<table><tr><td>Figarch Model</td><td>Embedding delay</td></tr><tr><td>Figarch d=0.05</td><td>2</td></tr><tr><td>Figarch d=0.15</td><td>1</td></tr><tr><td>Figarch d=0.25</td><td>2</td></tr><tr><td>Figarch d=0.35</td><td>4</td></tr><tr><td>Figarch d=0.45</td><td>6</td></tr><tr><td>Figarch d=0.55</td><td>7</td></tr><tr><td>Figarch d=0.65</td><td>8</td></tr><tr><td>Figarch d=0.75</td><td>8</td></tr><tr><td>Figarch d=0.80</td><td>6</td></tr><tr><td>Figarch d=0.90</td><td>8</td></tr></table>

Table 1: Mutual Information for each Figarch Model.

The nearest neighbor in phase space will be a vector;

\[ y ^ { N N } ( i ) = ( x _ { i } ^ { N N } , x _ { i + T } ^ { N N } , x _ { i + 2 T } ^ { N N } , . . . , x _ { i + ( d - 1 ) T } ^ { N N } ) \eqno ( 1 1 ) \]

If the vector \( y ^ { N N } ( i ) \) is a false neighbor of y(i) having arrived its neighborhood by projection from a higher dimension because the present dimension d doesn’t unfold the attractor, then by going to next dimension \( d + \) 1 this false neighbor may be removed out of the neighborhood of y(i).

By looking at every data point y(i) and asking at what dimension all false neighbors are removed, we will sequentially intersections of orbits of lower and lower dimension are removed until at last point intersections are removed. At thatj uncture d will have been identified where the attractor is unfolded.

Comparing the distance between the vectors y(i) and \( y ^ { N N } ( i ) \) in dimension d with the dis-tance between the same vectors in dimension \( d + \) 1, it can easily be established which are true neighbors and which false. It only needs to be compared \( x _ { ( i + d T ) } - x _ { ( i + d T ) } ^ { N N } \) with the Euclidian distance \|yi − yNNi\| between nearest neighbors in dimension d.

If the additional distance is large compared to the distance in dimension d between nearest neighbors, then we have a false neighbor.

The square of the Euclidian distance between the nearest neighbor points as seen in di-mension d is

\[ R _ { d } ( i ) ^ { 2 } = \sum _ { m = 1 } ^ { d } [ x _ { i + ( m - 1 ) T } - x _ { i + ( m - 1 ) T } ^ { N N } ] ^ { 2 } \eqno ( 1 2 ) \]

while dimension d+1 it is;

\[ R _ { d + 1 } ( i ) ^ { 2 } = \sum _ { m = 1 } ^ { d + 1 } [ x _ { i + ( m - 1 ) T } - x _ { i + ( m - 1 ) T } ^ { N N } ] ^ { 2 } \eqno ( 1 3 ) \]

\[ R _ { d + 1 } ( i ) ^ { 2 } = R _ { d } ( i ) ^ { 2 } + | x _ { i + d T } - x _ { i + d T } ^ { N N } | ^ { 2 } \eqno ( 1 4 ) \]

The distance between points when seen in dimension \( d + \) 1 relative to the distance in dimension d is;

\[ \sqrt { \frac { R _ { d + 1 } ( i ) ^ { 2 } - R _ { d } ( i ) ^ { 2 } } { R _ { d } ( i ) ^ { 2 } } } = \frac { x _ { i + d T } - x _ { i + d t } ^ { N N } } { R _ { d } ( i ) } > r _ { t o l } \eqno ( 1 5 ) \]

When this quantity is larger than some threshold, we have a false neighbor (Kennel M B, Brown R and Abarbanel H D 1992).

Plot of percentage of false neighbors show the unfolded geometry and where there is no unfolding any more. With the correct choice of d dimension, modelling the data in d number of dynamical degrees of freedom will be adequate to capture the properties of the source.

Figure 2: FNN embedding dimension result for FIGARCH d=0.90.

The figure 2 shows minimum embedding dimension where percentage of nearest neighbors goes to zero taken into account some threshold \( r _ { t o l } \). Disappearance of false neighbors indicates minimum embedding dimension \( r _ { t o l } \) is false neighbor Euclidian distance tolerance and \( a _ { t o l } \) is neighbor tolerance based on attractor size. The neighbors are declared false neighbors, when the ratio of the Euclidian distances between neighbor candidates in successive embedding dimensions exceeds \( r _ { t o l } \).