Contributions:StatisticsFilter - Revision history

Mellinger: /* Aim */

2022-05-20T11:23:02Z

Aim

← Older revision		Revision as of 11:23, 20 May 2022
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	==Methods==		==Methods==
	===Aim===		===Aim===
	To compute statistical measures from streams of data, such that at each point in time, estimates of those measure are available that depend on recent data more strongly than on old data.		To compute statistical measures from streams of data such that, at each point in time, estimates of those measure are available that depend on recent data more strongly than on old data.

	===Approach===		===Approach===
	Assigning weights to past data, such that a data point's weight only depends on its age, then doing "weighted statistics" on the data.		Assigning weights to past data, such that a data point's weight only depends on its age, then doing "weighted statistics" on the data.

Mellinger: /* Weight function */

2013-06-14T20:08:57Z

Weight function

← Older revision		Revision as of 20:08, 14 June 2013
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	At the same time, computation of the resulting power sums can be done very efficiently by an algorithm equivalent to a first-order low pass filter as follows:		At the same time, computation of the resulting power sums can be done very efficiently by an algorithm equivalent to a first-order low pass filter as follows:
	~~<code>~~

	1: Initialize nth power sum to 0 (all 0 entries in the multivariate case);		1: Initialize nth power sum to 0 (all 0 entries in the multivariate case);
			2: for each observed data vector do:
	2: for each observed data vector do:		3: multiply nth power sum with a decay factor <= 1;
			4: compute nth power of data vector (nth outer product with itself in the multivariate case);
	3: multiply nth power sum with a decay factor <= 1;		5: add the result of (4) to the power sum;
			6: done.
	4: compute nth power of data vector (nth outer product with itself in the multivariate case);

	5: add the result of (4) to the power sum;

	6: done.
	~~</code>~~

	In each step, an appropriate decay factor may be determined from the user-supplied time constant, and the time interval between the previous and the current observation. For data sampled at equally spaced time points, the decay factor is constant.		In each step, an appropriate decay factor may be determined from the user-supplied time constant, and the time interval between the previous and the current observation. For data sampled at equally spaced time points, the decay factor is constant.

Mellinger: /* Weight function */

2013-06-14T20:07:32Z

Weight function

← Older revision		Revision as of 20:07, 14 June 2013
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	2: for each observed data vector do:		2: for each observed data vector do:

	3: multiply nth power sum with a decay factor <= 1;		3: multiply nth power sum with a decay factor <= 1;

	4: compute nth power of data vector (nth outer product with itself in the multivariate case);		4: compute nth power of data vector (nth outer product with itself in the multivariate case);

	5: add the result of (4) to the power sum;		5: add the result of (4) to the power sum;

	6: done.		6: done.

Mellinger: /* Method: continously updated statistics */

2013-06-14T20:03:12Z

Method: continously updated statistics

← Older revision		Revision as of 20:03, 14 June 2013
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	*Known to compile under: 4164		*Known to compile under: 4164

	==~~Method: continously updated statistics~~==		==Methods==
	===Aim===		===Aim===
	To compute statistical measures from streams of data, such that at each point in time, estimates of those measure are available that depend on recent data more strongly than on old data.		To compute statistical measures from streams of data, such that at each point in time, estimates of those measure are available that depend on recent data more strongly than on old data.

Mellinger: /* Weight function */

2013-06-14T19:58:01Z

Weight function

← Older revision		Revision as of 19:58, 14 June 2013
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	====Weight function====		====Weight function====
	As a weight function, we choose a decaying exponential, and take the associated time constant to represent a measure of "past observation duration". The decaying exponential is not chosen arbitrarily, but is of special significance because it represents a one-parametric distribution of minimal entropy over the interval (-inf, 0], just as the Gaussian represents a one-parametric minimal-entropy distribution over the interval (-inf, inf).		As a weight function, we choose a decaying exponential, and take the associated time constant to represent a measure of "past observation duration". The decaying exponential is not chosen arbitrarily, but is of special significance because it represents a one-parametric distribution of minimal entropy over the interval (-inf, 0], just as the zero-mean Gaussian represents a one-parametric minimal-entropy distribution over the interval (-inf, inf).

	At the same time, computation of the resulting power sums can be done very efficiently by an algorithm equivalent to a first-order low pass filter as follows:		At the same time, computation of the resulting power sums can be done very efficiently by an algorithm equivalent to a first-order low pass filter as follows:

Mellinger: /* Weight function */

2013-06-14T19:57:20Z

Weight function

← Older revision		Revision as of 19:57, 14 June 2013
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	</code>		</code>

	In each step, an appropriate decay factor may be determined from the user-supplied time constant, and the time interval between the previous and the current observation. For data sampled at equally spaced time ~~intervals~~, the decay factor is constant.		In each step, an appropriate decay factor may be determined from the user-supplied time constant, and the time interval between the previous and the current observation. For data sampled at equally spaced time points, the decay factor is constant.

	====Higher order moments====		====Higher order moments====

Mellinger: /* Weight function */

2013-06-14T19:56:45Z

Weight function

← Older revision		Revision as of 19:56, 14 June 2013
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	====Weight function====		====Weight function====
	As a weight function, we choose a decaying exponential, and take the associated time constant to represent a measure of "past observation duration". The decaying exponential is not chosen arbitrarily, but is of special significance because it represents a one-parametric distribution of minimal entropy over the interval (-inf, 0], just as the Gaussian represents a one-parametric minimal-entropy distribution over the interval (-inf, inf). At the same time, computation of the resulting power sums can be done very efficiently by an algorithm equivalent to a first-order low pass filter as follows:		As a weight function, we choose a decaying exponential, and take the associated time constant to represent a measure of "past observation duration". The decaying exponential is not chosen arbitrarily, but is of special significance because it represents a one-parametric distribution of minimal entropy over the interval (-inf, 0], just as the Gaussian represents a one-parametric minimal-entropy distribution over the interval (-inf, inf).
	1: Initialize nth power sum to 0 (~~or an n-dimensional array with~~ all 0 entries in the multivariate case)
	2: for each observed data vector		At the same time, computation of the resulting power sums can be done very efficiently by an algorithm equivalent to a first-order low pass filter as follows:
	3: multiply nth power sum with a decay factor <= 1		<code>
	4: compute nth power of data vector (nth outer product with itself in the multivariate case)
	5: add the result of (4) to the power sum		1: Initialize nth power sum to 0 (all 0 entries in the multivariate case);

			2: for each observed data vector do:

			3: multiply nth power sum with a decay factor <= 1;

			4: compute nth power of data vector (nth outer product with itself in the multivariate case);

			5: add the result of (4) to the power sum;

			6: done.
			</code>

	In each step, an appropriate decay factor may be determined from the user-supplied time constant, and the time interval between the previous and the current observation. For data sampled at equally spaced time intervals, the decay factor is constant.		In each step, an appropriate decay factor may be determined from the user-supplied time constant, and the time interval between the previous and the current observation. For data sampled at equally spaced time intervals, the decay factor is constant.

Mellinger: /* obs.Quantile(number p) */

2013-06-14T19:53:54Z

obs.Quantile(number p)

← Older revision		Revision as of 19:53, 14 June 2013
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	=====obs.CentralMoment(n)=====		=====obs.CentralMoment(n)=====
	The n-th central moment for each dimension of observer "obs".		The n-th central moment for each dimension of observer "obs".
	=====obs.Quantile(~~number~~ p)=====		=====obs.Quantile(p)=====
	The value corresponding to the the given relative frequency in each dimension of observer "obs".		The value corresponding to the the given relative frequency p in each dimension of observer "obs". p is restricted by <math>0\leq p \leq 1</math>.

	=====obs.QQuantiles(n)=====		=====obs.QQuantiles(n)=====

Mellinger: /* obs.Quantile(0>=p>=1) */

2013-06-14T19:52:17Z

obs.Quantile(0>=p>=1)

← Older revision		Revision as of 19:52, 14 June 2013
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	=====obs.CentralMoment(n)=====		=====obs.CentralMoment(n)=====
	The n-th central moment for each dimension of observer "obs".		The n-th central moment for each dimension of observer "obs".
	=====obs.Quantile(~~0>=~~p~~>=1~~)=====		=====obs.Quantile(number p)=====
	The value corresponding to the the given relative frequency in each dimension of observer "obs".		The value corresponding to the the given relative frequency in each dimension of observer "obs".

	=====obs.QQuantiles(n)=====		=====obs.QQuantiles(n)=====
	For each dimension of observer "obs", a vector of values corresponding to the n-th q-quantile. This vector has n+1 entries, with the minimum in the first entry, and the maximum in the last entry.		For each dimension of observer "obs", a vector of values corresponding to the n-th q-quantile. This vector has n+1 entries, with the minimum in the first entry, and the maximum in the last entry.

Mellinger: /* General derivation of formulae */

2013-06-14T19:49:23Z

General derivation of formulae

← Older revision		Revision as of 19:49, 14 June 2013
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	In the multivariate case, powers are to be replaced by outer products of vectors with themselves, e.g. <math>P_1</math> is a vector, <math>P_2</math> is a square matrix, <math>P_3</math> is an array with 3 indices, etc.		In the multivariate case, powers are to be replaced by outer products of vectors with themselves, e.g. <math>P_1</math> is a vector, <math>P_2</math> is a square matrix, <math>P_3</math> is an array with 3 indices, etc.
	Then, covariance becomes (writing ** for the outer vector product):		Then, covariance becomes (writing ^ for the outer vector product):
	covariance matrix = <math>P_2/P_0-(P_1**P_1)/P_0^2</math>
			covariance matrix = <math>\frac{P_2}{P_0}-\frac{P_1\wedge P_1}{P_0^2}</math>.

	For statistical measures describing the relation between two sets of observables, e.g. the rsquared, or the z score, the same principle applies: the measure is expressed in terms of power sums over each of the two sets, and the weighted power sums from each set are inserted into the resulting expression.		For statistical measures describing the relation between two sets of observables, e.g. the rsquared, or the z score, the same principle applies: the measure is expressed in terms of power sums over each of the two sets, and the weighted power sums from each set are inserted into the resulting expression.