x’ forecast only on HP-32E

[Nigel (UK)]

I have to confess that I don't see I am committing any heresy by changing \(y = a + bx\) into \(x = (y-a)/b\)! In some cases - for example, unit conversions - both variables have the same "character". Any uncertainty in a temperature measurement in degrees Celsius corresponds to an exactly equivalent uncertainty when measuring the same temperature in degrees Fahrenheit. Inverting the formula in such cases is surely fine.

In other cases - for example, how population \(P\) changes with time \(t\) - the two variables are not equivalent. Not only is population harder to measure than what year it is, but its value cannot be chosen freely. It also depends on factors other than time. Even so, if I decide to fit a set of \((t,P)\) measurements to an equation \(P=a+bt+\epsilon\), where \(\epsilon\) is normally distributed with zero mean and fixed variance, I can get best estimates for \(a\) and \(b\) and then invert the equation again to get
\[t = {P - a- \epsilon\over b}.\]
This gives me a value for \(t\) along with an idea of the uncertainty in the answer.

Best of all, fire up a MCMC (Markov Chain Monte Carlo) Bayes engine and feed it with the \(t,P\) measurements and the model equation. In this situation, the "error" function \(\epsilon\) doesn't have to be normal and its parameters (e.g., variance) can be fitted from the data. You end up with a list of ten thousand (for example) sets of values for \(a\), \(b\), and the error function parameters. You can use each of these sets to calculate a spread of values for \(t\) from \(P\), and then combine them all into one. No current HP calculator does this out of the box, I fear!

Disclaimer: I am not a statistician, but I have read a few books.

Nigel (UK)