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3D plot of Akaike Information Criterion (AIC) for suitable ranges of Lˆ and k

Can the likelihood take values outside of the range [0, 1]?AIC & BIC number interpretationComputing a multi-sample (i.e., pooled) Akaike Information CriterionAkaike information criterion for categorical and numerical dataQuestion on practically using the Akaike information criterionIs the Akaike information criterion inversely proportional to the chi-squared statistic?How freqently are the information conditons for proper Akaike information criterion application actually met?What does the Akaike Information Criterion (AIC) score of a model mean?Combining AIC and BICAkaike Information criterion for k-meansRESET test and AIC

3

$begingroup$

Giving that Akaike Information Criterion (AIC) is as follow:

How can I Produce a 3D plot of AIC for suitable ranges of Lˆ and k.

In other words what could be a suitable ranges of L to try?

Moreover, what is the function of L^? I am struggling to find the equation to represent L^ so that I can plot it.

Thanks.

data-visualization model aic bic

share|cite|improve this question

edited Mar 16 at 3:17

Jan

asked Mar 16 at 3:11

JanJan

1515

$endgroup$

add a comment | 

3

$begingroup$

Giving that Akaike Information Criterion (AIC) is as follow:

How can I Produce a 3D plot of AIC for suitable ranges of Lˆ and k.

In other words what could be a suitable ranges of L to try?

Moreover, what is the function of L^? I am struggling to find the equation to represent L^ so that I can plot it.

Thanks.

data-visualization model aic bic

share|cite|improve this question

edited Mar 16 at 3:17

Jan

asked Mar 16 at 3:11

JanJan

1515

$endgroup$

add a comment | 

$begingroup$

Giving that Akaike Information Criterion (AIC) is as follow:

How can I Produce a 3D plot of AIC for suitable ranges of Lˆ and k.

In other words what could be a suitable ranges of L to try?

Moreover, what is the function of L^? I am struggling to find the equation to represent L^ so that I can plot it.

Thanks.

data-visualization model aic bic

share|cite|improve this question

edited Mar 16 at 3:17

Jan

asked Mar 16 at 3:11

JanJan

1515

$endgroup$

share|cite|improve this question

edited Mar 16 at 3:17

Jan

asked Mar 16 at 3:11

JanJan

1515

share|cite|improve this question

edited Mar 16 at 3:17

Jan

asked Mar 16 at 3:11

JanJan

1515

asked Mar 16 at 3:11

JanJan

1515

asked Mar 16 at 3:11

JanJan

1515

2 Answers
2

active

oldest

votes

1

$begingroup$

$hatL$ is the value of the assumed likelihood function evaluated at $hattheta$, i.e. at its maximum value for the observed data. If our likelihood function is $L(mathbfX;theta)$ then $hatL=L(mathbfX;hattheta)$. $k$ is the number of model parameters being estimated.

For comparing 2 models, the one with lower AIC is preferred. Higher values of the log-likelihood imply lower values of the AIC, holding $k$ constant, while fewer model parameters also imply lower values of AIC, holding $hatL$ constant. The idea is to reward higher likelihood and penalize each time you add a parameter to the model, as you are losing degrees of freedom.

Assuming you are intending to construct a 3-D plot using triplets (AIC,$hatL,k$) from various models you have estimated, I'm not sure the plot will give you much insight beyond simply looking at AIC. The problem with creating a surface (as mentioned in the answer from @Lucas Farias) is that $k$ alone does not tell us which regressors we are including. For example $y=a_0+a_1x+a_2z$ and $y=b_0+b_1w+b_2v$ both have $k=2$, but will yield different values of $hatL$ and AIC.

share|cite|improve this answer

edited Mar 16 at 3:56

answered Mar 16 at 3:47

dlnBdlnB

93712

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks dlnB for the clarification. In simple words, what do you suggest values to use to plot this 3D plot.
  $endgroup$
  – Jan
  Mar 16 at 7:54
  

  
  
  
  

add a comment | 

1

$begingroup$

According to wiki, for the specification you presented:

Let $k$ be the number of estimated parameters in the model. Let
$hatL$ be the maximum value of the function for the model.

While $k$ is always non-negative, the range and shape of the model likelihood function $hatL$ is different for each problem, since it depends on the densities and data you are working with.

For this reason, even if you can create the surface you want for different specifications of the same model, it's impossible to obtain an AIC surface that is representative for all types of models.

share|cite|improve this answer

edited Mar 16 at 9:19

answered Mar 16 at 3:52

Lucas FariasLucas Farias

8541522

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks Lucas, but I feel that I got confused with your answer. Can you please explain more?
  $endgroup$
  – Jan
  Mar 16 at 7:52
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @Jan what didn't you understand?
  $endgroup$
  – Lucas Farias
  Mar 16 at 9:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it possible to plot a Surface of AIC in a 3D space base on L and k inputs? I want to plot it based on any suitable range of L and K just to get a feel of it. I know that K cannot be negative (since it denotes the parameters size) so I can choose a range from 1 to 60 for example. However, since L is different for each problem, what could be a reasonable range (even if it is hypothetical)?
  $endgroup$
  – Jan
  Mar 16 at 12:39
  

  
  
  
  

add a comment | 

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1

$begingroup$

$hatL$ is the value of the assumed likelihood function evaluated at $hattheta$, i.e. at its maximum value for the observed data. If our likelihood function is $L(mathbfX;theta)$ then $hatL=L(mathbfX;hattheta)$. $k$ is the number of model parameters being estimated.

For comparing 2 models, the one with lower AIC is preferred. Higher values of the log-likelihood imply lower values of the AIC, holding $k$ constant, while fewer model parameters also imply lower values of AIC, holding $hatL$ constant. The idea is to reward higher likelihood and penalize each time you add a parameter to the model, as you are losing degrees of freedom.

Assuming you are intending to construct a 3-D plot using triplets (AIC,$hatL,k$) from various models you have estimated, I'm not sure the plot will give you much insight beyond simply looking at AIC. The problem with creating a surface (as mentioned in the answer from @Lucas Farias) is that $k$ alone does not tell us which regressors we are including. For example $y=a_0+a_1x+a_2z$ and $y=b_0+b_1w+b_2v$ both have $k=2$, but will yield different values of $hatL$ and AIC.

share|cite|improve this answer

edited Mar 16 at 3:56

answered Mar 16 at 3:47

dlnBdlnB

93712

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks dlnB for the clarification. In simple words, what do you suggest values to use to plot this 3D plot.
  $endgroup$
  – Jan
  Mar 16 at 7:54
  

  
  
  
  

add a comment | 

1

$begingroup$

$hatL$ is the value of the assumed likelihood function evaluated at $hattheta$, i.e. at its maximum value for the observed data. If our likelihood function is $L(mathbfX;theta)$ then $hatL=L(mathbfX;hattheta)$. $k$ is the number of model parameters being estimated.

For comparing 2 models, the one with lower AIC is preferred. Higher values of the log-likelihood imply lower values of the AIC, holding $k$ constant, while fewer model parameters also imply lower values of AIC, holding $hatL$ constant. The idea is to reward higher likelihood and penalize each time you add a parameter to the model, as you are losing degrees of freedom.

Assuming you are intending to construct a 3-D plot using triplets (AIC,$hatL,k$) from various models you have estimated, I'm not sure the plot will give you much insight beyond simply looking at AIC. The problem with creating a surface (as mentioned in the answer from @Lucas Farias) is that $k$ alone does not tell us which regressors we are including. For example $y=a_0+a_1x+a_2z$ and $y=b_0+b_1w+b_2v$ both have $k=2$, but will yield different values of $hatL$ and AIC.

share|cite|improve this answer

edited Mar 16 at 3:56

answered Mar 16 at 3:47

dlnBdlnB

93712

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks dlnB for the clarification. In simple words, what do you suggest values to use to plot this 3D plot.
  $endgroup$
  – Jan
  Mar 16 at 7:54
  

  
  
  
  

add a comment | 

$begingroup$

$hatL$ is the value of the assumed likelihood function evaluated at $hattheta$, i.e. at its maximum value for the observed data. If our likelihood function is $L(mathbfX;theta)$ then $hatL=L(mathbfX;hattheta)$. $k$ is the number of model parameters being estimated.

For comparing 2 models, the one with lower AIC is preferred. Higher values of the log-likelihood imply lower values of the AIC, holding $k$ constant, while fewer model parameters also imply lower values of AIC, holding $hatL$ constant. The idea is to reward higher likelihood and penalize each time you add a parameter to the model, as you are losing degrees of freedom.

Assuming you are intending to construct a 3-D plot using triplets (AIC,$hatL,k$) from various models you have estimated, I'm not sure the plot will give you much insight beyond simply looking at AIC. The problem with creating a surface (as mentioned in the answer from @Lucas Farias) is that $k$ alone does not tell us which regressors we are including. For example $y=a_0+a_1x+a_2z$ and $y=b_0+b_1w+b_2v$ both have $k=2$, but will yield different values of $hatL$ and AIC.

share|cite|improve this answer

edited Mar 16 at 3:56

answered Mar 16 at 3:47

dlnBdlnB

93712

$endgroup$

share|cite|improve this answer

edited Mar 16 at 3:56

answered Mar 16 at 3:47

dlnBdlnB

93712

answered Mar 16 at 3:47

dlnBdlnB

93712

answered Mar 16 at 3:47

dlnBdlnB

93712

1

$begingroup$

According to wiki, for the specification you presented:

Let $k$ be the number of estimated parameters in the model. Let
$hatL$ be the maximum value of the function for the model.

While $k$ is always non-negative, the range and shape of the model likelihood function $hatL$ is different for each problem, since it depends on the densities and data you are working with.

For this reason, even if you can create the surface you want for different specifications of the same model, it's impossible to obtain an AIC surface that is representative for all types of models.

share|cite|improve this answer

edited Mar 16 at 9:19

answered Mar 16 at 3:52

Lucas FariasLucas Farias

8541522

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks Lucas, but I feel that I got confused with your answer. Can you please explain more?
  $endgroup$
  – Jan
  Mar 16 at 7:52
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @Jan what didn't you understand?
  $endgroup$
  – Lucas Farias
  Mar 16 at 9:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it possible to plot a Surface of AIC in a 3D space base on L and k inputs? I want to plot it based on any suitable range of L and K just to get a feel of it. I know that K cannot be negative (since it denotes the parameters size) so I can choose a range from 1 to 60 for example. However, since L is different for each problem, what could be a reasonable range (even if it is hypothetical)?
  $endgroup$
  – Jan
  Mar 16 at 12:39
  

  
  
  
  

add a comment | 

1

$begingroup$

According to wiki, for the specification you presented:

Let $k$ be the number of estimated parameters in the model. Let
$hatL$ be the maximum value of the function for the model.

While $k$ is always non-negative, the range and shape of the model likelihood function $hatL$ is different for each problem, since it depends on the densities and data you are working with.

For this reason, even if you can create the surface you want for different specifications of the same model, it's impossible to obtain an AIC surface that is representative for all types of models.

share|cite|improve this answer

edited Mar 16 at 9:19

answered Mar 16 at 3:52

Lucas FariasLucas Farias

8541522

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks Lucas, but I feel that I got confused with your answer. Can you please explain more?
  $endgroup$
  – Jan
  Mar 16 at 7:52
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @Jan what didn't you understand?
  $endgroup$
  – Lucas Farias
  Mar 16 at 9:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it possible to plot a Surface of AIC in a 3D space base on L and k inputs? I want to plot it based on any suitable range of L and K just to get a feel of it. I know that K cannot be negative (since it denotes the parameters size) so I can choose a range from 1 to 60 for example. However, since L is different for each problem, what could be a reasonable range (even if it is hypothetical)?
  $endgroup$
  – Jan
  Mar 16 at 12:39
  

  
  
  
  

add a comment | 

$begingroup$

According to wiki, for the specification you presented:

Let $k$ be the number of estimated parameters in the model. Let
$hatL$ be the maximum value of the function for the model.

While $k$ is always non-negative, the range and shape of the model likelihood function $hatL$ is different for each problem, since it depends on the densities and data you are working with.

For this reason, even if you can create the surface you want for different specifications of the same model, it's impossible to obtain an AIC surface that is representative for all types of models.

share|cite|improve this answer

edited Mar 16 at 9:19

answered Mar 16 at 3:52

Lucas FariasLucas Farias

8541522

$endgroup$

share|cite|improve this answer

edited Mar 16 at 9:19

answered Mar 16 at 3:52

Lucas FariasLucas Farias

8541522

answered Mar 16 at 3:52

Lucas FariasLucas Farias

8541522

answered Mar 16 at 3:52

Lucas FariasLucas Farias

8541522