Time Series Toolbox

Welcome to the Time Series Toolbox

The Time Series Toolbox enables users to perform preliminary analysis on either user uploaded time series data or preloaded United States Geological Survey (USGS) streamflow gage data. Without programming expertise, users can deploy streamlined analysis pipelines, uncovering previously hidden data patterns and rapidly moving from data acquisition to analytic insight. Thus, the tool enables more consistent, repeatable, and efficient time series analysis.

This tool applies various statistical tests to facilitate a better understanding of the data, including tests for trend detection, change point analysis, breakpoint analysis, and time series modeling. In particular, the tool can detect nonstationarities in the historical record to help the user segment the record into datasets whose statistical properties can be considered stationary. Users can also explore their data through three different time series models: Auto-Regressive Integrated Moving Average (ARIMA), Exponential Smoothing (ETS), and Linear Models (TSLM). These models can be applied in the forecasting, error handling, interpretation, and decomposition of hydrologic and meteorologic data.

This tool was developed to detect nonstationarities in hydrometeorological time series. The stationarity assumption (i.e., that the statistical properties of a dataset are unchanging with time) is a fundamental concept underlying many different types of hydrological analysis. This tool is applied to evaluate the stationarity of hydrometeorological records analyzed in support of USACE planning and engineering decision-making.

This functionality is contained within four different sheets:

Explore Data — This sheet allows users to select a) their own data or b) preloaded USGS gage datasets (annual maximum and mean monthly streamflow and stage [gage height]). From there, the user can visualize that data for immediate inspection and evaluation, with further exploration on the Data Summary, Summary Statistics, and Seasonality tabs. If USGS preloaded data is selected, the user can also visually confirm the location of a gage and navigate to the USGS gage details and Water Year Summary websites.
Trend Analysis — The Trend Analysis Module applies three trend hypothesis tests, along with providing trend line coefficients (i.e., Traditional and Sen’s Slope).
Nonstationarity Detection — The Nonstationarity Detector sheet uses different statistical methods (i.e., changepoint tests, breakpoint analysis) to detect evidence of nonstationarity in the period of record.
Time Series Modeling — For deeper inspection, the last tab fits a time series model to the uploaded data. Users can choose between a Time Series Linear Model, Auto Regressive Integrated Moving Average Model, and Exponential Smoothing Model, extracting both model fit statistics and the model’s forecasts for the uploaded data.

Annual and monthly data updated as of December 2025

If you have any questions or comments, please let us know by contacting our team: iirsupport@usace.army.mil

1. Select Data Source

Select a preloaded data source or upload your own data

Seasonal cycle visualization is only available for monthly datasets. Daily data must first be aggregated to monthly data before this visualization can be used.

2. Select Search Method

Search by HUC, State, or Site Number

Filter gages by HUC8.

Filter gages by state.

Note: Not all site numbers are aligned to HUCs or States. Site state alignments are based on USGS data. If desired site number is not found under State or HUC, search by Site Number.

2. Upload Data Set

Define the path to the file you want to upload. It should be a csv file with two columns, the first of which is the date vector (mm/dd/yyyy, mm-dd-yyyy, or yyyy) and the second of which is the data for analysis. The first row should be column headers.

Use default file (Big Black River, monthly flow data)

Browse...

Title (opt) - carries on to future tabs

Data label (opt)

X-axis label (opt)

Y-axis label (opt)

3. Apply Preprocessing Methods

Select method for handling missing values:

Leave them blank

Last observation carried forward

Linear interpolation

Time Series Plot
Selected Site Map

Note: Zoom into any plot by clicking and dragging your mouse.

Y-axis Scale

Start at Zero Automatic Scaling

Missing Values

Total Data Points

Time Series Data Table

Start Date

End Date

Total Entries

Frequency

Missing Values

Autocorrelation (Lag 1)

** Years selected from the Data Upload tab will be applied to this data summary.

Summary Statistics*

AR Lag Correlation

Summary Statistics Description

Before formally testing a dataset for nonstationarity, Exploratory Data Analysis (EDA) is a crucial initial step. This involves examining key dataset properties relevant to subsequent analysis. The first seven summary statistics presented, often referred to as the "Magnificent Seven" (Archfield et al., 2013), provide a foundation for characterizing the data's statistical behavior. In addition to these, presented summary statistics include Anderson-Darling test results and an Autocorrelation Function (ACF) plot. These outputs help assess whether the dataset meets the underlying assumptions required for statistical methods applied in later stages of the analysis. Please note: summary statistics are generated using the time window specified within the data upload tab. The values below are linear combinations of order statistics (L-moments) rather than ordinary moments. Sample L-moments are defined in Hosking (1990).

1. L-Mean — The average value of the data--a measure of location. For reference, the normal distribution has an L-Mean of 0. Calculations use the algorithm given in Hosking (1996, p.14).

2. L-CV (Coefficient of L-Variance) — The ratio of standard deviation to the mean of the data. This coefficient takes values between 0 and 1. If the mean of the data set is zero, the coefficient of variation will approach infinity and hence cannot be calculated. Please see Hosking 1990 and 1996 for more information.

3. L-Skewness — The measure of asymmetry of the probability distribution. A normal distribution has a skew of zero, while a lognormal distribution, for example, would exhibit some degree of right-skew. L-skewness ranges from 0 to 1, with values greater than 0.300 indicative of large skewness. Please see Hosking 1990 and 1996 for more information.

4. L-Kurtosis — The measure of tail density of the probability distribution (i.e., how much of the distribution is contained in the tails). For reference, a uniform distribution has an L-Kurtosis of 0, while a normal distribution has one of 1/6. Please see Hosking 1990 and 1996 for more information.

5. AR1 — The autoregressive lag-one correlation coefficient (i.e., how predictive the previous value in the time series is of the next value). Long term monthly means are used to deseasonalize the data. The code normalizes the data and then applies a first order auto-regression function using the Yule Walker Method. Values can be positive or negative.

Please refer to the autocorrelation function (ACF) plot to visually assess this relationship: the AR1 value corresponds to the correlation at lag 1 in the correlogram.

6. Amplitude — A measure of the best fitting, annual sinusoidal curve height. Amplitude values are always positive numbers (e.g., 4, 1.5, 108). First, flows are standardized, then fitted to the linear model: cos(2*pi*t)+sin(2*pi*t) The final value is calculated as sqrt(cos(2*pi*t)^2+sin(2*pi*t)^2).

7. Phase — The measure, in radians, of the angle of the best fitting, annual sinusoidal curve at time zero. Using radians, each of the values will be between −π and π. The same pre-processing steps used for calculating Amplitude are used to calculate Phase. However, the final value is calculated as arcTan(-sin(2*pi*t)/cos(2*pi*t)).

8. Normality — The Anderson-Darling (AD) test is applied to determine whether the sample of data is drawn from a normal distribution. The Anderson-Darling test relies on the test statistic A*2 for detecting departures from normality. If A*2 exceeds a given critical threshold than the hypothesis of normality is rejected with some significance value. At a 5% significance level, normality is rejected if A*2 exceeds 0.754.

AR Lag Correlation Description

ACF: The autocorrelation function (acf) is depicted graphically using a correlogram for a given lag along with 95% confidence limits. Any output which falls outside these confidence limits can be considered autocorrelated at a 5% significance level. Autocorrelation is calculated by shifting a time series by a specific interval, known as a lag, and computing its correlation with the original, unshifted series. By definition, the autocorrelation at lag zero is always 1, as this represents the correlation of the series with itself. When autocorrelation at non-zero lags is statistically insignificant, the series can be considered random or statistically independent. In hydrology, this concept is critical. For instance, positive autocorrelation in annual peak streamflow indicates system 'memory' or persistence; it means a high-flow year is more likely to be followed by another. The most immediate interval, Lag-1, is often the most critical indicator of this short-term persistence. When examining many lags (e.g., 20 or more), it becomes statistically probable that at least one will appear significant purely by random chance.

References

1. Archfield, S. A., Kennen, J. G., Carlisle, D. M., and Wolock, D. M. (2014), AN OBJECTIVE AND PARSIMONIOUS APPROACH FOR CLASSIFYING NATURAL FLOW REGIMES AT A CONTINENTAL SCALE, River Res. Applic., 30, 1166- 1183, doi: 10.1002/rra.2710

2. Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Series B, 52, 105-124

3. Hosking, J. R. M. (1996). Fortran routines for use with the method of L-moments, Version 3. Research Report RC20525, IBM Research Division, Yorktown Heights, N.Y.

Seasonal Cycle Graph

The seasonal cycle graph shows aggregated monthly data throughout all of the years. This data includes monthly minimum, maximum, and average values. These values represent variance throughout the years for a particular month.

Analyze Trends

Use this page to detect the presence and severity of trends. It is recommended to have at least 25 years in the analysis periods. Please update years first and then perform analysis on your data.

Interpretation of Trend Significance & Strength

Kendall's Tau:

Kendall's Tau (τ) is a measure of the trend's direction and strength, ranging from -1 (perfect decreasing trend) to +1 (perfect increasing trend).

Range of p-values	Descriptors (Hirsch et al. 2015)	USACE Evaluation
>0.33	About as Likely as Not	Insufficient Evidence
>0.1 and ≤0.33	Likely	Weak Evidence
>0.05 and ≤0.1	Very Likely	Moderate Evidence
≥0 and ≤0.05	Highly Likely	Strong Evidence

Y-axis Scale

Start at Zero Automatic Scaling

Trend Line Coefficients

Trend Hypothesis Test

Identify Seasonality

Seasonal Decomposition refers to a series of statistical methods that attempt to separate out three of the different components embedded within time series data, where:

Data ~ Trend + Seasonality + Noise

These three components are defined as:

Trend – Consistent increases/decreases in the data as determined by a filter or smoother, such as a moving average
Seasonality – Recurring similarity of the data between time periods (i.e. months) after removing the trend
Noise – Any increases/decreases in the data after the trend and seasonality are removed

The tool uses two different methods to decompose the original data to better capture seasonality, trends, and random effects.

Moving Average Decomposition – Moving averages are fit to the data to identify both trends and seasonality
STL (Seasonal-Trend Decomposition by LOESS) – LOESS (Local Polynomial Regression) curves are fit to the data to identify both trends and seasonality

Note, using two methods in one analysis allows for a more robust analysis of the data and conclusions with higher analytical confidence.

Seasonal Decomposition: This tool uses a series of statistical methods to identify and define seasonal patterns in the data. These techniques take into account underlying trends in the data, as well as noise and natural variability

Define decomposition analysis:

Select method:

Moving Average

Seasonal-Trend Decomposition by LOESS (STL)

** Years selected from the trend analysis page will be applied to the seasonality analysis.

Seasonal Components (Download and Run Analysis)

Perform time series analysis with selected component:

Component:

Download data and graphs:

Download Seasonality Data

Download As One Image

Seasonal Decomposition of Uploaded Data

Original Series

Trend Component

Seasonality Component

Noise Component

Nonstationarity Analysis
Breakpoint Analysis

Detect Nonstationarities

The stationarity assumption enables the use of well-accepted statistical methods in water resources planning and design. It is the assumption that hydrometeorological processes are relatively unchanging in time. If nonstationarities are detected, one must be cautious in concluding any historical analysis is representative of future behavior.

Because each statistical test has strengths and weaknesses, multiple methods are recommended to establish evidence of nonstationarity in a dataset.

When testing for nonstationarity, researchers generally use three criteria to connotate strong evidence of nonstationarity:

Consensus implies that multiple methods (abrupt change point methods and breakpoint analysis) indicate evidence of change circa a given point in time (within a 5-year period). If three or more tests detect a change, the series is non-homogenous; if two tests detect a change, the series homogeneity is doubtful; and if one or no tests detect a change, the series is homogenous.
An operationally significant change in the magnitude of the record mean and standard deviation before and after the change point indicates a strong nonstationarity. Critical in application, as some changes may be statistically significant but may lack practical impact.
Confidence in results is described in terms of robustness. This includes communicating the significance level (i.e., p-value) associated with test results, considering the strengths, weaknesses, and underlying assumptions associated with the tests, sensitivity to record length, and field significance.

Even with no evidence of statistical change in the observed time series, the hydrometeorology could still reflect nonstationary conditions that have not materialized or been detected due to a limited sample size, confounding factors that mask the nonstationarity signal (e.g., changes in land use and water management over time), significant variability in observed records (i.e., small signal-to-noise ratio), and other sources of uncertainty.

The first category of tests in the TST are abrupt change point tests. Change points (or step changes) are abrupt changes in the structural patterns of a dataset, such that a dataset can be divided into two subsets where the statistical properties (e.g., mean, median, scale [variance]) of the two distributions are different. The TST also includes the Lombard Smooth change point detection tests. Unlike the abrupt change point tests, the Lombard model does not assume that the transition from one state to the next occurs discretely (i.e., from one year to the next). Instead, the transition is allowed to happen either smoothly over a certain number of years or abruptly. For more information related to change point tests, see Appendix D, ECB 2026-1. In addition to helping detect change points, the TST also supports breakpoint evaluation. For mor information related to breakpoint analysis results see the “Breakpoint Analysis” tab.

The only parametric and Bayesian changepoint test included in the TST is the Bayesian Changepoint Test. Because this test is parametric it should not be applied unless the assumption of normality is met (see Anderson-Darling Test statistics in the “Explore Data-Summary Statistics Tab”). Most of the change point tests included are nonparametric and take a frequentist approach to evaluating for evidence of statistical change. The included Bayesian approach characterizes the probability of a change point at each data point in the series being assessed in terms of a probability distribution.

The Stationarity Assumption: This tool uses regression techniques to fit trend lines to the data. The slopes produced by the regression techniques can be used as optimized trend measurements.

Nonstationarity Analysis: This tool uses statistical testing to detect the presence of nonstationarities in the uploaded data. These tests examine the data for nonstationarities (or changes) in the data mean, variance, or distribution.

Large Datasets and Computational Complexity: There are three computationally expensive tests: Energy Divisive, Smooth Lombard Wilcoxon, and Smooth Lombard Mood. When selecting the checkbox to remove computationally expensive tests, these tests will be removed from analysis and focus on the remaining nine methods.

*** If nonstationarities are not all appearing on the heatmap that appear in the time series graphic, please try expanding the graph by enlarging the application screen.

Remove Computationally Expensive Tests

The USGS streamflow gage sites available for assessment within this application include locations where there are discontinuities in USGS peak flow data collection throughout the period of record and gages with short records. Engineering judgment should be exercised when carrying out analysis where there are significant data gaps.

In general, a minimum of 30 years of continuous streamflow measurements must be available before this application should be used to detect nonstationarities in flow records.

Gage Drainage Area (SQM)

Plot Check

Method Explorer

If interested in individual test results, please select a test method to explore.

Note: Parameter values will be reset to default values after selecting new method.

**All tests are abrupt except for Smooth Lombard Mood and Smooth Lombard Wilcoxon.
** All tests are nonparametric except for the Bayesian Changepoint Tes.t
**All tests are change point tests except the Breakpoint method. For more details on detected breakpoints, see the Breakpoint tab.
** Begin analysis by clicking on 'Run Nonstationarity Analysis' button.

Sensitivity Parameters

(Sensitivity parameters are described in the manual. Engineering judgment is required if non-default parameters are seletected.)

Nonstationarity Detection

Y-axis Scale

Start at Zero Automatic Scaling

Test for Breakpoints

Approach: This tool uses linear regression, and the analysis of model errors with hypothesis testing, to identify points in the data that reflect sharp changes in behavior, suggesting the need for segmented analysis.

Missing Values: Breakpoint analysis models typically struggle to handle missing values. To facilitate robust analysis, this function concatenates data across missing values to smooth out gaps in the data and accurately detect breakpoints. The final visualization and analysis maintains your original selected approach to handle missing values, but the underlying methodology reduces that possibility that missing values do not misalign analytic insights.

Trend Significance: To aid in the understanding of the breakpoints, this tool has the option to show segment trend lines. For these individual segments, it also provides significance testing using the t-test. “Within the data table, the slope value will have an *, ** or *** if the p-value of these tests shows significance of trend at the 0.33 (*), 0.1(**), and 0.05 level (***)

In addition to helping detect change points, the TST also supports breakpoint evaluation. Breakpoints are detected when the character of the dataset is no longer behaving as anticipated given the behavior of the earlier part of the data. Breakpoint analysis is characterized by fitting models (i.e., linear regression) and comparing the error of fit. Unlike change point analysis, breakpoint analysis does not divide the data series into segments to test the difference in the statistics of those segments but iteratively adds data points to the series and identifies breakpoints based on an assessment of deviations in the ordinary least squares (OLS) linear regression model.

The Structural Change Framework:

This approach applies linear regression modeling to the data to define expected variance bounds for the regression model errors.

These expected variance bounds are known as empirical fluctuations (EFP).

To determine the presence of a structural change, a hypothesis test can be conducted on the regression model errors. Two different parameters can be used to detect a significant change in the pattern of the errors:

CUMSUM (Cumulative Sums of Squared Errors) — This method uses the running total of the squared errors as the test parameter.
MOSUM (Moving Sums of Squared Errors) — This method uses the moving sum of the squared errors as the test parameter.

These detected changes (known as breakpoints) in the regression errors can be used to segment the data into chunks that can be modeled and analyzed individually. Regression models fit to each of these segments individually yields improved performance over fitting one model to the entire data set.

Show Segment Trend

Metrics used in determining optimal breakpoints

Residual Sum of Squares (RSS):

the residual sum of squares. The residual value is a measure of how far the regression line is from the original data. This term is used to measure the amount of variance in a data set that is not explained by a regression model itself. Both RSS and BIC are directly used in the selection of breakpoints.

Bayesian Information Criterion (BIC):

an index, based on Bayesian statistics, that is used to determine what model is best for a given dataset. In this case, the criterion helps determine an optimal number of structural breaks. The BIC adds a penalty term, which favors more parsimonious models over more complex models. This penalty term helps prevent overfitting.

Root Mean Square Error (RMSE):

the standard deviation of the residual values. The residual value is a measure of how far the regression line is from the original data.

Y-axis Scale

Start at Zero Automatic Scaling

Breakpoint Segment Details

Instruction Documents

Installing DOD Certificates

Application Resources

To learn more about the tool, we strongly recommend reading the user guide.

If you have any questions or comments, please let us know by contacting our team: iirsupport@usace.army.mil

Time Series Toolbox (TST) User Guide

Welcome to the Time Series Toolbox

1. Select Data Source

Preloaded Data Information

2. Select Search Method

2. Upload Data Set

Default Dataset — Big Black River

3. Apply Preprocessing Methods

Time Series Data Table

Summary Statistics*

AR Lag Correlation

Summary Statistics Description

AR Lag Correlation Description

References

Seasonal Cycle Graph

Seasonal Cycle Data

Analyze Trends

Interpretation of Trend Significance & Strength

Trend Line Coefficients

Trend Hypothesis Test

Identify Seasonality

Understanding These Techniques

Define decomposition analysis:

Seasonal Components (Download and Run Analysis)

Perform time series analysis with selected component:

Conduct Time Series Analysis on Seasonal Component

Download data and graphs:

Seasonal Decomposition of Uploaded Data

Original Series

Trend Component

Seasonality Component

Noise Component

Detect Nonstationarities

Understanding These Techniques

Configure Testing Parameters

Nonstationarity Detection

Test for Breakpoints

Understanding These Techniques

Metrics used in determining optimal breakpoints

Breakpoint Segment Details

Build Time Series Models

Understanding These Techniques

Select a time series model:

Time Series Linear Model parameters:

Time Series Linear Model Parameter Descriptions

ARIMA Model parameters:

Configure ARIMA Model Parameters

Autoregressive, Differencing, and Moving Average Components

ARIMA Model Parameter Descriptions

ARIMA Components

ETS Model parameters:

Configure ETS Model Parameters

ETS Model Parameter Descriptions

Residuals Over Time

Residuals Autocorrelation

Instruction Documents

Application Resources

Time Series Toolbox (TST)

User Guide