How to know if a time series sequence is predictibale or just random (Univariate time series prediction)?

Question

I'm trying to predict a current value of a variable based on the its previous 10 values. I tried multiple time series approaches including ARIMA, LSTM and linear regression... None of them really performed well, so I'm starting to think that the sequence of data I have is just random and not predictable.

Please if you have any advice, let me know. Or if you know of any metrics I can compute to make sure that the sequences of data I have are not just random.

for LSTM I'm trying to use the Window Method in to do my prediction in the following link: https://machinelearningmastery.com/time-series-prediction-lstm-recurrent-neural-networks-python-keras/

Here's the auto-correlation plot and the plot and a part of the data sequence I'm using:

Answer 1

The metric to the time series forecastability is "the spectral entropy". I learned it from some talk of Rob Hyndman, so here is the description of his implementation for R tsfeatures package, entropy

The spectral entropy is the Shannon entropy −∫π−πf^(λ)logf^(λ)dλ, where f^(λ) is an estimate of the spectral density of the data. This measures the “forecastability” of a time series, where low values indicate a high signal-to-noise ratio, and large values occur when a series is difficult to forecast.

entropy(AirPassengers)
#>   entropy 
#> 0.2961049