mars.tensor.random.standard_t#
- mars.tensor.random.standard_t(df, size=None, chunk_size=None, gpu=None, dtype=None)[source]#
Draw samples from a standard Student’s t distribution with df degrees of freedom.
A special case of the hyperbolic distribution. As df gets large, the result resembles that of the standard normal distribution (standard_normal).
- Parameters
df (float or array_like of floats) – Degrees of freedom, should be > 0.
size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g.,
(m, n, k)
, thenm * n * k
samples are drawn. If size isNone
(default), a single value is returned ifdf
is a scalar. Otherwise,mt.array(df).size
samples are drawn.chunk_size (int or tuple of int or tuple of ints, optional) – Desired chunk size on each dimension
gpu (bool, optional) – Allocate the tensor on GPU if True, False as default
dtype (data-type, optional) – Data-type of the returned tensor.
- Returns
out – Drawn samples from the parameterized standard Student’s t distribution.
- Return type
Tensor or scalar
Notes
The probability density function for the t distribution is
\[P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}\]The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean.
The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student.
References
- 1
Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002.
- 2
Wikipedia, “Student’s t-distribution” http://en.wikipedia.org/wiki/Student’s_t-distribution
Examples
From Dalgaard page 83 1, suppose the daily energy intake for 11 women in Kj is:
>>> import mars.tensor as mt
>>> intake = mt.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \ ... 7515, 8230, 8770])
Does their energy intake deviate systematically from the recommended value of 7725 kJ?
We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value?
>>> s = mt.random.standard_t(10, size=100000) >>> mt.mean(intake).execute() 6753.636363636364 >>> intake.std(ddof=1).execute() 1142.1232221373727
Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1.
>>> t = (mt.mean(intake)-7725)/(intake.std(ddof=1)/mt.sqrt(len(intake))) >>> import matplotlib.pyplot as plt >>> h = plt.hist(s.execute(), bins=100, normed=True)
For a one-sided t-test, how far out in the distribution does the t statistic appear?
>>> (mt.sum(s<t) / float(len(s))).execute() 0.0090699999999999999 #random
So the p-value is about 0.009, which says the null hypothesis has a probability of about 99% of being true.