![]() The t-test is very sensible to extreme observation and will therefore compute a very high variance because of very few data points. The reason is that we have extremely skewed data. The t-test is much more conservative, telling us that the unlikeliness of the data is just $12%$ compared to the $1.1%$ of the permutation test. ![]() Ttest_ind(data_income.query('university=1'), data_income.query('university=0')) What would be the outcome of a standard t-test? from scipy.stats import ttest_ind In other words, it is very unlikely that university graduates earn the same income of non-university graduates. The permutation test is telling us that the difference is extremely unusual under the null hypothesis. T_perm = permutation_test(data_income, permute, 'university', compute_income_diff) Is this difference statistically different from zero? Let’s perform a permutation test. It looks like university graduates have higher income. The distribution is so skewed that we cannot actually visually perceive differences between the two groups. sns.kdeplot(data=data_income, x="income", hue="university")\ Let’s plot its density across the two groups. The distribution of income is very heavy tailed. Although its possible to understand antisymmetric tensors without discussing permutations and their parity, these concepts are invaluable to developing the theory. In this case we can directly compute this answer since we have a very little number of throws. In my next post, I would like to introduce a very special type of tensor whose properties are invaluable in many fields, most notably differential geometry. The question that permutation testing is trying to answer is “ how unlikely is the observed outcome under the null hypothesis that the coin is fair?”. Out of 10 coin throws, we got only 2 heads. You throw the coin 10 times and you count the number of times you get heads. Let’s start with an example: suppose you wanted to test whether a coin is fair. What does it mean? In this tutorial we are going to see in detail what this definition means, how to implement permutation tests, and their pitfalls. You therefore balance security as well.If you search “permutation test” on Wikipedia, you get the following definition:Ī permutation test (also called re-randomization test) is an exact statistical hypothesis test making use of the proof by contradiction in which the distribution of the test statistic under the null hypothesis is obtained by calculating all possible values of the test statistic under possible rearrangements of the observed data. So in summary, in order to mangle the bits better and for the algorithm's implementation onto commonly used hardware, you must compromise and perform a balancing act between S and P boxes. You've traded storage space for cpu cycles /time. And so you go round and round (in what are called rounds surprisingly) till you think that the original input is sufficiently encrypted. Somewhere inside there you might also stuff in some modular arithmetic and /or bitwise operations like SHA-1. If you're really good at it you might call it a Bent function as well. Putting the S and P boxes together creates another type of function so that now we have $x=f(x)$. You can lose bits, and you can duplicate bits. Any bit can map to any other so the input and output widths do not have to match. e-sushi's P box has a 1 - 1 mapping at the bit level, but it doesn't have to. Hence the P box to distribute the S box output as broadly as possible before doing it all again. You've sacrificed storage for less bit mangling and so less security and invert-ability. Problem is that a 4 bit S box isn't that dissimilar to a 1 bit input box, i.e. Amazing! Where did all that storage requirement go? That requires $2^4 \times 16$ nibbles or 128 bytes. ![]() ![]() e-sushi shows us 16 smaller S boxes with $2^4$ bit inputs. These things have to run on anything from mainframes to smart cards and iButtons for wide stream acceptance. This is one of the reasons that cryptography deals with large numbers. ![]() More than enough to fill a few floppy discs. As noted in this answer and this answer to another question, permutation is just a mathematical term for a function $\sigma:X$ bytes. ![]()
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