qfuncinv

Inverse Q function

Syntax

z = qfuncinv(y)

Description

z = qfuncinv(y) returns the input argument of the Q function for which the output value of the Q function is y. For more information, see Algorithms.

Examples

Recover Argument of Q Function

Recover the Q function input argument by using the inverse Q function. Show the inverse relationship between Q function and its inverse.

Calculate the Q function values for a real-valued input.

x1 = [0 1 2; 3 4 5];
y1 = qfunc(x1)

y1 = 2×3

    0.5000    0.1587    0.0228
    0.0013    0.0000    0.0000

Recover the Q function input argument by calculating the inverse Q function values for y1.

x1_recovered = qfuncinv(y1)

x1_recovered = 2×3

     0     1     2
     3     4     5

Confirm the original and recovered Q functions arguments are the same.

isequal (x1,x1_recovered)

ans = logical
   1

Calculate the inverse of values representing Q function output values.

y2 = 0:0.2:1;
x2 = qfuncinv(y2)

x2 = 1×6

       Inf    0.8416    0.2533   -0.2533   -0.8416      -Inf

Recover the Q function output argument by calculating the Q function values for x2.

y2_recovered = qfunc(x2)

y2_recovered = 1×6

         0    0.2000    0.4000    0.6000    0.8000    1.0000

Confirm the original values and recovered inverse Q functions arguments are the same.

isequal (y2,y2_recovered)

ans = logical
   1

Input Arguments

y — Q function output

scalar | vector | matrix | array

Q function output, specified as a scalar, matrix, or array. Input values must be in the range [0, 1].

Data Types: double

Output Arguments

z — Q function input argument

scalar | vector | matrix | N-D array

Q function input argument, returned as a real-valued scalar, matrix, or array. z has the same dimensions as input y.

Algorithms

For a scalar x, the Q function is (1 – f), where f is the result of the cumulative distribution function of the standardized normal random variable. The Q function is defined as

$$Q(x)=\frac{1}{\sqrt{2\pi }}{\displaystyle \underset{x}{\overset{\infty }{\int }}\exp (-t^2/2)dt}$$

The Q function is related to the complementary error function, erfc, according to

$$Q(x)=\frac{1}{2}\text{erfc}\left(\frac{x}{\sqrt{2}}\right)$$

Version History

Introduced before R2006a