Shape-constrained, changepoint additive models for time series omics data with cpam

Abstract

Time series omics experiments are critical for studying a range of biological processes, such as cell differentiation and developmental programs or responses to pathogens and environmental cues. While statistical tools for differential analysis across static conditions have matured, a comparable comprehensive methodology is lacking for time series data. Here, we introduce cpam, a novel method and user-friendly R package that performs temporal differential analysis of omics time series data, including pseudobulked single-cell sequence data. Powerful features include change-point detection and shape-constrained temporal trend estimation to allocate omics data into similar clusters. The software handles case-only and case-control designs, incorporates quantification uncertainty, and provides an interactive interface with customisable visualisations, offering graphical and statistical insight into molecular processes. Performance evaluation shows that cpam outperforms existing time series methods in terms of control of the false discovery rate versus power to detect temporal changes and accurate changepoint estimation. Application to published data illustrates RNA isoform-level modelling with high-resolution clustering during human embryogenesis and the identification of 910 novel genes that respond to excess-light in Arabidopsis.

Graphical Abstract

Graphical Abstract.

Introduction

Analysing temporal changes in genome-wide gene expression is fundamentally important for the study of many biological processes, such as the progression of developmental programs or responses to external factors such as nutrients, xenobiotics, pathogens, and environmental signals [1]. Omics technologies for the analysis of chromatin, transcriptome, proteome, and metabolome have developed rapidly, such that they are increasingly powerful and affordable, generating large amounts of data in single experiments [2]. Thus, a major limiting factor now lies in our ability to rapidly and effectively analyse such temporal data to visualise patterns of gene expression and to deduce underlying pathways and networks.

Methods for analysing time series omics data can be categorised based on the type of temporal trend being inferred. In the simplest case, trends are eschewed altogether, and each time point is treated independently using pairwise comparisons or treating time as a categorical factor. Despite the simplicity of the pairwise approach, it can be effective when there are few time points or when observations are sufficiently separated in time to be effectively independent [3].

The notion of temporal autocorrelation—the extent to which measurements that are recorded closely in time tend to be similar in value—is fundamental to the analysis of time series data. This concept is important because temporally autocorrelated data are not independent, and this dependence must be taken into account to ensure valid statistical inference. In the context of genomic networks, temporal autocorrelation is expected, and various approaches have been used to model it. These methods range from linear and polynomial trends implemented via commonly used software such as DESeq2 [4] and edgeR [5] to more complex approaches such as (linear) changepoint models [6], combinations of fixed-form trends [7, 8], and non-parametric techniques such as Gaussian processes and penalised splines that force trends to change smoothly over time [9–11].

In a broader context beyond time series, best practices in statistical genomics have continuously evolved over the past 20 years. The high-throughput revolution has produced a plethora of ‘small large ’ data where the number of samples () is low while the number of statistical comparisons () is extremely high [12]. This situation exacerbates the perennial challenge of discovery and replicability in the sciences, which rely on rigorous statistical methods to control type I error rates while maintaining sufficient power to detect true effects [13]. Variance estimation, which is critical for valid inference, is particularly challenging given the typical sample sizes of genomics data. Modern regularisation (also called shrinkage or moderation) techniques that allow the sharing of information across genes have emerged as an effective means to mitigate this issue [4, 14, 15].

A further challenge is that gene products can occur as different variants, particularly apparent in the case of transcripts and proteins, usually referred to as RNA and protein isoforms. In many cases in complex eukaryotes, a single gene can produce multiple RNA isoforms, potentially encoding protein isoforms. In the case of RNA isoforms, quantification is often uncertain, because of mapping ambiguities between (short) reads and the available reference transcriptome [16]. Recently developed methods estimate this quantification uncertainty [17, 18] and allow it to propagate into subsequent statistical analysis, improving inference [19–21]. A related consideration is that statistical analysis at the transcript level, followed by the aggregation of the values from the transcript to the gene level, has more power to discover differentially active genes than if the data themselves are aggregated to the gene level prior to analysis [22]. The former can detect important differences such as differential RNA isoform abundance (e.g. between experimental conditions or across a time series), which may be obscured if the data are aggregated.

Current time series omics methods struggle to accurately infer biological signals that can involve both gradual and sharp changes in abundance of gene products while maintaining statistical rigour. Pairwise comparisons treat timepoints as independent, ignoring temporal structure, which significantly reduces statistical power. Linear trends, even with changepoints, and other parametric models tend to impose overly simplistic functional forms that cannot capture complex regulatory dynamics [23]. Existing non-parametric approaches use smoothing constraints to improve on polynomial regression for nonlinear trends, but are unable to accommodate step changes in gene expression. Most time series methods do not incorporate quantification uncertainty or share information between genes.

To address these challenges, we present cpam, an R package for omics time series analysis that combines current best practices in statistical genomics with biologically motivated innovations. cpam introduces changepoint additive models that can detect sharp transitions using changepoints, while also modelling smooth expression changes. A particular strength of our approach is the use of changepoints to determine when a significant change in transcript abundance is first observed, capturing the onset of a response, not just its peak. The method includes shape-constrained trends that cluster genes or transcripts into biologically meaningful temporal shape classes. Here, we focus on its application to transcriptome data, but recognise that it can equally be adapted for other omics platforms. cpam performs gene- or transcript-level inferences while accounting for quantification uncertainty, aggregating -values at the gene level, and sharing information across genes when estimating dispersions. In comprehensive benchmarking, cpam demonstrates superior performance through well-calibrated -values, improved power to detect true positives while controlling false detection, and accurate changepoint detection compared to existing methods. cpam provides clear and interactive visualisations of temporal dynamics and changes in gene and transcript abundance, essential for the interpretation of complex genome-wide time series experiments. The cpam R package is freely available from the Comprehensive R Archive Network (CRAN), which includes detailed documentation and illustrative case studies.

Materials and methods

Overview of statistical model and inference

The cpam model sets out to analyse time series omics data in three stages so that the later computationally intensive stages focus on a subset of differentially expressed genes. Stage 1 identifies differentially expressed genes through analysis of their products, which, for current considerations, is provided by transcriptome data. In this case, the RNA products can be aggregated at the gene level or considered at the level of individual RNA isoforms. Before modelling, data are filtered to exclude zero-count and low-count genes. Stage 2 involves the determination of changepoints for the differentially expressed RNAs. Stage 3 involves selecting temporal shapes to which each RNA conforms to facilitate subsequent clustering analysis. In addition, unconstrained trends can be detected, visualised, and smoothed as required to capture complex trends such as cyclic expression patterns. Hereafter, we refer to isoforms (or genes for a gene-level analysis) as the ‘targets’ of computational analysis.

Stage 1: Temporal differential expression of genes and isoforms

In the first modelling stage, we fit penalised smoothing splines with a negative binomial response distribution. A spline is a smooth curve fitted to the observed data (see Fig. 1A for examples). For observed counts , indexed by target and sample , the Stage 1 statistical model is

Figure 1.

Spline-based models used in cpam. (A) Example smoothing splines. (B) Effect of the smoothing parameter on spline curvature. The middle plot shows the data-driven method used in cpam. (C) Example splines with a changepoint at time . For all plots, the x-axes denote time and the y-axes the response. The points are observed data, and the solid lines and filled envelopes are the estimated smoothing splines and their standard errors, respectively. The blue features denote a treatment group.

(1)

(2)

where  is the modelled mean, is the dispersion parameter, and NB is the negative binomial probability mass function. The optional term includes, by default, weights to account for unequal sampling depths and a possible scaling factor to account for uncertainty in the quantification of transcripts [21] (see Fig. 2A). The term includes an intercept and an optional treatment effect for case-control series, in addition to any user-provided covariates, such as batch identifiers. The term comprises one (for case-only) or two (for case-control) thin-plate temporal smoothing splines, each with an estimated smoothing parameter, which controls trend smoothness (see Fig. 1B and Model fitting). The dispersion estimates are regularised using an empirical Bayes approach [5].

Figure 2.

Data and trends for genes with multiple RNA isoforms. (A) Isoform quantification uncertainty. Mapping ambiguities generate uncertainty in the estimated number of counts at the isoform level. When estimated (e.g. using bootstrapping), cpam can plot these uncertainties and take them into account when fitting models and making inferences. (B) Visualising trends for multiple gene isoforms. cpam detects and plots temporal processes such as isoform switching and differential transcript usage. The points and vertical lines are the mean and the 63.8% bootstrapped interval of the observed data, respectively. The solid lines and envelopes are the estimated smoothing splines and their standard errors.

The statistical significance of the smoothing spline (1) for each target is assessed by comparison with a null model. In case-only series, the null model has a constant mean over time (i.e. a horizontal line), while in case-control series, the null model uses a single shared spline for both case and control, rather than the alternative of a separate spline for each (see Fig. 1A). Although accurately estimating -values for penalised additive models is challenging [24], using a negative binomial distribution instead of a Gaussian model of logged counts, along with a separate estimate of the smoothing parameter for each target, resulted in well-calibrated -values for the simulated time series data (see Simulation studies for performance benchmarking for calibration results and Test statistics for temporal effects for details).

There is a computational cost for estimating, rather than simply fixing, the smoothing parameter and also for using the negative binomial distribution, which is slower than a log-Gaussian additive model, but directly accommodates the overdispersion and zero counts typical of omics count data [4]. Despite these costs, the fitting times for cpam are competitive with comparable packages due to the use of efficient algorithms and parallel computation when available (see Runtime).

Modelling omics data at the RNA isoform level greatly increases the number of targets compared to gene-level counts, leading to more statistical comparisons and reduced power with classical -value adjustments. To address this, cpam employs -value aggregation, combining -values from isoforms into a single -value for each gene, which increases statistical power over gene-aggregated models while still allowing inference of differential transcript levels [22, 25]. Users can select either the Lancaster or the aggregated Cauchy association test (ACAT) method for aggregation; the latter is robust to statistical dependencies between isoforms [26]. The gene-level -values are then adjusted for multiple comparisons to assess differential expression at gene and RNA isoform levels over time and treatments. The default adjustment in cpam is the false discovery rate (FDR), or Benjamini–Hochberg (BH) method [13], with other options available. A subset of differentially expressed genes is identified by choosing a threshold for discovery; e.g. .

Stage 2: Changepoint detection

A changepoint is a point in a time series where there is an abrupt change in the statistical behaviour of the data, such as a change from constant to time-varying counts or a change in variability. Although gene networks typically exhibit smooth changes (e.g. due to self-regulating feedback), rapid changes can occur due to threshold effects (e.g. attainment of a critical concentration of a transcription factor) or switch-like dynamics (e.g. due to positive feedback), particularly in response to new conditions. An important instance is the transition of transcript abundance from a noisy baseline trend to a new temporal pattern.

We extend the spline models of the previous section to include an initial period of constant (case-only) or common (case-control) mean counts followed by smooth time-varying counts. Example trajectories are shown in Fig. 1C. The changepoint model augments (2) and is written as

(3)

for case-only designs and as

(4)

for case-control designs, where , is the time point of sample , and is the changepoint for target . We limit to the discrete observed values in the series, since, despite the use of smoothing constraints, there is generally insufficient information to infer the timing of changepoints beyond the temporal resolution of the data. Although we generally do not anticipate the true temporal trends to involve such abrupt changes in slope, changepoint models are a good approximation to rapidly changing continuous processes, given the limitation of discrete sampling.

To estimate changepoints, we fit models (3) or (4) for each value of and select among them using generalised cross-validation (GCV, default) or the Akaike Information Criterion (AIC). Model-selection uncertainty can be visualised by plotting the mean and standard errors of pointwise score differences with respect to the best-scoring model (see Fig. 5). These standard errors can be used to apply the one-standard-error rule, to avoid overfitting by selecting the least complex model (i.e. the largest ) within one standard error of the best model [27]. See Model Selection for details.

Figure 5.

Illustration of the changepoint model selection using the one-standard error rule. (A) Differences in predictive criterion () relative to the best-scoring model for each candidate changepoint, with vertical lines showing standard error. The one-standard error rule selects the latest changepoint (most parsimonious model) with a score within one standard error of the minimum (indicated by the horizontal dashed line), helping to protect against overfitting due to model-selection uncertainty. (B) Raw count data over time with fitted trajectory using the minimum scoring model, which places the changepoint at time 10. (C) The same data fitted using the one-standard error rule, which selects a more conservative changepoint at time 30.

Given the computational cost of fitting a separate model for each candidate changepoint, cpam only estimates changepoints for targets associated with ‘significant’ genes at the chosen threshold . In the limiting case , a changepoint is estimated for every target. These estimated changepoints capture transitions to provide a natural grouping variable, aiding downstream analyses such as clustering, gene-network inference [28, 29], or Granger causality [30].

Stage 3: Shape selection

The final modelling stage involves selecting shape-constrained trends to cluster targets based on their temporal trajectories. These shapes expand upon traditional up- and down-regulated classes by capturing trends such as rising then falling, or rising then stabilising. To mathematically characterise candidate shapes, we use shape-constrained additive models [31] which extend the class of changepoint additive models (3) and (4). Shape-constrained splines can be convex, concave, or unconstrained, and may be monotonically increasing, decreasing, or neither. For targets with a changepoint near the series end or a linear-time response, candidate shapes are increasing or decreasing linear. Fig. 3 displays examples of shape-constrained trends plotted on the log scale. Model estimation on the log scale means that shape selection and subsequent clustering is easily decoupled from the magnitude of the response.

Figure 3.

Prototypical temporal expression patterns modelled by cpam. Each panel shows a different shape-constrained pattern that can occur after a changepoint: concave and increasing concave (top row), convex and decreasing convex (middle row), and linear increasing/decreasing (bottom row).

We chose this class of shapes for their flexibility and ability to represent biologically significant trends. Instead of relying on fixed parametric forms such as exponentially decaying or sigmoidal functions, which may poorly approximate underlying trajectories [23], shape-constrained splines provide interpretable shapes by restricting only the sign of the first and second derivatives. These shapes were chosen to represent diverse biological dynamics for a wide range of biological processes. For example, transient up- and down-regulation (concave and convex shapes, respectively), transition to a different steady state under new conditions (monotonic convex or concave), or sustained up- or down-regulation throughout the time series (linear).

The changepoint estimated by cpam is an important component of the overall temporal shape. For instance, a constant trend changing to an increasing concave resembles a sigmoid, but estimates a specific time for the initial change and allows for more flexibility in the latter trajectory. The estimated changepoints can be used to cluster targets within a shape-constrained class. Fig. 4 illustrates clusters based on changepoint and shape characteristics using cpam.

Figure 4.

Examples of shape-based clustering using cpam. (A) Targets filtered by changepoint and shape (all increasing concave), and clustered by increasing stabilisation rate (time to reach 90% of the maximum response). (B) Targets filtered by changepoint, shape (concave), and time of maximum response (all the same), and clustered by the extent of down-regulation post maxima (ratio of the maximum to the final response). All trends are plotted on a logged vertical scale.

Using smoothed temporal trends, shape-constrained additive models de-noise the data and cluster targets into distinct, interpretable shapes [8]. In comparison, clustering unmodelled data (e.g. -means) often generates hard-to-interpret groups of erratic trajectories [32]. Even algorithmic clustering of modelled outputs, such as polynomial and smoothing spline regression coefficients, often produces clusters that mix divergent trend types.

Two shape selections for each target are made by cpam, the first selecting among linear, convex, and concave shape classes and their monotonic variants, and the second selecting among the first options plus an ‘unconstrained’ smooth. The inclusion of the ‘unconstrained’ type provides the flexibility to detect targets beyond simpler trends; for example, circadian rhythms. Shapes are selected using either generalised cross-validation (GCV, default) or the Akaike Information Criterion (AIC). For computational reasons, as per the changepoint estimation, cpam only selects shapes for those genes, or their isoforms, identified as significant at the chosen FDR threshold .

Statistical methods

Filtering low-count targets

To increase statistical power, low-count targets are filtered from the data prior to the estimation of normalisation factors and dispersions. The default filtering function retains a target if at least 60% (min_prop = 0.6) of the samples have a minimum of 5 (min_reads = 5) reads at one or more timepoints. Users can manually set min_prop and min_reads or provide their own filtering function.

Dispersion estimation

Dispersions are estimated using the approximate empirical Bayes method implemented in the R package edgeR [5]. The method fits a trend line to the relationship between the mean expression levels and a corresponding initial set of independent dispersion estimates. In cpam, the design matrix for these initial estimates is constructed from orthogonal quadratic polynomials in the time variable. The final dispersion estimate for each target is obtained by shifting, or regularising, the initial estimate towards the trend line, where the amount of regularisation depends on the number of samples, the mean count, the consistency of the target-wise dispersions, and the distance of the initial estimate from the trend line. The use of empirical Bayes regularisation can be optionally switched off.

Quantification uncertainty

Unlike gene counts, the quantification of isoform reads can be highly uncertain because of ambiguities in mapping reads to specific isoforms. To estimate the quantification uncertainty, computationally efficient pseudo-alignment tools such as kallisto [17] and salmon [18] allow for the rapid simulation of bootstrapped technical replicates. Although the use of these tools to generate isoform counts is common, the bootstrap replicates themselves are not often used to inform downstream analyses, perhaps due to limited software options. Here we apply the method recently developed by Baldoni et al. [21] to estimate a bootstrapped overdispersion coefficient for each target . The coefficient is used to rescale the counts, or ‘divide out’ the quantification uncertainty, before model fitting. We refer the reader to the original paper for details of the method.

When the data supplied are not a fixed count matrix but outputs from external quantification software, cpam imports them using the R package tximport [16] and applies, by default, the overdispersion scaling method if bootstrap replicates are present. When applied, the scaled counts are used for all model fitting stages, including the estimation of -values, changepoints, and shapes; however, the default plotting functions show the predicted trends and the observed data on the original scale.

Model fitting

The temporal smoothing splines introduced in Equations (3) and (4) are constructed using a set of basis functions, , , for which a single spline can be expressed as

where indexes the target and are the parameters to be estimated. For the regular smoothing splines used to estimate -values and changepoints in cpam, the basis functions are thin plate splines and is set to the number of timepoints in the series [33]. For shape-constrained models, the basis functions are derived from -splines, and the coefficients are subject to shape-specific sign and ordering constraints which are imposed via re-parametrisation [31]. For all models, the parameters are estimated using penalised regression, where the strength of the penalty depends on the curvature of the basis functions and an estimated smoothing parameter . The model can be understood from a mixed-effects or Bayesian perspective, where the penalisation is expressed through the prior distribution,

with elements of the penalty matrix , , derived from the second derivatives of the basis functions. All targets use the same set of basis functions, which depend only on the observed timepoints, and therefore share a common . The smoothing spline models in cpam are fit with the R packages mgcv [33] and scam [31] using either restricted maximum likelihood (REML) or generalised cross-validation (GCV) to estimate .

Test statistics for temporal effects

The Bayesian perspective on penalised smoothing provides a natural framework for hypothesis tests. Let denote the vector of values of the smooth evaluated at the observed timepoints. The posterior distribution is approximately where is the Bayesian posterior covariance matrix [34]. For case-only series, we test whether the temporal smooth differs from zero using a Wald-type statistic

where is a generalised inverse of whose rank and the corresponding reference distribution under the null hypothesis depend on the effective degrees of freedom (edf) [34]. For case-control series, we compare a model with separate treatment and control smooths against a null model with a shared smooth, using a generalised likelihood ratio test. The deviance difference approximately follows a distribution with degrees of freedom given by the difference in edf between models. All -values for smooth terms are obtained directly from mgcv [33], which implements these methods.

Model selection

Model selection for the estimation of changepoints and shapes makes use of the pointwise estimates of a predictive criterion , where is either GCV or AIC. For a set of models indexed by , the criterion estimate is written as

where is the number of samples and is the predictive log-likelihood of the th sample for model . Both criteria are derived from a single model fit by computing the pointwise log-likelihood of the data and applying a correction based on the effective degrees of freedom (EDF) of the model, as calculated by mgcv. For AIC, the correction is additive, while for GCV it is multiplicative [24]. An advantage of computing the criterion through pointwise estimates is that model comparisons based on differences can be accompanied by estimates of the standard error .

Fig. 5A shows a plot of estimates and their standard errors, where denotes the changepoint characterising each candidate model. For this target, the model with a changepoint at 10 has the best score (Fig. 5B), but the models with changepoints at 20 and 30 (Fig. 5C) have comparable scores when taking into account the uncertainty of the score differences (i.e. the standard errors of the differences overlap zero). To mitigate overfitting due to model-selection uncertainty, cpam applies the modified one-standard error rule [27] and chooses the simplest (smallest EDF) model within 1 of the best-scoring model.

For the changepoint models, the model with the latest changepoint will usually have the smallest EDF, where a changepoint at the final time point is simply the null model. For selection among shapes, convex and concave are the most complex, followed by their monotonic versions, and then linear trends. Within this nested hierarchy, model complexity is ranked by EDF. Instead of the one-standard error rule (default), users can opt to use the minimum scoring model.

Results

Simulation studies for performance benchmarking

We evaluated cpam against several alternative methods under diverse simulation conditions and goals [35]. Alternatives were selected for their past benchmark performance, relevance to research goals (here requiring the calculation of -values), widespread use in genomics applications, and the current availability of software for the R programming language. The alternative methods we evaluated were ImpulseDE2 [7], tradeSeq [36], TDEseq [37], maSigPro [9], and NBAMSeq [38]. We intended to evaluate two other spline-based methods: trendcatcher [39], which ran but did not produce sensible results, and lmms [10], which has approximately 50 citations, but is no longer available for download. Although TDEseq was designed for single-cell RNA-seq data with correlated cells and is not suggested for bulk analysis, we included it here because its methods most closely resemble those of cpam. Its performance in this setting should not be taken to reflect its merits for its intended use case. None of these alternative methods has parameters that need adjusting for the present analyses. We included time-as-a-factor (factor) and static pairwise comparisons (pairwise) as additional methods fitted using DESeq2 [4]. For the latter, the -value for each target was set to the lowest pairwise -value, thus marking targets as significant if any adjusted -value was equal to or below the nominal threshold at any timepoint. FDR correction was applied to the aggregated set of all -values.

We simulated omics count data from a human RNA-seq dataset [40] following the approach used in the comparative study of time series methods by Spies et al. [3]. The processed count data were obtained from the Recount resource [41]. Briefly, we randomly sampled 30,000 mean-dispersion pairs from an empirical distribution model. Replicates from a pair-specific NB distribution were simulated, averaging samples to minimise the influence of extreme values, generating biologically plausible data. Temporal patterns were created by multiplying simulated replicates by predicted values from shape-constrained additive models for randomly selected shapes. This method simulated case-only and case-control datasets for short (six time points) and long (12 time points) series. Additional simulation parameters included the number of replicates (baseline = 3) and the maximum expected log-base-2 fold change (LFC) with respect to the first time point for non-null targets, set to low (LFC = 1, baseline) or high (LFC = 2). The code for reproducing the simulated data is available in the Supplementary information.

Simulating data from empirically informed shape-constrained additive models produces biologically plausible time series trajectories. However, this shape-based approach does not favour cpam over alternative methods when benchmarking -value distributions, FDR control, and target discovery, since -value calculations and significance assessments in cpam occur before the use of shape-based modelling.

Distribution of p-values under the null hypothesis

Calculating -values and adjusting them for multiple comparisons is a fundamental aspect of many methods in statistical genomics. To mitigate inflated Type I error rates, -values should be at least conservative. However, the statistical power may increase if these values are well-calibrated, balancing the control of false positives with the detection of true effects. The non-adjusted -values computed by cpam are well-calibrated (i.e. predominantly within the uniformity envelope) for both short and long case-only time series (see Figs 6A and B), with similar results for case-control series (see Supplementary Fig. S1). For the pairwise models, we included both the complete set of -values for all pairwise comparisons (pairwise-all), and a reduced set with only the minimum -value per target (pairwise-min). Most methods are conservative (i.e. above the dashed line) in the range log (), but only cpam, TDESeq, and pairwise-all are consistently well-calibrated across all simulations.

Figure 6.

Quantile-quantile plots for time series methods comparing observed versus expected -values) from 20 case-only null simulations for experiments with 6 (A) and 12 (B) time points. The diagonal dashed line represents perfect calibration, where observed -values match their theoretical distribution under the null hypothesis. The blue envelope shows the 95% confidence region expected under uniformity. Coloured envelopes and vertical lines represent one standard error of the mean. cpam demonstrates well-calibrated -values across both experimental designs, largely remaining within the expected envelope, while several existing methods consistently deviate from the expected distribution.

Detection of temporal differential gene expression

Statistical models for detecting differential gene expression should have high true positive rates (TPR) and low false discovery rates (FDR) at a set nominal FDR threshold. TPR is the proportion of true positives identified, while FDR is the proportion of false positives among discoveries. Fig. 7 displays the TPR-FDR trade-off curves for increasing nominal FDR values, using simulated long series data with 10% true positives (characterised by maximal ) and 90% true negatives. See Supplementary Fig. S2 for short time series and Supplementary Fig. S3 for . Across all simulations and models, targets with adjusted -values below 0.05 were deemed significant (i.e. discovered).

Figure 7.

True positive rate (TPR) versus false discovery rate (FDR) for time series methods. The thin curves are computed from 20 sets of simulated data comprising 12 timepoints of three replicates for targets for case-only (A) and case-control (B) series. The thick curves are the aggregated results across all datasets. Around 10% of the targets are true positives. The filled shapes denote the actual aggregated TPR and FDR values that correspond to the nominal FDR thresholds 0.01, 0.05, and 0.1. Ideally, the shapes align with the corresponding reference lines of the observed FDR (vertical dashed lines). Curves that are closer to the top left corner have superior performance, finding more true positives while maintaining a low rate of false positives.

For case-only (Fig. 7A) and case-control (Fig. 7B) time series, cpam outperforms the other methods by maintaining a higher TPR across relevant FDR levels. There are fewer curves plotted in Fig. 7B as only a subset of the methods are applicable to case-control designs. For case-only, only cpam, TDESeq, and NBAMSeq adequately control the FDR (i.e. the filled shapes are close to the vertical dashed lines), and for case-control, only cpam.

Changepoint detection

To evaluate cpam’s changepoint detection, we simulated datasets with varying log-fold changes (LFC) after the changepoint. We generated 30 000 targets across six timepoints with three replicates, comprising 10% non-null targets with changepoints. Non-null targets followed a pattern of (0,0,1,2,3,4) (constant, then linear increase), scaled by LFC values ranging from 0.50 to 2.00 in increments of 0.25. This scaling determined the log-fold increase between adjacent timepoints after the changepoint.

We evaluated changepoint detection methods within a complete workflow, including low-count filtering, dispersion estimation, and for cpam, restricting changepoint analysis to targets meeting an FDR threshold. Performance was assessed by computing the bias (estimated minus true changepoint) and the proportion of correctly estimated changepoints for each LFC value. For bias calculations, the time between adjacent timepoints was set to one. For cpam, we tested both the one-standard error rule (1se, default) and the minimum scoring (min) methods for changepoint detection. We also tested the pairwise method, where the changepoint was assigned to the earliest timepoint showing significance () among all pairwise comparisons.

The results of the simulation study are shown in Fig. 8. At small log-fold changes (LFC ), where early transcriptional changes typically occur, both cpam methods achieve substantially lower bias and higher accuracy than the pairwise approach. While the pairwise method improves with larger LFC, it never matches the performance of the other methods. For cpam, the spline-based modelling of the temporal autocorrelation accurately detects emerging changes in expression. This sensitivity to early molecular events is important for understanding causal mechanisms in gene regulatory networks. Except for LFC = 0.5, the one-standard error rule outperforms the minimum scoring method, indicating that the default method, which aims to protect against overfitting, leads to improved inferences.

Figure 8.

Evaluation of changepoint detection accuracy across different log-fold changes (LFC). Data were simulated for 30 000 targets with 10% containing true changepoints, followed by linear increases scaled by different log-fold changes (LFC). (A) Bias in changepoint estimation (estimated minus true changepoint), where values closer to zero indicate more accurate detection of the true changepoint timing. (B) The proportion of correctly estimated changepoints demonstrates method accuracy at each effect size.

Runtime

We benchmarked runtimes for the computation of -values across 20 case-only and case-control simulations of 30 000 targets for both short (six time points) and long (12 time points) time series (Supplementary Table S1). cpam required 1.81–8.23 min, depending on design and series length. Importantly, cpam was 2–4 times faster than impulsede2, its closest competitor in statistical performance, and substantially faster than tradeseq and nbamseq. While fixed-basis methods (masigpro, tdeseq) were faster than cpam, they exhibited markedly inferior TPR-FDR curves. This demonstrates that the additional computational cost of smoothing splines, which flexibly adapt to each target’s temporal profile, is justified by superior detection power and false discovery control. Even for the longest case-control series (12 time points, 30 000 genes), cpam completed in under 10 min, representing a practical runtime for routine transcriptomic analyses while delivering the strongest overall statistical performance. Beyond -values, we note that cpam requires additional computation time to estimate changepoints (usually faster than the -value computations) and the shape selection (usually 1–3 times slower than the -value computations).

Novel features in human and Arabidopsis time course data revealed by application of cpam

We applied cpam to two publicly available datasets, which have previously been analysed using alternative methods, to demonstrate the novel statistical and biological insights that cpam provides. In the first example, we analysed RNA-seq data from a time series of human embryo development from zygote through 2, 4, and 8 cell, morula, and blastocyst stages [42]. The embryonic genome activation (EGA) transition occurs after the four-cell stage (t = 2.0 days). These data are ideally suited to highlight the powerful clustering capabilities of cpam from which we selected transcripts with an EGA changepoint (t = 2.0 days) and increasing concave shape. Crucially, the shape-constrained trends generated by cpam are easily sub-clustered based on user-defined factors such as rate of change and time to steady state (Fig. 9A). Next, we illustrate application to one specific gene, namely PTRH2 (ENSG00000141378), a peptidyl tRNA hydrolase, to show resolution at the RNA isoform level. This gene has eleven isoforms, six of which passed the initial filtering stages, and cpam can model the expression shape for each isoform (Fig. 9B). Importantly, different isoforms may belong to different expression clusters, showing the limitations of gene-level clustering and the importance of isoform-level clustering.

Figure 9.

Application of cpam to RNA-seq time series of human embryo development. (A) RNA isoforms all with EGA changepoints (t = 2.0 days) and increasing concave shapes clustered by the ratio of log-fold changes between and h. (B) Shape-constrained models of RNA isoforms from the PTRH2 (ENSG00000141378) gene.

In the second example, we applied cpam to RNA-seq data from an experiment in which Arabidopsis plants were transferred from low to very high (excess) light for 60 min (sampled at 0, 30, and 60 min) and returned to low light for 60 min (sampled at 67.5, 70, 90, and 120 min). Raw data were obtained from NCBI (Bioproject accession number: PRJNA391262) and trimmed based on quality using trimmomatic [43]. Kallisto [17] was used to quantify RNA isoform counts using 100 bootstrap replicates to estimate quantification uncertainty.

Application of cpam identified 11 594 differentially expressed transcripts (from 10 117 genes) compared to 13 572 gene-level DEGs previously identified [44]. The trend-based approach of cpam, which maintains a high true positive rate while accurately controlling the false discovery rate (Fig. 7), finds fewer putative DEGs overall, but here it identified 910 additional ones. Among these new DEGs were two transcription factor genes. One is a WRKY gene (AT1G64140) showing transient upregulation (Fig. 10A) as sought in the original analysis, while the other is bZIP11 (AT4G34590), which shows sustained upregulation (Fig. 10B) and has been previously characterised as responsive to energy status [45]. These two genes produce single transcripts, but we found other genes that demonstrate transcript switching, involving alternative RNA splicing to produce different protein products (Fig. 10C, D). The example in Fig. 10D encodes a ubiquitin ligase XBAT35 (AT1G26810) [46] in which only one isoform has a nuclear localisation signal. Figs 10A and B (red and blue lines) provide empirical examples of the ‘unconstrained’ shape class. These examples demonstrate that cpam can find potentially important new genes and gene functions.

Figure 10.

Application of cpam to RNA-seq time series of Arabidopsis plants treated with excess light. Plants were treated with excess light for 60 min, then returned to low light. (A) Expression profile of a WRKY gene (AT1G64140). (B) Expression profile of bZIP11 (AT4G34590). (C) Expression of a lipase gene (AT1G28610) showing the gene-level profile (black) and profiles of two isoforms (red and blue). (D) Expression of ubiqutin ligase XBAT35 (AT3G23280) showing gene-level profile (black) and two isoforms (red and blue). Gene structure schematics beneath panels C and D show the translated exon structures of the two isoforms in each case, illustrating that RNA isoforms produce different protein isoforms for each gene.

Discussion

As large-scale omics experiments become increasingly accessible due to decreasing costs, there is a clear shift towards time-series experiments. Such experiments provide insights into critical biological processes, including stem cell differentiation, embryogenesis, infection dynamics, ageing mechanisms, and responses to abiotic factors. These experiments span biological scales, from single cells to complex organisms and ecosystems, with implications for health care, agriculture, and environmental management. Here, we show that cpam enables comprehensive temporal analysis by combining smooth temporal trends, transcript- and gene-level inferences accounting for quantification uncertainty, and biologically-motivated shape classification and changepoint estimation for clustering and pathway inference. This unified approach provides greater statistical power than existing methods while maintaining false discovery control and enabling detailed characterisation of temporal omics programs.

From statistical principles, trend-based modelling approaches should outperform pairwise comparisons by sharing information across time points, reducing multiple testing burden, and better reflecting the continuous nature of gene expression trajectories. Our results (Fig. 7) confirm this expectation, in contrast to Spies et al. [3] who found pairwise comparisons to be one of the best performing approaches for time series with up to eight time points. This discrepancy may be explained by differences in simulated trends, where Spies et al. [3] appear to have used temporal patterns that include large log-fold changes (LFC ) that can benefit pairwise detection (e.g. see Fig. 8). Our simulations controlled the maximum LFC for all targets within a given simulation to assess model performance as a function of LFC.

Time series experiments aim to identify sequential gene expression patterns, triggered by environmental or developmental signals. The changepoint estimation of cpam identifies early gene expression changes with high confidence, even when the initial change is small in magnitude. This enables us to order change events in a chronological sequence, which in turn provides information about possible cause-and-effect relationships, and to infer pathways of responses.

The shape selection component of cpam provides a biologically interpretable framework to understand the dynamics of gene expression. By fitting predefined smoothed shapes that mirror common biological responses (e.g. sustained or transient changes), we create a standardised language for describing and comparing expression patterns. Combined with changepoint detection, this approach reveals not only the timing of changes but also captures the rate and persistence of responses (e.g. distinguishing between rapid versus gradual changes, or temporary versus sustained responses). Although previous methods have introduced shape-based analysis [7, 8, 37], our new method expands on these by incorporating the estimation of changepoints, target-specific smoothing, and transcript-level inferences that incorporate quantification uncertainty and permit the aggregation of -values to the gene level for improved inferential power [16].

The clustering approach of cpam can capture the temporal dynamics of gene expression by considering both the timing and pattern of gene expression changes. While two genes may share the same changepoint, they can exhibit different dynamics (e.g. a gene target of a transient transcription factor may show a slower rate of change; Fig. 9). The approach in cpam is based on clustering genes by the modelled trends rather than by the expression counts themselves. This is particularly important, as clustering purely on counts can allow spurious patterns to dominate the apparent dynamics of clustered expression patterns, even when they do not show statistically significant changes through time. The flexibility to choose the parameters by which gene expression clusters are defined enables cpam to create an almost limitless number of sub-clusters within each primary shape category. At the same time, cpam provides sensible clustering defaults (changepoint and shape) while allowing for user-defined customisation to explore more nuanced expression patterns.

An important application in modern omics programmes is the analysis of single-cell RNA sequencing (scRNA-seq) data. In this context, cells are first clustered into cell types based on their expression profiles, using methods such as PCA or UMAP [47], and then assigned to inferred trajectories corresponding to different biological processes. Each cell is assigned a ‘pseudo-time’ value, representing its progress along its trajectory [48]. As individual cell data are typically noisy and sparse, cells on a given trajectory can be grouped into pseudo-time intervals, and their expression counts aggregated, to create more robust ‘pseudobulked’ measurements. These pseudobulked counts can then be analysed by cpam to characterise the temporal dynamics of gene expression along each trajectory, providing insight into cellular differentiation and developmental processes.

It is becoming increasingly apparent that a large proportion of eukaryotic genes produce two or more transcripts, known as RNA isoforms. Analysing gene expression at the isoform level is crucial because RNA isoforms produced from a single gene often exhibit different expression patterns across biological conditions and can differ in their stability, translatability, and encoded protein products. We have shown that cpam can readily identify significant genes in cases of isoform switching (Fig. 10C, D). It is particularly important to undertake clustering analysis at the isoform level because clustering at the gene level may be misleading for the thousands of genes with more than one transcript.

For time series experiments with a fixed sample budget, we recommend prioritising temporal resolution over replication. The smoothing splines in cpam leverage temporal autocorrelation to share information across time points, enabling accurate inference even with fewer replicates. While we suggest a minimum of five time points to access the advantages of splines over pairwise comparisons, the optimal number of replicates depends on sampling frequency, where closely spaced measurements may need only 1–3 replicates, while widely spaced measurements may require 3 or more. The definition of ‘close’ versus ‘wide’ spacing ultimately depends on the rate of change in the biological process under study. An interesting investigation for the future may be to determine optimal sampling strategies for different types of temporal dynamics.

We envisage some extensions of the cpam method and software to be considered in future work. One interesting avenue has emerged from our investigations of changepoint additive models for RNA-seq data, which revealed that RNA isoforms often exhibit differential variance post-changepoint. For example, certain isoforms become more tightly regulated (i.e. less variable) post-changepoint despite an increase in their mean expression. This is in keeping with observations that changes in variance can be used to identify cancer-related genes in humans that are missed by standard differential expression analysis [49]. The present work could be extended to allow target-specific models of the variance (dispersion) parameter using the framework of generalised additive models for location, scale, and shape [GAMLSS, 24, 50]. Additionally, a current limitation of cpam is the restriction to a single changepoint. We imposed this constraint largely for computational reasons, but an additional changepoint may be helpful, for example, to identify when a target attains a stabilised expression level.

Our new package cpam provides a comprehensive framework for analysing time series omics data that combines statistical rigour with practical utility. The method leverages modern statistical approaches while remaining user-friendly, through sensible defaults and an interactive interface. Researchers can directly address key questions in time series analysis: when changes occur, what patterns they follow, and how responses are related. While we have focused on transcriptomics, the framework is applicable to other high-dimensional time series measurements, making it a valuable addition to the omics analysis toolkit.

Supplementary data

Supplementary data is available at NAR online.

Conflict of interest

None declared.

Funding

This research was funded by the Australian Research Council Centre of Excellence for Plant Success in Nature and Agriculture (grant CE200100015). Funding to pay the Open Access publication charges for this article was provided by the Australian Research Council Centre of Excellence for Plant Success in Nature and Agriculture (grant CE200100015).

Data availability

The code for reproducing the simulated data, statistical results, figures from the manuscript, and case studies is available at https://github.com/l-a-yates/cpam_manuscript. Source code for the cpam package is available at https://cran.r-project.org/package=cpam (https://doi.org/10.5281/zenodo.18705012).