2025-06-24 22:38:13
I saw a post online this morning that recomended the transformation
I could see how this could be very handy. Often you want something like a logarithmic scale, not for the exact properties of the logarithm but because it brings big numbers closer in. And for big values of x there’s little difference between log(x) and log(1 + x).
The function above is linear near the origin, literally linear for negative values and approximately linear for small positive values.
I’ve occasionally needed something like a log scale, but one that would handle values that dip below zero. This transformation would be good for that. If data were equally far above and below zero, I’d use something like arctangent instead.
2025-06-24 04:30:42
Suppose you have a system with n possible states. The entropy of the system is maximized when all states are equally likely to occur. The entropy is minimized when one outcome is certain to occur.
You can say more. Starting from any set of probabilities, as you move in the direction of more uniformity, you increase entropy. And as you move in the direction of less uniformity, you decrease entropy.
These statements can be quantified and stated more precisely. That’s what the rest of this post will do.
***
Let pi be the probability of the ith state and let p be the vector of the pi.
Then the entropy of p is defined as
If one of the p‘s is 1 and the rest of the p‘s are zero, then H(p) = 0. (In the definition of entropy, 0 log2 0 is taken to be 0. You could justify this as the limit of x log2x as x goes to zero.)
If each of the pi are equal, pi = 1/n, then H(p) = log2n. The fact that this is the maximum entropy, and that compromises between the two extremes always decrease entropy, comes from the fact that the entropy function H is concave (proof). That is, if p1 is one list of probabilities and p2 another, then
When we speak informally of moving from p1 in the direction of p2, we mean we increase the parameter λ from 0 to some positive amount no more than 1.
Because entropy is concave, there are no local maxima. As you approach the location of global maximum entropy, i.e. equal state probabilities, from any direction, entropy increases monotonically.
The post Uniformity increases entropy first appeared on John D. Cook.2025-06-23 20:38:55
The sinc function
sinc(x) = sin(x) / x
comes up continually in signal processing. If x is moderately small, the approximation
sinc(x) ≈ (2 + cos(x))/3
is remarkably good, with an error on the order of x4/180. This could be useful in situations where you’re working with the sinc function and the x in the denominator is awkward to deal with and you’d rather have a pure trig function.
Here’s a plot:
Of course the approximation is only good for small x. For large x the sinc function approaches zero while (2 + cos(x))/3 oscillates with constant amplitude forever.
When the approximation is good, it is very, very good, which reminds me of this nursery rhyme.
There was a little girl,
Who had a little curl,
Right in the middle of her forehead.
When she was good,
She was very, very good,
But when she was bad, she was horrid.
2025-06-23 08:09:55
Four years ago I wrote a blog post about simple solutions to client problems. The post opens by recounting a conversation with a friend that ended with my friend saying “So, basically you’re recommending division.”
That conversation came back to me recently when I had a similar conversation during a deposition. I had to explain that the highly sophisticated method I used to arrive at a number in my report was to take the ratio of two other numbers.
I don’t always do division for clients. Sometimes I do multiplication, as when a lawyer asked me to compute the momentum of a car. There I had to multiply mass and velocity. (I actually turned down that project because I had a sense there was a lot more to the problem than I was being told.)
Of course real-world problems do not always have trivial solutions. Sometimes getting to the solution is complicated even though the solution itself is simple. And sometimes the solution is complicated because that’s just how things are. You might need something complicated, like a cepstrum analysis, or you might just need arithmetic.
The post Arithmetic for fun and profit first appeared on John D. Cook.2025-06-23 01:31:01
A couple days ago I wrote about the the problem that Bitcoin requires to be solved as proof-of-work. In a nutshell, you need to tweak a block of transactions until the SHA256 double hash of its header is below a target value [1]. Not all cryptocurrencies use proof of work, but those that do mostly use hash-based puzzles.
Other cryptocurrencies use a different hashing problem, but they still use hashes. Litecoin and Dogecoin use the same proof-of-work problem, similar to the one Bitcoin uses, but with the scrypt (pronounced S-crypt) hash function. Several cryptocurrencies use a hashing problem based on Equihash. Monero uses its RandomX algorithm for proof-of-work, and although this algorithm has multiple components, it ultimately solves a hashing problem. [2]
Why do cryptocurrencies use hashing problems for proof of work? In principle they could use any computational problem that is hard to solve but easy to verify, such as numerous problems in computational number theory.
One reason is that computer scientists are confident that quantum computing would not reduce the difficulty of solving hash puzzles, even though it would reduce the difficulty of factoring-based puzzles. Also, there is general agreement that it’s unlikely a mathematical breakthrough will find a weakness in hashing functions.
Ian Cassels said “Cryptography is a mixture of mathematics and muddle, and without the muddle the mathematics can be used against you.” Hashing is much more muddle than mathematics.
Hash puzzles work well for demonstrating work done, but they’re otherwise useless. They keep the wheels of cryptocurrencies turning, but the solutions themselves are intrinsically worthless.
Wouldn’t it be nice if crypto miners were solving useful problems like protein folding? You could do that. In fact there is a cryptocurrency FoldingCoin that does just that. But FoldingCoin has a market cap seven orders of magnitude smaller than Bitcoin, on the order of $200,000 compared to Bitcoin’s market cap of $2T.
Cryptocurrencies that use proof of useful work have not taken off. This might necessarily be the case. Requiring practical work creates divergent incentives. If you base a currency on the difficulty of protein folding computations, for example, it would cause major disruption if a pharmaceutical company decided to get into mining protein folding cryptocurrency at a loss because it values the results.
Going back to Cassels’ remark about mathematics and muddle, practical real-world problems often have a mathematical structure. Which is a huge blessing, except when you’re designing problems to be hard. Hash-based problems have gradually become easier to solve over time, and cryptocurrencies have adjusted. But a mathematical breakthrough for solving a practical problem would have the potential to disrupt a currency faster than the market could adapt.
[1] You don’t change the transaction amounts, but you may change the order in which the transactions are arranged into a Merkle tree so that you get different hash values. You can also change a 32-bit nonce, and a timestamp, but most of the degrees of freedom you need in order to find an acceptable hash comes from rearranging the tree.
[2] Both scrypt and Equihash were designed to be memory intensive and to thwart the advantage custom ASIC mining hardware. However, people have found a way to use ASIC hardware to solve scrypt and Equihash problems. RandomX requires running a randomly generated problem before hashing the output in an attempt to frustrate efforts to develop specialized mining hardware.
The post Why use hash puzzles for proof-of-work? first appeared on John D. Cook.2025-06-21 21:16:55
Suppose you have a list of positive data points y1, y2, …, yn and you wanted to find a value α that minimizes the squared distances to each of the y‘s.
Then the solution is to take α to be the mean of the y‘s:
This result is well known [1]. The following variation is not well known.
Suppose now that you want to choose α to minimize the squared relative distances to each of the y‘s. That is, you want to minimize the following.
The value of alpha this expression is the contraharmonic mean of the y‘s [2].
[1] Aristotle says in the Nichomachean Ethics “The mean is in a sense an extreme.” This is literally true: the mean minimizes the sum of the squared errors.
[2] E. F. Beckenbach. A Class of Mean Value Functions. The American Mathematical Monthly. Vol. 57, No. 1 (Jan., 1950), pp. 1–6
The post Minimize squared relative error first appeared on John D. Cook.
