Comparing the second-order properties of spatial point processes

Testing the zonal stationarity of spatial point processes: applied to prostate tissues and trees locations

“…In this situation, the higher moments of the point pattern such as the second order intensity function must be compared together in different areas. If the restricted point patterns to disjoint sub-windows are assumed to be independent, their second order properties can be compared according to [20]. Without such independence, the two-factor analysis of variance model and the assumptions considered in the following are valid.…”

“…Model using 9 frequencies without ω9 T (2,1,6) 0.10 0.04 T (2,1,8) 0.13 0.08 T (5,0.25,4) 0.12 0.02 T (5,0.25, 6) 0.06 0.05 T (5,0.25, 8) 0.06 0.03 T (5,0.5,4) 0.02 0.01 T (5,0.5,6) 0.00 0.01 T (5,0.5, 8) 0.05 0.01 T (5,1,4) 0.03 0.02 T (5,1,6) 0.00 0.00 T (5,1,8) 0.00 0.00 T (7,0.25,4) 0.06 0.02 T (7,0.25, 6) 0.02 0.00 T (7,0.25, 8) 0.04 0.02 T (7,0.5,4) 0.01 0.00 T (7,0.5,6) 0.02 0.01 T (7,0.5,8) 0.00 0.00 T (7,1,4) 0.00 0.00 T (7,1,6) 0.00 0.00 T (7,1,8) 0.00 0.00 P (20) 0.00 0.00 P (30) 0.00 0.00 P (50) 0.00 0.00 P (0.1 exp(0.2 sin(4πx) + 0.1y)) 1.00 1.00 T (3, 0.2,exp(3 sin(4πxy) + 0.01x)) 1.00 1.00 y 0.98 1.00…”

“…The intensities of two point patterns are almost the same. According to the stationarity assumption considered by [11,12] to these patterns, the second order properties of healthy and cancerous tissues were compared [20] and concluded that cancer does not affect the first and second order properties of the locations of capillaries on the prostate tissue. All these researches assumed the corresponding point process to be stationary.…”

“…According to the new obtained evidence, the previous results could not be invoked. Nevertheless, we can use the local periodograms to extend the idea of [20] for comparing the spectral density functions of nonstationary point patterns. The asymptotic independence and the asymptotic distribution of local periodograms are used to compute the density function of the local periodograms and hence the likelihood function in terms of the local periodograms similar to [20].…”

“…Nevertheless, we can use the local periodograms to extend the idea of [20] for comparing the spectral density functions of nonstationary point patterns. The asymptotic independence and the asymptotic distribution of local periodograms are used to compute the density function of the local periodograms and hence the likelihood function in terms of the local periodograms similar to [20]. Let I h zi (ω j ) denotes to the local periodogram of healthy point pattern at the location z i and frequency ω j and there is similar notation, i.e., I c zi (ω j ), for cancerous one.…”