diffusion rate

[sue-shine]

Excuse me, I have meet a question about diffusion rate Can I calculate the rate of diffusion according to the parameters? Is there any formula Thank you in advance!

[andrewzm]

Hi Sue, The IDE has a kernel (see Chapter 5 in our book) which dictates how things move and diffuse across time. There are four parameters defining this kernel in 2D, and you can get the parameters from your fitted model using:

IDEmodel$get("k")

The first parameter is the amplitude of the kernel, the second the scale (the 'l' in exp(-h^2 / l)), and the third and fourth parameters are the offset parameters from the origin (which determine the direction of flow). Hope this helps.

Andrew

On Wed, Jul 22, 2020 at 1:06 PM sue-shine notifications@github.com wrote:

Excuse me, I have meet a question about diffusion rate Can I calculate the rate of diffusion according to the parameters? Is there any formula Thank you in advance!

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[sue-shine]

Yeah,I have read the book, and got parameters. The shift can be caculated by the third and fourth parameters((θ3)^2+(θ4)^2 ). But what is the meaning of the formula(exp(-h^2 / l))? Is it diffusion rate, h is θ1, and l is θ2?

[andrewzm]

exp(-h^2 / l) is the shape of the IDE kernel (e.g., Figure 5.3 right panel). Then "l" (theta2) is the "width" of the kernel and theta1 is the amplitude of the kernel (see the definition for m() in Lab 5.1 for the 1D case), i.e., it's what you multiply the kernel by. There is no straightforward "diffusion" rate you get out, both theta1 and theta2 control the rate of "diffusion" by modifying the kernel shape.

On Fri, Jul 24, 2020 at 3:09 PM sue-shine notifications@github.com wrote:

Yeah,I have read the book, and got parameters. The shift can be caculated by the third and fourth parameters((θ3)^2+(θ4)^2 ). But what is the meaning of the formula(exp(-h^2 / l))? Is it diffusion rate, h is θ1, and l is θ2?

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[sue-shine]

Thank you so much!

[sue-shine]

Hi Andrewzm, I also want to know more about what the first and second parameters mean. For example, if the amplitude is higher, whether the rate of "diffusion" is higher? Such as theta1 is 698.9058 , theta2 is 0.0004. Which parameter has the greater influence on diffusion? How to evaluate or explain the rate of diffusion by these two parameters?

[andrewzm]

Hi Sue, I suggest you run Lab 5.1 in our book and experiment a bit with the effect of these parameters.

There is a relationship between the IDE and the diffusion equation, when theta3 and theta4 are zero than the IDE with the squared exponential kernel is a solution to the diffusion equation. You can see the relationship in many papers, for example in this one here:

de Bezenac, E., Pajot, A., Gallinari, P., 2018. Deep learning for physical processes: Incorporating prior scientific knowledge. In: Proceedings of ICLR 2018. Vancouver, Canada.

This should also give you an idea on the behaviour of theta1 and theta2.

Andrew

On Sat, Jul 25, 2020 at 5:40 PM sue-shine notifications@github.com wrote:

Hi Andrewzm, I also want to know more about what the first and second parameters mean. For example, if the amplitude is higher, whether the rate of "diffusion" is higher? Such as theta1 is 698.9058 , theta2 is 0.0004. Which parameter has the greater influence on diffusion? How to evaluate or explain the rate of diffusion by these two parameters?

— You are receiving this because you commented. Reply to this email directly, view it on GitHub https://github.com/andrewzm/IDE/issues/4#issuecomment-663823808, or unsubscribe https://github.com/notifications/unsubscribe-auth/AABEU3M5YHCRZYPRHL3DVGDR5KD77ANCNFSM4PEILFLA .

[sue-shine]

Thank you again!

[sue-shine]

Hi Andrewzm, If I put covariates in IDEmodel, can I get the relationship between the covariates and the diffusion based on the coefficient estimates. The results The results are as follows: intercept and covariate1 are 30 and -0.05 respectively. The four theta are 20.11, 0.04, 0.18, 0.22. How to identify the promotion or inhibition of diffusion by covariates?

[andrewzm]

Hi Sue, There is no direct way to do that. Maybe fit the model with and without covariates and look at how the diffusion coefficient changes.

Andrew

On Sun, Aug 2, 2020 at 5:15 PM sue-shine notifications@github.com wrote:

Hi Andrewzm, If I put covariates in IDEmodel, can I get the relationship between the covariates and the diffusion based on the coefficient estimates. The results The results are as follows: intercept and covariate1 are 30 and -0.05 respectively. The four theta are 20.11, 0.04, 0.18, 0.22. How to identify the promotion or inhibition of diffusion by covariates?

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[stevenhuyi]

The same puzzle with Sue. According to your reply, andrewzm, the cofficients of covariates only explain the correlation between covariates and the repsonse variable, but can not imply facilitating or impeding the spread (and/or advection) of the response variable, right? I also have a further issue. You mentioned that "fit the model with and without covariates and look at how the diffusion coefficient changes", what dose it mean if the theta 3 and theta 4 approximate zero after taking covariates into account while they are away from zero without considering covariates. Would this suggest that these covariate can explain spatio-temporal variation within the response variable? Looking forward to your reply.

With regards steven

[andrewzm]

Hi Steven, The IDE just models the "residual" after discounting what the covariates explain. It is perfectly possible, if one has a spatio-temporal covariate, that the residual process changes behaviour completely once it is included. Indeed, I think even the inclusion of a spatial-only covariate can change the spatio-temporal residual process quite substantially, and the parameters that are estimated. The IDE parameters are most easily interpreted when the covariates are very simple, such as an "intercept" or a latitudinal trend, for example.

Andrew

On Mon, Aug 3, 2020 at 9:02 PM stevenhuyi notifications@github.com wrote:

The same puzzle with Sue. According to your reply, andrewzm, the cofficients of covariates only explain the correlation between covariates and the repsonse variable, but can not imply facilitating or impeding the spread (and/or advection) of the response variable, right? I also have a further issue. You mentioned that "fit the model with and without covariates and look at how the diffusion coefficient changes", what dose it mean if the theta 3 and theta 4 approximate zero after taking covariates into account while they are away from zero without considering covariates. Would this suggest that these covariate can explain spatio-temporal variation within the response variable? Looking forward to your reply.

With regards steven

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