# Modeling non-linear (short) time series and cross-validate them

beginner data scientist here. Time series analysis is a completly new area for me, so please correct me if i write something that makes no sense.

I have many multivariante short time series, between 6 and 16 total times of observations, each describing the lifetime of a tested electrical part. All series describe the test of different models under different conditions, but all tested models are from the same type.(eg. type=CPU, model=i3,i5,i7)

Ultimately i want to find a model which describes all series from those lifetime test and is able to predict long-term.

In the first step i am trying to find models which fit only on one series and are using only one column of the series.

One of those series could look like this(x(t=0) will always be 100):

x[99.86, 99.34, 98.63, 98.19, 97.38, 96.55] observed val
t[1000,2000,3000,4000,5000,6000] time in hours


Another one could look like:

x[99.98, 99.61, 99.16, 98.75, 98.36, 98.02, 97.70, 97.41, 97.11] observed val
t[1000,2000,3000,4000,5000,6000,7000,8000,9000] time in hours


We are interested in forecasting the time t when x will reach a threshold eg x<=80.00

From underlying process we know that the decay in x is not constant, but growing over time. Also there is no sesonality.

My question is how to find appropriate models and in the next step compare them?

After reading some textbooks, my first guess was to check if an ARIMA or ETS model would fit. I realized i need to first check if my series is stationary, which clearly is not. I tried to difference the series or taking the log, but could not stationarize it. Do i need to find a model for the trend and subtract this model from my observed data to get a stationary series? If so, how would i find this trend function and would the result of the difference (observed - trend) be called residuals?

My supervisor suggested that i maybe have to look into non-linear time series and suggested me some books. The problem is i am not even familiar with regular linear time series. Does anyone know if there is a test to check if i indeed have to use a non linear model? And what would be the first non linear models to try?

My last question is how would i compare different models and their forecasts? I have seen there is time series cross validation, but aren't my series to short to divide them?

• Thank you for your answer. I included some plot, but i don't think one can guess that this resembles expotential decay. We know from the underlying process, that some physical phenomenon causes our test object to degrade and that this phenomenon further accelerates the degradation, thus we assume the trend will follow expotential decay. – elmo1113 Jul 16 '19 at 17:42
• I noticed that your response starts at 100 and decreases monotonically, is this per chance a percentage / proportion? – user2974951 Jul 17 '19 at 7:51
• @user2974951 yes it it is. It is the relative power of the tested unit. Thus will always be 100 (%) at the t=0. – elmo1113 Jul 17 '19 at 7:57
• Does it also decrease monotonically then through time? Eventually leading to 0? – user2974951 Jul 17 '19 at 8:01
• yes it decreases monotonically. There can be sometimes local optima, but this are measurement errors which we can ignore. The knowledge about the physical process suggest that it the UUT will degrade slowly but continuous and falls faster with time. It is very likely that the process will eventually reach 0 – elmo1113 Jul 17 '19 at 8:09