How small is Zero?

Question

Ahmed Ramadan 2023-3-3

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1922665-how-small-is-zero

回答： Walter Roberson 2023-3-3

I run some statistical analyses which result in p-values=0. Are they < the minimum positive value of a double precision (<2e-308), which is odd to me?

I am worried if another precision is used under the hood in the Statistics and Machine Learning Toolbox (e.g., corr).

0 个评论
显示 -2更早的评论隐藏 -2更早的评论

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

John D'Errico 2023-3-3

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1922665-how-small-is-zero#answer_1184800

编辑：John D'Errico 2023-3-3

在 MATLAB Online 中打开

It is easy for that to happen. Of course, you don't actually tell us WHAT test you did. But this just means you have an event out in the tails. And it need not be even that far out.

For example, what is the probability of a standard Normally distributed number being as far out as -40?

format long g
Z = (-40:-35)';
[Z,normcdf(Z)]
ans = 6×2
                       -40                         0
                       -39                         0
                       -38     2.88542835100396e-316
                       -37     5.72557122252514e-300
                       -36     4.18262406579739e-284
                       -35     1.12491070647255e-268

So below 38 sigma, the probability just underflows. Actually, in context, -39*sigma really is a long way out. But outliers exist. It may mean your assumptions of normality may be in question.

But it is not at all impossible for a statistical test to yield an underflow as you have reported. You just need some data that justifies a result out in the tails.

4 个评论
显示 2更早的评论隐藏 2更早的评论

Walter Roberson 2023-3-3

编辑：Walter Roberson 2023-3-3

在 MATLAB Online 中打开

smallest_normal_number = realmin
smallest_normal_number = 2.2251e-308
smallest_representable_number = eps(smallest_normal_number)
smallest_representable_number = 4.9000e-324

IEEE 754 Double Precision has two ranges.

In the range that is used nearly all of the time, the representation is sign * 2^(exponent+ bias) * (2^53 + mantissa) . So for example, 1.0 exactly is stored as [0, -53+bias, 0] meaning (-1)^0*2^(-53) * (2^53 + 0) -- a mantissa of 0 and an exponent that cancels out the 2^53 giving 1. And 2.0 exactly is stored as [0, -52+bias, 0] meaning (-1)^0*2^(-52)*(2^53 + 0) canceling out to 2^(53-52) giving 2^1 == 2.0 . In this "normalized" range, in each case, the mantissa is an integer at most 2^53-1 and the representation is always such that doubling the number just increases the exponent by 1 without changing the mantissa.

The second range applies only to numbers that are less than 2^(-1022) . For such numbers, the representation is instead a fixed 2^(-1074) * mantissa where mantissa is at most 2^53-1 . In that range, doubling the number leaves the exponent fixed and requires doubling the mantissa itself. The increment between adjcent representable numbers in this "denormalized" range is fixed at 2^(-1074), whereas the increment numbers in the "normalized" range varies with floor(log2(abs()) of the number.

Ahmed Ramadan 2023-3-3

Thank you, Les and Walter!

I do appreciate it.

请先登录，再进行评论。

Answer 2

Walter Roberson 2023-3-3

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1922665-how-small-is-zero#answer_1184925

在 MATLAB Online 中打开

format long g

A = sym(floor(randn(5,2) * 16)) / 16

A =

cord = corr(double(A))

cord = 2×2

1 -0.864423766518445 -0.864423766518445 1

cors = corr(A)

cors =

corsv = vpa(cors, 16)

corsv =

corsd = double(cors)

corsd = 2×2

1 -0.864423766518445 -0.864423766518445 1

That is, the corr() function happens to be able to run on symbolic numbers, and will provide exact results over a rather wide range -- 10^-10000 not being a problem for example. So you could test your "exact 0"