Extrapolation - Revision history

Guillaume: /* Gaussian Weighted Average */

2012-02-13T16:34:39Z

‎Gaussian Weighted Average

Guillaume at 18:22, 10 February 2012

2012-02-10T18:22:20Z

Guillaume: /* Gaussian Weighted Average */

2012-02-10T18:21:27Z

‎Gaussian Weighted Average

Guillaume: /* Gaussian Weighted Average */

2012-02-10T16:35:26Z

‎Gaussian Weighted Average

Guillaume: /* Gaussian Weighted Average */

2012-02-10T16:33:23Z

‎Gaussian Weighted Average

Guillaume: /* Gaussian Weighted Average */

2012-02-10T16:12:17Z

‎Gaussian Weighted Average

Guillaume: /* Gaussian Weighted Average */

2012-02-10T16:08:38Z

‎Gaussian Weighted Average

Guillaume: /* Gaussian Weighted Average */

2012-02-10T13:07:28Z

‎Gaussian Weighted Average

Guillaume: /* Squared Weighted Average */

2012-02-10T12:58:11Z

‎Squared Weighted Average

Guillaume at 12:54, 10 February 2012

2012-02-10T12:54:27Z

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 ==Gaussian Weighted Average==
-A gaussian weighted average allows to determine characteristic radius of influence, <math>\sigma</math>, where, within that radius, the points of <math>I</math> will have a total weight of 68% in the average relative to the other points of <math>I</math>. This radius of influence parameter is useful to parameterize the interpolation/extrapolation. Choices can be made such as choosing a shorter radius when the interpolating dataset shows greater variability or such as choosing a larger radius when the interpolating dataset displays a smoother variability. The isotropic gaussian function is given by:
+A [http://en.wikipedia.org/wiki/Gaussian_function gaussian] weighted average allows to determine characteristic radius of influence, <math>\sigma</math>, where, within that radius, the points of <math>I</math> will have a total weight of 68% in the average relative to the other points of <math>I</math>. This radius of influence parameter is useful to parameterize the interpolation/extrapolation. Choices can be made such as choosing a shorter radius when the interpolating dataset shows greater variability or such as choosing a larger radius when the interpolating dataset displays a smoother variability. The isotropic gaussian function is given by:
 <math> w_{ki} = \e{\left(-\frac{ d^2\left(X_i,\;Y_k\right)}{\sigma^2}\right)} </math>

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-Extrapolation is often required when interpolation techniques cannot be used; often when mapping information from one dataset to a grid. Extrapolation methods vary, yielding different results both in quality and in performance. This article  proposes a couple of simple extrapolation techniques that use all the information available to extrapolate. The techniques inspire from the weighted average principle.
+Extrapolation is often required when interpolation techniques cannot be used; often when mapping information from one dataset to a grid. Extrapolation methods vary, yielding different results both in quality and in performance. This article  proposes a couple of simple extrapolation techniques that use all the available information to extrapolate. The techniques are inspired from the weighted average principle.
 ==Weighted Average==
-Suppose you have a discrete dataset <math>I</math>, with coordinates <math>X</math> on a normed space, <math>d\left(.\,,\,.\right)</math>, and want to map the dataset to another dataset, <math>O</math>, with a new set of coordinates <math>Y</math>. The operation is performed by constructing mapping coefficients <math>w</math>, also called "weights", and by performing a linear combination of the elements of <math>I</math> with the weights, <math>w</math>, to compute each element of <math>O</math>.
+Suppose that you have a discrete dataset <math>I</math>, with coordinates <math>X</math> on a normed space, <math>d\left(.\,,\,.\right)</math>, and that you want to map the dataset to another dataset, <math>O</math>, with a new set of coordinates <math>Y</math>. The operation is performed by constructing mapping coefficients <math>w</math>, also called "weights", and by performing a linear combination of the elements of <math>I</math> with the weights, <math>w</math>, to compute each element of <math>O</math>.
 <math>O_k = \frac{ \sum_i w_{ki} \, I_i }{\sum_i w_{ki}}; </math>

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 <math> w_{ki} = \e{\left(-\frac{ d^2\left(X_i,\;Y_k\right)}{\sigma^2}\right)} </math>
-Of course, non-isotropic elliptical gaussian could be used instead, but being more complex, a physical rationale to use them would also be expected. The rationale behind the use of the simple isotropic gaussian is that the user intuitively wants to define a circle of influence of the extrapolating field in the extrapolated points. And this radius is a function of the specific characteristics of field to extrapolate (smooth, constrasted, periodic, etc...).
+Of course, non-isotropic elliptical gaussian could be used instead, but being more complex, a physical rationale to use them would also be expected. The rationale behind the use of the simple isotropic gaussian is that the user intuitively wants to define a circle of influence of the extrapolating field in the extrapolated points. And this radius is a function of the specific characteristics of the field to extrapolate (smoothness, constrast, periodicity, etc...).
 <gallery widths=400px heights=250px perrow=2 caption="Extrapolation methods">

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 <math> w_{ki} = \e{\left(-\frac{ d^2\left(X_i,\;Y_k\right)}{\sigma^2}\right)} </math>
-Of course, non-isotropic elliptical gaussian could be used instead, but being more complex, a physical rationale to use them would also be expected. The rationale behind the simple isotropic gaussian is that the user intuitively wants to define a circle of influence of the extrapolating field in the extrapolated points. This radius will depend in the region to extrapolate.
+Of course, non-isotropic elliptical gaussian could be used instead, but being more complex, a physical rationale to use them would also be expected. The rationale behind the use of the simple isotropic gaussian is that the user intuitively wants to define a circle of influence of the extrapolating field in the extrapolated points. And this radius is a function of the specific characteristics of field to extrapolate (smooth, constrasted, periodic, etc...).
 <gallery widths=400px heights=250px perrow=2 caption="Extrapolation methods">

@@ Line 23: / Line 23: @@
 <math> w_{ki} = \e{\left(-\frac{ d^2\left(X_i,\;Y_k\right)}{\sigma^2}\right)} </math>
-Of course, non-isotropic elliptical gaussian could be used instead, but being more complex, a physical rationale would also be expected. The rationale behind the simple isotropic gaussian is that the user intuitively wants to define a circle of influence of the extrapolating field in the extrapolated points. This radius will depend in the region to extrapolate.
+Of course, non-isotropic elliptical gaussian could be used instead, but being more complex, a physical rationale to use them would also be expected. The rationale behind the simple isotropic gaussian is that the user intuitively wants to define a circle of influence of the extrapolating field in the extrapolated points. This radius will depend in the region to extrapolate.
 <gallery widths=400px heights=250px perrow=2 caption="Extrapolation methods">

← Older revision		Revision as of 16:08, 10 February 2012
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	<math> w_{ki} = \e{\left(-\frac{ d^2\left(X_i,\;Y_k\right)}{\sigma^2}\right)} </math>		<math> w_{ki} = \e{\left(-\frac{ d^2\left(X_i,\;Y_k\right)}{\sigma^2}\right)} </math>
		+
		+	Of course, non-isotropic elliptical gaussian could be used instead, but being more complex, a physical rationale would also be expected.

	<gallery widths=400px heights=250px perrow=2 caption="Extrapolation methods">		<gallery widths=400px heights=250px perrow=2 caption="Extrapolation methods">

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 ==Squared Weighted Average==
-The same as above, except that the squared norm is used instead of the norm. This gives an even bigger weight to the points that are closer when compared to the geometric average, proportionally to the inverse of the squared distance.
+The same as above, except that the squared norm is used instead of the norm. This gives an even bigger weight to the points that are closer when compared to the geometric average, proportionally to the inverse of the squared distance. Using the euclidean norm, this is the fastest of the three algorithms proposed in this text.
 <math> w_{ki} = \frac{1}{d^2\left(X_i,\;Y_k\right)} </math>