Calculating the variance and validate prediction interval equivalent to kriging?

[Jingyi Huang]

Dear R-inla group,

Can someone give me some advice on calculating the confidence interval in INLA-SPDE? Basically, I am comparing a hierarchical model estimated by INLA-SPDE with a linear mixed model (LMM) estimated by REML.

1) As the LMM-REML uses a kriging variance to show the prediction uncertainty, I wonder how to get a similar variance by INLA-SPDE? 

I have enabled R-INLA to calculate the quantiles and the SD value. However, I find that the SD value (squared) is too smaller compared with the kriging variance. Is there a way to calculate a similar kriging variance in R-INLA?

2) How to get a prediction interval equivalent to the kriging variance? 

I need to know this is because I calculated the prediction interval coverage probability (PICP) versus 100(1-a)% (see attached figure) to validate the prediction intervals.

However, as shown in the attache figure for soil pH, the LMM-REML shows good performance on the validation data (quantiles are obtained by calculating SD using kriging variance and multiplying a significant value assuming a Gaussian distribution) while the INLA-SPDE shows very small PICP (quantiles are obtained using the estimated quantiles from inla() when applying the INLA-SPDE model onto the calibration points). I suspect that the interval given by INLA is credible intervals due to parameter uncertainty only, thus it is quite small and different from LMM (or kriging). Am I correct? But how can I calculate a prediction interval which follows the 1:1 line in the PICP plot?

Thanks very much for your help.

Best regards,

Jingyi Huang

[Finn Lindgren]

Hi,

I would guess that you include the measurement variance as a "nugget" variance in the non-INLA method?

You can get an approximate version of that from INLA by adding the posterior mean of 1/precision.

To include the full uncertainty you need to use inla.posterior.sample().

Finn

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<pH.jpg>

<PICP.jpg>

[Jingyi Huang]

Hello Finn,

Thanks very much for your reply.

Best regards,

Jingyi

On 12 February 2017 at 18:03, Finn Lindgren <finn.l...@gmail.com> wrote:

Hi,

I would guess that you include the measurement variance as a "nugget" variance in the non-INLA method?

You can get an approximate version of that from INLA by adding the posterior mean of 1/precision.

To include the full uncertainty you need to use inla.posterior.sample().

Finn

On 12 Feb 2017, at 05:50, Jingyi Huang <xueha...@gmail.com> wrote:

Dear R-inla group,

Can someone give me some advice on calculating the confidence interval in INLA-SPDE? Basically, I am comparing a hierarchical model estimated by INLA-SPDE with a linear mixed model (LMM) estimated by REML.

1) As the LMM-REML uses a kriging variance to show the prediction uncertainty, I wonder how to get a similar variance by INLA-SPDE? 

I have enabled R-INLA to calculate the quantiles and the SD value. However, I find that the SD value (squared) is too smaller compared with the kriging variance. Is there a way to calculate a similar kriging variance in R-INLA?

2) How to get a prediction interval equivalent to the kriging variance? 

I need to know this is because I calculated the prediction interval coverage probability (PICP) versus 100(1-a)% (see attached figure) to validate the prediction intervals.

However, as shown in the attache figure for soil pH, the LMM-REML shows good performance on the validation data (quantiles are obtained by calculating SD using kriging variance and multiplying a significant value assuming a Gaussian distribution) while the INLA-SPDE shows very small PICP (quantiles are obtained using the estimated quantiles from inla() when applying the INLA-SPDE model onto the calibration points). I suspect that the interval given by INLA is credible intervals due to parameter uncertainty only, thus it is quite small and different from LMM (or kriging). Am I correct? But how can I calculate a prediction interval which follows the 1:1 line in the PICP plot?

Thanks very much for your help.

Best regards,

Jingyi Huang

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<pH.jpg>

<PICP.jpg>