Oke so i've done some research myself and hope that I can help someone with answering my questions. I also want to thank the Algo Trading and Quant Crew Discord groups for helping me answering the questions. I also want to thank Ramon for discussing some answers with me.
Question 1: Are the log returns of some fixed time period just used for simplicity or is there something else?
This is pretty simple, I won't be giving an in depth explanation answering this question. I am going to try to explain each one in a single sentence, for a more extensive explanation please visit the sources or reference. There are four main reasons why someone would want to use log returns for some fixed time period.
- Log-normality, this assumes that the returns are distributed log-normal as much of classic statistics presumes normality.
- Approximate raw-log equality, which ensures they are close in value to raw returns.
- Time-additivity, the individual log returns can just be added which reduces the algorithmic complexity.
- Mathematical ease, calculus identities from the e^x which is useful as much of financial mathematics is build upon continuous time stochastic processes.
Question 2: Am I allowed to model on the log returns of a different kind of data into a homogeneous series like volume bars?
Yes, I am allowed to do that as I can use the log returns of any kind of data as an Alpha can be found anywhere. The goal is to generate an Alpha not absolute statistical purity. A personal thing I am seeing is that often time based returns are being used. I think the motive is that with the fixed time period, the length in time (t) is a constant and only the frequency (f) of the signal needs to be analyzed. With a non-fixed time series or every other series that doesn't involve time, the length in time (t) and the frequency (f) of the signal needs to be estimated. This is because we need to predict the future of the model with time (t) as we might want to use the efficient frontier to split our portfolio between the different models. The alternative bars might need some calculation time so, the model doesn't have as fast of a reaction time as models that just use raw data. But on the brighter side, alternative bars might have better statistical properties as Marcos Lopez De Prado explained in his book Advances in Financial Machine Learning. Time bars also assume that markets are chronological, which they are not, where alternative bars doesn't assume that trades occur at a constant time interval.
So, modeling on the log returns of a different kind of data into a homogeneous series like volume bars is allowed. This might result in better statistical properties and fixes the chronological market problem. When doing this remember that you might want to estimate not only the frequency of the signal but also the time of the signal and to calculate the latency of the signal.
Question 3: Am I allowed to model on the log returns of a simple model that has an input of non-stationary time series data that has constant signals, like the log returns of the crossing of two MA's on non-stationary data (time candles)?
Yes, as mentioned in the answer of question 2, you can model any kind of data as an Alpha can be found anywhere. The goal is to generate an Alpha not absolute statistical purity.
This is basically what you're doing when taking a couple indicators and layering them above each other. But what about the insertion of non-stationary data? The insertion of non-stationary data is irrelevant as the signal needs to be stationary.
When modeling on top of another model, the margin of error of the first model needs to be considered in the statistics of the 2nd model. So when the model is made on top of another model the chances of success are getting smaller and smaller.
Question 4: Am I allowed to model on the log returns of a more complex model that has an input of non-stationary time series data that has non-constant signals, like the log returns of when a RSI on non-stationary data (time candles) hits 70 or 30 till a 50 RSI?
The answer to this question is the same as question 3. Everything can be put inside a model to generate an Alpha, the goal is to generate an Alpha not absolute statistical purity.
Everything can be modeled, it might not follow certain statistical rules but that doesn't matter. The thing that matters is that the Alpha signals follows the statistical rules. When using some kind of homogeneous series derived from regular sampling, the statistical properties might be better and might solve the chronological market problem. Modeling on homogeneous data has pros and cons as every other existing choice.
We are going to build our model on inhomogeneous and homogeneous data to make sure we include everything in our model. We need to find an equillibruim in inserting the data to the different features as we don't have unlimited computer power.