From the Kalman Filter to the Particle Filter: A Geometrical Perspective of the Curse of Dimensionality

Figure 10

Illustration of the hyperspheres that contain the typical samples of the prior and a posteriori Gaussian distributions valid for the EnKF and the PF in the Gaussian setting, where + denotes the background , denotes the analysis state , the circle without mark denotes the background distribution, the circle with triangle marks denotes the analysis distribution, and the circle with diamond marks is centered on the analysis state with a radius of the typical distance between the analysis state and the background samples. These hyperspheres are reproduced for the various length-scale parameters (a)  km, (b)  km, (c)  km, and (d)  km, for the dimension set to .

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