Hi all,
I have river run-off annual data and a catchment composite mean value of PDSI values (drought index). A simple regression model tells me that the PDSI well predicts the run-off (what can be physiologically expected). I checked the residuals, normality, homocedasticity, p-value. All perform fine, all is significant. The data spans 1869-2012.
Because I have the PDSI values for the period 0-2012, would it be possible to use the results of the model to reconstruct the run-off for the same period?
the model reads:
runoff_1869_2012 = ß0 + ß*PDSI_1869_2012
-> summary(model): ß0 = 1055.363 and ß=55.668
hence, would this make sense:
runoff_0_2012 = 1055.363 + 55.668*PDSI_0_2012
Eventually, I get the reconstructed run-off for the whole period with a certain uncertainty?
Many thanks for suggestions on this very basic question!
all best,
Michael
It depends. This is essentially what the model is for: predicting y from x. The prediction might not be very good (R^2 will help give a sense of that) and may well break down in some situations - especially outside of the observed ranges of x. There may be other variables that are important that are omitted and so on.
You can't really reconstruct the run-off but if the predictions are very good and the model is appropriate for the context then the predictions might be reasonable for that purpose.
Michael -
Are you saying that each year gives you a data point, so 144 data points, and assuming all things else unchanged, you use those points in a simple linear regression, regardless of dates? (I don't know why you have 1869-2012 one place and 0-2012 another???) I would expect heteroscedasticity based on size, as measured by predicted-y, b0 + b1*PDSI_year, with years out of order. Anyway, did you use a "graphical residual analysis" to check fit? "Cross-validation?" What does your intercept represent, or is it noise, and you should use a ratio estimator?
Time series are not my interest, but I have to wonder if you should be considering that. Perhaps there is autocorrelation.
Perhaps I do not understand exactly what you are doing. If you can show a graph, or graphs, that might be helpful for everyone.
If you can demonstrate that what you have works by dropping data you have and seeing how close you come to predicting it, that could be helpful.
Cheers - Jim
Dear Thom Baguley and James R Knaub many thanks for your quick reply!
To clarify a bit:
I have a set of data that ranges 1869-2012 and represents both, river run off data and PDSI at an annual temporal resolution. And I have PDSI that ranges from 0-2012 but no run off (that is of course prior to instrumental measurements). My research interest is the period 100-800 AD and my goal is to use the PDSI as an indicator for river run off behavior at one particular site.
Physiologically, geographically etc, the PDSI can be used for that reason because it is a complex index including precipitation, soil moisture, temperature etc....
From the training data 1869-2012, I get the model output and I 'reconstructed' the run off by just using the coefficient and the intercept on the 0-2012 PDSI values.
- there is no heteroscedasticity in the training data
- residuals of the lm are normally distributed
I have been diving into time series a bit but the data is probably tricky, first, as you said, the training data period is too short to use e.g. ARIMA models for forecasting. And there is no cyclicity in the annually resolved data. The PDSI is June-August composites (by 'nature') and the best-fit model of run off was the annual mean (compared to seasonal, MAM, JJA; SON, DJF and monthly resolved subsets). I checked the data for lags using Granger test, I also check auto- and crosscorrelation.
As I said, I would like to use the predictor variable as an 'indicator' of run off rather than to reconstruct single year events.
Here is a plot of the PDSI and the 'reconstruction' so you get an idea of how that looks.
Many thanks again for your suggestions and looking forward to critical comments!
all my best,
Michael
Michael -
I'm not clear on PDSI, but it seems questionable that you could have good data on that so far back in history. But even if you do, you have no runoff data until more than 1000 years after your PDSI data. Also, as Thom Baguley said, "There may be other variables that are important that are omitted ...." There may be such factors back in history, important then, but not now, and vice versa. So, if you find a model that works for the data you have, there are reasons to doubt how well it would work for your period of interest. You may find a flexible model that apparently works well enough for all segments and samples from data you do have, but you will need to note caveats as to its use in the past. Perhaps too many variables would have been worse than too few, as PDSI may hopefully be the major predictor, with less important variables, if any, coming and going, unknown to you. ??? So you don't want to ovetfit.
If you try to apply results that far back, then in addition to describing the situation, you might also rely on your subject matter knowledge to describe what other factors might influence results, and how they would influence those results.
[By the way, were the regression model to have been really good, I would have expected heteroscedasticity, as explained in https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, with bounds on the coefficient of heteroscedasticity as explained by Ken Brewer. If you are saying homoscedasticity 'passed' a test of 'significance,' that is not very convincing. Heteroscedasticity is a matter of degree, not to be addressed as "yes" or "no." Also, there is good reason that you would have it.]
This is really a time series problem, likely with autocorrelation, but with such a large gap in time between data collected on runoff and the period of interest, I can see why you would give regression, without considering time, a look. But then you would need your caveats to address the historical gap issue based on subject matter knowledge.
Best wishes - Jim
cheers Jim!
for the PDSI, see Cook et al, Old World megadroughts and pluvials during the Common Era. Science Advances, 1, doi: 10.1126/sciadv.1500561
of course, the runoff data is not available, that is why I wanted to estimate the hydrological behavior from the model. Thing is, it makes sense geographically, and of course, there are many factors that do change over time, however, the PDSI acknowledges that using the respective proxy, e.g. tree ring.
I will do some more research on that and probably there will be other suggestions that contribute to the discussion!
many thanks again for your comments!
Michael
James R Knaub has answered this very fully. This in particular: " ... I can see why you would give regression, without considering time, a look. But then you would need your caveats to address the historical gap issue based on subject matter knowledge."
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