[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (168.86, 52.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (168.96, 44.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (168.96, 62.48) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (169.08, 43.07) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (169.41, 62.95) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (169.52, 69.90) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (169.60, 42.93) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (169.68, 53.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (169.68, 64.27) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (169.96, 64.69) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (170.03, 48.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (170.35, 64.05) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (170.43, 62.37) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (170.81, 73.93) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (171.35, 65.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (172.26, 52.45) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (172.54, 47.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (172.68, 60.33) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (173.36, 48.49) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (173.53, 43.18) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (173.61, 66.09) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (173.83, 48.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (173.99, 53.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (174.00, 69.17) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (174.46, 44.23) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (174.61, 73.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (175.39, 56.83) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (175.43, 62.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (175.49, 64.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (175.77, 72.11) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (175.92, 58.66) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (176.04, 63.69) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (176.38, 62.72) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (176.73, 61.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (178.06, 62.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (178.28, 50.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (178.76, 53.57) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (179.25, 42.90) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (179.38, 64.51) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (179.83, 55.01) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (180.24, 53.89) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (180.25, 47.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (180.48, 64.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (180.63, 41.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (183.64, 42.65) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (183.80, 42.03) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (183.93, 45.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (185.09, 49.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (185.35, 48.99) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (185.38, 45.90) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (186.70, 49.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (188.22, 44.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (188.48, 46.06) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (188.51, 43.48) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (188.97, 51.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (189.59, 42.17) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (192.85, 44.63) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (193.37, 46.76) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (194.02, 46.94) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (195.05, 44.71) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (195.32, 47.97) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (195.44, 42.86) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (195.47, 50.20) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (196.56, 51.85) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (196.92, 53.01) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (198.06, 51.95) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (198.17, 44.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (198.59, 41.43) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (199.94, 52.61) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (200.31, 47.81) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (200.38, 48.10) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (202.33, 42.49) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (202.44, 51.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (202.44, 53.07) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (203.12, 51.61) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (203.33, 49.99) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (203.52, 46.22) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (203.86, 53.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (204.79, 52.20) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (205.85, 51.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (206.62, 47.82) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (206.67, 41.81) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (207.97, 41.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (208.96, 43.35) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (209.74, 41.89) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (211.83, 47.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (214.78, 46.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (217.16, 43.73) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (219.45, 44.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (219.67, 43.78) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (219.70, 47.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (222.19, 44.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (223.50, 44.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (226.08, 45.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (226.11, 43.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (226.91, 41.37) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (227.40, 46.57) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (227.93, 47.72) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (230.49, 44.19) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (230.92, 44.49) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (236.62, 41.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (236.66, 42.97) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (237.84, 42.25) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (240.38, 46.15) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (241.61, 42.11) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (243.71, 45.68) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (244.48, 44.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (245.54, 40.94) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (247.27, 45.37) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (247.92, 46.27) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (249.63, 43.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (251.61, 44.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (251.78, 41.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (255.60, 42.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (258.74, 45.23) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (259.50, 41.93) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (259.95, 42.70) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (260.22, 46.81) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (260.31, 41.56) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (263.18, 46.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (263.47, 46.09) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (263.59, 46.89) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (264.31, 43.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (264.85, 46.03) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (265.15, 41.71) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (265.52, 43.92) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (267.49, 42.82) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (268.09, 45.81) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (269.24, 46.78) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (270.39, 46.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (270.61, 42.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (271.42, 42.94) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (272.27, 46.30) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (273.22, 41.14) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (273.51, 49.49) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (274.41, 47.51) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (274.81, 42.74) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (275.25, 51.71) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (275.79, 47.77) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (276.09, 53.97) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (277.59, 41.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (277.82, 51.81) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (279.17, 41.88) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (279.53, 53.03) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (279.65, 45.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (280.97, 49.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (281.17, 50.07) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (281.52, 50.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (282.43, 49.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (282.97, 41.06) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (283.23, 43.78) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (283.74, 51.33) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (287.19, 49.07) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (288.21, 48.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (288.89, 40.99) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (290.48, 41.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (294.26, 41.57) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (294.37, 49.53) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (294.50, 46.03) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (295.16, 48.73) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (295.24, 43.45) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (295.42, 49.19) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (296.75, 46.10) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (297.09, 52.91) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (297.09, 44.69) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (298.06, 42.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (299.43, 48.29) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (299.82, 50.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (300.38, 42.75) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (302.92, 48.16) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (304.83, 49.05) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (304.91, 42.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (306.64, 50.94) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (308.04, 48.50) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (309.05, 41.06) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (309.39, 49.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (310.04, 52.44) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (310.28, 55.63) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (310.47, 56.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (311.21, 47.04) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (311.58, 41.47) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (313.10, 54.66) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (313.44, 44.27) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (314.95, 54.64) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (315.59, 55.17) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (316.68, 45.58) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (317.61, 51.05) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (318.39, 49.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (319.62, 51.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (321.23, 52.15) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (321.33, 43.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (321.58, 46.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (323.63, 49.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (324.46, 48.38) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (324.80, 54.61) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.41, 47.94) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.45, 51.23) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.55, 42.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.55, 45.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.59, 43.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.62, 57.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.72, 41.34) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.78, 62.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.94, 60.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (326.06, 57.18) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (326.40, 41.62) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (326.45, 47.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (326.85, 44.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (327.48, 58.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (327.87, 42.63) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (328.27, 63.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (328.31, 55.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (328.38, 43.67) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (328.86, 53.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (329.33, 44.66) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (329.70, 51.92) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (330.29, 58.58) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (330.30, 58.86) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (331.42, 48.80) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (332.11, 48.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (332.32, 42.62) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (332.72, 42.85) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (333.59, 42.04) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (334.55, 42.75) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (334.60, 51.04) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (334.74, 62.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (335.02, 50.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (335.26, 55.61) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (336.09, 57.45) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (336.28, 56.59) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (336.37, 44.69) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (336.80, 58.37) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (337.38, 55.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (337.50, 62.57) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (337.91, 59.88) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (338.00, 54.68) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (338.48, 68.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (339.23, 47.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (341.27, 56.50) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (341.55, 45.51) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (342.20, 55.27) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (342.46, 46.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (342.66, 45.50) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (343.15, 48.92) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (343.34, 52.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (343.54, 56.69) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (343.86, 48.29) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.74, 49.25) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.76, 58.25) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.79, 48.75) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.80, 42.50) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.86, 42.10) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.92, 44.49) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.94, 53.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (345.26, 51.34) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (345.70, 59.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (345.72, 43.60) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (345.85, 45.54) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (346.13, 68.51) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (346.13, 53.45) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (347.52, 63.60) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (347.57, 46.93) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (347.76, 66.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (347.81, 44.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (347.88, 62.43) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (348.07, 68.22) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (348.16, 54.82) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (348.31, 55.10) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (348.43, 46.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (349.33, 57.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (349.47, 47.50) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (349.49, 61.06) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (349.69, 54.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (349.78, 47.64) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (350.12, 54.05) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (350.29, 41.62) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (350.43, 43.76) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (351.07, 43.71) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (351.88, 50.76) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (352.08, 59.22) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (352.38, 42.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (352.53, 41.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (352.59, 44.86) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (353.88, 62.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (354.69, 59.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (357.90, 46.85) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (357.90, 43.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (358.59, 46.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (359.23, 55.40) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (360.22, 50.09) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (361.75, 46.67) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (362.40, 45.77) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (363.26, 51.92) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (363.92, 52.70) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (364.11, 50.88) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (364.22, 52.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (364.56, 45.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (364.73, 54.70) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (364.96, 47.80) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (365.18, 43.74) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (367.64, 47.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (369.38, 47.64) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (369.90, 45.33) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (371.14, 44.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (373.08, 46.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (373.69, 43.02) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (375.46, 45.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (375.50, 45.59) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (376.86, 43.44) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (379.53, 44.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (380.45, 41.17) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (381.22, 44.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (384.45, 44.07) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (385.84, 42.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (387.18, 43.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (388.55, 46.37) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (389.32, 45.01) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (391.70, 45.64) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (394.52, 41.70) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (396.30, 43.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (396.85, 48.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (397.43, 48.73) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (398.35, 46.02) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (398.84, 41.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (398.86, 41.67) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (399.77, 42.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (400.05, 55.20) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (400.44, 43.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (400.45, 53.04) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (401.06, 52.45) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (401.33, 56.18) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (401.46, 53.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (402.13, 44.60) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (404.25, 53.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (404.61, 52.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (404.79, 51.38) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (405.13, 54.89) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (405.85, 47.10) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (406.80, 42.29) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (408.55, 42.47) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (408.58, 47.01) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (409.05, 45.16) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (409.13, 55.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (409.46, 52.61) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (409.54, 42.83) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (409.66, 55.86) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (410.50, 59.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (410.70, 58.96) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (410.86, 56.50) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (410.93, 59.54) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (411.61, 44.53) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (411.76, 42.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (412.25, 58.73) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (412.65, 63.96) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (412.66, 46.78) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (413.03, 54.89) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (413.23, 42.95) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (413.54, 55.78) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (414.02, 52.47) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (414.16, 47.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (414.35, 55.88) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (414.51, 59.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (414.78, 66.66) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (414.92, 63.38) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (415.42, 67.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (415.48, 53.48) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (416.37, 44.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (416.42, 51.82) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (416.71, 61.89) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (416.88, 46.58) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (417.22, 45.75) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (418.45, 56.97) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (418.79, 64.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (418.82, 51.40) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (419.83, 56.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (420.83, 49.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (422.80, 47.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (423.13, 43.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (423.16, 41.30) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (423.18, 43.38) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (425.10, 48.45) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (425.36, 53.57) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (425.72, 41.85) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (425.77, 44.99) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (426.35, 41.49) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (426.77, 56.91) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (427.56, 48.59) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (427.72, 46.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (428.44, 56.19) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (428.74, 46.81) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (430.00, 46.48) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (430.15, 60.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (430.38, 44.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (430.99, 41.70) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (431.22, 60.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (431.35, 47.77) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (431.66, 50.18) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (431.72, 59.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (431.96, 49.26) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (432.05, 43.45) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (432.28, 60.16) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (432.38, 44.17) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (432.68, 46.01) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (432.75, 44.62) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (433.07, 46.74) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (433.34, 47.09) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (433.79, 60.54) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (434.37, 43.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (434.56, 58.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (434.62, 65.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (435.08, 57.77) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (435.14, 50.20) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (435.55, 63.51) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (435.96, 64.56) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (436.11, 54.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (436.29, 56.17) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (436.39, 56.99) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (436.58, 44.62) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (436.70, 51.84) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (436.79, 65.19) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (437.26, 55.90) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (438.38, 64.64) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (438.60, 56.58) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (438.73, 61.18) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (439.12, 47.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (439.49, 54.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (439.88, 60.92) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (440.21, 44.66) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (441.09, 53.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (441.57, 54.22) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (441.63, 45.15) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (441.79, 55.68) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (442.09, 42.48) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (442.36, 59.04) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (442.42, 48.78) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (442.75, 52.83) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (443.11, 50.23) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (443.54, 48.71) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (443.56, 65.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (443.61, 63.53) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (443.81, 48.77) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (444.02, 45.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (444.61, 54.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (444.69, 44.46) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (446.01, 55.15) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (446.35, 46.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (447.45, 42.85) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (448.17, 41.33) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (448.20, 58.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (448.28, 43.88) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (448.65, 49.25) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (448.76, 42.91) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (449.15, 43.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (449.40, 48.94) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (449.54, 44.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (449.74, 57.01) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (449.81, 43.83) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (450.04, 58.25) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (452.14, 57.03) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (452.23, 42.56) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (452.70, 41.70) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (453.88, 50.19) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (454.03, 52.58) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (454.13, 54.72) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (454.65, 50.30) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (455.08, 51.85) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (455.18, 52.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (456.97, 53.34) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (457.40, 43.62) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (458.06, 41.28) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (459.09, 48.63) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (460.17, 54.75) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (460.30, 41.30) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (461.20, 50.09) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (461.74, 48.95) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (461.99, 55.94) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (462.47, 41.30) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (463.39, 51.02) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (463.49, 51.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (463.62, 56.51) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (463.87, 50.60) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (465.80, 51.31) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (465.94, 47.43) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (466.64, 53.44) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (466.83, 51.73) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (466.90, 42.72) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (466.90, 43.40) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (467.46, 43.65) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (468.53, 42.88) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (468.60, 44.87) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (469.10, 48.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (469.12, 52.95) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (470.41, 46.18) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (470.42, 49.60) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (472.72, 51.84) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (473.29, 52.59) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (473.93, 50.05) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (474.01, 52.50) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (474.73, 45.29) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (476.17, 50.62) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (477.37, 51.74) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (477.93, 41.37) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (478.22, 46.43) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (478.34, 44.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (479.09, 47.22) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (479.68, 49.16) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (480.56, 43.86) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (481.58, 52.96) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (481.59, 44.71) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (482.14, 51.88) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (482.14, 54.90) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (486.54, 51.01) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (487.29, 51.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (487.45, 48.70) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (489.89, 48.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (491.31, 47.38) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (491.32, 46.69) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (494.25, 45.57) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (496.45, 41.18) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (497.16, 50.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (497.45, 46.00) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (498.55, 45.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (499.01, 55.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (499.38, 52.99) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (500.06, 58.16) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (500.76, 43.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (501.33, 48.37) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (503.36, 50.40) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (503.92, 46.61) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (504.03, 46.79) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (504.64, 48.84) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (505.16, 42.07) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (506.62, 41.47) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (506.86, 53.52) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (507.96, 41.02) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (508.01, 51.65) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (508.96, 47.22) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (509.63, 54.27) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (509.74, 52.75) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (509.82, 47.10) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (510.03, 47.23) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (510.83, 42.96) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (510.98, 46.98) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (511.20, 44.41) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (512.46, 46.10) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (512.53, 50.64) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (513.05, 55.55) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (513.26, 42.63) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (513.31, 56.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (513.90, 57.33) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (514.19, 44.96) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (514.39, 44.65) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (514.59, 55.58) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (514.65, 41.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (514.69, 51.54) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (515.14, 41.39) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (516.51, 57.84) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (516.93, 41.42) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (517.59, 44.07) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (517.71, 45.93) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (517.82, 61.97) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (518.13, 49.82) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (519.35, 42.30) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (520.03, 45.48) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (520.37, 48.47) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (520.51, 48.90) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (521.17, 43.13) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (521.25, 41.09) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (521.49, 56.71) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (522.20, 45.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (522.79, 42.40) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (523.80, 47.24) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (524.01, 48.15) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (524.79, 49.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (524.84, 53.66) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (525.35, 53.95) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (525.45, 49.05) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (525.45, 57.20) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (526.69, 42.19) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (527.83, 46.08) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (528.44, 46.03) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (528.91, 51.56) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (529.33, 53.66) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (529.55, 52.21) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (531.45, 42.38) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (531.71, 49.16) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (532.03, 48.12) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (532.52, 49.47) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (533.10, 41.09) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (533.23, 45.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (533.34, 41.75) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (533.68, 47.25) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (535.42, 51.72) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (536.57, 44.47) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (536.94, 49.03) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (537.09, 49.67) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (540.56, 43.32) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (541.43, 45.36) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (541.95, 42.30) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (543.15, 41.54) circle (  1.96);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (545.12, 46.89) circle (  1.96);

[draw=drawColor,line width= 0.6pt,line join=round] (  6.92,239.01) – (572.66,239.01);

[draw=drawColor,line width= 0.6pt,line join=round] (  6.92,172.98) – (572.66,172.98);

[draw=drawColor,line width= 0.6pt,line join=round] (  6.92,106.95) – (572.66,106.95);

[draw=drawColor,line width= 0.6pt,line join=round] (  6.92, 40.91) – (572.66, 40.91);

[draw=drawColor,line width= 0.1pt,line join=round] (  6.92, 29.36) –
(572.66, 29.36);

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 31.16, 19.68) $0$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (548.02, 19.68) $T$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.10] at (289.79,  7.64) Time;

at (0,-2.8);
[draw, circle, text = ggplot4, scale = 0.8] (b) at (0,-2) $\mathbf{j}$;
[draw, circle, text = ggplot3, scale = 0.8] (h) at (0,-0.333) $\mathbf{h}$;
[draw, circle, text = ggplot2, scale = 0.8] (c) at (0,1.333) $\mathbf{c}$;
[draw, circle, text = ggplot1, scale = 0.8] (a) at (0,3) $\mathbf{k}$;

[->] (c) edge (a);
[->] (c) edge [bend right = 40] (b);
[->] (h) edge (c);
[->] (h) edge (b);
[->] (a) edge [loop right] (a);
[->] (b) edge [loop right] (b);
[->] (c) edge [loop right] (c);
[->] (h) edge [loop right] (h);

(left): Data from a $4$-dimensional Hawkes process. The vertical position of points reflect the local frequency of points. (right): Local independence graph (see <ref>) of the process that generated the data.

Let $V = \{1, \ldots, d\}$ and let $N = (N^k)_{k \in V}$ denote a collection of point processes on $\mathbb{R}$ indexed by $V$. Each mark $k\in V$ represents a particular type of event, and $N$ is also referred to as a marked point process [30].
More formally, if for each $k = 1, \ldots, d$, we let $\{\ldots, \tau_{-1}^k, \tau_{0}^k, \tau_1^k, \tau_2^k,\ldots\}$ be a series of event times, the point process $N^k$ is defined as the random measure

\begin{equation*}
N^k(A) = \sum_{i} \delta_{\tau^{k}_i}(A),
\end{equation*}

where $\delta_t(A)$ is a Dirac measure.
We associate with the  point process the counting process $N_t^k = N^k((0,t])$. We will assume that $N$ is simple and non-exploding, meaning that event times are distinct and any finite interval only has finitely many points.

For a point process $N$, the intensity $\lambda_t = (\lambda_t^1, \ldots, \lambda^d_t)$ describes the conditional rate of new events at time $t$,

\begin{align*}
\lambda^k_t = \mathbb{E}(N^k(\dd t) \mid \mathcal{F}_{t-}^V),
\end{align*}

where $\F^V_{t-}$ is the predictable filtration generated by $N^1, \ldots, N^d$, i.e. $\F^V_{t-}$ is the history of events strictly prior to time $t$ of any type $j \in V$.

Let $g^{jk} : [0,\infty) \rightarrow [0,\infty)$ for $j,k \in V$ be integrable
functions, which we call kernels. We introduce the intensity process

\begin{align}
\lambda_t^k &= \beta_0^k + \sum_{j \in V} \sum_{i: \tau^j_i < t} g^{jk}(t-\tau^j_i) \nonumber \\
&=\beta_0^k + \sum_{j \in V} \int_{-\infty}^{t-} g^{jk}(t-s) N^{j}(\mathrm{d} s)\label{eq:Hawkes_int}
\end{align}

where we call $\beta_0^k \geq 0$ baseline intensities.

A $d$-dimensional point process with intensity processes
$\lambda^k$, $k \in V$, as defined by (<ref>) is called a multivariate linear Hawkes process with kernels $g^{jk}$ and
baseline intensities $\beta_0^k$.

In this paper, we only consider stationary Hawkes processes. If we define

\begin{equation}\label{eq:integrated-kernel}
\mathbf{g}^{jk} = \int_{0}^{\infty} g^{jk}(t) \mathrm{d} t,
\end{equation}

and define the matrix $G = (\mathbf{g}^{jk})_{j,k}$, stationarity for the linear Hawkes process is achieved if the spectral radius of $G$ is strictly smaller than $1$, see
[31] and Chapter 6 in [30].

The linear Hawkes process can be extended to the nonlinear Hawkes process using a link function $\eta$:

\begin{equation*}
\eta(\lambda_t^k) = \beta_0^k + \sum_{j \in V} \int_{-\infty}^{t-} g^{jk}(t-s) N^{j}(\mathrm{d} s).
\end{equation*}

Useful alternatives to $\eta(x) = x$ are $\eta(x)=\log(x)$ or  In both cases, $\eta^{-1}$ maps $\R$ into $[0, \infty)$, which ensures that $\lambda^k_t \geq 0$ even if the kernels are allowed to take negative values. In the following subsection, we only discuss marginalization in the linear Hawkes process. However, the approximation result in <ref> extends readily to nonlinear processes, and so we state that result in generality.

§.§ Marginalization in Hawkes processes

If we only observe events corresponding to marks in $V' \subset V$,
the distribution of the $V'$-events is a marginalization of the
distribution of $V$-events. Even if all event types of a system are observed, the local independence statement $j \not\rightarrow k \mid C$ relates to the marginal distribution of $N^j, N^k$ and $N^C$, so when $\{j,k\}\cup C \neq V$, we test local independence in a marginalized distribution.

This creates a problem for testing, because many model classes, including Hawkes processes, are not closed under marginalization, i.e. the marginalized distribution need not be in the same model class as before.
We explore the case of marginalized Hawkes processes in more detail.

For $C \subseteq V$ let $\mathcal{F}^C_{t-} := \cup_{s<t}\F_s^C$ denote the predictable filtration generated by $N^j$ for $j \in C$, and let $E( \cdot \mid \mathcal{F}^C_{t-})$ denote expectations given only information about events strictly prior to $t$ of types $C$.[Technically, $E( \cdot \mid \mathcal{F}^C_{t-})$ is the predictable projection operator in order to have regular sample paths of the resulting stochastic process. See the remark in [32] for a discussion of this.] Suppose that $k \in C$, then by the innovation theorem (see e.g. [33]) the $\mathcal{F}^C_{t-}$-intensity of $N^k$ is

\begin{equation*}
\lambda_t^{k,C} = E( \lambda_t^{k} \mid \mathcal{F}^C_{t-}).
\end{equation*}

We will refer to this as the $C$-intensity.
For the linear Hawkes process we have from (<ref>) that

\begin{align}
\lambda_t^{k,C} &= \beta_0^k + \sum_{j \in C} \int_{-\infty}^{t-}
g^{jk}(t-s) N^{j}(\mathrm{d} s) \label{eq:Hawkes_filt}\\
&\quad + \sum_{l \in C^c} \int_{-\infty}^{t-} g^{lk}(t-s) E(N^{l} \mid \mathcal{F}^C_{t-}) (\mathrm{d} s), \nonumber
\end{align}

thus for a complete computation of the $C$-intensity we need to
compute $E(N^{l} \mid \mathcal{F}^C_{t-})$, which is a classical
filtering problem. The solution can be characterized via general
filtration equations, see [34] and [35].

We could approximate the solution of the filtering problem by a linear filter

\begin{align*}
&E(N^{l} \mid \mathcal{F}^C_{t-})(\mathrm{d}s) \simeq \\
& \quad \bigg(\gamma^l_0 \quad + \sum_{j \in C} \int_{-\infty}^{t-} h^{jl}(t - u, t - s) N^j(\mathrm{d}u) \bigg) \mathrm{d}s,
\end{align*}

for a choice of kernels $h^{jl}$. Using this, we arrive at the following approximate $C$-intensity

\begin{equation} \label{eq:Hawkes_filt_approx}
\tilde{\lambda}_t^{k,C} = \tilde{\beta}_0^{k,C} + \sum_{j \in C} \int_{-\infty}^{t-}\tilde{g}^{jk, C}(t-s) N^{j}(\mathrm{d} s)
\end{equation}

$$\tilde{\beta}_0^{k, C} = \beta_0^k + \sum_{l \in C^c} \int_0^{\infty}
\gamma_0^l g^{lk}(t) \mathrm{d} t$$

$$\tilde{g}^{jk, C}(t) = g^{jk}(t) + \sum_{l \in C^c} \int_{0+}^{\infty}
h^{jl}(t, s) g^{lk}(s) \mathrm{d}s$$

for $j \in C$. We recognise (<ref>) as being the intensity for a linear Hawkes process over event types indexed by $C$.
However, this is only an approximation, and the marginalized process will generally not be a linear Hawkes process. Thus some effects of this model misspecification should be expected if we fit a model of the form
(<ref>) to marginalized data.

§.§ Local independence hypotheses

Following [26], we define local independence for a point process $N$ by saying that $N^k$ is locally independent of $N^j$ given $N^C$ if $\lambda^{k, C\cup\{j\}}$ has an $\mathcal{F}^{C}_{t}$-predictable version. Intuitively that means that $\lambda^{k, C\cup\{j\}}$ only depends on events in $N^C$ and not $N^j$. In this case, we write $j \not \rightarrow k \mid C$, and else (if $\lambda_t^{k,C}$ is not a version of $\lambda_t^{k,C\cup \{j\}}$) we write $j \rightarrow k \mid C$.

Our goal is to test the local independence hypothesis,

\begin{align*}
H_0: j \not \rightarrow k \mid C.
\end{align*}

In the approximate $C$-intensity from <ref> this hypothesis corresponds  to $\tilde{g}^{jk, C \cup \{j\}}$ being 0.
However, a test of $\tilde{g}^{jk, C \cup \{j\}} = 0$ as a surrogate for $H_0$ comes with no guarantee on the level due to the model misspecification of $\tilde{\lambda}^{k,C}$.

Instead of relying on the first-order approximation in <ref> for $\lambda^{k,C}$, we propose to base
the test on the approximation

\begin{equation}\label{eq:approximate_intensity}
\overline{\lambda}^{k, C \cup \{j\}}_t = \lambda^{k, C}_t + \int_{-\infty}^{t-}
\overline{g}^{jk}(t - s) N^j(\mathrm{d} s)
\end{equation}

of the $ C \cup \{j\}$-intensity $\lambda^{k, C\cup\{j\}}$. This
approximation only uses a linear filter to model the contribution from $j$
under the alternative, and under $H_0$ the model with $\overline{g}^{jk} = 0$ is, in fact, correctly specified.
Thus we will carry out tests of $H_0$ by testing $\overline{g}^{jk} = 0$. A major practical and technical challenge is to approximate and fit $\lambda^{k, C}$ sufficiently well for the test to maintain level, and we dedicate <ref> to developing methods for appropriately fitting $\lambda^{k, C}$.

§.§ Local independence graphs

In this paper, we consider tests for local independence, with the motivation of learning graphical representations of causal relations in point processes.
In particular, we consider the local independence graph for point processes, introduced by Didelez in [18], where the absence of an edge $j\not\rightarrow k$ in the graph corresponds to the local independence $j\not\rightarrow k \mid V\setminus \{j,k\}$.

For the linear Hawkes process, the local independence graph is a graph with vertices $V$ and an edge $j \rightarrow k$ if and only if $\mathbf{g}^{jk} > 0$, where $\mathbf{g}^{jk}$ is defined in <ref>. That is, there is an edge from $j$ to $k$ if and only if the kernel $g^{jk}$ is not constantly equal to 0. <Ref> displays data from a Hawkes process and the underlying local independence graph that was used to generate the data.

§ HIGHER-ORDER EXPANSIONS

§.§ Motivating higher-order interactions

In the following, we propose a general expansion of point process intensities, which we show to converge to the true intensity as the degree of the expansion approaches infinity. We intend to apply this to marginalized Hawkes processes, in order to remove the model misspecification discussed above, but the result does not rely on the process being Hawkes, and applies to any point process model.

The expansion utilizes iterated integrals, which already [9] used for specifying models with higher-order interactions. [36] showed that the chaos expansion of point processes initiated at zero can approximate any measurable variable arbitrarily well, by integrals over random intervals. Similar to [36] our proof relies on martingale convergence, but uses integrals over deterministic intervals.

Iterated integrals are also used in the theory of Volterra series [28], where the dynamics of a time-homogeneous system over variables $x$ and $y$ is approximated by the $\mathit{L^{\textrm{th}}}$-order expansion:

\begin{align*}
y_t &\approx \beta^0 + \\&
\sum_{n=1}^L \int_{-\infty}^t \! \! \cdots \int_{-\infty}^t h^n(s_1, \ldots, s_n) x_{t-s_1} \cdots x_{t-s_n} \dd s_1 \cdots \dd s_n
\end{align*}

Under various regularity conditions, including continuity and finite memory of the system, this approximation will converge, that is, the right hand side converges to $y_t$ for all $t$ when $L$ tends to infinity [37, 29]. Although point process systems are very different in nature to continuous systems, we show a similar expansion for point processes below.

§.§ Intensity representations

We consider a fixed subset $C \subseteq V$, and a stationary process $N$.
Let $C_n$ be the set of tuples $\alpha = (j_1, \ldots, j_n)$ of length $n$ where $j_i \in C$ and $j_{i_1} \leq j_{i_2}$ for $i_1 < i_2$.
Further define

\begin{equation*}
E_n = \mathcal{L}([0, \infty)^{n}, \mathbb{R})^{C_n},
\end{equation*}

where $\L([0,\infty)^n, \R)$ is the set of measurable functions $h: [0,\infty)^n \to \R$.
That is, every element $(h^\alpha)_{\alpha\in C_n}$ in $E_n$ is a collection of functions, indexed by the distinct combinations of $C$.
Also define the functional $\varphi_n^{t}$ on $E_n$:

\begin{equation*}
\varphi_n^{t}: (h^\alpha)_{\alpha \in C_n} \mapsto \sum_{\alpha \in C_n} \int_{(-\infty, t)} h^\alpha (t-s^n) N^\alpha(\dd s^n)
\end{equation*}

\begin{align*}
&\int_{(-\infty, t)} g(t-s^n) N^\alpha(\mathrm{d}s^n) := \\ &\quad \int_{-\infty}^{t-} \cdots \int_{-\infty}^{t-} g(t - s_1, \ldots, t-s_n) N^{j_1}(\dd s_1) \cdots N^{j_n}(\dd s_n).
\end{align*}

Note that $\varphi_n^{t}$ maps into $\mathcal{L}(\mathcal{F}_{t-}^C)$ because for any $h:= (h^{\alpha})_{\alpha \in C_n}$, the filter $\varphi_n^{t}(h)$ is $\mathcal{F}_{t-}^C$-measurable.

For example, if $V = \{1, 2\}$ and $n = 2$, we have $C_n = \{(1,1), (1,2), (2,2)\}$, $E_n = \{(h^{(1,1)},h^{(1,2)}, h^{(2,2)})\mid h^{(i,j)}: [0, \infty)^2 \to \R \textrm{ measurable}\}$ and

\begin{align*}
&\varphi_n^{t}(h^{(1,1)},h^{(1,2)}, h^{(2,2)}) \\
& = \int_{-\infty}^{t-}\int_{-\infty}^{t-} h^{(1,1)}(t-s_1, t-s_2) N^1(\dd s_1)N^1(\dd s_2) \\
& + \int_{-\infty}^{t-}\int_{-\infty}^{t-} h^{(1,2)}(t-s_1, t-s_2) N^1(\dd s_1)N^2(\dd s_2) \\
& + \int_{-\infty}^{t-}\int_{-\infty}^{t-} h^{(2,2)}(t-s_1, t-s_2) N^2(\dd s_1)N^2(\dd s_2),
\end{align*}

is the evaluation of the kernels $h^{(1,1)}, h^{(1,2)}$ and $h^{(2,2)}$ in all combinations of points in the respective event types $N^1$ and $N^2$.

We now show that we can approximate point process intensities by such sums of iterated integrals.
We first show this for $t=0$ and then extend the result to all $t \in \R$ using time homogeneity.
At $t=0$, we define the set $W_n$ of all $\mathcal{F}_{t-}^C$-measurable random variables, that can be written as a $n$-fold iterated integral and are almost surely finite:

\begin{align*}
W_n = \{X \in \varphi_n^{0}(E_n) \mid |X| < \infty \text{ a.s} \}
\end{align*}

This allows us to state the following theorem, which is proven in the appendix.

With $\mathcal{F}_{0-}^C = \sigma(\cup_{s<0}\F_s^C)$ it holds that
$\bigoplus_{n\in\mathbb{N}} W_n$ is dense in $\{X \in \mathcal{L}(\mathcal{F}_{0-}^C) \mid |X| < \infty \text{ a.s.}\}$ in the topology of convergence in probability. [i.e. the topology induced by the Ky Fan metric
$$d(X, Y) = \inf\{\epsilon > 0 \mid P(|X - Y| > \epsilon) \leq \epsilon\}.$$

That is, every finite $\mathcal{F}_{0-}^C$-measurable variable can be approximated arbitrarily well by iterated integrals, over the past events of the processes in $C$.

Consider now the case of a point process intensity $\lambda^{k,C}_t$, and let $\eta$ link function.
Assume further that the intensity is time homogeneous: if $\eta(\lambda_t^{k,C})(\tau_1, \tau_2, \ldots)$ denotes the mechanism with which $\eta(\lambda^{k,C}_t)$ depends on the event times prior to time $t$, we say that $\eta(\lambda^{k,C}_t)$ is time-homogeneous if for $s \geq 0$,

\begin{equation*}
\eta(\lambda^{k,C}_t)(\tau_1, \tau_2, \ldots) = \eta(\lambda^{k,C}_{t-s})(\tau_1 -s, \tau_2-s, \ldots).
\end{equation*}

If $\eta(\lambda^{k,C}_t)$ is a time homogeneous point process intensity, $\eta(\lambda^{k,C}_t)$ can at all times be arbitrarily well approximated by iterated integrals in the topology of convergence in probability.

Take $\epsilon > 0$ and any $t \in \mathbb{R}$. Since the intensity is $\F^C_t$-predictable, $\eta(\lambda_0^{k,C}) \in \F_{0-}^C$ at time $t=0$.
Thus take $\phi^0 \in \bigoplus_{n\in \mathbb{N}} W_n$ such that $P(|\eta(\lambda^{k,C}_0) - \phi^0| > \epsilon) < \epsilon$, which is possible by <ref>.
Since $\bigoplus_{n\in \mathbb{N}} W_n$ is a sum of images, we can choose $h_1 \in E_1, h_2 \in E_2, \ldots, h_L \in E_L$ such that $\phi^0 = \sum_{n=1}^L \varphi_n^{0}(h_n)$. Let $\phi$ be the process $t \mapsto \sum_{n=1}^L \varphi_n^{t}(h_n)$, and observe that $\phi$ is time homogeneous.

Conclusively, the process $\eta(\lambda^{k,C}_t) - \phi^t$ is time homogeneous, and by the assumed stationarity, the distribution of $\eta(\lambda^{k,C}_t) - \phi^t$ is invariant over $t$. In particular $P(|\eta(\lambda^{k,C}_t) - \phi^t| > \epsilon) < \epsilon$ for all $t \in \mathbb{R}$.

Observe that it is the same kernels $h_1, \ldots, h_L$ that
enter into the approximation of $\lambda^{k,C}_t$ for all $t$. In <ref>, there is nothing special about $t=0$, and one could as well have proven that $\bigoplus_{n\in\mathbb{N}} \varphi_n^{t}(E_n)$ is dense in $\mathcal{L}(\mathcal{F}_{t-}^C)$. However, only by the time-homogeneity can one be ensured that the same kernels can be used for all $t$.

§.§ Approximate intensities

The fully observed (nonlinear) Hawkes process has intensity given by sums of first-order terms

\begin{equation*}
\eta(\lambda^k_t) = \eta(\lambda^{k,V}_t) = \beta^0 + \sum_{j\in V} \int_{-\infty}^{t-} g^{jk}(t-s) N^j(\dd s).
\end{equation*}

As discussed in <ref>, when $C \neq V$, $\lambda^{k, C}$ cannot in general be represented by sums of first-order terms. However, by <ref> the intensity can be approximated by including interaction terms of higher orders, and so one could approximate $\lambda^{k,C}$ by the $\mathit{L^{\textrm{th}}}$-order expansion

\begin{align*}
\eta(\lambda_t^{k,C}) \approx \beta_0^k + \sum_{n=1}^L \sum_{\alpha \in C_n} \int_{(-\infty,t)} h_n^{\alpha}(t-s^n) N^{\alpha}(\dd s^n)
\end{align*}

for some sequence of kernels $h_n^\alpha$, $1\leq n \leq L$, $\alpha \in C_n$. For $L = 2$ we obtain the approximate intensity:

\begin{align}\label{eq:second_order}
&\eta(\lambda_t^{k, C}) \approx
\beta_0^k \\
&+ \sum_{j_1 \in C}\int_{-\infty}^{t-} h^{j_1}(t-s_1) N^{j_1}(\dd s_1) \notag \\
&+ \sum_{j_1, j_2 \in C}\int_{-\infty}^{t-} \int_{-\infty}^{t-} h^{j_1, j_2}(t-s_1, t-s_2) N^{j_1}(\dd s_1) N^{j_2}(\dd s_2) \notag
\end{align}

The class of models described by (<ref>) contains the class of linear Hawkes processes (corresponding to $h^{j_1, j_2} = 0$) but also encompasses more complicated models, such as a model where the intensity boosts only when two events occur very close to each other.

§ TESTING LOCAL INDEPENDENCE

We now return to the question of developing a test for local independence $j \not\rightarrow k \mid C$. We consider the approximation of $\lambda^{k,C\cup\{j\}}$ in <ref>, and use the higher-order interactions from <ref> together with basis splines to approximate $\lambda^{k,C}$. We fit this approximation from data and test significance of the contribution from $j$.

§.§ Approximating kernel functions

We consider the question of approximating the intensities $\int_0^{t-} \overline{g}^{jk} N^j(\dd s)$ and $\lambda_t^{k, C}$ from (<ref>).

To approximate the intensity $\lambda_t^{k, C}$, we utilize the $W_0 \oplus W_1 \oplus W_2$-approximation from <ref>. We approximate the kernels $h^{j_1}(s_1)$ and $h^{j_1, j_2}(s_1, s_2)$ by spline expansions

\begin{equation*}
h^{j_1} \approx \sum_{i} \beta_i^{j_1} b_i \quad \textrm{and}\quad  h^{j_1, j_2} \approx \sum_{i_1, i_2} \beta_{i_1, i_2}^{j_1, j_2} b_{i_1} \otimes b_{i_2},
\end{equation*}

for some class of basis functions $\{b_i\}_{i}$ such as B-splines [38].
Due to the linearity in $\beta$, the coefficient terms can be collected into one vector $\beta^C$ and we can write $\lambda_t^{k, C}\approx \left(\beta^C\right)^T x_t^C$.
Each entry of $x_t^C$ corresponds to one basis function integrated with respect to either a single event type or a pair of event types. For instance the entry corresponding to $\beta_{i_1, i_2}^{j_1, j_2}$ would be

\begin{equation*}
\int_{-\infty}^{t-} \int_{-\infty}^{t-} b_{i_1}(t- s_1) b_{i_2}(t-s_2) N^{j_1}(\dd s_1) N^{j_2}(\dd s_2).
\end{equation*}

Similarly, we approximate the kernel $\overline{g}^{jk}$ by $\sum_i \overline{\beta}^j_i b_i$, and collect the coefficients to $\overline{\beta}^j$ and $\overline{x}_t^j$. Conclusively, the intensity (<ref>) can be approximated by

\begin{equation*}
\eta\left(\overline{\lambda}_t^{k, C\cup\{j\}}\right) = \left(\beta^C\right)^T x_t^C + \left(\overline{\beta}^j\right)^T \overline{x}_t^j =: \left(\beta^{C \cup\{j\}}\right)^T x_t^{C\cup \{j\}}
\end{equation*}

for some choice of $\beta^C$ and $\overline{\beta}^j$.

§.§ Maximum likelihood

[>=latex,font=,every node/.style=inner sep = 4pt]

[draw, circle] (A) at (4, 0) $j$;
[draw, circle] (C) at (5,0) $c$;
[draw, circle] (B) at (6,0) $k$;

[->] (A) edge [loop below] (A);
[->] (B) edge [loop below] (B);
[->] (C) edge [loop below] (C);
[->] (C) edge (A);
[->] (C) edge (B);

[draw, circle] (A) at (8, 0) $j$;
[draw, circle] (C1) at (9,0) $c$;
[draw] (C2) at (10,0) $h$;
[draw, circle] (B) at (11,0) $k$;

[->] (A) edge [loop below] (A);
[->] (B) edge [loop below] (B);
[->] (C1) edge [loop below] (C1);
[->] (C2) edge [loop below] (C2);
[->] (A) edge (C1);
[->] (C1) edge (C2);
[->] (C2) edge (B);

[draw, circle] (A) at (13, 0) $j$;
[draw, circle] (C1) at (14,0) $c$;
[draw] (C2) at (15,0.5) $h$;
[draw, circle] (B) at (16,0) $k$;

[->] (A) edge [loop below] (A);
[->] (B) edge [loop below] (B);
[->] (C1) edge [loop below] (C1);
[->] (C2) edge [loop below] (C2);
[->] (C1) edge (A);
[->] (C2) edge (C1);
[->] (C2) edge (B);
[->] (C1) edge[bend right=40] (B);

at (3.2, 0) $L_1$:;
at (7.2, 0) $L_2$:;
at (12.2, 0) $L_3$:;

[draw, circle] (A) at (4, -2) $j$;
[draw, circle] (B) at (6, -2) $k$;

[->] (A) edge [loop below] (A);
[->] (B) edge [loop below] (B);
[->] (A) edge (B);

[draw, circle] (A) at (8, -2) $j$;
[draw] (C2) at (9.5,-2) $h$;
[draw, circle] (B) at (11, -2) $k$;

[->] (A) edge [loop below] (A);
[->] (B) edge [loop below] (B);
[->] (C2) edge [loop below] (C2);
[->] (A) edge (C2);
[->] (C2) edge (B);

[draw, circle] (A) at (13, -2) $j$;
[draw, circle] (C1) at (14.5, -2.5) $c$;
[draw] (C2) at (14.5, -1.5) $h$;
[draw, circle] (B) at (16, -2) $k$;

[->] (A) edge [loop below] (A);
[->] (B) edge [loop below] (B);
[->] (C1) edge [loop below] (C1);
[->] (C2) edge [loop below] (C2);
[->] (A) edge (C1);
[->] (A) edge (C2);
[->] (C1) edge (B);
[->] (C2) edge (B);

at (3.2, -2) $P_1$:;
at (7.2, -2) $P_2$:;
at (12.2, -2) $P_3$:;
at (5, -3.5);

[draw=drawColor,line width= 0.3pt,line join=round] ( 39.80, 63.67) –
( 39.80, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] ( 52.15, 63.67) –
( 52.15, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] ( 76.84, 63.67) –
( 76.84, 86.33);

[draw=drawColor,line width= 0.6pt,line join=round] ( 37.33, 69.85) –
( 91.66, 69.85);

[draw=drawColor,line width= 0.6pt,line join=round] ( 37.33, 80.15) –
( 91.66, 80.15);

[fill=fillColor] ( 39.80, 77.58) rectangle ( 42.27, 82.73);

[fill=fillColor] ( 39.80, 67.28) rectangle ( 41.88, 72.43);

[fill=fillColor] ( 42.27, 77.58) rectangle ( 89.19, 82.73);

[fill=fillColor] ( 41.88, 67.28) rectangle ( 89.19, 72.43);

[draw=drawColor,line width= 0.6pt,dash pattern=on 2pt off 2pt ,line join=round] ( 42.27, 63.67) – ( 42.27, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] ( 39.80, 18.94) –
( 39.80, 41.60);

[draw=drawColor,line width= 0.3pt,line join=round] ( 52.15, 18.94) –
( 52.15, 41.60);

[draw=drawColor,line width= 0.3pt,line join=round] ( 76.84, 18.94) –
( 76.84, 41.60);

[draw=drawColor,line width= 0.6pt,line join=round] ( 37.33, 25.12) –
( 91.66, 25.12);

[draw=drawColor,line width= 0.6pt,line join=round] ( 37.33, 35.42) –
( 91.66, 35.42);

[fill=fillColor] ( 39.80, 32.84) rectangle ( 72.50, 37.99);

[fill=fillColor] ( 39.80, 22.54) rectangle ( 70.82, 27.69);

[fill=fillColor] ( 72.50, 32.84) rectangle ( 89.19, 37.99);

[fill=fillColor] ( 70.82, 22.54) rectangle ( 89.19, 27.69);

[draw=drawColor,line width= 0.3pt,line join=round] ( 99.63, 63.67) –
( 99.63, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] (111.97, 63.67) –
(111.97, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] (136.67, 63.67) –
(136.67, 86.33);

[draw=drawColor,line width= 0.6pt,line join=round] ( 97.16, 69.85) –
(151.48, 69.85);

[draw=drawColor,line width= 0.6pt,line join=round] ( 97.16, 80.15) –
(151.48, 80.15);

[fill=fillColor] ( 99.63, 77.58) rectangle (104.07, 82.73);

[fill=fillColor] ( 99.63, 67.28) rectangle (102.39, 72.43);

[fill=fillColor] (104.07, 77.58) rectangle (149.01, 82.73);

[fill=fillColor] (102.39, 67.28) rectangle (149.01, 72.43);

[draw=drawColor,line width= 0.6pt,dash pattern=on 2pt off 2pt ,line join=round] (102.10, 63.67) – (102.10, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] ( 99.63, 18.94) –
( 99.63, 41.60);

[draw=drawColor,line width= 0.3pt,line join=round] (111.97, 18.94) –
(111.97, 41.60);

[draw=drawColor,line width= 0.3pt,line join=round] (136.67, 18.94) –
(136.67, 41.60);

[draw=drawColor,line width= 0.6pt,line join=round] ( 97.16, 25.12) –
(151.48, 25.12);

[draw=drawColor,line width= 0.6pt,line join=round] ( 97.16, 35.42) –
(151.48, 35.42);

[fill=fillColor] ( 99.63, 32.84) rectangle (127.28, 37.99);

[fill=fillColor] ( 99.63, 22.54) rectangle (126.00, 27.69);

[fill=fillColor] (127.28, 32.84) rectangle (149.01, 37.99);

[fill=fillColor] (126.00, 22.54) rectangle (149.01, 27.69);

[draw=drawColor,line width= 0.3pt,line join=round] (159.45, 63.67) –
(159.45, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] (171.80, 63.67) –
(171.80, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] (196.49, 63.67) –
(196.49, 86.33);

[draw=drawColor,line width= 0.6pt,line join=round] (156.98, 69.85) –
(211.31, 69.85);

[draw=drawColor,line width= 0.6pt,line join=round] (156.98, 80.15) –
(211.31, 80.15);

[fill=fillColor] (159.45, 77.58) rectangle (168.74, 82.73);

[fill=fillColor] (159.45, 67.28) rectangle (165.28, 72.43);

[fill=fillColor] (168.74, 77.58) rectangle (208.84, 82.73);

[fill=fillColor] (165.28, 67.28) rectangle (208.84, 72.43);

[draw=drawColor,line width= 0.6pt,dash pattern=on 2pt off 2pt ,line join=round] (161.92, 63.67) – (161.92, 86.33);

[draw=drawColor,line width= 0.3pt,line join=round] (159.45, 18.94) –
(159.45, 41.60);

[draw=drawColor,line width= 0.3pt,line join=round] (171.80, 18.94) –
(171.80, 41.60);

[draw=drawColor,line width= 0.3pt,line join=round] (196.49, 18.94) –
(196.49, 41.60);

[draw=drawColor,line width= 0.6pt,line join=round] (156.98, 25.12) –
(211.31, 25.12);

[draw=drawColor,line width= 0.6pt,line join=round] (156.98, 35.42) –
(211.31, 35.42);

[fill=fillColor] (159.45, 32.84) rectangle (182.47, 37.99);

[fill=fillColor] (159.45, 22.54) rectangle (178.52, 27.69);

[fill=fillColor] (182.47, 32.84) rectangle (208.84, 37.99);

[fill=fillColor] (178.52, 22.54) rectangle (208.84, 27.69);

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 64.50, 46.85) $P_1$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (124.32, 46.85) $P_2$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (184.15, 46.85) $P_3$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 64.50, 91.59) $L_1$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (124.32, 91.59) $L_2$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (184.15, 91.59) $L_3$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 64.50,  7.93) 50%;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 89.19,  7.93) 100%;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (124.32,  7.93) 50%;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (149.01,  7.93) 100%;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (184.15,  7.93) 50%;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (208.84,  7.93) 100%;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 32.38, 66.82) Second;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 32.38, 77.12) First;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 32.38, 22.09) Second;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 32.38, 32.39) First;

[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  1.10] at ( 53.85, -3.43) Test result;

[fill=fillColor] (111.23, -2.61) rectangle (118.34,  4.50);

[fill=fillColor] (157.16, -2.61) rectangle (164.27,  4.50);

[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at (124.55, -2.08) Accept;

[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at (170.48, -2.08) Reject;

(left) Graphical structures used for testing local independence. Square nodes indicate unobserved event types. For each, we simulate $500$ samples from a Hawkes process with this true local independence graph, and evaluate the test $j \not\rightarrow k \mid C$ for with $C$ being $\{c,k\}$ or (in the absence of a node $c$) $\{k\}$.
(right) $H_0$ acceptance rates ($p < 0.05$ level) for the 500 repetitions of the test $j \rightarrow k \mid \{k, c\}$ in each of the structures using both a first- (1) and second- (2) order approximation of $\lambda^{k, C}$. The colors indicate the proportion of tests accepted and rejected, and the dashed line marks $5\%$ rejection rate (only relevant for graphs $L_1$-$L_3$).

Given an observation of a point process over the interval $[0, T]$, we compute maximum likelihood estimates $\hat\beta^{C \cup \{j\}}$ using the penalized log-likelihood

\begin{equation*}
\int_0^T  \log \overline{\lambda}_t^{k, C\cup \{j\}} N^k(\dd t) - \int_0^T \overline{\lambda}_t^{k, C\cup \{j\}} \dd t - \rho(\beta^{C\cup\{j\}}),
\end{equation*}

where $\rho(\beta) = \kappa_0 \beta^T \Omega \beta$ is a quadratic penalization, and where $\kappa_0 > 0$ and $\Omega$ is the roughness penalty matrix, which penalizes curvature of the kernel estimates (see Chapter 5 in [38]).

Assuming that the true model belongs to the model class, with parameter $\beta_0^{C\cup \{j\}}$, it follows from [14] that the distribution of the maximum likelihood estimate $\hat\beta^{C\cup \{j\}}$ is approximately normal with mean

$$\mu = (I + 2\kappa_0 \hat{J}_T^{-1}\Omega)\beta_0^{C\cup \{j\}}$$

and covariance matrix

$$\Sigma = \hat J_T^{-1} \hat K_T \hat J_T^{-1}$$

\begin{align*}
\hat K_T &= \int_0^T x_t^{C\cup \{j\}} {x_t^{C\cup \{j\}}}^T \frac{\left((\eta^{-1})'(\hat \beta x_t^{C\cup \{j\}})\right)^2}{\eta^{-1}(\hat \beta x_t^{C\cup \{j\}})} \dd t \\
\hat J_T &= \hat K_T - 2 \kappa_0 \Omega.
\end{align*}

If $\mu_j, \Sigma_j$ denotes the respective subvector and -matrix which corresponds to the entries of $\overline\beta^j$, the approximate distribution of the estimated parameter $\hat{\overline\beta}^j$ is known and can be used for testing.

§.§ Hypothesis testing

We can now test the hypothesis $H_0: \overline g^{jk} = 0$ by testing whether $\hat{\overline\beta^j} = 0$.
In the setting of testing $g = 0$ for a function $g = \sum_i \beta_i b_i$, [39] show that directly testing $\hat \beta^j = 0$ can lead to loss of power. Instead,  [39] proposes to evaluate the function in a grid $\mathbb{X} = (x_1, \ldots, x_M)$ and perform the hypothesis test that the resulting vector $g(\mathbb{X}):= (g(x_m))_{1 \leq m \leq M}$ is $0$.

Let $\mathbb{B} = (b_i(x_m))_{m, i}$ be the matrix where the $i$-th column is the evaluation of the $i$-th basis function evaluated in $\mathbb{X}$. Then $\overline g^{jk}(\mathbb{X}) = \mathbb{B}\hat{\overline\beta}^{jk}$ is the evaluation of $\overline g^{jk}$ in $\mathbb{X}$, which is then approximately $\mathcal{N}\left(\mathbb{B}\mu_j, \mathbb{B} \Sigma_j \mathbb{B}^T \right)$-distributed.
This allows for testing the hypothesis $\overline g^{jk} = 0$ by the Wald-test statistic:

$$T = \left[\mathbb{B}\hat{\overline\beta}^{jk}\right]^T \left(\mathbb{B} \Sigma_j \mathbb{B}^T \right)^{-1} \left[\mathbb{B}\hat{\overline\beta}^{jk}\right]$$
which is approximately $\chi^2_{(M)}$-distributed. By comparing $T$ to the theoretical quantiles of $\chi^2_{(M)}$, we can test for significance of the contribution of $j$ to the intensity $\lambda^{k, C\cup\{j\}}$.
The test is implemented in  and is available online.[Code available at <https://github.com/nikolajthams/LIPP>.]

§ SIMULATION EXPERIMENTS

We evaluate our test using simulated data. First we explore the level and power for several graphical structures. Second we apply the test in a causal discovery algorithm to learn the local independence graph from an observed data set.
In both experiments, we compare our method to the first-order method in <ref>, where also the $\lambda^{k, C}$ intensity is approximated by basis expansions using only first-order interaction terms.

§.§ Level and power

In <ref> we argued that the misspecification from using only first-order terms may lead to a loss of level.
To validate this, for each of the graphs $\mathcal{G}$ in <ref>, we sample $n=500$ point processes from the Hawkes process with kernel $g^{i_1i_2}(s) = \alpha_{i_1i_2}\beta_{i_1i_2}e^{-\beta_{i_1i_2} s}$ if $(i_1, i_2) \in \mathcal{G}$ and otherwise $g^{i_1i_2}(s)=0$.
Simulation details are in <ref>.

For each sample, we test the hypothesis  with $C = \{c, k\}$ (or $C = \{k\}$ in the graphs with no node $c$). The hypothesis $H_0$ is true in structures $L_1$–$L_3$ (and thus we here evaluate level) and false in structures $P_1$–$P_3$ (and so we here evaluate power).

The nodes $h$ represent an unobserved event type, and so is not included in the conditioning set $C$. Due to the latent events, we expect the first-order test to loose level compared to the second-order test.
We conduct the test of $H_0$ from <ref> on a nominal $5\%$ level and display in <ref> the proportion of $p$-values below $5\%$ for each structure, with red indicating a rejected test of $H_0$.

In the structure $L_1$, we observe that the both the first- and second order tests maintain level in the structure $L_1$. This is as expected, because the ground truth structure $L_1$ has no
latent events, and so the effect $c \rightarrow k$ is truly a first-order interaction.
In the structure $L_2$, our proposed second order test has a rejection rate around $5\%$, while the first-order test exceeds the nominal level by rejecting in around $9\%$ of the simulations. This indicates that due to the latent process $N^h$ being marginalized out, the dependence between $N^c$ and $N^k$ is not fully captured by first-order interactions, and so when fitting only first-order interactions, there is some residual information which mistakenly is then captured in the fitted kernel $\overline{g}^{jk}$.
By introducing second-order interactions, this residual information is reduced, and the false negative link $j\rightarrow k$ becomes less likely.
In $L_3$ both the first- and second-order tests reject in more than $5\%$ of cases, however with the level of the second-order test being closer to the nominal $5\%$ level. This indicates that the marginalization of $h$ induces a model misspecification which is partly captured by the second-order interaction.

For the graphs $P_1, P_2$ and $P_3$, where truly $j \rightarrow k \mid C$, we observe that both the first- and second-order approaches have substantial power. For the structure $P_3$, we observe that the first-order test has more power than the second-order test, possibly due to the fewer parameters that need to be estimated to use the first-order test.

§.§ Causal Structure Learning

We also evaluate the proposed test in the context of the Causal Analysis (CA) algorithm proposed by Meek [24], which is similar to the PC-algorithm [27] but applies to local independence graphs (see <ref> for an illustration of the algorithm). For $d \in \{3, \ldots, 7\}$, we simulate $n=60$ graphs of dimension $d$ and with each edge occurring with a fixed probability of $0.2$. We then simulate a Hawkes process with the simulated graph as causal graph. Simulation details are in <ref>.

[draw, circle,red] (A) at (0, 0) $a$;
[draw, circle, red] (B) at (2, 0) $b$;
[draw, circle] (C) at (2, 2) $c$;
[draw, circle] (D) at (0, 2) $d$;

[->, dashed, red] (A) edge [bend left = 15] (B);
[->] (B) edge [bend left = 15] (A);
[->] (A) edge [bend left = 15] (C);
[->] (C) edge [bend left = 15] (A);
[->] (A) edge [bend left = 15] (D);
[->] (D) edge [bend left = 15] (A);
[->] (B) edge [bend left = 15] (C);
[->] (C) edge [bend left = 15] (B);
[->] (B) edge [bend left = 15] (D);
[->] (D) edge [bend left = 15] (B);
[->] (C) edge [bend left = 15] (D);
[->] (D) edge [bend left = 15] (C);
[->] (A) edge [loop left] (A);
[->] (B) edge [loop right] (B);
[->] (C) edge [loop right] (C);
[->] (D) edge [loop left] (D);

[->] (2.5, 1) edge node[above, midway,scale=0.6] $a \not\to b \mid \{b, c, d\}$ (3.5, 1);

[draw, circle] (A) at (4, 0) $a$;
[draw, circle] (B) at (6, 0) $b$;
[draw, circle] (C) at (6, 2) $c$;
[draw, circle] (D) at (4, 2) $d$;

[->] (B) edge [bend left = 15] (A);
[->] (A) edge [bend left = 15] (C);
[->] (C) edge [bend left = 15] (A);
[->] (A) edge [bend left = 15] (D);
[->] (D) edge [bend left = 15] (A);
[->] (B) edge [bend left = 15] (C);
[->] (C) edge [bend left = 15] (B);
[->] (B) edge [bend left = 15] (D);
[->] (D) edge [bend left = 15] (B);
[->] (C) edge [bend left = 15] (D);
[->] (D) edge [bend left = 15] (C);
[->] (A) edge [loop left] (A);
[->] (B) edge [loop right] (B);
[->] (C) edge [loop right] (C);
[->] (D) edge [loop left] (D);

[->] (6.5, 1) edge node[above,midway,scale=0.6] $\cdots$ (7.5, 1);

[draw, circle] (A) at (8, 0) $a$;
[draw, circle] (B) at (10, 0) $b$;
[draw, circle] (C) at (10, 2) $c$;
[draw, circle] (D) at (8, 2) $d$;

[->] (D) edge [bend left = 15] (A);
[->] (C) edge [bend left = 15] (B);
[->] (B) edge [bend left = 15] (D);
[->] (C) edge [bend left = 15] (D);
[->] (D) edge [bend left = 15] (C);
[->] (A) edge [loop left] (A);
[->] (B) edge [loop right] (B);
[->] (C) edge [loop right] (C);
[->] (D) edge [loop left] (D);

Illustration of constrained based learning algorithms like the Causal Analysis algorithm [24] or the PC-algorithm [27]. The algorithm starts with the fully connected graph (left), and removes the edge $a \rightarrow b$ if there exist a set $C$ of current parents of $b$, such that $a \not\rightarrow b \mid C$ (middle). This is then done repeatedly for all nodes and for sets $C$ of increasing size. The algorithm terminates, when no more edges can be removed, that is when no more local independences can be found (right).

[draw=drawColor,line width= 0.3pt,line join=round] ( 31.71, 84.33) –
(211.31, 84.33);

[draw=drawColor,line width= 0.3pt,line join=round] ( 31.71,108.88) –

[draw=drawColor,line width= 0.3pt,line join=round] ( 31.71,133.43) –

[draw=drawColor,line width= 0.3pt,line join=round] ( 31.71,157.99) –

[draw=drawColor,line width= 0.6pt,line join=round] ( 31.71, 72.05) –
(211.31, 72.05);

[draw=drawColor,line width= 0.6pt,line join=round] ( 31.71, 96.60) –
(211.31, 96.60);

[draw=drawColor,line width= 0.6pt,line join=round] ( 31.71,121.16) –

[draw=drawColor,line width= 0.6pt,line join=round] ( 31.71,145.71) –

[draw=drawColor,line width= 0.6pt,line join=round] ( 31.71,170.26) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 45.96, 72.05) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 45.96, 72.05) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 45.96, 96.60) circle (  0.68);

[draw=drawColor,line width= 0.6pt,line join=round] ( 45.96, 86.78) – ( 45.96, 91.69);

[draw=drawColor,line width= 0.6pt,line join=round] ( 45.96, 81.87) – ( 45.96, 76.96);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] ( 40.13, 86.78) –
( 40.13, 81.87) –
( 51.79, 81.87) –
( 51.79, 86.78) –
( 40.13, 86.78) –

[draw=drawColor,line width= 1.1pt,line join=round] ( 40.13, 81.87) – ( 51.79, 81.87);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.91, 72.05) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.91, 72.05) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.91, 96.60) circle (  0.68);

[draw=drawColor,line width= 0.6pt,line join=round] ( 58.91, 86.78) – ( 58.91, 91.69);

[draw=drawColor,line width= 0.6pt,line join=round] ( 58.91, 81.87) – ( 58.91, 76.96);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] ( 53.08, 86.78) –
( 53.08, 81.87) –
( 64.74, 81.87) –
( 64.74, 86.78) –
( 53.08, 86.78) –

[draw=drawColor,line width= 1.1pt,line join=round] ( 53.08, 81.87) – ( 64.74, 81.87);

[draw=drawColor,line width= 0.6pt,line join=round] ( 80.50, 96.60) – ( 80.50,111.34);

[draw=drawColor,line width= 0.6pt,line join=round] ( 80.50, 86.78) – ( 80.50, 76.96);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] ( 74.67, 96.60) –
( 74.67, 86.78) –
( 86.33, 86.78) –
( 86.33, 96.60) –
( 74.67, 96.60) –

[draw=drawColor,line width= 1.1pt,line join=round] ( 74.67, 91.69) – ( 86.33, 91.69);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 93.45,111.34) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 93.45, 76.96) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 93.45, 76.96) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 93.45,101.51) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 93.45,101.51) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 93.45,101.51) circle (  0.68);

[draw=drawColor,line width= 0.6pt,line join=round] ( 93.45, 91.69) – ( 93.45, 96.60);

[draw=drawColor,line width= 0.6pt,line join=round] ( 93.45, 86.78) – ( 93.45, 81.87);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] ( 87.62, 91.69) –
( 87.62, 86.78) –
( 99.28, 86.78) –
( 99.28, 91.69) –
( 87.62, 91.69) –

[draw=drawColor,line width= 1.1pt,line join=round] ( 87.62, 91.69) – ( 99.28, 91.69);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (115.04,130.98) circle (  0.68);

[draw=drawColor,line width= 0.6pt,line join=round] (115.04,111.34) – (115.04,121.16);

[draw=drawColor,line width= 0.6pt,line join=round] (115.04,100.29) – (115.04, 86.78);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] (109.21,111.34) –
(109.21,100.29) –
(120.86,100.29) –
(120.86,111.34) –
(109.21,111.34) –

[draw=drawColor,line width= 1.1pt,line join=round] (109.21,106.43) – (120.86,106.43);

[draw=drawColor,line width= 0.6pt,line join=round] (127.99,106.43) – (127.99,121.16);

[draw=drawColor,line width= 0.6pt,line join=round] (127.99, 96.60) – (127.99, 86.78);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] (122.16,106.43) –
(122.16, 96.60) –
(133.82, 96.60) –
(133.82,106.43) –
(122.16,106.43) –

[draw=drawColor,line width= 1.1pt,line join=round] (122.16,101.51) – (133.82,101.51);

[draw=drawColor,line width= 0.6pt,line join=round] (149.57,126.07) – (149.57,140.80);

[draw=drawColor,line width= 0.6pt,line join=round] (149.57,111.34) – (149.57, 91.69);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] (143.74,126.07) –
(143.74,111.34) –
(155.40,111.34) –
(155.40,126.07) –
(143.74,126.07) –

[draw=drawColor,line width= 1.1pt,line join=round] (143.74,121.16) – (155.40,121.16);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (162.53, 86.78) circle (  0.68);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (162.53,135.89) circle (  0.68);

[draw=drawColor,line width= 0.6pt,line join=round] (162.53,117.47) – (162.53,130.98);

[draw=drawColor,line width= 0.6pt,line join=round] (162.53,106.43) – (162.53, 96.60);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] (156.70,117.47) –
(156.70,106.43) –
(168.35,106.43) –
(168.35,117.47) –
(156.70,117.47) –

[draw=drawColor,line width= 1.1pt,line join=round] (156.70,111.34) – (168.35,111.34);

[draw=drawColor,line width= 0.6pt,line join=round] (184.11,145.71) – (184.11,170.26);

[draw=drawColor,line width= 0.6pt,line join=round] (184.11,126.07) – (184.11, 96.60);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] (178.28,145.71) –
(178.28,126.07) –
(189.94,126.07) –
(189.94,145.71) –
(178.28,145.71) –

[draw=drawColor,line width= 1.1pt,line join=round] (178.28,135.89) – (189.94,135.89);

[draw=drawColor,line width= 0.6pt,line join=round] (197.06,135.89) – (197.06,160.44);

[draw=drawColor,line width= 0.6pt,line join=round] (197.06,115.02) – (197.06, 96.60);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] (191.23,135.89) –
(191.23,115.02) –
(202.89,115.02) –
(202.89,135.89) –
(191.23,135.89) –

[draw=drawColor,line width= 1.1pt,line join=round] (191.23,126.07) – (202.89,126.07);

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 26.76, 69.02) 0;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 26.76, 93.57) 5;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 26.76,118.13) 10;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 26.76,142.68) 15;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 26.76,167.23) 20;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 52.44, 56.13) 3;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 86.97, 56.13) 4;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (121.51, 56.13) 5;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (156.05, 56.13) 6;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (190.59, 56.13) 7;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.10] at (121.51, 44.09) System dimension;

[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.10] at ( 13.08,121.16) SHD;

[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  1.10] at ( 27.70, 14.44) Test type;

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round] ( 85.70, 12.45) –
( 85.70, 14.61);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round] ( 85.70, 21.84) –
( 85.70, 24.01);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] ( 80.28, 14.61) rectangle ( 91.12, 21.84);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round] ( 80.28, 18.23) –
( 91.12, 18.23);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round] (152.70, 12.45) –
(152.70, 14.61);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round] (152.70, 21.84) –
(152.70, 24.01);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round,fill=fillColor] (147.28, 14.61) rectangle (158.12, 21.84);

[draw=drawColor,line width= 0.6pt,line join=round,line cap=round] (147.28, 18.23) –
(158.12, 18.23);

[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 98.42, 15.20) First order;

[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at (165.42, 15.20) Second order;

Structural Hamming Distances (SHD) between the true graph that simulated data and the graphs estimated by using either first or second order tests in the experiment in <ref>.

Constrained based causal learning algorithms, such as the CA-algorithm, estimate the causal graph by sequentially testing local independence $j\not\rightarrow k\mid C$ for nodes $j,k$ given conditioning sets $C\subset V\setminus\{j\}$ of increasing size.
If at some point, a local independence $j\not\rightarrow k\mid C$ is found, the edge $j\rightarrow k$ is removed from the graph.

For each simulated Hawkes process, we run the CA-algorithm using either the first- or the second-order tests and obtain a resulting estimated graph. We then compare the estimated graphs to the true graph that generated the Hawkes process by the Structural Hamming Distance (SHD), which measures the number of edge additions, removals or flips that is needed to convert the estimated graph into the true graph. That is, the SHD measures how far the estimated graph is from the true graph.
<ref> shows the resulting Structural Hamming Distances for the different dimensions. We observe that for all dimensions, the second-order approach performs as well or better than the first-order approach. Notably, this is more outspoken as dimensions increase: In larger systems, more processes are marginalized away when testing $j\not\rightarrow k \mid C$, and so the effect of model misspecification is more severe for larger dimensions.

§ NEURON FIRING DATA

We employ a causal discovery algorithm using our proposed tests to a data set of neuron firing in turtles.[Data provided by Associate Professor Rune W. Berg, University of Copenhagen.] The turtles were exposed to a stimuli in a period of $10$ seconds, in which the activity of $d=6$ channels were measured. The experiment was repeated $5$ times.

For each repetition, we employ the Causal Analysis (CA) algorithm from [24] to learn the causal structure, using either first- or second-order tests. <ref> shows data from the first repetition of the experiment and the resulting learned graphs (repetitions 2–5 are shown in <ref> in the appendix). The graph estimated using second-order tests is sparser than the one using first-order tests. This concurs with our motivation for including second-order terms: when level is lost due to misspecification, the edge $j \rightarrow k$ will too often remain in the graph, even though $j \not\rightarrow k \mid C$ for some $C$. Using first order tests results in a denser and less informative graph.
This effect is more outspoken in the neuron firing data than in the simulated data in <ref>: While the synthetic data was truly simulated from a Hawkes process, and so the misspecification would only be due to marginalization, there may be additional misspecification in the real data if the full process is not truly a Hawkes process.

Since ground truth graphs for the neural connections are not available, we cannot directly evaluate which test provides estimated graphs closer to ground truth. Instead, we compare the first- and second-order tests by their consistency across the $5$ repetitions, i.e. how similar the estimated graphs are from the $5$ repetitions.
For each repetition, a separate graph is learned using the CA-algorithm, with a test using either first- or second-order terms.
In <ref> we display the proportion of edges where either i) all $5$ graphs agree on the presence or absence of the edges and ii) at least $4$ of $5$ graphs agree.
As a baseline, we include the theoretical proportions, if in each graph, an edge would appear randomly with a probability of $1/2$. Self-edges, which are easy to detect, and hence inflates consistency, are excluded from all numbers.
We observe that the second-order approach is more consistent in terms of both agreement between all $5$ repetitions and agreement between at least $4$ repetitions.

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 33.97) –
(139.04, 33.97);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 47.10) –
(139.04, 47.10);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 60.23) –
(139.04, 60.23);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 73.36) –
(139.04, 73.36);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 86.49) –
(139.04, 86.49);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 99.62) –
(139.04, 99.62);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 32.27, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 32.48, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 32.59, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 33.83, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 34.20, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 34.73, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 36.30, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 37.07, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 38.60, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 42.61, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 42.99, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 43.21, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 43.69, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 45.67, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 46.73, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 47.76, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 48.54, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 53.21, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 54.88, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 55.08, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 55.44, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 56.46, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 57.12, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.95, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 59.28, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 59.74, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 60.03, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 61.46, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 61.62, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 64.72, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 66.31, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 66.56, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 71.91, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 72.70, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 74.23, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 75.06, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 76.53, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 78.65, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 79.50, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 79.58, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 80.23, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 81.14, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 85.40, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 85.64, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 86.85, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 87.39, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 88.68, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 88.90, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 92.34, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 92.68, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 95.37, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 97.27, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.08, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.40, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 99.83, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (100.72, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (101.18, 33.97) circle (  0.57);