Suppose I have data on my child C's height measured every week. Presumably there is a positive trend, due to growth, and some noise due to measurement errors, and maybe even seasonality (winter boots or haircuts add extra height). If want to predict C's height into the future (at say t+s), I might use an ARIMA model, which would represent C's height next week as a function of his height in previous weeks.
Now suppose that in the past, I had 2 other children, A and B, who are older than C. Thus I have prior information about the distribution my children's heights at time t+s. How can I incorporate this into my prediction?
For instance, consider the following data. Observe that while C6 (the height of C at t=6), is unknown, A6 and B6 are known.
set.seed(1) A = seq(1,6,.5) + rnorm(11)/4 B = seq(1,6,.5) + rnorm(11)/4 C = c((seq(1,3,.5) + rnorm(5)/4), rep(NA, 6)) df = data.frame(A, B, C) > df A B C 1 0.8433865 1.097461 1.018641 2 1.5459108 1.344690 1.002662 3 1.7910928 1.446325 2.154956 4 2.8988202 2.781233 2.485968 5 3.0823769 2.988767 2.961051 6 3.2948829 3.495952 NA 7 4.1218573 4.235959 NA 8 4.6845812 4.705305 NA 9 5.1439453 5.148475 NA 10 5.4236529 5.729744 NA 11 6.3779453 6.195534 NA
I considered using 2NN (2 nearest neighbors) to predict C6. 2NN estimates C6 as the avg of 2.485968 and 2.961051, which is not great (although we get better results if we difference the data).
Alternatively we might say C6 is the expected value of the prior observations so (3.2948829, 3.495952)/2 in this case.
What are some other methods I could use to incorporate this prior information into my forecast? Could I combine my ARIMA forecast with this prior information somehow to form an ensemble forecast?
I've also started looking into Bayesian time series, dynamic linear models, and state space models, but don't have much experience in these areas and could use a pointer.