# Analysis of time series data with peaks for counts of occurrences

I have time series data that measures the occurrence of insects ona weekly basis during a certain period of several month during several years at several locations. Depending on the insect species and the developmental stage that was surveyed the data in general has 1 to 4 peaks. Between the peaks the counts of the insects fall down to 0. To keep things simple, I focus now on data with a single peak. I am interested in the mechanistic of the peak. I want to know which mathematical description of the course of the peak fits best, When the peak occurs and also when the first specimens are to be captured. Moreover, I am interested which environmental variables influence the key elements mentioned before. That is, does temperture (day degrees) shape the course of the occurrence of the insect. However up to now I mainly worked with linear regession, which does not fit the rather bell-shaped time courses.

Does anybody know, which statistical methods I need to study to reach my goal? Or can anybody even help with an example, where I can get Input on how i could handle my data?

Here is a subset of my data, created with dput() in R:

• field_ID = identifier for investigated site
• date = sampling date
• response = number of insect specimens on traps
• temp_int = average temperature during sampling interval
• prec_int = average precipitation during sampling interval
• cumsum_temp = daily sum of mean daily temperatures above 5°C during the course of the year
• age = age of the investigation site

df <- structure(list(field_ID = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L), .Label = c("F29", "F31", "F36"), class = "factor"), date = structure(c(17259, 17265, 17273, 17279, 17287, 17289, 17293, 17296, 17300, 17303, 17307, 17311, 17314, 17322, 17324, 17328, 17336, 17343, 17350, 17356, 17364, 17371, 17377, 17385, 17623, 17629, 17636, 17643, 17652, 17657, 17672, 17678, 17685, 17692, 17699, 17706, 17713, 17720, 17727, 17734, 17741, 17993, 18000, 18008, 18014, 18021, 18028, 18035, 18042, 18049, 18057, 18063, 18070, 18077, 18084, 18091, 17259, 17265, 17273, 17279, 17287, 17289, 17293, 17300, 17307, 17314, 17322, 17324, 17328, 17336, 17343, 17350, 17356, 17364, 17371, 17377, 17385, 17623, 17629, 17636, 17639, 17643, 17652, 17657, 17664, 17672, 17678, 17685, 17692, 17699, 17706, 17713, 17720, 17727, 17734, 17741, 17993, 17995, 18000, 18002, 18008, 18014, 18021, 18028, 18035, 18042, 18049, 18057, 18063, 18070, 18077, 18084, 18091, 17259, 17265, 17273, 17279, 17287, 17293, 17300, 17307, 17314, 17322, 17324, 17328, 17336, 17344, 17350, 17356, 17364, 17371, 17377, 17385, 17623, 17629, 17636, 17643, 17652, 17657, 17664, 17667, 17672, 17685, 17692, 17699, 17706, 17713, 17720, 17727, 17734, 17741, 17993, 18000, 18008, 18014, 18021, 18028, 18035, 18042, 18049, 18057, 18063, 18070, 18077, 18084, 18091, 18098), class = "Date"), response = c(0L, 0L, 0L, 0L, 0L, 0L, 0L, 4L, 7L, 4L, 3L, 3L, 4L, 3L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 2L, 18L, 30L, 22L, 11L, 3L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 3L, 1L, 17L, 2L, 3L, 0L, 1L, 5L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 9L, 12L, 6L, 5L, 0L, 0L, 1L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 18L, 12L, 34L, 19L, 6L, 2L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 1L, 9L, 8L, 1L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 2L, 6L, 10L, 44L, 0L, 3L, 6L, 6L, 2L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 10L, 21L, 62L, 106L, 54L, 113L, 13L, 5L, 2L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 6L, 5L, 15L, 15L, 28L, 74L, 62L, 0L, 1L, 3L, 0L, 0L, 0L, 0L), temp_int = c(12.08, 9.63, 7.27, 6.19, 7.62, 9.82, 10.18, 8.94, 15.71, 19.71, 15.44, 15.99, 21.35, 17.38, 14.36, 18.05, 18.52, 17.94, 16.76, 18.95, 16.64, 18.65, 19.39, 18.73, 5.37, 11.9, 12.43, 16.21, 11.82, 13.75, 15.16, 20.82, 21.34, 20.7, 17.3, 15.47, 19.01, 18.51, 17.6, 21.22, 25.51, 9.39, 5.17, 13.43, 12.88, 8.41, 9.96, 13.22, 14.56, 17.58, 19.17, 19.23, 21.17, 21.9, 15.73, 15.75, 12.08, 9.64, 7.26, 6.18, 7.62, 9.81, 10.18, 12.8, 17.27, 18.28, 17.38, 14.36, 18.05, 18.52, 17.94, 16.75, 18.94, 16.64, 18.66, 19.4, 18.74, 5.36, 11.88, 12.45, 15.98, 16.38, 11.82, 13.74, 18.47, 15.21, 20.82, 21.33, 20.7, 17.3, 15.47, 19, 18.54, 17.61, 21.23, 25.5, 9.4, 6.17, 4.75, 10.46, 14.4, 12.88, 8.42, 9.97, 13.21, 14.57, 17.6, 19.2, 19.25, 21.18, 21.91, 15.74, 15.75, 12.08, 9.63, 7.27, 6.19, 7.62, 10.05, 12.81, 17.27, 18.29, 17.39, 14.36, 18.04, 18.52, 18.02, 16.45, 18.95, 16.64, 18.64, 19.39, 18.73, 5.37, 11.89, 12.43, 16.22, 11.81, 13.75, 18.48, 15.03, 15.21, 21.32, 20.69, 17.29, 15.46, 19, 18.51, 17.59, 21.22, 25.51, 9.39, 5.16, 13.44, 12.86, 8.41, 9.96, 13.2, 14.56, 17.58, 19.16, 19.22, 21.16, 21.9, 15.73, 15.74, 19.03), prec_int = c(1.19, 0.15, 1.36, 0.32, 1.6, 0.95, 0, 0.53, 0.38, 2.53, 3.23, 0, 0.1, 1.5, 2.1, 1.5, 0.79, 2.36, 5.09, 1.43, 2.2, 9.87, 2.25, 0.49, 2.35, 0.75, 2.39, 0.23, 0.77, 0, 0.32, 0, 1.84, 0.23, 1.09, 1.5, 0, 0, 4.41, 0, 1.47, 0.74, 0.24, 0, 2.07, 1.11, 1.99, 1.2, 0.2, 0.64, 2.34, 5.02, 1.66, 0.03, 0.57, 1.39, 1.4, 0.1, 1.49, 0.32, 1.86, 1.15, 0, 0.36, 3.21, 0.19, 1.51, 2.55, 1.23, 1.12, 2.6, 5.67, 1.85, 2.42, 10.8, 2.5, 0.61, 2.26, 0.77, 2.7, 0, 0.52, 0.78, 0, 0.04, 0, 0, 2.39, 0.21, 1.14, 1.57, 0, 0, 3.57, 0.01, 1.77, 0.84, 0, 0.48, 0, 0, 2, 1.26, 1.69, 1.31, 0.31, 0.87, 2.65, 7.18, 1.33, 0.03, 0.56, 1.79, 1.07, 0.15, 1.7, 0.35, 1.75, 0.4, 0.44, 2.87, 0.09, 1.09, 2.5, 1.18, 1.11, 3.33, 5.87, 1.45, 2.35, 10.01, 2.07, 0.5, 2.54, 0.63, 2.16, 0.24, 0.64, 0, 0.03, 0.87, 0, 1.76, 0.24, 1.01, 1.66, 0, 0, 4.53, 0.01, 1.56, 0.74, 0.21, 0, 2.08, 1.09, 1.71, 1.1, 0.11, 0.6, 2.48, 3.28, 1.56, 0.03, 0.54, 1.4, 1.43 ), cumsum_temp = c(344.83, 402.61, 456.84, 486.48, 547.43, 567.08, 607.8, 634.61, 697.45, 756.59, 818.34, 882.3, 946.34, 1085.42, 1114.14, 1186.33, 1334.51, 1460.07, 1577.38, 1691.05, 1824.2, 1954.76, 2071.12, 2220.99, 173.35, 244.73, 331.73, 445.2, 551.55, 620.29, 870.99, 995.93, 1145.28, 1290.18, 1411.29, 1519.55, 1652.61, 1782.19, 1905.41, 2053.97, 2232.53, 461.96, 482.72, 590.16, 667.41, 726.27, 796.01, 888.52, 990.44, 1113.52, 1266.9, 1382.29, 1530.47, 1683.74, 1793.86, 1904.11, 344.61, 402.43, 456.64, 486.27, 547.19, 566.82, 607.55, 697.14, 818, 945.98, 1085.04, 1113.76, 1185.98, 1334.12, 1459.7, 1576.96, 1690.62, 1823.77, 1954.37, 2070.76, 2220.69, 173.11, 244.36, 331.48, 379.43, 444.96, 551.37, 620.07, 749.39, 870.55, 995.45, 1144.79, 1289.7, 1410.81, 1519.12, 1652.13, 1781.92, 1905.16, 2053.77, 2232.29, 461.64, 473.98, 482.37, 503.3, 589.72, 667, 725.93, 795.71, 888.15, 990.13, 1113.31, 1266.91, 1382.39, 1530.63, 1684.03, 1794.2, 1904.46, 345.03, 402.8, 457.04, 486.66, 547.6, 607.92, 697.58, 818.44, 946.45, 1085.57, 1114.28, 1186.43, 1334.58, 1478.75, 1577.43, 1691.12, 1824.27, 1954.77, 2071.14, 2220.95, 173.51, 244.85, 331.89, 445.42, 551.74, 620.49, 749.84, 794.93, 871, 1145.19, 1290.03, 1411.04, 1519.27, 1652.29, 1781.85, 1904.96, 2053.53, 2232.08, 461.99, 482.75, 590.25, 667.39, 726.23, 795.95, 888.38, 990.28, 1113.34, 1266.62, 1381.93, 1530.07, 1683.35, 1793.45, 1903.6, 2036.83), age = c(2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L)), row.names = c(NA, -167L), class = c("tbl_df", "tbl", "data.frame"))