EC.3D {AnalyzeFMRI} R Documentation

## Expected Euler Characteristic for a 3D Random Field

### Description

Calculates the Expected Euler Characteristic for a 3D Random Field thesholded a level u.

### Usage

```EC.3D(u, sigma, voxdim = c(1, 1, 1), num.vox, type = c("Normal", "t"), df = NULL)
```

### Arguments

 `u` The threshold for the field. `sigma` The spatial covariance matrix of the field. `voxdim` The dimensions of the cuboid 'voxels' upon which the discretized field is observed. `num.vox` The number of voxels that make up the field. `type` The marginal distribution of the Random Field (only Normal and t at present). `df` The degrees of freedom of the t field.

### Details

The Euler Characteristic χ_u (Adler, 1981) is a topological measure that essentially counts the number of isolated regions of the random field above the threshold u minus the number of 'holes'. As u increases the holes disappear and χ_u counts the number of local maxima. So when u becomes close to the maximum of the random field Z_{\textrm{max}} we have that

P( \textrm{reject} H_0 | H_0 \textrm{true}) = P(Z_{\textrm{max}}) = P(χ_u > 0) \approx E(χ_u)

where H_0 is the null hypothesis that there is no signicant positive actiavtion/signal present in the field. Thus the Type I error of the test can be controlled through knowledge of the Expected Euler characteristic.

### Value

The value of the expected Euler Characteristic.

J. L. Marchini

### References

Adler, R. (1981) The Geometry of Random Fields.. New York: Wiley. Worlsey, K. J. (1994) Local maxima and the expected euler characteristic of excursion sets of χ^2, f and t fields. Advances in Applied Probability, 26, 13-42.

`Threshold.RF`
```EC.3D(4.6, sigma = diag(1, 3), voxdim = c(1, 1, 1), num.vox = 10000)