Glossary - Revision history

Mellinger at 13:29, 30 January 2013

2013-01-30T13:29:15Z

← Older revision		Revision as of 13:29, 30 January 2013
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	A glossary of general terms		A glossary of general terms

	==<math>r^2</math> ~~(RSquared, Coefficient of Determination)~~==		== Coefficient of Determination, Determination Coefficient, RSquared, <math>r^2</math>==
	The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ in relation to variance. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition. It is a measure of how well the original task condition ("user intent") may be inferred from a brain signal.		The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ in relation to variance. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition. It is a measure of how well the original task condition ("user intent") may be inferred from a brain signal.

Mellinger at 13:27, 30 January 2013

2013-01-30T13:27:52Z

← Older revision		Revision as of 13:27, 30 January 2013
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	A glossary of general terms		A glossary of general terms

	==<math>r^2</math> (Coefficient of Determination)==		==<math>r^2</math> (RSquared, Coefficient of Determination)==
	The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ in relation to variance. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition. It is a measure of how well the original task condition ("user intent") may be inferred from a brain signal.		The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ in relation to variance. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition. It is a measure of how well the original task condition ("user intent") may be inferred from a brain signal.

Mellinger at 13:13, 23 July 2012

2012-07-23T13:13:43Z

← Older revision		Revision as of 13:13, 23 July 2012
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	==<math>r^2</math> (Coefficient of Determination)==		==<math>r^2</math> (Coefficient of Determination)==
	The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ in relation to variance. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition.		The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ in relation to variance. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition. It is a measure of how well the original task condition ("user intent") may be inferred from a brain signal.

	===Computation===		===Computation===
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	where we have introduced <math>G:=\frac{(s_1+s_2)^2}{n_1+n_2}</math> to arrive at the last expression, which is useful for efficient computation of <math>r^2</math>.		where we have introduced <math>G:=\frac{(s_1+s_2)^2}{n_1+n_2}</math> to arrive at the last expression, which is useful for efficient computation of <math>r^2</math>.

			===Relation to Classifier Training, and Linear Discriminant Analysis (LDA)===
			LDA is a training method for linear classifiers. A linear classifier is a classification algorithm that infers the condition (class, task, label) under which a point of data was recorded, simply by computing a weighted sum of the data point's dimensions, and comparing the result to a threshold. "Training" a classifier means that recorded data is used in order to determine a weighting that is "optimally suited" to do so. In high-dimensional space, weighting dimensions corresponds to the geometrical operation of projecting a data point onto a line. Thus, one also speaks of a "linear projection".

			In case of LDA, the criterion of optimality is chosen as the ratio of "inter-class variance" and "within-class variance". The LDA algorithm determines a projection from high-dimensional data space onto a single dimension such that, for the resulting numbers, the ratio of mean difference between classes, and variance within classes, is as large as possible. Comparing this ratio to the above formulae, you may see that the numerator of the <math>r^2</math> value is proportional to the squared mean difference between classes, and the denominator is proportional to the squared mean difference within each class (when assuming equal covariances, as LDA does in its simplest form).

			Thus, we may say that LDA classifier training chooses a linear classifier such that the classifier's output has the highest possible <math>r^2</math> value, given the data available.

	===Significance Test===		===Significance Test===

Mellinger: /* $r^2$ (Coefficient of Determination) */

2012-07-11T15:58:36Z

<math>r^2</math> (Coefficient of Determination)

← Older revision		Revision as of 15:58, 11 July 2012
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	==<math>r^2</math> (Coefficient of Determination)==		==<math>r^2</math> (Coefficient of Determination)==
	The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition.		The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ in relation to variance. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition.

	===Computation===		===Computation===

Mellinger: /* Computation */

2012-01-09T16:54:15Z

Computation

← Older revision		Revision as of 16:54, 9 January 2012
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	===Computation===		===Computation===
	The coefficient of determination is in fact the squared correlation coefficient for a distribution constructed from two sets of univariate data as follows:		The coefficient of determination is in fact the squared correlation coefficient for a single bivariate distribution constructed from two sets of univariate data as follows:
	*There are <math>n_1</math> values <math>x^{(1)}_i</math> measured under condition 1, and <math>n_2</math> values <math>x^{(2)}_i</math> measured under condition 2.		*There are <math>n_1</math> values <math>x^{(1)}_i</math> measured under condition 1, and <math>n_2</math> values <math>x^{(2)}_i</math> measured under condition 2.
	*From these two data sets, a single, two-dimensional data set is constructed, consisting of points <math>(x,y)</math> by assigning the measured value to <math>x</math>, and setting <math>y=+1</math> if the value has been measured under condition 1, and <math>y=-1</math> for condition 2. The actual label values assigned to <math>y</math> do not matter, as long as they are distinct, but choosing +1 and -1 simplifies further computation.		*From these two data sets, a single, two-dimensional data set is constructed, consisting of points <math>(x,y)</math> by assigning the measured value to <math>x</math>, and setting <math>y=+1</math> if the value has been measured under condition 1, and <math>y=-1</math> for condition 2. The actual label values assigned to <math>y</math> do not matter, as long as they are distinct, but choosing +1 and -1 simplifies further computation.

Mellinger: /* Computation */

2012-01-09T16:53:28Z

Computation

← Older revision		Revision as of 16:53, 9 January 2012
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	<math>r^2 = \frac{1}{n_1n_2} \frac{(s_1n_2-s_2n_1)^2}{(n_1+n_2)(q_1+q_2)-(s_1+s_2)^2} = \frac{{s_1^2}/{n_1}+{s_2^2}/{n_2}-G}{q_1+q_2-G}</math>		<math>r^2 = \frac{1}{n_1n_2} \frac{(s_1n_2-s_2n_1)^2}{(n_1+n_2)(q_1+q_2)-(s_1+s_2)^2} = \frac{{s_1^2}/{n_1}+{s_2^2}/{n_2}-G}{q_1+q_2-G}</math>

	where we have ~~defined~~ <math>G=\frac{(s_1+s_2)^2}{n_1+n_2}</math> to arrive at the last expression, which is useful for efficient computation of <math>r^2</math>.		where we have introduced <math>G:=\frac{(s_1+s_2)^2}{n_1+n_2}</math> to arrive at the last expression, which is useful for efficient computation of <math>r^2</math>.

	===Significance Test===		===Significance Test===

Mellinger at 16:45, 9 January 2012

2012-01-09T16:45:47Z

@@ Line 2: / Line 2: @@
 ==<math>r^2</math> (Coefficient of Determination)==
-The statistical measure <math>r^2</math> calculates the fraction of the total signal variance that is accounted for by the task.
+The coefficient of determination, or <math>r^2</math> value, is a statistical measure computed over a pair of sample distributions, giving a measure of how strongly the means of the two distributions differ. In a BCI context, the coefficient of determination is computed over signals that have been measured under two different task conditions, and represents the fraction of the total signal variance that is accounted for ("determined by") by the task condition.
 For small numbers of trials (<30 per label), determination coefficients can become rather large even if there is no significant effect. To assess significance under the (robust) assumption of Gaussianity, determination coefficients may be converted into t-values according to
-<math>t^2=(n-2) \frac{r^2}{1-r^2}</math>
+<math>t^2=(n_1 + n_2 -2) \frac{r^2}{1-r^2}</math>.
-and used in a one-tailed t-test against the "no correlation" null hypothesis.
+Such a t-value may be used in a one-tailed t-test against the "no correlation" null hypothesis.
 ==Sensorimotor Rhythm (SMR)==

Mellinger at 10:26, 27 February 2008

2008-02-27T10:26:21Z

← Older revision		Revision as of 10:26, 27 February 2008
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	and used in a one-tailed t-test against the "no correlation" null hypothesis.		and used in a one-tailed t-test against the "no correlation" null hypothesis.

			==Sensorimotor Rhythm (SMR)==
			A general term referring to idling activity from the sensory and motor cortices.
			The [[User Tutorial:Introduction to the Mu Rhythm\|mu rhythm]] is the most prominent SMR activity.

	[[Category:Glossary]]		[[Category:Glossary]]

Mellinger: /* $r^2$ (Coefficient of Determination) */

2008-01-16T17:33:40Z

<math>r^2</math> (Coefficient of Determination)

← Older revision		Revision as of 17:33, 16 January 2008
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	For small numbers of trials (<30 per label), determination coefficients can become rather large even if there is no significant effect. To assess significance under the (robust) assumption of Gaussianity, determination coefficients may be converted into t-values according to		For small numbers of trials (<30 per label), determination coefficients can become rather large even if there is no significant effect. To assess significance under the (robust) assumption of Gaussianity, determination coefficients may be converted into t-values according to

	<math>t^2=(n-2) frac{r^2}{1-r^2}</math>		<math>t^2=(n-2) \frac{r^2}{1-r^2}</math>

	and used in a one-tailed t-test against the "no correlation" null hypothesis.		and used in a one-tailed t-test against the "no correlation" null hypothesis.

	[[Category:Glossary]]		[[Category:Glossary]]

Mellinger at 16:36, 16 January 2008

2008-01-16T16:36:01Z

← Older revision		Revision as of 16:36, 16 January 2008
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	==<math>r^2</math> (Coefficient of Determination)==		==<math>r^2</math> (Coefficient of Determination)==
	The statistical measure <math>r^2</math> calculates the fraction of the total signal variance that is accounted for by the task.		The statistical measure <math>r^2</math> calculates the fraction of the total signal variance that is accounted for by the task.

			For small numbers of trials (<30 per label), determination coefficients can become rather large even if there is no significant effect. To assess significance under the (robust) assumption of Gaussianity, determination coefficients may be converted into t-values according to

			<math>t^2=(n-2) frac{r^2}{1-r^2}</math>

			and used in a one-tailed t-test against the "no correlation" null hypothesis.

	[[Category:Glossary]]		[[Category:Glossary]]

Glossary - Revision history

Mellinger at 13:29, 30 January 2013

Mellinger at 13:27, 30 January 2013

Mellinger at 13:13, 23 July 2012

Mellinger: /* r^2 (Coefficient of Determination) */

Mellinger: /* Computation */

Mellinger: /* Computation */

Mellinger at 16:45, 9 January 2012

Mellinger at 10:26, 27 February 2008

Mellinger: /* r^2 (Coefficient of Determination) */

Mellinger at 16:36, 16 January 2008

Mellinger: /* $r^2$ (Coefficient of Determination) */

Mellinger: /* $r^2$ (Coefficient of Determination) */