unexpected intra-variant variances estimation in disaggregation spatial regression model using INLA()

[何博文]

I used an intercept, a multivariate normal distribution with Matérn covariance function and a nugget effect sigmae to simulate a spatial autocorrelated 100*100 grids value as simulation data. Then, I averaged 100 grids to one polygon, resulting  10*10 polygons as coarser observations. The goal is to predict the 10000 underlying grids level data when only observing those coarser a100 real/polygon observations. While INLA() can reveal the kappa, marignal variances of the Matérn covariance function as well as the intercept with a reasonable accuracy, the nugget effect sigmae is way off from the true value. I am wondering what might be the reason that costs this issue?

[Finn Lindgren]

Hi,

there isn't enough information provided to know for sure. If the nugget variance from the 100x100 grid is sigma^2, then the nugget variance on the 10x10averaged grid is sigma^2 / 100 (average of 100 independent iid variables).

So in a perfect world your nugget variance estimate should be 1/100 times the fine resolution nugget variance.
In practice, it will likely differ from that due to interaction with the numerical representation of the random field itself, and due to non-identifiability of the covariance parameters (with only a single field realisation, the parameters are not fully identifiable).

Finn

On Thu, 2 Dec 2021 at 05:08, 何博文 <steven...@gmail.com> wrote:

I used an intercept, a multivariate normal distribution with Matérn covariance function and a nugget effect sigmae to simulate a spatial autocorrelated 100*100 grids value as simulation data. Then, I averaged 100 grids to one polygon, resulting  10*10 polygons as coarser observations. The goal is to predict the 10000 underlying grids level data when only observing those coarser a100 real/polygon observations. While INLA() can reveal the kappa, marignal variances of the Matérn covariance function as well as the intercept with a reasonable accuracy, the nugget effect sigmae is way off from the true value. I am wondering what might be the reason that costs this issue?

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Finn Lindgren
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[何博文]

Thanks so much! It is so nice that you have answered.

Yes, I did two interesting experiments. The first one is that I used an arbitrarily chosen simple normal nugget effect (N(0, sigma)) to generate one single 100*100 grid realization, without intercept and any covariates, just the nugget effect (N(0, sigma)). Then, I averaged those 100*100 grids to 10*10 coarser averaged areal value, use INLA() to function to fit those coarser 10*10 averaged areal values. Even for this one single realization, the precision for the gaussian observation is 1/100 times the sigma of the finer resolution (100*100) nugget variance that I arbitrarily chose in the first place. This is aligned with a perfect world without any spatial random fields or SPDE model or numerical representation involved.

However, if I added a Matérn covariance spatial random field in the simulation model, plus a simple intercept and a simple normal nugget effect like above, and fit the model on the composite averaged observations. The INLA() function cannot give accurate precision for the observation value for the averaged coarser areal observations (very different from 1/100 times the fine resolution nugget variance, which is expected). Thus, I am wondering if it is caused by the interaction with the numerical representation of the spatial random field itself, since the INLA() function can derive a good estimation on the pure composite nugget effects without any spatial random fields involved. Looking forward to your reply!

Thanks,

Bowen

[Finn Lindgren]

It is extremely likely that the model simply isn't identifiable. Even a model with _only_ a Matern field and no nugget has non-identifiable covariance parameters (only the variance of the driving noise process is consistently estimable under infill asymptotics) when only a single realisation is available.  It is therefore quite possible that INLA has actually computed an accurate estimate of the true posterior distribution for the problem presented to it, and that that posterior distribution isn't as close to the true parameter values as one might want it to be.

Finn

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[何博文]

Thank you so much for your explanation. 

Yes, it helps a lot to understand this phenomenon.

Another interesting thing I found that is INLA() function can derive a very good estimation for all the covariance parameters (marginal variance for the process, scale/range) associated with the Matern field as well as the nugget effect's sigma^2 value even one realization is available under the condition that the original 100*100 finer grids values are input as observational values in the function, not the 10*10 coarser areal observational values. When the 10*10 coarser areal values that are averaged from the original finer grids values are used as input as observational values in the function, the INLA() function can also derive good estimation for all the covariance parameters (marginal variance for the process, scale/range) associated with Matern field, however, the nugget effect's sigma is off from the expected one (1/100 times the fine resolution nugget variance). Thus, in the context of the areal interpolation process which is to derive the original model parameters from coarser averaged composite observations (lower resolution to predict higher resolution), the model seems still able to derive the covariance parameters for a reasonable precision but only lose the ability to derive the nugget variances (precision for the gaussian observations). I am wondering what could be the possible reason for the INLA() function to lose the ability to reveal the underlying true nugget variances only in the context of the areal interpolation process using a single realization. Your thoughts would be extremely helpful! Thanks!

Best,

Bowen   

[Finn Lindgren]

Hi,

From my previous answer:

“ If the nugget variance from the 100x100 grid is sigma^2, then the nugget variance on the 10x10averaged grid is sigma^2 / 100 (average of 100 independent iid variables).”

So it sounds like your actually getting the appropriate estimate, if your nugget variance estimate on the aggregated scale is 1/100 times that on the fine scale.

Finn

On 20 Dec 2021, at 06:05, 何博文 <steven...@gmail.com> wrote:



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[何博文]

Hi,

Thanks for your reply!

I apologize for the miscommunications in the previous question. I summarized two experiments here:

(1)  A very simple model with nugget effects only (variance of the only driving noise process). The INLA() function can reveal the true/appropriate nugget variances value in both the original 100*100 grid's value as observations and the composite averaged 10*10 coarser areal values as observations. If the original 100*100 grids simulation values are input to the INLA() function as observations, true nugget variances sigma^2 can be revealed and if the 10*10 composite averaged areal values are input to the INLA() function as observations, 1/100 times the true sigma^2 that I chose for simulating the finer grids value can also be successfully revealed. This simple nugget effect variance-only model is consistently estimable when a single realization is available.

(2) Matern random field involved spatial model. The INLA() function can reveal a good estimation for all the covariance parameters (marginal variance for the process, scale/range) associated with the Matern field as well as the nugget effect's variance value when only one realization is available only when the original 100*100 finer grids values are input as observational values in the function. If the composite averaged areal values like 10*10 composite averaged areal values are input to the function as observations, the INLA() cannot derive the supposed 1/100 times the finer scale nugget effect sigma^2 value but all the other Matern covariance parameters.  

Thus, I am wondering why the INLA() function can successfully derive all the model parameters including the true nugget effect variance when observing the finer resolution grids value. However, the function failed to derive the 1/100 times the finer resolution nugget effect variance (precision for the gaussian observation calculated in this case is very different from 1/100 times the original nugget effect variance for finer grids) when observing aggregated scale values but still capable of deriving a good estimate for all the other Matern covariance parameters? Thanks!

Thanks,

Bowen

[Finn Lindgren]

Hi,

I explained some of the possible reasons in a previous reply; the parameters are not consistently estimable even under infill asymptotics and known nugget variance, so you should not expect the nugget variance estimate to be close to the true data-generating value, for any estimation method.

Finn

On 20 Dec 2021, at 18:29, 何博文 <steven...@gmail.com> wrote:



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[何博文]

Thanks for your reply!

As I am doing the simulation study, I am curious that is the INLA() function cannot estimate model parameters consistenly under any scenarios or I can have certain confidence in believing some of the model scenarios such as the SPDEtoy exmaple? For instance, the SPDE toy in the book exhibits that the INLA() can successfully reveal the underlying true model parameters (including the covariance model parameters as well as the gaussian variances/nugget variances) from a single simple realisation. Is that mean in this model scenarios, the model parameters might not be consistenly estimable when changing the model parameters from only one model realization? What is the true purpose of this SPDE toy example when the parameters might not be consistently estimable from single realization that is sampled from some known model parameters? Thanks again for your reply!

[Elias T. Krainski]

Hi,

One way to go is to fix one of the parameters, if you don't have both, infill and increased domain. In the spde-book we always fixed the smoothness parameter and considered increased domain for the simulated examples in order to be able to fit the other parameters.

Elias

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[何博文]

Hi,

My question is whether it is meaningful to use the INLA() function to estimate the underlying model parameters from a single realization that is generated by those model parameters to show that the INLA() function has the ability to reveal the true data-generating parameters. Since I was told that the parameters might not be consistently estimable given one single realization, I am wondering what is the purpose of the SPDE-toy example in the SPDE book and what is the takeaway of this example shown in the book?

Thanks,

Bowen

[Finn Lindgren]

Like I explained before, this is a fundamental property of even the simplest Gaussian Matern random field model, and has very little specific to do with either inla or the spde representations for Matern fields. Even in the case of a directly observed field (I.e. with no observation noise), infill asymptotics aren’t enough to consistently estimate all the model parameters. The best one can hope for is that the parameter estimates are reasonable, and that the resulting field properties (e.g. spatial prediction/infill) produces appropriate results with appropriate uncertainty estimates. One reference about the non-consistency of parameter estimates is  https://www.jstor.org/stable/20441245

Non-consistency however doesn’t mean that the parameter estimates are _necessarily_ far from the true values. It just means that there is no guarantee that they will be close, and the choice of prior distributions will affect the estimates; in some applications that means one needs to think more carefully about those priors.

In order to be able to identify the parameters consistently in a model with observation noise, one would at least need multiple realisations of the random field (which for some applications is fundamentally impossible).

Finn

On 3 Jan 2022, at 18:55, 何博文 <steven...@gmail.com> wrote:



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[何博文]

Thanks for your explanation! From multiple infill asymptotics simulations (100 realizations), I found there is a strong correlation between the marginal variances and the kappa estimates of those posterior estimates for the spde representation of the Matern field model from those 100 realizations. Based on this, I have two questions:

(1) Is it appropriate to derive a relationship between those two model parameters (marginal variance and kappa/range) from the simulation data and apply the derived relationship in the real-world data to improve the prediction accuracy in the real-world data scenario? If it is doable, is there a way to constrain the model parameters and apply the derived relationship between the parameters in the inla() model to improve the prediction accuracy in the real-world data scenario?

(2) Or the correlation and the uncertainty estimates of the spde Matern model parameters will not affect the prediction performance of the model with single realization data no matter what constrain/relationship of the model parameters are applied in the model? Since our main goal in the real-world data is to predict the missing values with only one observation and not to estimate the underlying model parameters, this could be helpful. Thanks!!

Bowen

[Elias T. Krainski]

There is a recent paper with some new results on this matter https://rss.onlinelibrary.wiley.com/doi/10.1111/rssb.12472

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[何博文]

Thanks for your response. 

Is there a way to calculate the posterior distribution of this consistently identifiable parameter theta = sigma^2*phi^(2*nu) in the inla() function based on the posterior distribution of the model parameters computed by the function?

Thanks,

Bowen