, and median.
The optional inputs method and option can be used to specify the type of normalization performed on x. Note that only the scale and center options may be specified together using any of the methods defined below. Valid normalization methods are:
zscore(Default) Normalizes the elements in x to the scaled distance from a central value. Valid Options:
std(Default) Data is centered at mean (x) and scaled by the
standard deviation.
robustData is centered at median (x) and scaled by the median
absolute deviation.
normz is the general vector norm of x, with option being the normalization factor p that determines the vector norm type according to:
z = [sum (abs (x) .^ p)] ^ (1/p)
p can be any positive scalar, specific values being:
p = 1x is normalized by sum (abs (x)).
p = 2(Default) x is normalized by the Euclidian norm, or vector magnitude, of the elements.
P = Infx is normalized by max (abs (x)).
scalex is scaled by a factor determined by option, which can be a numeric scalar or one of the following:
std(Default) x is scaled by its standard deviation.
madx is scaled by its median absolute deviation.
firstx is scaled by its first element.
iqrx is scaled by its interquartile range.
rangex is scaled to fit the range specified by option as a two element scalar row vector. The default range is [0, 1].
centerx is shifted by an amount determined by option, which can be a numeric scalar or one of the following:
mean(Default) x is shifted by mean (x).
medianx is shifted by median (x).
medianiqrx is shifted by median (x) and scaled by the
interquartile range.
Known MATLAB incompatibilities:
See also: zscore, iqr, norm, rescale, std, median, mean, mad.
n = histc (x, edges) ¶n = histc (x, edges, dim) ¶[n, idx] = histc (…) ¶Compute histogram counts.
When x is a vector, the function counts the number of elements of
x that fall in the histogram bins defined by edges. This
must be a vector of monotonically increasing values that define the edges
of the histogram bins.
n(k)
contains the number of elements in x for which
edges(k) <= x < edges(k+1).
The final element of n contains the number of elements of x
exactly equal to the last element of edges.
When x is an N-dimensional array, the computation is carried out along dimension dim. If not specified dim defaults to the first non-singleton dimension.
When a second output argument is requested an index matrix is also returned. The idx matrix has the same size as x. Each element of idx contains the index of the histogram bin in which the corresponding element of x was counted.
See also: hist.
unique function documented at unique is often
useful for statistics.
c = nchoosek (n, k) ¶c = nchoosek (set, k) ¶Compute the binomial coefficient of n or list all possible combinations of a set of items.
If n is a scalar then calculate the binomial coefficient of n and k which is defined as
/ \ | n | n (n-1) (n-2) ... (n-k+1) n! | | = ------------------------- = --------- | k | k! k! (n-k)! \ /
This is the number of combinations of n items taken in groups of size k.
If the first argument is a vector, set, then generate all
combinations of the elements of set, taken k at a time, with
one row per combination. The result c has k columns and
nchoosek (length (set), k) rows.
For example:
How many ways can three items be grouped into pairs?
nchoosek (3, 2) ⇒ 3
What are the possible pairs?
nchoosek (1:3, 2)
⇒ 1 2
1 3
2 3
Programming Note: When calculating the binomial coefficient nchoosek
works only for non-negative, integer arguments. Use bincoeff for
non-integer and negative scalar arguments, or for computing many binomial
coefficients at once with vector inputs for n or k.
P = perms (v) ¶P = perms (v, "unique") ¶Generate all permutations of vector v with one row per permutation.
Results are returned in reverse lexicographic order if v is in ascending
order. If v is in a different permutation, then the result is permuted
that way too. Consequently, an input in descending order yields a result in
normal lexicographic order. The result has size
factorial (n) * n, where n is the length of v.
Any repeated elements are included in the output.
If the optional argument "unique" is given then only unique
permutations are returned, using less memory and taking less time than calling
unique (perms (v), "rows").
Example 1
perms ([1, 2, 3]) ⇒ 3 2 1 3 1 2 2 3 1 2 1 3 1 3 2 1 2 3
Example 2
perms ([1, 1, 2, 2], "unique") ⇒ 2 2 1 1 2 1 2 1 2 1 1 2 1 2 2 1 1 2 1 2 1 1 2 2
Programming Note: If the "unique" option is not used, the length of
v should be no more than 10-12 to limit memory consumption. Even with
"unique", there should be no more than 10-12 unique elements in
v.
y = ranks (x) ¶y = ranks (x, dim) ¶y = ranks (x, dim, rtype) ¶Return the ranks (in the sense of order statistics) of x along the first non-singleton dimension adjusted for ties.
If the optional dim argument is given, operate along this dimension.
The optional parameter rtype determines how ties are handled. All
examples below assume an input of [ 1, 2, 2, 4 ].
"fractional" (default) for fractional ranking (1, 2.5,2.5, 4);
"competition" for competition ranking (1, 2, 2, 4);"modified" for modified competition ranking (1, 3, 3, 4);"ordinal" for ordinal ranking (1, 2, 3, 4);"dense" for dense ranking (1, 2, 2, 3).cnt = run_count (x, n) ¶cnt = run_count (x, n, dim) ¶Count the upward runs along the first non-singleton dimension of x of length 1, 2, …, n-1 and greater than or equal to n.
If the optional argument dim is given then operate along this dimension.
See also: runlength.
count = runlength (x) ¶[count, value] = runlength (x) ¶Find the lengths of all sequences of common values.
count is a vector with the lengths of each repeated value.
The optional output value contains the value that was repeated in the sequence.
runlength ([2, 2, 0, 4, 4, 4, 0, 1, 1, 1, 1]) ⇒ 2 1 3 1 4
See also: run_count.