Courses
Structured learning pathways for computational and quantitative methods in language research
LADAL Courses are curated sequences of tutorials, readings, and practical exercises for learners progressing from foundational knowledge to advanced skills. All courses are free, open, and built around reproducible R workflows. Whether you are a complete beginner or an experienced analyst, there is a pathway here for you.
By following a LADAL course, you will develop practical skills in R — data management, visualisation, statistics, and text analytics — and learn to apply them to real research questions in linguistics, the humanities, and the social sciences.
All Courses at a Glance
Introduction to Language Technology
A conceptual and practical first introduction to language technology — from text processing and regex to OCR and NLP overview.
Introduction to Corpus Linguistics
Core corpus methods — concordancing, collocations, keyness, and frequency analysis — using R and reproducible workflows.
Introduction to Text Analysis
From text processing basics to topic modelling, sentiment analysis, and network analysis of text collections.
Data Visualisation for Linguists
Publication-quality visualisation with ggplot2 — histograms, scatter plots, maps, Likert charts, and more.
Introduction to Statistics
Statistical literacy from the ground up — descriptive statistics, hypothesis testing, t-tests, chi-square, and simple regression.
Introduction to Learner Corpus Research
Learner corpus methods — frequency comparison, collocations, lexical diversity, readability, and error analysis with ICLE and LOCNESS.
Natural Language Processing with R
NLP pipeline in R — preprocessing, TF-IDF, classification, NER, dependency parsing, and introduction to word embeddings.
Introduction to Digital Humanities with R
Full semester course: DH methods from data literacy and text processing through corpus analysis, topic modelling, networks, and mapping.
Corpus Linguistics and Text Analysis with R
Corpus construction through concordancing, collocations, keywords, topic modelling, sentiment analysis, and network analysis.
Introduction to Statistics in the Humanities
From probability and descriptive statistics through regression and mixed-effects modelling, using R throughout.
Advanced Statistics in the Humanities
Multivariate modelling, classification trees, random forests, clustering, correspondence analysis, and survey data analysis.
Short Courses
Introduction to Language Technology
Introduction to Language Technology
A first introduction to language technology — what it is, what it can do, and how to get started
Language technology encompasses the computational tools and methods used to analyse, generate, and interact with human language. This short course introduces learners to the landscape of language technology with hands-on practice in R. By the end, learners will understand the key methods and be equipped to explore more specialised pathways.
Introduction to Text Analysis
What text analysis is, how it relates to corpus linguistics and NLP, and key concepts: corpus, token, type, frequency, and concordance.
Getting Started with R
Installing packages, loading data, working with vectors and data frames, and writing simple functions in R and RStudio.
Loading and Saving Data
Importing text from plain text files, CSV, Excel, and web URLs — and saving results for later use.
String Processing
Pattern matching, substitution, splitting, and the core string operations (using stringr) that underpin all text analysis.
Regular Expressions
Character classes, quantifiers, anchors, and look-arounds with worked linguistic examples — the pattern language for searching and transforming text.
Converting PDFs to Text
Extracting machine-readable text with pdftools (digital PDFs) and tesseract (scanned documents), including post-OCR spell-checking.
Introduction to Corpus Linguistics
Introduction to Corpus Linguistics
Core corpus methods — concordancing, collocations, keyness — using R and reproducible workflows
Corpus linguistics uses large, principled collections of authentic text to investigate patterns of language use. This short course takes learners from a conceptual introduction through hands-on practice with the most widely used corpus methods, culminating in a case-study showcase integrating all techniques into a full corpus-based analysis.
quanteda, tidytext, and related toolsIntroduction to Text Analysis
Key concepts — corpus, concordance, collocation, keyword, frequency — used throughout the course.
Getting Started with R
First introduction to R and RStudio. Focus on the first four sections (up to Working with Tables).
String Processing
Essential string manipulation: pattern matching, substitution, tokenisation preparation, and whitespace management.
Concordancing (Keywords-in-Context)
KWIC concordance search and display in R — sorting, filtering, and interpreting concordance output.
Collocation and N-gram Analysis
Statistically significant collocations and n-gram sequences — PMI, log-likelihood, t-score, and visualisation.
Keyness and Keyword Analysis
Comparing two corpora to identify words that are statistically more or less frequent — the foundation of contrastive corpus analysis.
Corpus Linguistics with R
Capstone showcase: complete case studies integrating concordancing, frequency analysis, collocations, and keyness.
Introduction to Text Analysis
Introduction to Text Analysis
From text processing basics to topic modelling, sentiment analysis, and network analysis
Text analysis uses computational methods to extract patterns, topics, sentiment, and relational structure from large collections of text. This course builds from foundational R skills through to topic modelling, sentiment analysis, and network analysis. By the end, learners will be able to apply a range of text-analytic methods to their own research texts.
Introduction to Text Analysis
Overview of the field, key concepts, and situating text analysis within computational humanities research.
Getting Started with R
First introduction to R and RStudio. Focus on the first four sections.
String Processing
Core string manipulation skills for preparing raw text for analysis.
Practical Overview of Text Analytics Methods
Frequency analysis, TF-IDF, and basic classification workflows using R.
Topic Modelling
Latent Dirichlet Allocation (LDA) for discovering thematic structure in document collections — theory and R implementation.
Sentiment Analysis
Lexicon-based and machine-learning approaches to opinion and emotion extraction, including dictionary methods and valence shifting.
Network Analysis
Representing relational structure in textual and social data — node and edge construction, centrality measures, and visualisation.
Data Visualisation for Linguists
Data Visualisation for Linguists
Publication-quality visualisation with ggplot2 — from frequency plots to maps
Effective visualisation is one of the most transferable skills in quantitative research. This course builds from visualisation principles through the mechanics of ggplot2, covering the graph types most commonly needed in linguistics: frequency distributions, scatter plots, heat maps, geographic maps, and interactive visualisations. Special attention is given to colour accessibility, annotations, and formatting for publication.
Getting Started with R
Introduction to R with a focus on data structures and workflow needed for visualisation.
Introduction to Data Visualisation
Visualisation philosophy, perceptual principles, grammar of graphics, and when to use which chart type.
Descriptive Statistics
Summary statistics — means, medians, distributions, variance — that underpin most visualisations of linguistic data.
Data Visualisation with R
In-depth ggplot2: histograms, density plots, box plots, bar charts, scatter plots, and line graphs with worked linguistic examples.
Visualising and Analysing Survey Data
Cumulative density plots, diverging stacked bar charts, and Likert scale visualisation for questionnaire data.
Maps and Spatial Visualisation
Dialect maps, distribution maps, and choropleth maps of linguistic data using ggplot2 and sf.
Introduction to Statistics
Introduction to Statistics in the Humanities and Social Sciences
Statistical literacy and practical quantitative skills from the ground up, using R throughout
This course provides a conceptual and practical introduction to statistics for researchers whose background is in the humanities or social sciences. It begins with the philosophical foundations of quantitative reasoning and builds through descriptive statistics, visualisation, and inferential testing. By the end, learners will be able to conduct and interpret basic statistical analyses and communicate their results clearly.
Introduction to Quantitative Reasoning
Scientific thinking, the logic of hypothesis testing, and the role of quantitative methods in humanities and social science research.
Basic Concepts in Quantitative Research
Variables, data types, sampling, populations, reliability, and validity.
Getting Started with R
Introduction to R and RStudio. Focus on the first four sections.
Handling Tables in R
Importing, cleaning, reshaping, and summarising data frames using dplyr and tidyr.
Descriptive Statistics
Means, medians, standard deviations, distributions, and frequency tables in R.
Introduction to Data Visualisation
Visualisation principles and hands-on practice creating and customising graphs in R.
Basic Inferential Statistics
Hypothesis testing, p-values, confidence intervals, t-tests, chi-square, correlation, and simple linear regression with R exercises.
Introduction to Learner Corpus Research
Introduction to Learner Corpus Research
Corpus methods for studying learner language — from frequency comparison to error analysis
Learner corpus research uses collections of authentic language produced by second-language learners to investigate the structure, development, and distinctiveness of interlanguage. This course covers the major analytical methods — concordancing, frequency comparison, collocation, POS tagging, lexical diversity, and error analysis — using the ICLE and LOCNESS corpora as running examples.
Introduction to Text Analysis
Key concepts — corpus, frequency, concordance, collocation — underpinning learner corpus research.
Getting Started with R
Data structures and workflow for corpus analysis.
String Processing
Cleaning, normalising, splitting, and extracting character patterns from raw learner corpus texts.
Concordancing (Keywords-in-Context)
Extracting and inspecting KWIC concordances from learner texts to investigate how learners use specific words or constructions.
Collocation and N-gram Analysis
Comparing collocational patterns between learner and native-speaker corpora for studying collocational competence and L1 transfer.
Analysing Learner Language with R
Frequency comparison, POS tagging, lexical diversity, readability scores, and spelling error detection with ICLE and LOCNESS examples.
Keyness and Keyword Analysis
Words systematically over- or under-used by learners relative to native-speaker norms — one of the most informative methods in LCR.
Natural Language Processing with R
Natural Language Processing with R
Text preprocessing, feature extraction, classification, NER, and transformer-based representations
NLP builds on corpus and statistical methods to develop computational pipelines for understanding and generating language at scale. This course introduces the NLP workflow in R using real linguistic datasets, progressing from text preprocessing and feature engineering to supervised classification, topic models, and an introduction to working with large language model embeddings and APIs.
Introduction to Text Analysis
Situating NLP within corpus linguistics and computational linguistics.
String Processing
Foundation string manipulation — essential for all preprocessing steps in NLP pipelines.
Regular Expressions
Regex as the primary pattern-matching tool in text preprocessing and feature extraction.
Practical Overview of Text Analytics Methods
Document-feature matrices, TF-IDF, and basic classification workflows in R.
Topic Modelling
Probabilistic topic models as an unsupervised NLP method for discovering thematic structure.
Analysing Learner Language with R
POS tagging with udpipe, sequence analysis, and lexical diversity measures — key NLP tasks applied to real corpus data.
Network Analysis
Representing relational structure in language data — semantic networks, co-occurrence graphs, and social networks of linguistic interaction.
Long Courses
Introduction to Digital Humanities with R
Introduction to Digital Humanities with R
Computational methods for humanistic inquiry — from data literacy through corpus analysis, networks, and mapping
Digital humanities applies computational methods to humanistic inquiry: analysing large literary corpora, mapping cultural data geographically, tracing discourse patterns across historical archives, or modelling networks of social interaction. This 12-week course introduces students to the core DH toolkit through R, with weekly tutorials grounded in real humanities datasets. No prior programming experience is assumed.
Week 1What Is Digital Humanities?▾
Overview of digital humanities — history, debates, and current landscape; relationship to corpus linguistics, text analysis, and data science; what counts as DH research.
- Burdick et al. (2012). Digital humanities. MIT Press, Ch. 1
- Drucker (2021). The digital humanities coursebook. Routledge, Ch. 1
Week 2Reproducible Research and Data Management▾
Why reproducibility matters in DH; introduction to R and RStudio; file organisation, project workflows, and version control basics.
- Flanagan, J. (2025). Reproducibility, replicability, robustness, and generalizability in corpus linguistics. International Journal of Corpus Linguistics. doi:10.1075/ijcl.24113.fla
Week 3Getting Started with R▾
R syntax, data types, vectors, and data frames; the tidyverse ecosystem; reading and writing data.
- Wickham & Grolemund (2016). R for data science. Ch. 1–3. r4ds.had.co.nz
Week 4Working with Text Data▾
How text is represented computationally; encoding, tokenisation, and the document-feature matrix; from raw text to structured data.
- Jockers, M. L. (2014). Text analysis with R for students of literature. Springer, Ch. 1–3
Week 5Building and Exploring Digital Corpora▾
What is a corpus? Sampling principles, metadata, corpus design for humanities research; downloading and preparing digital texts.
- Biber, Conrad & Reppen (1998). Corpus linguistics. Cambridge University Press, Ch. 1–2
Week 6Frequency Analysis and Visualisation▾
Zipf's law and frequency distributions; word counts, type-token ratios, and dispersion; principles of effective visualisation for humanities data.
- Jockers (2014), Ch. 4–5
Week 7Concordancing, Collocations, and Keywords▾
Searching corpora; KWIC concordances and their interpretation; collocation and association measures; keyness and corpus comparison.
- Baker, P. (2006). Using corpora in discourse analysis. Continuum, Ch. 3–4
Week 8Topic Modelling and Thematic Analysis▾
Latent Dirichlet Allocation (LDA); interpreting topics; applications in literary and historical research; limitations and critical perspectives.
- Blei, D. M. (2012). Probabilistic topic models. Communications of the ACM, 55(4), 77–84
- Maier et al. (2021). Applying LDA topic modeling in communication research. In Computational methods for communication science (pp. 13–38). Routledge
Week 9Sentiment Analysis and Opinion Mining▾
Lexicon-based and machine learning approaches to sentiment; subjectivity, valence, and emotion; applications in literary and media studies.
- Liu, B. (2012). Sentiment analysis and opinion mining. Ch. 1–2
Week 10Network Analysis for Humanities Research▾
Graphs and networks as representations of humanistic data; character networks in fiction; citation and social networks; centrality and community detection.
- Moretti, F. (2011). Network theory, plot analysis. New Left Review, 68, 80–102
Week 11Maps, Space, and Geographic Visualisation▾
Spatial thinking in digital humanities; mapping literary geography, dialect distribution, and historical events; choropleth maps and point maps in R.
- Drucker (2021), Ch. 8
Week 12Project Workshop and Critical Reflections▾
Critical DH — bias in corpora and algorithms, data ethics, representation, and positionality; communicating DH research; the future of digital humanities.
Student project presentations and peer feedback.
📚 Core Reading List
- Baker, P. (2006). Using corpora in discourse analysis. Continuum.
- Biber, D., Conrad, S., & Reppen, R. (1998). Corpus linguistics. Cambridge University Press.
- Burdick, A., et al. (2012). Digital humanities. MIT Press.
- Drucker, J. (2021). The digital humanities coursebook. Routledge.
- Flanagan, J. (2025). Reproducibility in corpus linguistics. International Journal of Corpus Linguistics. doi:10.1075/ijcl.24113.fla
- Jockers, M. L. (2014). Text analysis with R for students of literature. Springer.
- Liu, B. (2012). Sentiment analysis and opinion mining. Synthesis Lectures on Human Language Technologies, 5(1), 1–167.
- Wickham, H., & Grolemund, G. (2016). R for data science. O'Reilly. r4ds.had.co.nz
Corpus Linguistics and Text Analysis with R
Introduction to Corpus Linguistics and Text Analysis with R
Corpus construction through concordancing, keywords, topic modelling, and network analysis
Week 1Introduction to Corpus Linguistics and Text Analytics▾
What is corpus linguistics? Key concepts, history, and applications; corpus vs. introspective and experimental methods; overview of the course.
- McEnery & Hardie (2012). Corpus linguistics: Method, theory and practice. CUP, Ch. 1–2
Week 2Working with Digital Data and Reproducibility▾
Principles of reproducible research; introduction to R Notebooks; file management and workflow.
- Flanagan (2025). Reproducibility in corpus linguistics. doi:10.1075/ijcl.24113.fla
Week 3Getting Started with R▾
R and RStudio; installing packages; basic syntax; workflow setup.
- Wickham & Grolemund (2016), Ch. 1–3
Week 4Corpus Compilation and Preparation▾
Types of corpora; sampling principles and representativeness; metadata and annotation; legal and ethical issues in corpus construction.
- Biber, Conrad & Reppen (1998), Ch. 1–2
Week 5Frequency and Dispersion▾
Counting words and n-grams; Zipf's law; normalised frequencies; dispersion measures and why they matter; type-token ratio.
- McEnery & Hardie (2012), Ch. 3
- Gries (2024). Frequency, dispersion, association, and keyness. Ch. 1–2
Week 6Concordancing and KWIC▾
Searching corpora; concordance displays and their interpretation; sorting and filtering; from examples to patterns.
- Baker (2006), Ch. 3
Week 7Collocations and N-grams▾
Association measures (MI, t-score, log-likelihood, Dice); phraseology and formulaic sequences; n-gram extraction and analysis.
- Gries (2024), Ch. 2
Week 8Keywords and Keyness▾
Reference corpora and keyness; log-likelihood and log ratio as keyness measures; interpretation and applications in discourse analysis.
- Gries (2024), Ch. 3
Week 9Advanced Text Analytics I — Topic Modelling▾
Unsupervised text classification; LDA and its assumptions; interpreting and validating topic models; applications in linguistics and discourse analysis.
- Maier et al. (2021). Applying LDA topic modeling in communication research (pp. 13–38)
Week 10Advanced Text Analytics II — Sentiment and Network Analysis▾
Sentiment lexicons; opinion mining; co-occurrence networks and semantic networks from corpus data.
- Liu (2012), Ch. 1–2
Week 11Case Studies in Corpus Linguistics▾
Corpus-based studies of grammar, lexis, and discourse; from method to interpretation; writing up corpus research.
- Baker (2006), Ch. 7
Week 12Project Workshop and Presentations▾
Ethics in corpus research; future directions; communicating corpus findings to non-specialist audiences.
Student project work.
📚 Core Reading List
- Baker, P. (2006). Using corpora in discourse analysis. Continuum.
- Biber, D., Conrad, S., & Reppen, R. (1998). Corpus linguistics. Cambridge University Press.
- Flanagan, J. (2025). Reproducibility in corpus linguistics. International Journal of Corpus Linguistics. doi:10.1075/ijcl.24113.fla
- Gries, S. T. (2024). Frequency, dispersion, association, and keyness (Studies in Corpus Linguistics, Vol. 115). John Benjamins.
- Liu, B. (2012). Sentiment analysis and opinion mining. Synthesis Lectures on Human Language Technologies, 5(1), 1–167.
- McEnery, T., & Hardie, A. (2012). Corpus linguistics: Method, theory and practice. Cambridge University Press.
- Wickham, H., & Grolemund, G. (2016). R for data science. r4ds.had.co.nz
Introduction to Statistics in the Humanities
Introduction to Statistics in the Humanities and Social Sciences
Probability and descriptive statistics through regression and mixed-effects modelling, using R throughout
Week 1Introduction to Quantitative Research▾
The role of quantitative methods in humanities and social sciences; an overview of statistical thinking; the research cycle; types of research questions.
- Field, Miles & Field (2012). Discovering statistics using R. Ch. 1
- Baayen (2008). Analyzing linguistic data. Ch. 1
Week 2Basic Concepts in Quantitative Research▾
Data types and measurement scales; variables, operationalisation, and construct validity; sampling and representativeness; reliability and validity.
- Gries (2013). Statistics for linguists. Ch. 1–2
Week 3Getting Started with R — Part 1▾
Introduction to R and RStudio; installing and loading packages; basic syntax and data structures; the tidyverse ecosystem.
- Wickham & Grolemund (2016), Ch. 1–3
Week 4Loading and Handling Data▾
Importing datasets from CSV, Excel, and text files; data cleaning and transformation; working with factors and missing values.
- Baayen (2008), Ch. 2
Week 5R Basics for Statistical Analysis▾
Vectors, factors, data frames, indexing, and subsetting; writing functions; applying operations across groups with dplyr.
- Field, Miles & Field (2012), Ch. 2–3
Week 6Descriptive Statistics▾
Measures of central tendency and dispersion; frequency distributions; skewness and kurtosis; the normal distribution; summarising grouped data.
- Baayen (2008), Ch. 3
- Winter (2019). Statistics for linguists. Ch. 2
Week 7Visualising Data▾
Principles of effective visualisation; histograms, box plots, scatter plots, and bar charts; ggplot2 grammar of graphics.
- Wickham & Grolemund (2016), Ch. 14
Week 8Hypothesis Testing and Power Analysis▾
The logic of null hypothesis significance testing; t-tests, ANOVA; p-values and their interpretation; effect sizes; statistical power and sample size planning.
- Field, Miles & Field (2012), Ch. 4
- Gries (2013), Ch. 3
Week 9Correlation and Simple Regression▾
Pearson and Spearman correlation; simple linear regression; interpreting intercepts and slopes; assumptions and diagnostics.
- Baayen (2008), Ch. 4
Week 10Multiple Regression and Model Diagnostics▾
Multiple regression; multicollinearity; residual analysis; model comparison with AIC/BIC; stepwise and theory-driven model building.
- Winter (2019), Ch. 5
Week 11Logistic Regression▾
Binary and ordinal outcomes; logistic regression model fitting and interpretation; odds ratios and predicted probabilities; the proportional odds model.
- Baayen (2008), Ch. 5
- Winter (2019), Ch. 6
Week 12Mixed-Effects Models▾
Why mixed effects? Random intercepts and random slopes; by-participant and by-item random effects; fitting and interpreting mixed models with lme4.
- Gries (2013), Ch. 6
- Field, Miles & Field (2012), Ch. 12
📚 Core Reading List
- Baayen, R. H. (2008). Analyzing linguistic data. Cambridge University Press.
- Field, A., Miles, J., & Field, Z. (2012). Discovering statistics using R. Sage.
- Gries, S. T. (2013). Statistics for linguists. De Gruyter Mouton.
- Wickham, H., & Grolemund, G. (2016). R for data science. O'Reilly. r4ds.had.co.nz
- Winter, B. (2019). Statistics for linguists: An introduction using R. Routledge.
Advanced Statistics in the Humanities and Social Sciences
Advanced Statistics in the Humanities and Social Sciences
Multivariate modelling, classification, clustering, and survey data analysis using R
Week 1Advanced Data Management and Reproducible Workflows▾
Organising complex datasets; reproducibility in advanced research; scripting and automating analysis pipelines; version control with Git.
- Flanagan (2025), Ch. 1
Week 2Review of Descriptive and Inferential Statistics▾
Quick review of key concepts: distributions, t-tests, correlations, confidence intervals, effect sizes, and power.
- Field, Miles & Field (2012), Ch. 1–4
Week 3Advanced Regression — Multiple and Hierarchical Models▾
Multiple regression; interaction terms; hierarchical (nested) models; mixed-effects models with random intercepts and slopes.
- Baayen (2008), Ch. 4–5
- Winter (2019), Ch. 5–6
Week 4Logistic Regression and Generalised Linear Models▾
Binary and multinomial outcomes; model fitting and interpretation; goodness-of-fit; GLMs as a unified framework.
- Winter (2019), Ch. 6
Week 5Classification — Decision Trees▾
Decision trees; recursive partitioning; overfitting and pruning; interpreting tree outputs; applications in linguistic classification problems.
- Gries (2013), Ch. 6
Week 6Classification — Random Forests and Ensemble Methods▾
Ensemble learning; bagging and boosting; random forests; variable importance; improving prediction accuracy and generalisability.
- James, Witten, Hastie & Tibshirani (2021). An introduction to statistical learning. Ch. 8
Week 7Clustering and Correspondence Analysis▾
Unsupervised classification; k-means and hierarchical clustering; choosing the number of clusters; correspondence analysis for categorical data.
- Gries (2013), Ch. 7
Week 8Survey and Questionnaire Data Analysis I▾
Preparing survey data; dealing with missing values; Likert scales and their properties; descriptive analysis and visualisation of survey items.
- Field, Miles & Field (2012), Ch. 10
- Baayen (2008), Ch. 6
Week 9Survey and Questionnaire Data Analysis II▾
Reliability (Cronbach's α, McDonald's ω); factor analysis and scale validation; cross-tabulations and chi-square; ordinal regression for Likert outcomes.
- Field, Miles & Field (2012), Ch. 11
Week 10Dimension Reduction and Multivariate Techniques▾
Principal Component Analysis (PCA); multidimensional scaling (MDS); detecting latent variables; applications to linguistic and social science data.
- Gries (2013), Ch. 8
Week 11Model Evaluation, Diagnostics, and Advanced Visualisation▾
Residual analysis and outlier detection; model comparison and selection criteria (AIC, BIC, cross-validation); visualisation for multivariate data.
- Winter (2019), Ch. 7
Week 12Applications and Student Mini-Projects▾
Integrating advanced methods into humanities and social science research; ethical considerations; communicating complex statistical results; reproducibility revisited.
Student project work applying classification, clustering, and survey analysis to real datasets.
- Baayen (2008), Ch. 7
- Field, Miles & Field (2012), Ch. 12
📚 Core Reading List
- Baayen, R. H. (2008). Analyzing linguistic data. Cambridge University Press.
- Field, A., Miles, J., & Field, Z. (2012). Discovering statistics using R. Sage.
- Flanagan, J. (2025). Reproducibility in corpus linguistics. International Journal of Corpus Linguistics. doi:10.1075/ijcl.24113.fla
- Gries, S. T. (2013). Statistics for linguists. De Gruyter Mouton.
- James, G., Witten, D., Hastie, T., & Tibshirani, R. (2021). An introduction to statistical learning. Springer.
- Winter, B. (2019). Statistics for linguists: An introduction using R. Routledge.