Mode - Maple Help

Statistics

 Mode
 compute the mode

 Calling Sequence Mode(A, ds_options) Mode(X, rv_options)

Parameters

 A - X - algebraic; random variable or distribution ds_options - (optional) equation(s) of the form option=value where option is one of ignore, weights, type, bandwidth, bins, left, right, result; specify options for computing the mode of a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the mode of a random variable

Description

 • The Mode function computes the mode of the specified random variable or computes the kernel or sample mode for a data set.
 • The first parameter can be a data set (for example, a Vector), a Matrix data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

Computation

 • By default, all computations involving random variables are performed symbolically (see option numeric below).
 • All computations involving data are performed in floating-point; therefore, all data provided must have type/realcons and all returned solutions are floating-point, even if the problem is specified with exact values.
 • For more information about computation in the Statistics package, see the Statistics[Computation] help page.

Data Set Options

 The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.
 • ignore=truefalse -- This option controls how missing data is handled by the Mode command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Mode command will return undefined. If ignore=true all missing items in A will be ignored. The default value is false.
 • weights=rtable -- Vector of weights (one-dimensional rtable). If weights are given, the Mode function will scale each data point to have given weight. Note that the weights provided must have type/realcons and the results are floating-point, even if the problem is specified with exact values. Both the data array and the weights array must have the same number of elements.
 • type=sample or kernel -- The type options indicates what mode type should be calculated on the data set (by default this is set to sample'.  If sample' is set, the sample mode (defined as the most frequent data item in the set) will be calculated.  If kernel is set, the kernel mode (defined as the mode of a kernel density estimate based on the data set) will be calculated.
 • kernel=gaussian, biweight, epanechnikov, triangular or rectangular -- If type='kernel' then this option specifies the type of kernel used in the kernel density estimate (by default this is gaussian).  This parameter is ignored otherwise.
 • bandwidth=realcons -- If type='kernel' then this options specifies the bandwidth of the kernel density estimate (by default this is 1/4).  This parameter is ignored otherwise.
 • bins=posint -- If type='kernel' and result='discrete' then this option represents the number of bins used in the operation of calculating a discrete kernel density estimate (by default this is 512).  This parameter is ignored otherwise.
 • left=realcons -- If type='kernel' then this option represents the lower (left) bound on elements of the data sample used in calculating the kernel density estimate.  This parameter is ignored otherwise.
 • right=realcons -- If type='kernel' then this option represents the upper (right) bound on elements of the data sample used in calculating the kernel density estimate.  This parameter is ignored otherwise.
 • method=discrete or exact -- If type='kernel' then this parameter specifies the output method for the kernel density estimate (by default this is discrete).  This parameter is ignored otherwise.

Random Variable Options

 The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.
 • numeric=truefalse -- By default, the mode is computed using exact arithmetic. To compute the mode numerically, specify the numeric or numeric = true option.

Notes

 • This function is only guaranteed to return one potential mode. In cases where multiple modes exist, only the first detected mode is guaranteed to be returned. If a single mode is detected, it is returned as a number, whereas if multiple modes are detected, they are returned as a set or range.
 • Note that discrete kernel density estimation removes results that do not fall within the generated range, including missing data.  Hence, if you do not specify result='exact' option, missing data will be automatically ignored.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$

Compute the mode of the Weibull distribution with parameters p and q.

 > $\mathrm{Mode}\left(\mathrm{Weibull}\left(p,q\right)\right)$
 $\left\{\begin{array}{cc}{0}& {q}{<}{1}\\ {p}{}{\left({1}{-}\frac{{1}}{{q}}\right)}^{\frac{{1}}{{q}}}& {\mathrm{otherwise}}\end{array}\right\$ (1)

Use numeric parameters.

 > $\mathrm{Mode}\left(\mathrm{Weibull}\left(3,5\right)\right)$
 $\frac{{3}{}{{4}}^{{1}}{{5}}}{}{{5}}^{{4}}{{5}}}}{{5}}$ (2)
 > $\mathrm{Mode}\left(\mathrm{Weibull}\left(3,5\right),\mathrm{numeric}\right)$
 ${2.869057499}$ (3)

Determine the mode of a set of sample data (most frequently occurring item).

 > $A≔⟨1,2,2,2,3,4,5,5,5,6,7,7,7,7,7,7,8,8,1,1,2,3,4,5⟩:$
 > $\mathrm{Mode}\left(A\right)$
 ${7.}$ (4)

Generate a random sample of size 100000 drawn from the above distribution and compute the sample mode.

 > $A≔\mathrm{Sample}\left(\mathrm{Weibull}\left(3,5\right),{10}^{5}\right):$
 > $\mathrm{Mode}\left(A,\mathrm{type}=\mathrm{kernel}\right)$
 ${2.86291689708702}$ (5)

Compute the mode of a sum of two random variables.

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right)$
 ${X}{≔}{\mathrm{_R3}}$ (6)
 > $Y≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(10,1\right)\right):$
 > $\mathrm{Mode}\left(X+Y\right)$
 $\left\{{15}\right\}$ (7)

Verify this using simulation.

 > $C≔\mathrm{Sample}\left(X+Y,{10}^{5}\right):$
 > $\mathrm{Mode}\left(C,\mathrm{type}=\mathrm{kernel}\right)$
 ${14.8615463667490}$ (8)

Compute the mode of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{Mode}\left(V,\mathrm{weights}=W\right)$
 ${67.}$ (9)
 > $\mathrm{Mode}\left(V,\mathrm{type}=\mathrm{kernel},\mathrm{weights}=W\right)$
 ${67.0284486386156}$ (10)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,1527,88464\right],\left[2,878,96484\right],\left[4,995,96484\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {1527}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {96484}\end{array}\right]$ (11)

We compute the mode of each of the columns.

 > $\mathrm{Mode}\left(M\right)$
 $\left[\begin{array}{ccc}\left\{{3.}{,}{4.}\right\}& {1527.}& {96484.}\end{array}\right]$ (12)

References

 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

Compatibility

 • The A parameter was updated in Maple 16.