## Overview

### Precip

A positive trend in precipitation means that the number of days within the specified “season” that have reached the threshold precipitation amount have increased annually over time, with the trend starting from the user’s selected start year. A negative trend in precipitation means that the number of days within the specified “season” that have reached the threshold precipitation amount have decreased annually over time, with the trend starting from the user’s selected start year. No trend would mean that the annual number of days within the specified “season” that have reached the threshold precipitation amount have not significantly changed starting from the user’s start year. The “significance” value refers to the two-tailed alpha (1 minus p-value) for the Mann-Kendall trend test (please see the section below for more on this)

Example: The user selects “precip” for index, 1970 for start year, 0.05 for significance, “spring” for season, and 0.1 for precipitation threshold. This means that trends will be calculated from 1970 to the most recent data on the number of days in March, April and May each year that accumulate at least 0.10 inches. The trend (or lack thereof) reported will be calculated with a 95% confidence interval, given the significance selection of 0.05. “Positive trend” would indicate that, from 1970 onward, the annual number of spring days with at least a tenth of an inch of precipitation has tended to increase. “Negative trend” would indicate that the annual number of spring days with at least a tenth of an inch of precipitation has tended to decrease over that time period. “No trend” would indicate that there wasn’t a statistically significant trend.

For reference, the seasons we are using are defined as: 1) “Calendar Year” = January-December, 2) “Water Year” = October-September, 3) “Growing Season” = April-September, 4) “Winter” = December-February, 5) “Spring” = March-May, 6) “Summer” = June-August, 7) “Fall” = September-November.

### Dry Streaks

The “dry streaks” trend refers to the average length of streaks of days without measurable precipitation in a year, and how that average annual number has changed over time. As a simple hypothetical example, in 2005, suppose Omaha had 55 dry events, defined as one or multiple consecutive days without precipitation (35 of those a single day, 15 of them 5 days long, and 5 of them 20 days long, 210 total dry days). For 2005, the annual average dry streak length for Omaha would be the average of 35 single days, 15 5-day streaks, and 5 20-day streaks, which would end up as 4.2 days on average. A positive trend would indicate that, from the picked start year to the most recent year, those annual average dry streak lengths would have increased over time at Omaha (i.e., a tendency toward longer dry streaks). A negative trend would indicate that, from the picked start year to the most recent year of data, the annual average dry streak lengths at Omaha would have decreased over time (i.e., a tendency toward shorter dry streaks). No trend would indicate that the average annual length of dry streaks at Omaha didn't significantly change over time (starting the trend at the specified start year), at the specified statistical significance level.

Please note that, in theory, a year with more total dry days could have a shorter average dry streak length than one with fewer total dry days. As an extreme (unrealistic) example to demonstrate this, suppose that in 2011, Colorado Springs recorded two (separate) streaks of 50 consecutive days without precipitation, but on the other 265 days, measurable precipitation fell every day. The average annual dry streak length for 2011 at Colorado Springs would be 50 days (two streaks of 50 days average to 50 days). In 2012, suppose that precipitation fell on 183 days, and the other 182 were dry. Now suppose that the dry and wet days alternated, such that days 1,3,5,7, etc. were wet, and days 2,4,6, and 8 were dry, and so on. Despite 2012 having 182 dry days at Colorado Springs and 2011 only having 100, the average annual dry streak length for 2012 would be just one day, compared to 50 days for 2011.

### Dry Days

Unlike the “dry streaks” option, “dry days” is referring to the total annual number of days without measurable precipitation for a location. A positive trend in dry days would indicate that, starting from the start year, the number of days without measurable precipitation per year tends to increase over time. A negative trend in dry days would indicate that, starting from the start year, the number of days without measurable precipitation per year tends to decrease over time. No trend in dry days would indicate that, starting from the start year, the number of days without measurable precipitation per year did not trend either up or down over time in a statistically significant way.

Please note that this doesn’t necessarily mean that if a location has increased dry days, the total annual precipitation has also decreased, or vice versa. As a hypothetical example, suppose that the precipitation in St. Louis was 26 inches in 2002 and 23 inches in 2019. It’s possible that St. Louis’s precipitation was spread out across fewer days in 2002 than in 2019, such that the number of dry days was greater in 2002 than in 2019, despite 2002’s total precipitation being higher than 2019’s.

### SPI

For more information on the Standardized Precipitation Index (SPI), please see the Handbook of Drought Indicators and Indices, page 13. An increasing trend in SPI would indicate that, for the user-selected season, annual SPI values (going from the start year to the most recent year) for that season tend to increase at a statistically significant level (depending on the user’s selected statistical significance). Said another way, the data trend toward wetter weather in the season in question. A decreasing trend in SPI would indicate the opposite (i.e., a trend toward drier weather in the season in question). No trend would indicate that there was not a statistically significant trend in wetness or dryness in either direction. For each season, the following SPI values are used to make a time series from which trends are then calculated: 1) “Calendar Year”: 12-month SPI values from December, 2) “Water Year”: 12-month SPI values from September, 3) “Growing Season”: 6-month SPI values from September, 4) “Winter”: 3-month SPI values from February, 5) “Spring”: 3-month SPI values from May, 6) “Summer”: 3-month SPI values from August, 7) “Fall”: 3-month SPI values from November. These specific values are used for these time periods because SPI is a backward-looking index. As a generic example, for SPI-N, N is the number of months (inclusive of the current month) that the index includes when compiling monthly precipitation data. So, if a user wanted to have a time series of SPI values that cover the winter months of December through February, they would use SPI-3 for February for this purpose, because for any SPI-3_february in the data record, the monthly precipitation used to calculate that value would be the sum of precipitation for that February, and the January and December totals immediately preceding it (making a total of 3 months). SPI-12 in September represents an SPI for a full water year, because the monthly precipitation data going into it would include everything from that September going back through the previous October.

### SPEI

For more information on the Standardized Precipitation Evapotranspiration Index (SPEI), please see the Handbook of Drought Indicators and Indices, page 13. An increasing trend in SPEI would indicate that, for the user-selected season, annual SPEI values (going from the start year to the most recent year) for that season tend to increase at a statistically significant level (depending on the user’s selected statistical significance). Said another way, the data trend toward wetter and/or cooler weather in the season in question. A decreasing trend in SPEI would indicate the opposite (i.e., a trend toward drier and/or warmer weather in the season in question). No trend would indicate that there was not a statistically significant trend in either direction. For each season, the following SPEI values are used to make a time series from which trends are then calculated: 1) “Calendar Year”: 12-month SPEI values from December, 2) “Water Year”: 12-month SPEI values from September, 3) “Growing Season”: 6-month SPEI values from September, 4) “Winter”: 3-month SPEI values from February, 5) “Spring”: 3-month SPEI values from May, 6) “Summer”: 3-month SPEI values from August, 7) “Fall”: 3-month SPEI values from November. These specific values are used for these time periods because SPEI is a backward-looking index. As a generic example, for SPEI-N, N is the number of months (inclusive of the current month) that the index includes when compiling monthly precipitation data. So, if a user wanted to have a time series of SPI values that cover the winter months of December through February, they would use SPI-3 for February for this purpose, because for any SPEI-3_february in the data record, the monthly precipitation used to calculate that value would be the sum of precipitation for that February, and the January and December totals immediately preceding it (making a total of 3 months). SPEI-12 in September represents an SPEI for a full water year, because the monthly precipitation data going into it would include everything from that September going back through the previous October.

### PDSI

For more information on the Palmer Drought Severity Index (PDSI), please see the Handbook of Drought Indicators and Indices, page 20. A positive trend in PDSI would mean that, over the period from the selected start year to the most recent data, there has been a tendency in the PDSI data toward less drought/less severe drought. A negative trend in PDSI would mean that, over the period from the selected start year to the most recent data, there has been a tendency in the PDSI data toward more drought/more severe drought. No trend would mean that there is not a statistically significant trend in PDSI data toward either more drought/more severe drought or less drought/less severe drought. Users should be aware that, by using the Mann-Kendall (MK) test to determine trend significance in PDSI, some statistical assumptions behind the MK test are violated; specifically, the serial correlation in the PDSI time series is ignored. Here, serial correlation refers to the correlation of a time series with itself (at some specific time lag). The PDSI has a time lag built into its calculations, so the assumption that there is no serial correlation when applying the MK test is always violated.

### Self-calibrated PDSI

For more information on the self-calibrated Palmer Drought Severity Index (sc-PDSI), please see the Handbook of Drought Indicators and Indices, page 22. A positive trend in sc-PDSI would mean that, over the period from the selected start year to the most recent data, there has been a tendency in the sc-PDSI data toward less drought/less severe drought. A negative trend in sc-PDSI would mean that, over the period from the selected start year to the most recent data, there has been a tendency in the sc-PDSI data toward more drought/more severe drought. No trend would mean that there is not a statistically significant trend in sc-PDSI data toward either more drought/more severe drought or less drought/less severe drought. Users should be aware that, by using the Mann-Kendall (MK) test to determine trend significance in sc-PDSI, some statistical assumptions behind the MK test are violated; specifically, the serial correlation in the sc-PDSI time series is ignored. Here, serial correlation refers to the correlation of a time series with itself (at some specific time lag). The sc-PDSI has a time lag built into its calculations, so the assumption that there is no serial correlation when applying the MK test is always violated.

## Mann-Kendall Trend Test

The Mann-Kendall Test was used to determine trend significance, or lack there-of, for the trends calculated in SPI, SPEI, PDSI, and precipitation data. To paraphrase the description of the Mann-Kendall test found in Hamed and Rao 1998: The Mann-Kendall test ranks the data points in a time series, compares the ranks of all points in the time series to each other, and calculates a test statistic from a summation of these comparisons. For a large sample size, the test statistic tends towards a normal distribution, so statistical significance is determined by comparing a standardized test statistic (dependent on the variance of the test statistic) to a standard normal distribution. The null hypothesis for the test is that data are randomly ordered, while the alternative hypothesis is that there is a monotonic trend in the data, either positive or negative. The different statistical significance levels that the user can pick are used to determine statistical significance for the Mann-Kendall test, which is run as a two-tailed test.

While interpreting the results of the trend tests from the time series and national maps, the following should be kept in mind:

- Shorter time periods (particularly the 35- and 30-year analysis periods) are likely sensitive to over-estimating the presence of a long-term trend (or lack thereof) due to single events.
- The presence of serial correlation in a time series (a time series correlating with itself at some time lag) can violate the assumption that all data points in a time series are independent samples, which can lead to mis-evaluation of trend significance. The SPI and SPEI trend significance calculations should be less sensitive to this problem because of the way those data are sub-sampled, but the serial correlation problem may be present with trend significance calculations for PDSI trends.