Growth rate patterns through time: ARMA an appropriate approach? The Data
Total Biomass and Annual Growth Rates for 37 permanent sample (i.e., repeated sample) forest plots that have been resampled at different intervals (and sometimes in different years) for 80 years. 


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*Note: additional confounding variables (e.g., soil nutrients, forest age, etc.) do exist and would likely need to be incorporated in my final model. 


Scientific question
Are forest growth rates increasing through time?
Data Structure
The growth rates are clearly more complicated than a linear trend -- they're highly variable within plots and between plots. See growth rates for 2 of the 37 plots below:

Stats Question
I know that ARMA models are often used for time series data, but Ives et al. (2010) don't mention this type of data as a candidate for ARMA analyses; instead they mention using ARMA for population densities. 
My question: Would an ARMA(p,q) model be an appropriate approach for determining an increasing trend in my time series data??


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*If not, what alternative analysis approach would be more appropriate/valid for my data?


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*Some sort of mixed model, perhaps? 



 Update 
Actually, is it true that an ARMA model is only appropriate if the observations are equally spaced in time? 
If so, what options do I have for my data which are not evenly spaced in time???

Cited:
Ives, A.R., Abbott, K.C. and Ziebarth, N.L., 2010. Analysis of ecological time series with arma (p, q) models. Ecology, 91(3), pp.858-871.
 A: You are trying to detect a trend in the data, ARMA won't work for this because ARMA specifically requires that the data should have no trend in it, or more specifically that the time series is [stationary], for it to be modeled as an ARMA process. When representing data as an ARMA process you first have to remove the trend using differencing, then you model it as an ARMA process. The combined model is then called an ARIMA model. 
In your case, I think your best estimate is to model that data as a linear function of time. 
That is use simple linear regression to try to fit a model of the type: 
$GR= at+b$  
with $GR$ the growth rate and $t$ the time of the measurement to your data , and then test the goodness of fit of your model and its statistical significance. 
This would allow you to see how closely your data follows a linear trend, and has the added advantage of nor requiring evenly spaced measurements. 
The paper by Ives et al. (2010)  is weird, they mention the stationarity requirement, but I don't see how population density time series are necessarily stationary. They seem to be using the ARMA model itself as a test of stationarity, but there are better ways of doing that, i.e. the Dickey-Fuller test. 
They might be using ARMA because of the Auto-Regressive nature of population densities (i.e the current population density is an obvious function of the previous population density) - Is your forest plot data similar to that? 
Either way, you have so few data points, and as you mentioned your data points are not equally spaced, so your best option is a linear regression I think. 

A few days after I posted the reply, I came across this paper from the Facebook research team. They are using a variation on GAMs (General Additive Models) to model time series. 
This line 

Unlike with ARIMA models, the measurements do not need to be regularly spaced, and we do not need to interpolate missing values e.g. from removing outliers.

in their paper caught my attention and reminded me of your post. 
It should be noted that their approach won't allow you to recover any auto-regressive aspects of your time series, but it will definitely help you in establishing a trend. 
Moreover, their API which works in both R and Python is very easy to use. 
A: you can compute the acf but it can't be interpreted in a standard way as  your interval is not fixed. You might jerry-rig the data by obtaining estimates for the unobserved values e.g. in series 1 if you take a smoothed value between 1952 and 1958  to estimate the missing 1953 value via a linear filter and then discard the 1952 value. Repeat this until you have a "value" every 5 periods .
This is a possible way to alleviate the non-fixed interval issue.
