Statistical Parametric Mapping - Historial de revisiones

Alexsavio en 15:02 13 ene 2009

2009-01-13T15:02:49Z

← Revisión anterior		Revisión del 17:02 13 ene 2009
Línea 1:		Línea 1:
			== Topological inference and the theory of random fields ==

	<center>[[Imagen:Random_Field_Theory_SPM.png]]</center>		<center>[[Imagen:Random_Field_Theory_SPM.png]]</center>

Alexsavio en 15:01 13 ene 2009

2009-01-13T15:01:07Z

@@ Línea 1: / Línea 1: @@
 <center>[[Imagen:Random_Field_Theory_SPM.png]]</center>
-Schematic illustrating the use of Random Field Theory in making inferences about SPMs. If one knew precisely where to look, then inference can be based on the value of the statistic at the specified location in the SPM.  However, generally, one does not have a precise anatomical <i>prior</i>, and an adjustment for multiple dependent comparisons has to be made to the <i>p</i>-values. These corrections use distributional approximations from RFT. This schematic deals with a general case of <i>n</i> SPM{<i>t</i>} whose voxels all survive a common threshold <img class='tex' src="/wiki/images/math/1e0fbb458ac1325b073be7a4fad98fc8.png" alt="u" /> (<i>i.e.</i> a conjunction of <img class='tex' src="/wiki/images/math/30c2a24cd38b0414fae013c0d44b0ca9.png" alt="n" /> component SPMs). The central probability, upon which all peak, cluster or set-level inferences are made, is the probability <img class='tex' src="/wiki/images/math/802c4d6b0b186aa9ab1853dfa96590b2.png" alt="P(u,c,k)" /> of getting <img class='tex' src="/wiki/images/math/2d043a3ca0862ca27fb2ff3a1eb2f1cf.png" alt="c" /> or more clusters with <img class='tex' src="/wiki/images/math/524eecc1d5922a435e8e848094405caa.png" alt="k" /> or more <i>RESELS</i> (resolution elements) above this threshold. By assuming that clusters behave like a multidimensional Poisson point-process (<i>i.e.</i>, the Poisson clumping heuristic), <img class='tex' src="/wiki/images/math/802c4d6b0b186aa9ab1853dfa96590b2.png" alt="P(u,c,k)" /> is determined simply: the distribution of <img class='tex' src="/wiki/images/math/2d043a3ca0862ca27fb2ff3a1eb2f1cf.png" alt="c" /> is Poisson with an expectation that corresponds to the product of the expected number of clusters, of any size, and the probability that any cluster will be bigger than <img class='tex' src="/wiki/images/math/524eecc1d5922a435e8e848094405caa.png" alt="k" /> RESELS. The latter probability depends on the expected number of RESELS per cluster <img class='tex' src="/wiki/images/math/3dc2e0d31e939dc8e285bb3aa56234db.png" alt="\eta" />. This is simply the expected supra-threshold volume, divided by the expected number of clusters. The expected number of clusters <img class='tex' src="/wiki/images/math/b0ca0d9df95d65286632112b2c3f9cc1.png" alt="\psi_0" /> is estimated with the Euler characteristic (EC) (effectively the number of blobs minus the number of holes). This depends on the EC density for the statistic in question (with  degrees of freedom <img class='tex' src="/wiki/images/math/959822faa03ebcea9210021baf302c54.png" alt="\nu" />) and the RESEL counts. The EC density is the expected EC per unit of <img class='tex' src="/wiki/images/math/5616208fafc6b96190c6b46e0b649a10.png" alt="D" />-dimensional volume of the SPM where the volume of the search is given by the RESEL counts. RESEL counts are a volume measure that has been normalized by the smoothness of the SPMs component error fields (<img class='tex' src="/wiki/images/math/444bc4156297541dbcd45dc59237d731.png" alt="\epsilon" />), expressed in terms of the full width at half maximum (<i>FWHM</i>). In this example equations for a sphere of radius <img class='tex' src="/wiki/images/math/444bc4156297541dbcd45dc59237d731.png" alt="\epsilon" /> are given. <img class='tex' src="/wiki/images/math/398b8c3e58e961cdea0fe626771f3243.png" alt="\Psi" /> denotes the cumulative density function for the statistic in question.</div></div></div></a>
+Schematic illustrating the use of Random Field Theory in making inferences about SPMs. If one knew precisely where to look, then inference can be based on the value of the statistic at the specified location in the SPM. However, generally, one does not have a precise anatomical prior, and an adjustment for multiple dependent comparisons has to be made to the p-values. These corrections use distributional approximations from RFT. This schematic deals with a general case of n SPM{t} whose voxels all survive a common threshold u (i.e. a conjunction of n component SPMs). The central probability, upon which all peak, cluster or set-level inferences are made, is the probability P(u,c,k) of getting c or more clusters with k or more RESELS (resolution elements) above this threshold. By assuming that clusters behave like a multidimensional Poisson point-process (i.e., the Poisson clumping heuristic), P(u,c,k) is determined simply: the distribution of c is Poisson with an expectation that corresponds to the product of the expected number of clusters, of any size, and the probability that any cluster will be bigger than k RESELS. The latter probability depends on the expected number of RESELS per cluster \eta. This is simply the expected supra-threshold volume, divided by the expected number of clusters. The expected number of clusters \psi_0 is estimated with the Euler characteristic (EC) (effectively the number of blobs minus the number of holes). This depends on the EC density for the statistic in question (with degrees of freedom \nu) and the RESEL counts. The EC density is the expected EC per unit of D-dimensional volume of the SPM where the volume of the search is given by the RESEL counts. RESEL counts are a volume measure that has been normalized by the smoothness of the SPMs component error fields (\epsilon), expressed in terms of the full width at half maximum (FWHM). In this example equations for a sphere of radius \epsilon are given. \Psi denotes the cumulative density function for the statistic in question.
-<p>Classical inference using SPMs can be of two sorts, depending on whether one knows where to look in advance. With an anatomically constrained hypothesis, about effects in a particular brain region, the uncorrected <i>p</i>-value associated with the height or extent of that region in the SPM can be used to test the hypothesis. With an anatomically open hypothesis (<i>i.e.</i> a null hypothesis that there is no effect anywhere in a specified volume) a correction for multiple dependent comparisons is necessary. The theory of random fields provides a way of adjusting the <i>p</i>-value that takes into account the fact that neighbouring voxels are not independent, by virtue of continuity in the original data. Provided the data are smooth the RFT adjustment is less severe (<i>i.e.</i> is more sensitive) than a Bonferroni correction for the number of voxels. As noted above RFT deals with the multiple comparisons problem in the context of continuous, statistical fields, in a way that is analogous to the Bonferroni procedure for families of discrete statistical tests. There are many ways to appreciate the difference between RFT and Bonferroni corrections. Perhaps the most intuitive is to consider the fundamental difference between an SPM and a collection of discrete <i>t</i>-values. When declaring a peak or cluster of the SPM to be significant, we refer collectively to all the voxels associated with that feature. The false positive rate is expressed in terms of peaks or clusters, under the null hypothesis of no activation. This is not the expected false positive rate of voxels. If the SPM is smooth, one false positive peak may be associated with hundreds of voxels. Bonferroni correction controls the expected number of false positive <i>voxels</i>, whereas RFT controls the expected number of false positive <i>peaks</i>. Because the number of peaks is always less than the number of voxels, RFT can use a lower threshold, rendering it much more sensitive.  In fact, the number of false positive voxels is somewhat irrelevant because it is a function of smoothness. The RFT correction discounts voxel size by expressing the search volume in terms of smoothness or resolution elements (<i>RESELS</i>), see Fig.<a href="#F2">2</a>. This intuitive perspective is expressed formally in terms of differential topology using the <i>Euler characteristic</i> (Worsley et al. 1992). At high thresholds the Euler characteristic corresponds to the number peaks above threshold.
+== Reference ==
-</p><p>There are only two assumptions underlying the use of the RFT:
+Information copied from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]
 == More info ==
 [http://www.ehu.es/ccwintco/uploads/4/4c/Spm-rft-slides-poirrier06.pdf Random Field Theory on fMRI]

Alexsavio en 14:59 13 ene 2009

2009-01-13T14:59:46Z

@@ Línea 1: / Línea 1: @@
 <center>[[Imagen:Random_Field_Theory_SPM.png]]</center>
-Schematic illustrating the use of Random Field Theory in making inferences about SPMs. If one knew precisely where to look, then inference can be based on the value of the statistic at the specified location in the SPM. However, generally, one does not have a precise anatomical prior, and an adjustment for multiple dependent comparisons has to be made to the p-values. These corrections use distributional approximations from RFT. This schematic deals with a general case of n SPM{t} whose voxels all survive a common threshold u (i.e. a conjunction of n component SPMs). The central probability, upon which all peak, cluster or set-level inferences are made, is the probability P(u,c,k) of getting c or more clusters with k or more RESELS (resolution elements) above this threshold. By assuming that clusters behave like a multidimensional Poisson point-process (i.e., the Poisson clumping heuristic), P(u,c,k) is determined simply: the distribution of c is Poisson with an expectation that corresponds to the product of the expected number of clusters, of any size, and the probability that any cluster will be bigger than k RESELS. The latter probability depends on the expected number of RESELS per cluster \eta. This is simply the expected supra-threshold volume, divided by the expected number of clusters. The expected number of clusters \psi_0 is estimated with the Euler characteristic (EC) (effectively the number of blobs minus the number of holes). This depends on the EC density for the statistic in question (with degrees of freedom \nu) and the RESEL counts. The EC density is the expected EC per unit of D-dimensional volume of the SPM where the volume of the search is given by the RESEL counts. RESEL counts are a volume measure that has been normalized by the smoothness of the SPMs component error fields (\epsilon), expressed in terms of the full width at half maximum (FWHM). In this example equations for a sphere of radius \epsilon are given. \Psi denotes the cumulative density function for the statistic in question.
+Schematic illustrating the use of Random Field Theory in making inferences about SPMs. If one knew precisely where to look, then inference can be based on the value of the statistic at the specified location in the SPM.  However, generally, one does not have a precise anatomical <i>prior</i>, and an adjustment for multiple dependent comparisons has to be made to the <i>p</i>-values. These corrections use distributional approximations from RFT. This schematic deals with a general case of <i>n</i> SPM{<i>t</i>} whose voxels all survive a common threshold <img class='tex' src="/wiki/images/math/1e0fbb458ac1325b073be7a4fad98fc8.png" alt="u" /> (<i>i.e.</i> a conjunction of <img class='tex' src="/wiki/images/math/30c2a24cd38b0414fae013c0d44b0ca9.png" alt="n" /> component SPMs). The central probability, upon which all peak, cluster or set-level inferences are made, is the probability <img class='tex' src="/wiki/images/math/802c4d6b0b186aa9ab1853dfa96590b2.png" alt="P(u,c,k)" /> of getting <img class='tex' src="/wiki/images/math/2d043a3ca0862ca27fb2ff3a1eb2f1cf.png" alt="c" /> or more clusters with <img class='tex' src="/wiki/images/math/524eecc1d5922a435e8e848094405caa.png" alt="k" /> or more <i>RESELS</i> (resolution elements) above this threshold. By assuming that clusters behave like a multidimensional Poisson point-process (<i>i.e.</i>, the Poisson clumping heuristic), <img class='tex' src="/wiki/images/math/802c4d6b0b186aa9ab1853dfa96590b2.png" alt="P(u,c,k)" /> is determined simply: the distribution of <img class='tex' src="/wiki/images/math/2d043a3ca0862ca27fb2ff3a1eb2f1cf.png" alt="c" /> is Poisson with an expectation that corresponds to the product of the expected number of clusters, of any size, and the probability that any cluster will be bigger than <img class='tex' src="/wiki/images/math/524eecc1d5922a435e8e848094405caa.png" alt="k" /> RESELS. The latter probability depends on the expected number of RESELS per cluster <img class='tex' src="/wiki/images/math/3dc2e0d31e939dc8e285bb3aa56234db.png" alt="\eta" />. This is simply the expected supra-threshold volume, divided by the expected number of clusters. The expected number of clusters <img class='tex' src="/wiki/images/math/b0ca0d9df95d65286632112b2c3f9cc1.png" alt="\psi_0" /> is estimated with the Euler characteristic (EC) (effectively the number of blobs minus the number of holes). This depends on the EC density for the statistic in question (with  degrees of freedom <img class='tex' src="/wiki/images/math/959822faa03ebcea9210021baf302c54.png" alt="\nu" />) and the RESEL counts. The EC density is the expected EC per unit of <img class='tex' src="/wiki/images/math/5616208fafc6b96190c6b46e0b649a10.png" alt="D" />-dimensional volume of the SPM where the volume of the search is given by the RESEL counts. RESEL counts are a volume measure that has been normalized by the smoothness of the SPMs component error fields (<img class='tex' src="/wiki/images/math/444bc4156297541dbcd45dc59237d731.png" alt="\epsilon" />), expressed in terms of the full width at half maximum (<i>FWHM</i>). In this example equations for a sphere of radius <img class='tex' src="/wiki/images/math/444bc4156297541dbcd45dc59237d731.png" alt="\epsilon" /> are given. <img class='tex' src="/wiki/images/math/398b8c3e58e961cdea0fe626771f3243.png" alt="\Psi" /> denotes the cumulative density function for the statistic in question.</div></div></div></a>
 Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]

Alexsavio en 14:57 13 ene 2009

2009-01-13T14:57:44Z

← Revisión anterior		Revisión del 16:57 13 ene 2009
Línea 1:		Línea 1:
	<center>[[Imagen:Random_Field_Theory_SPM.png]]</center>		<center>[[Imagen:Random_Field_Theory_SPM.png]]</center>

			Schematic illustrating the use of Random Field Theory in making inferences about SPMs. If one knew precisely where to look, then inference can be based on the value of the statistic at the specified location in the SPM. However, generally, one does not have a precise anatomical prior, and an adjustment for multiple dependent comparisons has to be made to the p-values. These corrections use distributional approximations from RFT. This schematic deals with a general case of n SPM{t} whose voxels all survive a common threshold u (i.e. a conjunction of n component SPMs). The central probability, upon which all peak, cluster or set-level inferences are made, is the probability P(u,c,k) of getting c or more clusters with k or more RESELS (resolution elements) above this threshold. By assuming that clusters behave like a multidimensional Poisson point-process (i.e., the Poisson clumping heuristic), P(u,c,k) is determined simply: the distribution of c is Poisson with an expectation that corresponds to the product of the expected number of clusters, of any size, and the probability that any cluster will be bigger than k RESELS. The latter probability depends on the expected number of RESELS per cluster \eta. This is simply the expected supra-threshold volume, divided by the expected number of clusters. The expected number of clusters \psi_0 is estimated with the Euler characteristic (EC) (effectively the number of blobs minus the number of holes). This depends on the EC density for the statistic in question (with degrees of freedom \nu) and the RESEL counts. The EC density is the expected EC per unit of D-dimensional volume of the SPM where the volume of the search is given by the RESEL counts. RESEL counts are a volume measure that has been normalized by the smoothness of the SPMs component error fields (\epsilon), expressed in terms of the full width at half maximum (FWHM). In this example equations for a sphere of radius \epsilon are given. \Psi denotes the cumulative density function for the statistic in question.

	Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]		Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]

Alexsavio: /* More info */

2009-01-13T14:56:53Z

More info

← Revisión anterior		Revisión del 16:56 13 ene 2009
Línea 3:		Línea 3:
	Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]		Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]

	=== More info ===		== More info ==

	[http://www.ehu.es/ccwintco/uploads/4/4c/Spm-rft-slides-poirrier06.pdf Random Field Theory on fMRI]		[http://www.ehu.es/ccwintco/uploads/4/4c/Spm-rft-slides-poirrier06.pdf Random Field Theory on fMRI]

Alexsavio: /* More info */

2009-01-13T14:56:21Z

More info

← Revisión anterior		Revisión del 16:56 13 ene 2009
Línea 5:		Línea 5:
	=== More info ===		=== More info ===

	[[http://www.ehu.es/ccwintco/uploads/4/4c/Spm-rft-slides-poirrier06.pdf \| Random Field Theory on fMRI]]		[http://www.ehu.es/ccwintco/uploads/4/4c/Spm-rft-slides-poirrier06.pdf Random Field Theory on fMRI]

Alexsavio en 14:56 13 ene 2009

2009-01-13T14:56:00Z

← Revisión anterior		Revisión del 16:56 13 ene 2009
Línea 2:		Línea 2:

	Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]		Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]

			=== More info ===

			[[http://www.ehu.es/ccwintco/uploads/4/4c/Spm-rft-slides-poirrier06.pdf \| Random Field Theory on fMRI]]

Alexsavio en 14:53 13 ene 2009

2009-01-13T14:53:38Z

← Revisión anterior		Revisión del 16:53 13 ene 2009
Línea 1:		Línea 1:
			<center>[[Imagen:Random_Field_Theory_SPM.png]]</center>

	[~~[Imagen~~:~~Random_Field_Theory_SPM~~.~~png]~~]		Information extracted from [http://www.scholarpedia.org/article/Statistical_parametric_mapping_(SPM) here]

Alexsavio en 14:50 13 ene 2009

2009-01-13T14:50:59Z

Página nueva

[[Imagen:Random_Field_Theory_SPM.png]]