Dynamic signatures: a mathematical approach to analysis

Abstract

This study evaluates mathematical tools (principal component analysis, dynamic time warping, and the Kolmogorov–Smirnov hypothesis test) to analyse global and local data from dynamic signatures to reduce subjectivity and increase the reproducibility of handwriting examination using a two-step approach. A dataset composed of 1 800 genuine signature samples, 870 simulated signatures, and 60 disguises (30 formally similar or “autosimulated” and 30 random but different from usual) provided by 30 volunteers was collected. The first step involved global data analysis using principal component analysis and a hypothesis test performed for 62 global characteristics, and associations of these characteristics were analysed through calculations of multivariate distance followed by a hypothesis test. The second step involved the analysis of local characteristics including vertical and horizontal positions, speed, pressure gradient, acceleration, and jerk point-to-point, by using dynamic time warping followed by a hypothesis test. Optimization of sensitivity and specificity metrics of the hypothesis test was explored by varying its stringency and observing accuracy rates for the simulated and genuine groups. A P-value threshold of 1 × 10⁻¹⁰ was found to be optimal, making the test more restrictive and yielding accuracy rates of 96.7% for genuine global data and 88.9% for simulated data. The same cut-off value for local characteristics provided an average accuracy rate of 95.4% for genuine samples and 94.7% for simulated samples, demonstrating high accuracy for both simulated and genuine samples. However, the method did not offer reasonable accuracy rates for disguises, consistent with observations in traditional handwriting examination. Our approach provided satisfactory results for forensic examination use. The visualization of graphs and signatures and analysis of all identifying elements of handwriting by the examining expert are still essential. In future studies, we plan to perform blind tests to validate our approach and propose a rigorous methodology.

Introduction

The transition from physical to digital technology has significantly impacted the forensic scrutiny of documents and manuscripts. The increased use of electronic signatures has, in turn, heightened the need for forensic examinations, especially concerning digitally captured handwritten signatures [1, 2]. Experts can only quantitatively assess static characteristics in traditional forensic handwriting and static electronic signature analysis, while dynamic traits remain inferential. However, in biometric signatures, the availability of quantitative data enables experts to apply mathematical techniques and statistical analysis [1–6].

While existing studies primarily focus on distinguishing between genuine and nongenuine signatures, they often do not address disguises, which refers to a deliberately altered one’s signature to conceal their true identity, in depth. Despite the abundant literature on dynamic signatures, a standardized approach to data treatment and examination remains elusive [4, 5, 7, 8]. Moreover, there are challenges associated with averaging analysis across simulation samples, and there is a need for more transparency regarding the application of statistical tests and their adherence to test conditions.

Although numerous automated programmes exist for signature classification and handwriting recognition, they are unsuitable for forensic handwriting analysis. Most of these programmes require comprehensive data visualization, crucial for forensic analysis, and are not trained for disguise classification. Conversely, forensic analysis software often presents data without any statistical treatment [1, 2, 8].

In this study, we aim to analyse dynamic signatures of various styles (legible, mixed, and stylized) to differentiate between genuine and nongenuine samples. Our objectives include assessing the efficacy of the proposed data approach in distinguishing simulated samples from genuine ones and disguises, exploring the forensic potential of utilizing biometric data with both global and local features, employing mathematical tests to evaluate statistical significance, and determining the relevance of specific global and local characteristics in expert signature examination.

Methodology

Materials

The same hardware configuration was used to capture all signature samples. The configuration consisted of a high-end Avell laptop (Manaus, Brazil) connected to a Wacom tablet STU540 (Tokyo, Japan). This tablet has electromagnetic resonance capture technology at a frequency of 200 Hz and a pressure-sensitive stylus with visual feedback. The Wacom Signature Scope software version 1.48.1.93 × 64 (Wacom Co., Ltd, Santa Clara, CA, USA) was used for global and local data capture. The analysis of raw data, calculations, and graph generation were performed using a Python algorithm implemented by the authors.

Participants

Thirty volunteers (28 right-handed and two left-handed writers, 16 females and 14 males) with no expertise in forensic handwriting examinations provided the signature samples. All the participants provided their informed consent, and ethical approval was not required for this noninterventional study. Participants were chosen randomly, without the use of any advertisement or remuneration, up to the number of samples determined for each signature style. The participants were acquaintances verbally invited to participate in the study. Those who accepted the invitation were provided with the consent forms to confirm their willingness to volunteer.

All participants possessed automated handwriting and had completed a higher degree. The age range of the participants was 28–65 years. At the time of data collection, none of them had any conditions or had used any substances that could influence writing. There were 10 writers for each signature style: legible, mixed, and stylized. The style classification was based on responses from 10 forensic handwriting examiners to mitigate cognitive bias.

Samples

All samples were acquired using the same software and hardware. Each volunteer produced 60 repetitions of their genuine (“natural”) signature; two samples of disguises, one in a free-form manner (“random disguise”) and another formally resembling their signature (“autosimulation”); and 29 simulations, one of each signature of the other participants. In total, 2 730 instances were obtained (1 800 genuine signatures, 60 disguises, and 870 simulations). The tablet was placed on a horizontal table, and writers assumed a comfortable seated position. They could adjust the tablet angle for wrist comfort, but the tablet remained on the surface. The volunteers provided their genuine samples in three sets of 20 repetitions each, with a rest interval of at least 10 min between sets. Next, the disguises were acquired, with participants allowed to train as desired at least for 2 min. Only one sample of each type was collected. For the free-form disguise, volunteers created a signature as they pleased, which they could later deny. For the formal disguise, each subject made a signature, the authenticity of which they could deny later, but with the condition that it closely resembled their own and could be accepted in an institution. For the simulations, the volunteers visualized a sample with a size identical to the digital tablet screen on a laptop. They were allowed practice using the model on the tablet until they achieved a satisfactory simulation.

Global characteristics

Global characteristics represent averaged quantities for each dynamic signature. The Wacom Signature Scope software automatically exports 62 global characteristics in a comma-separated values (CSV) file (Supplementary Material 1), such as total time, pen-down duration, pen-up duration, average pen force, and root mean square pen speed.

Local characteristics

Local characteristics are point-to-point numerical values in the signature. The local characteristics analysed included relative pressure, horizontal and vertical axis coordinates, velocity, acceleration, and jerk. The raw data included time, horizontal and vertical position, and pressure. The velocity, acceleration, jerk, and relative pressure were calculated using the raw data. For the relative pressure (point force relative to the maximum tablet measurement) data, in accordance with the recommendations of ISO/IEC 19794-7 [9], the force values were transformed into a percentage of the capturing range.

Data analysis

As the first step, an initial exploratory analysis using global features was conducted to compare genuine and nongenuine signature groups (simulated, free-form disguise, and formal disguise) using two-dimensional principal component analysis (PCA), followed by descriptive statistics analysis through box plots of each mean feature [10–12]. PCA was performed on all subjects with all 62 global characteristics to observe if the method could successfully group genuine, simulated, formal disguise, and random disguise samples. To confirm if the trends found during exploratory analysis were representative of actual data, evaluations of the characteristics were conducted using box plots.

The difference between genuine, simulated, and disguised samples for each global characteristic was assessed using a hypothesis test. The hypothesis test was selected after the normality of each characteristic data was determined using frequency histograms, distribution curves graphs, and the Kolmogorov–Smirnov (KS) test. In this study, neither global nor local characteristics for an individual followed a normal distribution in any sample group. Consequently, analysis of variance (ANOVA), as sometimes used in studies, was not recommended [5]. Hence, the nonparametric KS hypothesis test was used. For the statistical test, the null hypothesis (H0) considered was “The nongenuine sample (simulated or disguised) belongs to the pattern group”, and the alternative hypothesis (H1) was “The nongenuine sample does not belong to the pattern group”. Initially, a significance level of P-value less than 0.05 was used.

In addition, a multivariate distance test was conducted by associating all global features. This test compared the distances between simulated and disguised samples against those with genuine ones using box plots and the KS test, with the same null hypothesis. The accuracy rate was defined by the method’s ability to correctly classify simulated samples as not belonging to the pattern distribution (true negative) and disguised as belonging to the pattern group (true positive).

In the second step, analyses of the local features (relative pressure, positions in x and y coordinates, velocity, acceleration, and jerk) were conducted to test whether it was possible to differentiate simulated and disguised (nongenuine) signatures from genuine samples. A dissimilarity measure between genuine and nongenuine samples was evaluated using dynamic time warping (DTW) with Euclidean distance [11, 13]. DTW cost matrix values were assessed by comparing the groups of genuine and nongenuine signatures using a KS test. The accuracy rate for local features was the same as for global features.

The DTW analysis used curves of each studied variable versus the relative position in the stretched signature, calculated at point n, |${dS_n}^{\prime }$| = |${\sum}_{i=0}^n\left({dS}_n\right)$|⁠, where |${dS}_n=\sqrt{{\left({x}_n-{x}_{\left(n-1\right)}\right)}^2+\left({y}_n-{y}_{\left(n-1\right)}\right)^2}$|⁠. This approach, analogous to trace analysis, was used to enhance the comparability of similar regions between nongenuine and genuine samples. Because simulations do not follow the isochrony principle, this approach reduces the chance of comparing the final region of a standard signature to the start region of a simulation. In addition, this approach maintains time dependence so that DTW can be used.

Accuracy rates analysis

It was observed that using a P = 0.05 in the KS test resulted in error rates unacceptable for natural samples (average of 47.4% for local features and 57.3% for global features) despite being accurate for simulated samples (average of 99.6% for local features and 100% for global features). Therefore, we explored how accuracy rates responded to increases in test rigor by varying the P value cut-off. To do this, the number of correct classifications was tallied for each sample group at different KS P-value thresholds to determine the optimal P-value that could provide a reasonable accuracy rate for both natural and simulated samples. We calculated accuracy rates using P values of 0.05, 0.01, 1 × 10⁻³, 1 × 10⁻⁴, 1 × 10⁻⁵, 1 × 10⁻⁶, 1 × 10⁻⁷, 1 × 10⁻⁸, 1 × 10⁻⁹, 1 × 10⁻¹⁰, 1 × 10⁻¹¹, 1 × 10⁻¹², 1 × 10⁻¹³, 1 × 10⁻¹⁴, and 1 × 10⁻¹⁵.

Results

Global characteristics

PCA, when using all 62 mean characteristics, achieved the best clustering of natural, disguised, and simulated signatures. The accuracy rate for genuine samples was 99.8%; for simulated samples, it was 97.8%; and for the disguises, 46.7% of formal disguise samples and 43.3% of random disguise samples were accurately clustered near natural samples. Figure 1 illustrates an example of PCA clustering on the data of one subject. The genuine samples were clustered closely together, and no simulated or disguised samples were grouped near the natural cluster. However, this PCA clustering method was unable to differentiate all disguises from simulations for this individual.

In the analysis of each global characteristic separately by using a KS test (with a P-value of 0.05), a sample was considered to belong to the natural cluster (genuine signatures) if more than half of the characteristics were within the same distribution. Although 50% is a subjective criterion, this value was used for exploratory analysis, and it does not imply that all mean characteristics hold the same relevance. Consequently, the accuracy rates were 100% for genuine samples, 68.4% for simulated samples, 53.3% for formal disguises, and 30.0% for random disguises. This method’s predictions were more accurate for disguises that resembled natural signature forms. The characteristics with the highest accuracy rates varied according to the sample group (Supplementary Material 2). For genuine samples and disguises, the number of pauses was, among all characteristics, correlated with the highest accuracy rates. In contrast, for simulated samples, characteristics related to timing had the highest percentage of accuracy, aligning with the theory of isochrony in genuine samples [14].

Multivariate distance analysis associated with the KS test (P value of 0.05) yielded an accuracy rate of 42.7% for genuine samples, 100% for simulations, 36.7% for formal disguises, and 63.3% for random disguises.

Local characteristics

The DTW normality data of costs for characteristics and individuals showed a non-normal distribution, so to analyse the statistical significance of the observed differences between groups (genuine, disguises, and simulations), we used the KS test (P ≤ 0.05). Table 1 shows the accuracy rates for each feature and group.

Accuracy rates analysis

For different P values, the multivariate analysis yielded much better results than isolated characteristics for global characteristics analysis. For isolated characteristics, any value analysed returned a 0% accuracy rate. In the multivariate analysis, a P-value of 1 × 10⁻⁵ showed the best accuracy rates (Table 2).

For local characteristics analysis, a P value of 1 × 10⁻¹⁰ yielded the best combination of accuracy rates for both genuine and simulated samples (an average of 94.7% for simulated and 95.4% for genuine samples). For disguises, a P value of 1 × 10⁻¹⁰ resulted in average accuracy rates below 17%. Table 3 summarizes the accuracy rates for each P-value assessed and each local characteristic, as well as the average accuracy rates considering all attributes for each group.

Discussion

Global characteristics

For global characteristics, the PCA technique successfully grouped natural and simulated samples, with only a 2.2% error rate for simulated samples. Most of the erroneously grouped samples consisted of stylized models that corroborate traditional handwriting analysis observations, e.g. that stylized models are easier to simulate than legible models [14]. PCA is a technique that compares a single questioned sample against a group of known standards, which is a situation that occurs in many forensic examinations. The PCA error rate for disguises was ~50%, which would be unacceptable in a forensic scenario. This result agrees with traditional handwriting analysis observations where error rates for differentiating simulations from disguises are typically higher than those for differentiating genuine from nongenuine samples [3, 5]. It was impracticable to affirm if the error rates for classifying formal disguises were lower than those for random disguises by using PCA. The characteristics that exhibited higher score values for PCA varied from person to person, which allowed us to infer that some global characteristics are more relevant than others, depending on what factors are more constant in a person’s handwriting.

Our analysis of global characteristics demonstrated that prediction for disguises that resembled natural signatures (formal disguises) was more accurate than that for random disguises (free form ones). However, when generalizing to all disguises, the accuracy rate remained extremely low, and it is only possible to infer that formal disguises are more likely to be identified as disguises and, in some cases, may be helpful to improve forensic expert conviction.

The global characteristics with higher accuracy rates varied for each group sample. For simulations, the factors related to the time it takes to execute the signature are consistent with the theory of isochrony in natural samples and previous findings.

Multivariate analysis was less accurate than global analysis of characteristics when the P-value was 0.05. However, different cut-off values yielded better results for multivariate analysis that considers all global characteristics simultaneously, so it would have been more appropriate to use multivariate distance followed by the KS test for global characteristics associated with PCA.

Local characteristics

Using the KS statistical test for local variables, we achieved a high accuracy rate for simulated samples with a P value of 0.05. Although this method performed better with formal disguises than with random disguises, the error rate for disguises using this criterion was still remarkably high. The increase in test rigor and analysis of various P value cut-off values showed that for a P value of 1 × 10⁻¹⁰, the accuracy rate was ~95% for both the genuine and simulated samples, demonstrating that this method may be relevant for forensic examination.

Accuracy rates analysis

When the accuracy rates were varied with the P value cut-off in the KS test, we obtained the best results for multivariate analysis of global characteristics with a P value of 1×10⁻⁵ (91.8% for genuine samples and 96.9% for simulation samples).

For analysis of local characteristics, the best P value for the accuracy rates in the simulated and genuine samples group was 1 × 10⁻¹⁰ (95.4% for simulation samples and 94.7% for genuine samples). Using a P value of 1 × 10⁻¹⁰ for the KS test associated with multivariate distance applied to global characteristics as well led to an accuracy rate of 96.7% for genuine samples and 88.9% for simulation samples. These values still provide high success rates for examinations.

Limitations

The following limitations of our study are of note: all samples were collected with the same software and hardware, and the volunteers were not pressured to produce good simulations, which may not reflect the behaviour of experienced forgers. If the simulation was not good, there were no losses; conversely, there was also no compensation for participants’ efforts. The same person collected and analysed the samples, even though mathematical tools were used to reduce subjectivity and cognitive bias. The relatively small number of disguised samples is also a limitation in that there were significantly fewer data points for disguised samples than other types.

Additionally, because all the natural signatures were collected in effectively one sitting, the range of variation exhibited is likely to be lower than in samples collected over a longer period (e.g. weeks to months). In future studies, different people should collect samples, and another person should analyse the samples using the proposed methods. In this work, we explored cut-off values for better accuracy rates for simulated and genuine samples. In the future, we plan to conduct a blind test with defined cut-off points.

Conclusion

To classify various types of signatures, we propose a first-step analysis of global characteristics using PCA, followed by an analysis of multivariate distances associated with a KS test. We also propose a second-step analysis of local characteristics using DTW associated with a KS test. The proposed sequence of steps yielded reasonably accurate rates for both genuine and simulated samples and is promising for forensic signature analysis. It was not possible to satisfactorily classify disguises as belonging to the same individual as genuine samples or to differentiate disguises from simulations, a problem that has been described in the literature. For disguises where the shape resembles an individual’s natural signature, the proposed analysis may be interesting from an expert’s point of view to help bolster the evidence for conviction. In future studies, we plan to validate the proposed approach using blind tests.

Acknowledgements

We thank the Brazilian National Police Academy for the specialization programme in forensic document examinations, in which the study was initiated.

Authors' contributions

Jessica Baleiro Okado carried out the conceptualization, data curation, formal analysis, investigation, methodology, project administration, resources, software, validation, visualization, roles/writing original draft, and writing review and editing. Priscila Dias Sily carried out conceptualization, supervision, methodology, project administration, writing review and editing. Erick Simões da Câmara e Silva carried out supervision, project administration, and writing review and editing. All authors contributed to the final text and approved it.

Compliance with ethical standards

All participants provided their informed consent, and ethical approval was not required for this noninterventional study.

Disclosure statement

The authors report that there are no competing interests to declare.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

References