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Residual

In statistics, a residual is defined as the difference between an observed value of a dependent variable and the value predicted by a fitted model, typically expressed as e_i = y_i - \hat{y}_i, where y_i is the observed response and \hat{y}_i is the model's prediction.^[1] This measure quantifies the unexplained variation in the data after accounting for the effects of the independent variables in regression analysis.^[2] Residuals play a crucial role in model diagnostics, enabling researchers to assess the adequacy of a statistical model by examining assumptions such as independence, homoscedasticity, and normality of errors.^[3] Pioneering work by F. J. Anscombe and John W. Tukey in the 1960s emphasized the importance of residual analysis through graphical methods, such as residual plots against fitted values, to detect outliers, non-linearity, heteroscedasticity, or influential observations that could indicate model misspecification.^[3] Under the assumption of normally distributed errors with constant variance, the sum of squared residuals provides an unbiased estimate of the error variance, scaled by the degrees of freedom, and forms the basis for tests like the F-statistic in analysis of variance.^[1] Beyond statistics, the term residual broadly denotes a remainder or leftover after a process, part, or quantity has been removed or accounted for, with applications in fields like economics (e.g., residual income as earnings after deducting costs of capital) and engineering (e.g., residual stress in materials).^[4] In machine learning, residual connections—skip connections that add input to output in deep neural networks—mitigate vanishing gradient problems and enable training of very deep architectures, as introduced in the ResNet framework. These diverse uses underscore the concept's foundational role in quantifying discrepancies and remainders across scientific disciplines.

Mathematics, Statistics, and Econometrics

General Definition

In statistics, a residual represents the discrepancy between an observed value and the value predicted by a statistical model, serving as a measure of prediction error. Formally, for the i-th observation, it is expressed as e_i = y_i - \hat{y}_i, where y_i denotes the observed response, and \hat{y}_i is the fitted or predicted value from the model.^[5] This concept assumes familiarity with basic statistical modeling, where the goal is to approximate underlying patterns in data while quantifying unexplained variation through these errors.^[6] The origins of residuals trace back to the early 19th century, emerging from the development of the least squares method for handling measurement errors in scientific observations. Adrien-Marie Legendre introduced the method of least squares in 1805, formalizing the minimization of the sum of squared residuals to fit models to data, initially applied in astronomy for determining planetary orbits.^[7] Carl Friedrich Gauss independently developed and published the approach between 1809 and 1821, extending it to weighted least squares that accounted for varying measurement precisions, further refining its use in celestial mechanics and geodesy.^[7] These foundational contributions established residuals as essential for assessing model adequacy in overdetermined systems, where observations exceed the parameters to estimate.^[8] Residuals come in several basic forms, each scaled to facilitate interpretation and analysis. Raw residuals are the unadjusted differences e_i, directly reflecting the magnitude of errors but varying in scale across datasets. Standardized residuals normalize these by dividing by an estimate of the residual standard deviation, yielding values comparable to a standard normal distribution for detecting unusually large errors.^[9] Studentized residuals further adjust for the leverage or influence of individual observations, typically by using a leave-one-out estimate of the standard deviation, which helps identify outliers more robustly in regression contexts.^[10]

In Regression Analysis

In ordinary least squares (OLS) regression, residuals represent the differences between observed values and those predicted by the fitted model, serving as estimates of the error term in the linear regression equation y = X\beta + e, where y is the vector of observed responses, X is the design matrix of predictors (including a column of ones for the intercept), \beta is the vector of estimated parameters, and e is the vector of residuals.^[11] The OLS estimator minimizes the sum of squared residuals to obtain \hat{\beta} = (X^T X)^{-1} X^T y, after which residuals are computed as e = y - X\hat{\beta}. Under the classical linear model assumptions, the residuals sum to zero, ensuring that the fitted model passes through the mean of the data, as \sum e_i = 0.^[12] Key assumptions underlying the validity of OLS residuals include linearity in parameters, independence of errors, homoscedasticity (constant variance of residuals across predictor levels), and normality of residuals for inference purposes.^[13] Violations of these, such as heteroscedasticity or autocorrelation, can lead to inefficient or biased parameter estimates and invalid standard errors, compromising the reliability of hypothesis tests and confidence intervals.^[12] For instance, if residuals exhibit non-constant variance, weighted least squares may be employed as a remedy, but the core OLS framework relies on these properties for the residuals to be unbiased and consistent estimators of the true errors.^[13] In econometric applications, residuals extend to time-series models like ARIMA, where they capture the unexplained variation after adjusting for autocorrelation in the data through autoregressive (AR), integrated (I), and moving average (MA) components. In an ARIMA(p,d,q) model, the residuals are the innovations or white noise process remaining after differencing the series d times and fitting AR(p) and MA(q) terms, ideally exhibiting no serial correlation to validate the model specification.^[14] This allows residuals to quantify model adequacy in capturing temporal dependencies, such as in economic forecasting where past errors influence current predictions. Residuals are calculated for each point as e_i = y_i - \hat{y}_i, where \hat{y}_i is obtained from the fitted model. A residual plot, graphing residuals against fitted values or predictors, visualizes model fit: a random scatter around zero indicates adherence to assumptions, while patterns suggest issues like nonlinearity or heteroscedasticity.^[11]

Diagnostic Applications

In regression analysis, residuals serve as a primary tool for diagnosing model adequacy by revealing violations of key assumptions such as linearity, homoscedasticity, normality, and independence. Visual inspection through residual plots is a foundational diagnostic technique, where plotting residuals against fitted values or predictors helps identify non-linearity if patterns like curves or funnels appear, and heteroscedasticity if the spread of residuals varies systematically across the range of predictors or fitted values. For assessing normality, quantile-quantile (Q-Q) plots compare the ordered residuals to theoretical quantiles from a standard normal distribution; points aligning closely along the reference line indicate approximate normality, while deviations in the tails suggest skewness or heavy tails. Formal statistical tests complement these graphical methods to quantify specific issues. The Durbin-Watson test detects first-order autocorrelation in residuals, computed as DW = \frac{\sum_{i=2}^{n} (e_i - e_{i-1})^2}{\sum_{i=1}^{n} e_i^2}, where e_i are the residuals; values near 2 indicate no serial correlation, while values below 1.5 or above 2.5 (with the test statistic ranging from 0 to 4) suggest positive or negative autocorrelation, respectively.^[15] For heteroscedasticity, the Breusch-Pagan test regresses the squared residuals on the predictors and examines the significance of the resulting chi-squared statistic, providing evidence against constant variance if the p-value is low.^[16] Outlier detection and influence assessment rely on measures derived from residuals and the hat matrix. Leverage values h_{ii} from the diagonal of the hat matrix identify high-leverage points, and Cook's distance quantifies their overall influence as D_i = \frac{e_i^2 h_{ii}}{(p+1) s^2 (1 - h_{ii})}, where e_i is the studentized residual, p is the number of parameters, and s^2 is the mean squared error; values exceeding $4/p flag influential observations that disproportionately affect coefficient estimates.^[17] When diagnostics reveal problems, remediation strategies often involve data transformations or alternative estimation methods to restore assumption validity. For heteroscedasticity or non-normality, the Box-Cox transformation y^{(\lambda)} = \begin{cases} \frac{y^\lambda - 1}{\lambda} & \lambda \neq 0 \\ \log y & \lambda = 0 \end{cases} stabilizes variance and approximates normality by estimating an optimal power parameter \lambda, applied to the response variable when it is positive. Robust regression methods, such as those minimizing Huber's loss function—which applies quadratic loss for small residuals and linear loss for large ones—downweight outliers to produce more stable estimates under contamination.^[18] Additionally, quantile regression extends diagnostics by estimating conditional quantiles rather than means, using asymmetric absolute residuals to yield robust inference across the response distribution, particularly useful when tail behavior is of interest.^[19]

Business and Finance

Residual Income

Residual income (RI) in finance is a performance measure that calculates the profit generated by a business or investment after deducting the cost of capital employed. It is formally defined as RI = NOPAT - (WACC × IC), where NOPAT represents net operating profit after taxes, WACC is the weighted average cost of capital, and IC denotes invested capital.^[20]^[21] This metric highlights whether an entity is earning returns above the required rate to compensate capital providers, effectively isolating economic profit from accounting profit.^[22] The concept of residual income gained prominence in the 1980s through the work of Stern Stewart & Co., who developed and trademarked Economic Value Added (EVA) as a refined version of RI to align managerial incentives with shareholder value creation.^[23] EVA applies the RI formula with specific accounting adjustments to better reflect economic reality.^[21] Unlike return on investment (ROI), which can discourage managers from accepting profitable projects that temporarily dilute returns due to increased capital base, RI promotes decisions that exceed the cost of capital by focusing on absolute dollar value added.^[24] In corporate applications, RI underpins EVA frameworks to evaluate divisional performance and guide capital allocation, with firms using it to incentivize managers to prioritize value-creating initiatives.^[22] For instance, a company with $100 million in invested capital, a 10% WACC, and $15 million in NOPAT would have RI of $5 million, indicating positive economic value after covering capital costs ($15M - (0.10 × $100M) = $5M). In personal finance, RI refers to disposable income remaining after deducting essential expenses and debts, serving as a key indicator for affordability in debt repayment strategies such as the debt snowball method, where excess funds are systematically applied to accelerate payoff of obligations starting with the smallest balances.^[25]^[26]

Residual Value

Residual value, also known as salvage value, refers to the estimated monetary worth of a fixed asset at the conclusion of its useful life or lease term, after accounting for depreciation.^[27]^[28] This projection serves as a key input in financial calculations, particularly for determining the depreciable base in methods such as straight-line depreciation, where the residual value is subtracted from the initial cost to spread the expense over the asset's lifespan.^[29] In the straight-line approach, the book value at the end of the useful life equals the residual value, expressed as RV = \text{Initial Cost} - \text{Accumulated Depreciation}.^[30] In financial applications, residual value plays a central role in leasing agreements, where it influences periodic payments by representing the portion of the asset's value not depreciated during the lease period.^[31] For instance, in automotive leasing, the residual value is typically calculated as a percentage of the manufacturer's suggested retail price (MSRP), guiding the lessee's monthly obligations based on the expected resale value at term end.^[32] Similarly, in balloon mortgages or financing structures, the residual value manifests as a substantial lump-sum payment due at the loan's maturity, reflecting the outstanding principal after amortized portions.^[33] Several factors inform the estimation of residual value, including prevailing market trends, the asset's physical condition and maintenance history, rates of economic and functional depreciation, and broader economic conditions like supply-demand dynamics.^[27]^[34] For tax purposes, the Internal Revenue Service (IRS) requires taxpayers to estimate salvage value as the anticipated amount recoverable at the end of an asset's useful life, with guidelines allowing it to be disregarded if below 10% of the asset's cost in certain depreciation scenarios.^[35]^[36] A practical example illustrates its impact: Consider a vehicle with an MSRP of $30,000 leased for three years at a projected residual value of 50%, equating to $15,000 at the end of the term; this residual percentage reduces the depreciable amount, thereby lowering monthly lease payments compared to a lower residual scenario.^[27]^[31]

Residual Payments in Entertainment

Residual payments in entertainment, often simply called residuals, are supplementary compensations provided to actors, writers, directors, and other production contributors for the reuse of their work beyond its initial exhibition or broadcast. These payments recognize the enduring value of creative contributions in formats such as television reruns, syndication deals, home video releases, and digital streaming. Administered through union-managed funds, residuals ensure fair distribution based on contractual formulas, with SAG-AFTRA overseeing payments for performers and the Writers Guild of America (WGA) handling those for writers.^[37]^[38] The practice emerged in the 1950s amid television's rise, as studios repurposed theatrical films and produced new content for broadcast, leading SAG to secure initial agreements for performer payments on TV licenses starting with select studios like Monogram in 1948. Pivotal expansions occurred in 1960 through separate WGA (approximately 148 days) and SAG (42 days) strikes, which established profit participation for reuse of pre-1960 films and set precedents for future residuals on television and emerging media.^[39]^[40] The 2023 SAG-AFTRA strike, lasting 118 days, further modernized the system by addressing streaming inadequacies, introducing viewership-based bonuses and increasing overall residual contributions by $697.6 million over the contract term (2023-2026) to adapt to on-demand platforms, with annual wage increases of 7% in year one, 4% in year two, and 3.5% in year three as of November 2025.^[41] Residual structures differ by medium, platform, and union, typically calculated as percentages of license fees, distributor gross receipts, or fixed multiples of initial compensation, with payments due within specified timelines post-reuse. For network television reruns, WGA residuals for credited writers amount to 1.2% of the distributor's gross receipts, while SAG-AFTRA performer residuals are based on the performer's session fee adjusted by contract-specific factors for reruns and syndication. In streaming, the 2023 SAG-AFTRA agreement reformed high-budget subscription video-on-demand (SVOD) residuals to include a base rate of approximately 0.4% of the distributor's gross for qualifying features, plus bonuses for titles exceeding 100 million domestic viewing hours—up to 75% uplift for hits like Netflix series, scaled by subscriber tiers (e.g., 150% factor for major platforms) and higher for broadcast than video-on-demand. Foreign residuals are generally 35% of domestic amounts, and all structures prioritize principal cast and key creatives, with unions auditing compliance.^[42]^[43]^[44] A representative example (using approximate 2023 rates, adjusted annually) involves a principal performer on a network half-hour sitcom who receives around $6,000 in initial weekly compensation per episode; for reruns and syndication, earnings vary based on contract terms and can range from 20-50% of the session fee or more for popular shows, whereas on a streaming service, the same actor might receive base payments plus view-based bonuses totaling thousands more for globally popular seasons, reflecting platform-specific scaling.^[45]^[43]^[46]

Engineering and Physics

Residual Stress

Residual stress refers to self-equilibrating internal stresses that persist in a material or component after the removal of external loads and under uniform temperature conditions.^[47] These stresses arise from non-uniform plastic deformation or thermal expansion during manufacturing and are critical in engineering because they influence the structural integrity and performance of components.^[48] Residual stresses are classified into three types based on their spatial scale and influence: Type I (macroscopic) stresses, which act over large distances comparable to the component size and result from body forces or thermal gradients; Type II (microscopic) stresses, which operate at the scale of individual grains or phases and self-equilibrate across grain boundaries; and Type III (sub-microscopic or surface) stresses, which occur within grains due to dislocations or atomic-scale defects.^[49]^[50] This classification helps in understanding how residual stresses affect material behavior at different levels, from overall distortion to local crack initiation. Common causes of residual stress include manufacturing processes such as welding, machining, and heat treatment. In welding, non-uniform heating leads to localized expansion followed by contraction upon cooling, generating tensile stresses in the weld zone and compressive stresses in surrounding areas.^[51]^[52] Machining induces stresses through plastic deformation and friction at the surface, often resulting in tensile residual stresses that can alter part dimensions.^[53] Heat treatment processes, like quenching, create thermal gradients that cause differential expansion and contraction, leading to internal stresses if not properly controlled.^[54] Residual stresses are measured using techniques such as X-ray diffraction (XRD), which determines lattice strain non-destructively on the surface by analyzing diffraction peak shifts, providing high-resolution data for near-surface stresses.^[55] Another common method is the hole-drilling technique with strain gauges, a semi-destructive approach where a small hole is drilled into the material, and the resulting strain relaxation is measured using a rosette of strain gauges to calculate through-thickness stresses.^[56]^[57] The effects of residual stress can be detrimental or beneficial depending on their nature and distribution. Tensile residual stresses promote warping, cracking, and accelerated fatigue by increasing the effective stress intensity at crack tips and facilitating crack propagation under cyclic loading.^[58] In contrast, compressive residual stresses, such as those introduced by shot peening—a process that bombards the surface with spherical media to induce plastic deformation—enhance durability by retarding fatigue crack initiation and growth, thereby extending component life in high-stress applications.^[59]^[60] In aircraft components, such as turbine blades or structural joints, residual tensile stresses approaching or exceeding 200 MPa can significantly increase the risk of premature failure through mechanisms like stress corrosion cracking or fatigue, particularly in high-performance alloys.^[61] Mitigation strategies include stress-relief annealing, a heat treatment process that heats the material to a temperature below its critical range (typically 500–650°C for steels) to allow viscoelastic relaxation of stresses without altering the microstructure, often reducing residual stresses by up to 70–90% while preserving mechanical properties.^[62]^[54] This technique is widely applied post-welding or machining in aerospace to ensure reliability under operational loads.^[63]

Residual Current and Magnetism

In electrical engineering, residual current refers to the vector sum of currents in a circuit that is not zero, typically arising from an imbalance between the outgoing phase current and the returning neutral current due to leakage paths to ground or other faults. This fault current to ground can result from insulation failures, accidental contact, or equipment malfunctions, posing risks of electric shock or fire in power systems.^[64] The residual current is calculated as I_{\text{residual}} = I_{\text{phase}} - I_{\text{neutral}} in single-phase systems, where the device monitors the difference to detect anomalies.^[65] Residual current devices (RCDs), also known as ground fault circuit interrupters (GFCIs) in some regions, are protective switches that automatically disconnect the circuit when residual current exceeds a preset threshold, thereby preventing electrocution or equipment damage. These devices trip at sensitivities of 30 mA or less for personnel protection in household and similar installations, as specified in the international standard IEC 61008-1, which outlines requirements for RCDs without integral overcurrent protection.^[66] The standard mandates tripping within 300 ms at the rated residual operating current to ensure rapid response to faults.^[67] In practice, high-quality RCDs may activate as low as 15-18 mA for a 30 mA rating, providing an additional safety margin. For household circuits, residual currents above 5 mA often indicate leakage that could lead to perceptible shocks or hazards, prompting diagnostic checks even if below the RCD trip threshold; this level aligns with safety guidelines where ground fault protection devices interrupt at 5 mA to mitigate risks in wet or conductive environments.^[68] For example, in a typical residential setup, minor leaks from faulty appliances might produce 1-5 mA under normal conditions, but exceeding 5 mA signals a need for isolation and repair to prevent escalation.^[69] Residual magnetism, also termed remanence, describes the persistent magnetic flux density in ferromagnetic materials after the external magnetizing field is removed, stemming from the material's hysteresis behavior where magnetic domains align and retain orientation.^[70] This phenomenon is depicted in the B-H curve, a hysteresis loop plotting magnetic flux density B against magnetizing field strength H; the curve's intercept on the B-axis at H = 0 represents the remanence B_r, quantifying the retained flux.^[71] Coercivity, denoted as H_c, measures the reverse magnetic field intensity required to demagnetize the material and reduce B to zero, serving as a key indicator of a ferromagnet's resistance to demagnetization and its suitability for permanent magnet applications.^[72] In the B-H curve, H_c is the point where the descending branch crosses the H-axis, reflecting energy losses in domain wall motion and rotation during hysteresis cycles.^[73] Materials with high coercivity, such as neodymium-iron-boron alloys, exhibit strong residual magnetism, while soft magnetic materials like silicon steel have low coercivity for efficient flux reversal.^[74] In transformer applications, residual magnetism in the iron core can induce severe inrush currents—up to 10-20 times the rated value—upon re-energization due to flux saturation, potentially causing mechanical stress or relay maloperation.^[75] Demagnetization techniques mitigate this by applying a decaying alternating or polarity-reversing DC voltage to the windings, gradually reducing the core flux to near zero before operation; one established method uses an ultra-low-frequency source to control flux decay, ensuring residual levels below 5% of saturation. These procedures comply with utility standards to enhance reliability in power distribution.^[76] Residual magnetism also underpins data storage in hard disk drives, where microscopic ferromagnetic domains on platter surfaces are magnetized to represent binary bits (0 or 1), relying on high remanence for stable retention over years without power.^[77] For instance, in perpendicular magnetic recording used in modern drives, the material's coercivity exceeds 5,000 Oe to prevent unintended demagnetization from thermal fluctuations or stray fields, enabling densities over 1 Tb/in².^[78] This controlled hysteresis ensures data integrity, though secure erasure requires degaussing to disrupt remanence completely.

Geomorphology and Geology

Residual Landforms

Residual landforms, such as inselbergs and monadnocks, are isolated hills or mountains that persist as erosional remnants above surrounding lowlands after prolonged denudation has stripped away less resistant materials. These features represent the enduring cores of more extensive rock masses that resist breakdown, often rising abruptly from flat or gently undulating plains. In the Davisian cycle of erosion, developed by geomorphologist William Morris Davis in the late 19th century, residual landforms emerge during the mature to old age stages of landscape evolution, where base-level erosion reduces the terrain toward a peneplain, leaving behind these resistant outliers as the final topographic elements.^[79] Formation of residual landforms occurs primarily through differential weathering and erosion, where variations in rock resistance lead to the selective removal of surrounding softer strata, exhuming the harder cores over extended periods. Resistant lithologies, such as granite, sandstone, or quartzite, weather more slowly than adjacent weaker rocks, resulting in the isolation of these prominences. A classic example is Uluru (also known as Ayers Rock) in central Australia, an inselberg of coarse-grained arkosic sandstone formed from Precambrian sediments dating back approximately 550 million years, which has been exposed and shaped by episodic uplift and subsequent erosion events.^[80] The primary processes driving their development include mechanical exfoliation, where unloading from overlying material causes expansion and spalling of outer rock sheets parallel to the surface, and chemical weathering, which dissolves or alters minerals in less resistant rocks through hydrolysis and oxidation, facilitating their erosion by water and wind. These mechanisms preferentially degrade the pediments and regolith around the residuals, often under deep weathering mantles in tropical or subtropical environments, with the entire process unfolding over millions of years as evidenced by the ancient origins of many such features.^[81] Residual landforms hold significant value as proxies for past climates, recording episodes of intense chemical weathering linked to humid, tropical conditions that promoted deep regolith formation and subsequent stripping under drier regimes. For instance, the morphology and weathering profiles of inselbergs often reflect paleoenvironments with high rainfall and temperature, serving as witnesses to climatic shifts during the Cenozoic era.^[82]

Residual Soils

Residual soils are formed in situ through the chemical and physical weathering of underlying bedrock, without significant transportation of materials, distinguishing them from transported or alluvial soils. These soils develop directly atop the parent rock, retaining much of its mineralogical composition while undergoing transformation over time. For instance, laterites, a common type of residual soil, form from the intense weathering of basalt in tropical environments, resulting in iron- and aluminum-rich profiles that can reach depths of several meters.^[83]^[84] Key characteristics of residual soils include their increasing depth with prolonged weathering, often exhibiting a stratified profile with distinct horizons: the A-horizon near the surface, rich in organic matter and humus; the B-horizon below, where clays, iron oxides, and other materials accumulate through eluviation and illuviation; and the C-horizon transitioning to weathered bedrock. These soils typically have high clay content due to the breakdown of primary minerals into secondary clays like kaolinite, leading to fine textures and variable permeability. Their engineering properties, such as shear strength and compressibility, often decrease with depth as cementation effects diminish, which can pose challenges for foundation stability in construction, requiring site-specific investigations to assess load-bearing capacity and potential settlement.^[84]^[85]^[86] The formation of residual soils is influenced primarily by climate, parent rock type, and time, with topography and biota playing secondary roles. Tropical and subtropical climates with high rainfall and temperature promote deep chemical weathering, producing thick residual profiles up to 30 meters or more, whereas temperate climates yield shallower soils. The parent rock determines the soil's chemistry; for example, felsic rocks like granite produce sandy, quartz-dominated residuals, while mafic rocks like basalt yield clay-rich, fertile ones. In the United States Department of Agriculture (USDA) Soil Taxonomy system, residual soils are classified across various orders (e.g., Ultisols or Oxisols in tropical settings) based on diagnostic horizons, mineralogy, and degree of weathering, emphasizing their pedogenic evolution from the underlying bedrock.^[84]^[87] A prominent example is terra rossa, a red, clayey residual soil found in karst regions such as the Mediterranean and parts of the southern United States, formed by the dissolution of limestone bedrock, leaving behind insoluble iron oxides and clays as a residual mantle. These soils, often 0.5 to 2 meters thick, overlie karstic landscapes and exhibit high plasticity, which can lead to shrink-swell behavior under moisture changes, impacting foundation design by necessitating deeper pilings or soil stabilization to prevent differential settlement.^[88]^[89]

Popular Culture and Media

Music

In music, the term "residual" manifests in artist and band names, record labels, song titles, and lyrical themes denoting lingering sonic or emotional aftereffects. One prominent example is the American rock band Residual Kid, formed in 2009 in Austin, Texas, at Rock Camp USA, featuring Deven Ivy on vocals and guitar, Ben Redman on drums, and Max Redman on bass; the group draws from garage rock and indie influences, gaining recognition through their energetic live performances and debut EP Box (2011).^[90] Similarly, Residual Echoes, a psychedelic noise rock outfit from Santa Cruz, California, active since 2003 and involving key members like Adam Payne and Ethan Miller (of Comets on Fire), explores experimental soundscapes in their self-titled debut album released in 2004 on The Holy Mountain label, where tracks like "Slant" and "The Diamond Drops" evoke sonic "remainders" through layered feedback, echoing distortions, and improvised textures that mimic fading reverberations.^[91]^[92] Record labels bearing the name further highlight "residual" in music production. Residual Recordings, established in fall 1998 by producer and DJ Titonton Duvante in Columbus, Ohio, focuses on electronic, techno, and funky house genres, issuing influential releases such as Duvante's own Futurity EP (1998) and later reissues that revived late-1990s sounds, emphasizing subtle, persistent grooves and atmospheric builds.^[93]^[94] Song titles incorporating "residual" often address themes of persistence and aftermath. For instance, Allan Evans' Electronic Release 07 (2007) on Residual Recordings uses minimalist synth layers to convey echoing, unresolved tones typical of experimental ambient music.^[95] More mainstream, Chris Brown's R&B single "Residuals" from the 2024 deluxe edition of 11:11, co-written with Dewain Whitmore Jr. and others, delves into post-breakup motifs with lyrics portraying "residuals" as haunting emotional remnants ("All these residuals, they keep comin' back"), blending smooth production with introspective vulnerability; the official video was released on January 23, 2025.^[96] Such themes recur in indie and alternative lyrics, symbolizing enduring impacts like faded relationships or memories, as seen in Trash.'s punk-infused "Residual" (2022) from their album Ghosts, where raw energy underscores leftover turmoil.^[97] Culturally, "residual" in music appears in both niche indie, experimental, and underground electronic scenes as well as mainstream R&B, fostering explorations of impermanence and echo, as evidenced by Residual Echoes' role in Santa Cruz's early-2000s psych revival.^[98] This contrasts briefly with broader media contexts but underscores music's unique auditory emphasis on persistence.

Film and Television

In film and television, the term "residual" commonly evokes concepts of lingering supernatural remnants or psychological aftereffects, particularly within the horror genre as a narrative device for non-interactive hauntings. A residual haunting, theorized as an energetic imprint of past events replaying like a recording on locations or objects, has influenced storytelling since the 1972 BBC teleplay The Stone Tape, which popularized the Stone Tape theory of ghostly playback.^[99] Post-2010 productions have adapted this motif to explore themes of inescapable trauma and supernatural echoes, often in low-budget shorts and episodic formats that emphasize atmospheric repetition over direct confrontation.^[100] Notable examples include the 2015 short film Residual Haunting, directed by Glen Barnhart, which portrays a family's torment by the looped reenactment of a child's murder, trapped in temporal isolation without awareness of the living observers.^[101] Similarly, the 2014-2015 web series The Residuals, a mockumentary comedy created by Michael Paul Smith and Gillian Pensavalle, employs the term to depict the absurdities of commercial acting and auditions, satirizing the entertainment industry's hidden hierarchies and fleeting opportunities.^[102] These works highlight "residual" as a metaphor for overlooked or enduring presences in both literal and figurative senses. In production contexts, "residual effects" refer to visual techniques in VFX that simulate persistence and continuity, such as motion blur to mimic the human eye's retention of moving images. For instance, in action-heavy films like X2: X-Men United (2003), VFX teams used residual particle trails and blur simulations to convey Nightcrawler's teleportation, blending high-speed motion with realistic afterimages for immersive realism.^[103] Streaming-era horror series, such as those on Netflix, have increasingly integrated these VFX elements with residual haunting themes to heighten psychological tension, though traditional references often overlook such modern applications.

References

[1]
[PDF] EXAMINATION OF RESIDUALS
Residuals (zi) are calculated as Yi - Yi. Examination involves considering their empirical distribution and dependence on fitted values (Yi).
[2]
Residual Value - an overview | ScienceDirect Topics
Residual values refer to the difference between the observed and predicted values in a regression analysis, representing unexplained variation after fitting ...
[3]
The Examination and Analysis of Residuals - Taylor & Francis Online
Apr 30, 2012 · The Examination and Analysis of Residuals. F. J. Anscombe Princeton University and Bell Telephone Laboratories. & ... View PDF (open in a new ...
[4]
RESIDUAL Definition & Meaning - Merriam-Webster
Oct 26, 2025 · The meaning of RESIDUAL is remainder, residuum. How to use residual in a sentence.Missing: encyclopedia - | Show results with:encyclopedia -
[5]
4.4.4. How can I tell if a model fits my data?
The residuals from a fitted model are the differences between the responses observed at each combination values of the explanatory variables and the ...
[6]
What Is a Residual in Stats? - Outlier Articles
Feb 3, 2022 · A residual (or error) is the difference between the predicted value of your data and the actual value of your data.What Is A Residual? · Linear Model · Residuals And Linear...
[7]
A tutorial history of least squares with applications to astronomy and ...
This article surveys the history, development, and applications of least squares, including ordinary, constrained, weighted, and total least squares.
[8]
Gauss and the Invention of Least Squares - jstor
The most famous priority dispute in the history of statistics is that between Gauss and Legendre, over the discovery of the method of least squares.
[9]
9.3 - Identifying Outliers (Unusual Y Values) | STAT 462
The good thing about standardized residuals is that they quantify how large the residuals are in standard deviation units, and therefore can be easily used to ...
[10]
9.4 - Studentized Residuals | STAT 462
That is, a studentized residual is just a deleted residual divided by its estimated standard deviation (first formula). This turns out to be equivalent to the ...Missing: source | Show results with:source
[11]
4.1 - Residuals | STAT 462
The basic idea of residual analysis, therefore, is to investigate the observed residuals to see if they behave “properly.”
[12]
Key Assumptions of OLS: Econometrics Review - Albert.io
Jul 13, 2021 · The necessary OLS assumptions, which are used to derive the OLS estimators in linear regression models, are discussed below.
[13]
7 Classical Assumptions of Ordinary Least Squares (OLS) Linear ...
In this post, I cover the OLS linear regression assumptions, why they're essential, and help you determine whether your model satisfies the assumptions.
[14]
3.3 Residual diagnostics | Forecasting: Principles and Practice (2nd ...
Residuals are what's left after fitting a model, used to check if a model captured data well. Good residuals are uncorrelated and have zero mean.
[15]
4 Linear Regression for Continuous Outcomes
It looks like at least one of our residuals is pretty large. Residuals of $y=1.2x + 5$ for our 10 observations. Figure 4.5: Residuals of ...
[16]
Testing for Serial Correlation in Least Squares Regression: I - jstor
410 Testing for serial correlation in least squares regression. I. The bounds are applicable to all cases in which the independent variables in the ...
[17]
A Simple Test for Heteroscedasticity and Random Coefficient Variation
BY T. S. BREUSCH AND A. R. PAGAN. A simple test for heteroscedastic disturbances in a linear regression model is developed using the framework of the ...
[18]
Detection of Influential Observation in Linear Regression - jstor
Dennis Cook. Department of Applied Statistics. University of Minnesota. St. Paul, Minnesota 55108. A new measure based on confidence ellipsoids is developed for ...
[19]
Robust Regression: Asymptotics, Conjectures and Monte Carlo
Huber. "Robust Regression: Asymptotics, Conjectures and Monte Carlo." Ann. Statist. 1 (5) 799 - 821, September, 1973. https://doi.org/10.1214/aos/1176342503 ...
[20]
Regression Quantiles - jstor
REGRESSION QUANTILES'. BY ROGER KOENKER AND GILBERT BASSETT, JR. A simple minimization problem yielding the ordinary sample quantiles in the location model ...
[21]
How to Value a Company Using the Residual Income Method
Residual income is the income left after accounting for the cost of capital. It provides a way to estimate a firm's intrinsic value using financial statements.The Residual Income Concept · Using Residual Income
[22]
Economic Value Added (EVA) - Definition, Formula
EVA adopts almost the same form as residual income, and the EVA formula can be expressed as follows: EVA = NOPAT – (WACC * Invested Capital). Where: NOPAT = Net ...
[23]
Residual Income Valuation | CFA Institute
Conceptually, residual income is net income less a charge (deduction) for common shareholders' opportunity cost in generating net income.Introduction · Learning Outcomes · Summary
[24]
EVA as Superior Performance Measurement Tool
Economic Value Added (EVA) was introduced and advocated by Stern Stewart and Co. in 1982. This study intended to identify why EVA should be used as financial ...
[25]
Residual Income: A Superior Alternative to ROI for Measuring ...
Apr 24, 2024 · Unlike ROI, which may deter managers from pursuing high-capital projects with lower initial returns, RI emphasizes absolute profit generation.
[26]
Residual Income: What It Is, Types, and How to Make It - Investopedia
In personal finance, residual income is synonymous with monthly disposable income. It is the total income that remains after paying all monthly debts. Thus, ...What Is Residual Income? · How It Works · Types · How to Generate Residual...
[27]
7 Of The Best Ways To Build Residual Income | Bankrate
Aug 14, 2024 · Residual income has several meanings but in personal finance, it is what's left after you've paid all of your bills and other expenses.
[28]
Residual Value Explained, With Calculation and Examples
Residual value is an asset's estimated worth at the end of its useful life. · Depreciation calculations and lease terms are based on residual value.
[29]
Residual Value: Meaning, Examples, How to Calculate
Mar 23, 2025 · Residual value, also referred to as salvage value, is the estimated remaining worth of an asset at the end of its expected useful life. It ...
[30]
Straight Line Depreciation - Formula, Definition and Examples
It is calculated by simply dividing the cost of an asset, less its salvage value, by the useful life of the asset.
[31]
Straight Line Depreciation | Formula + Calculator - Wall Street Prep
Dec 28, 2023 · To calculate the straight line depreciation rate for a fixed asset, subtract the salvage value from the asset cost to compute the total ...
[32]
What Is Residual Value? (Plus How to Calculate It) - Car and Driver
Jul 12, 2023 · A vehicle's residual value is its total value at the end of the lease or the amount you'll pay to purchase it at the conclusion of its lease term.
[33]
Is a Car Lease Residual Based on MSRP or Sales Price? | RefiJet
May 31, 2025 · What is the meaning of residual value? ... The residual value is the estimated worth of a leased vehicle at the end of the lease term. It's ...
[34]
Residual Value and Balloon Payments Explained - Finlease
Sep 17, 2025 · A Residual Value and Balloon Payments are the same. Both refer to a pre-agreed payment due at the end of a loan for a vehicle or machine.
[35]
Residual Value in Equipment Leasing: How It's Calculated & Why It ...
Feb 26, 2025 · Residual value, also referred to as the salvage value, represents the estimated fair market value of equipment at the end of its lease period or useful life.
[36]
[PDF] 2024 Publication 946 - IRS
Apr 15, 1976 · discusses the rules for determining the depreciation de- duction ... Salvage value: An estimated value of property at the end of its ...
[37]
26 CFR § 1.167(f)-1 - Reduction of salvage value taken into account ...
A taxpayer may reduce the amount taken into account as salvage value in computing the allowance for depreciation under section 167(a) with respect to “personal ...<|control11|><|separator|>
[38]
SAG-AFTRA TV and Theatrical Residuals Quick Guide
Jan 28, 2021 · What is a residual? Residuals are additional compensation paid to performers when a production is shown beyond the original use covered by ...
[39]
What are residuals and how do they work? A strike explainer
Sep 20, 2023 · Residuals are payments a writer, actor or director can earn when their work gets reused: for instance, if a movie featuring them airs on cable or a show they ...
[40]
1950s | SAG-AFTRA
Residuals: Monogram Studios makes agreement with SAG to pay residuals on 70 feature films made since August 1, 1948, licensed for TV – members told “Guild Sets ...
[41]
Residuals Are Key to Nearly Every Strike in Hollywood History
videocassettes, basic cable, premium cable, the internet, and now ...
[42]
2023 TV/Theatrical Contracts | SAG-AFTRA
Nov 10, 2023 · This $1.014 billion includes an increase of $697.6 million to be paid to performers in the form of wages and residuals over the term of the ...
[43]
Residuals Survival Guide - WGA
The residuals due the credited writer are 1.2% of the distributor's gross receipts, except for residuals due for DVD/videocassette usage, which are calculated ...<|control11|><|separator|>
[44]
What Are Residuals? How Much Actors Make in Royalties | Backstage
Jun 13, 2024 · For television, SAG-AFTRA pays residuals to principal performers for episode reruns starting with the program's second broadcast or transition ...
[45]
New Streaming Residual Model For WGA & SAG-AFTRA Explained
Nov 30, 2023 · For SAG-AFTRA, any title that surpasses the viewership threshold will trigger a payment to each principal performer equivalent to 100% of the ...
[46]
[PDF] Streaming Residuals Gains - sag-aftra
High Budget SVOD shows that qualify for the streaming bonus will see a 75% increase to the residuals due for any exhibition year that the show qualifies. Page 2 ...
[47]
[PDF] INTRODUCTION TO RESIDUAL STRESS - shotpeener.com
Residual stresses are self-equilibrating stresses existing in materials or components under uniform temperature conditions. It is well established that.
[48]
Types of Residual Stress - Proto Manufacturing
There are three types of residual stress: Type I (macro), Type II (micro), and Type III (sub-micro), based on the scale of the stress.Missing: engineering definition
[49]
[PDF] A Review of Residual Stress Measurement Methods
Type III are micro residual stresses that exist within a grain, essentially as a result of the presence of dislocations and other crystalline defects. Types II ...
[50]
Residual Stress Measurement Techniques for Metal Joints, Metallic ...
Dec 27, 2022 · Long range stresses arise from macrostrains and are classified as Type I. Type II refers to stresses that self-equilibrate over a length ...
[51]
Residual Stress in Weld Joints - Pulstec USA, Inc.
Oct 8, 2024 · Heating during welding causes expansion, followed by contraction, as the welding torch withdraws. Stress only starts to build in the metal after ...
[52]
Welding Deformation and Residual Stress Prevention - GORDAK
Aug 24, 2023 · One of the primary causes is the non-uniform heating and cooling during the welding process. When the welding torch applies heat to the metal, ...
[53]
Induced Residual Stress - an overview | ScienceDirect Topics
Induced residual stresses are internal stresses in a material, existing without external loads, caused by processes like machining, and can lead to deformation.
[54]
Heat Treatment: Stress Relief - Eagle Group Blog
Feb 6, 2024 · Stress relieving is a heat treatment that eliminates residual stresses while ensuring the material's microstructure and mechanical properties ...
[55]
Residual Stress Measurement + Testing - Lambda Technologies
X-Ray Diffraction (XRD) is the method of choice for the highest resolution measurement of surface and near-surface residual stress, where most stress-related ...
[56]
[PDF] Measurement of Residual Stresses by the Hole-Drilling* Strain Gage ...
The hole-drilling method involves installing a strain gage, drilling a small hole, and measuring relaxed strains to calculate residual stresses.
[57]
Hole drilling method for residual stress measurement SONATS
The hole drilling method, also called incremental hole-drilling strain-gauge method, is a semi-destructive method used to measure residual stresses. It is fast ...
[58]
Effect of Residual Stresses on Fatigue Crack Growth - NIH
Mar 15, 2022 · Residual stresses affect the fatigue behavior, given that compressive stresses delay the phenomenon, while tensile stresses accelerate it.
[59]
Effect of shot peening residual stresses and surface roughness on ...
SP can induce work hardening and a compressive residual stress on the surface of metal materials, which suppresses the initiation of fatigue crack [12–14].Shot Peening Process · Fretting Fatigue Fem Model · Fretting Fatigue Life Model
[60]
[PDF] SHOT PEENING AND ITS IMPACT ON FATIGUE LIFE OF ...
The high compressive residual stress may enhance the crack propagation life by decreasing the effective stress intensity range. Optimization of Peening ...
[61]
Residual stress and its role in failure - IOP Science
Nov 27, 2007 · There are many ways in which failure can occur including brittle fracture, plastic collapse, fatigue, creep and stress corrosion cracking. While ...Missing: aircraft | Show results with:aircraft
[62]
Why is the Post-Weld Heat Treatment Process Important for Stress ...
Residual stress can cause distortion and cracking in the material, potentially resulting in the failure of the welded structure. PWHT can help to reduce stress ...<|separator|>
[63]
Review of Residual Stress Modification Techniques for Extending ...
The induced residual stresses are mainly effective in retarding surface fatigue crack formation and early crack growth.Cold Expansion · Shot Peening · Laser Shock PeeningMissing: warping | Show results with:warping
[64]
Behavior of Residual Current Devices at Earth Fault Currents ... - NIH
Nov 1, 2022 · An RCD is a protective device, the use of which is recommended for protection against electric shock, and in some installations its use is even ...Missing: definition | Show results with:definition
[65]
Electrical Safety | Division of Safety and Risk Services
It works by comparing the amount of current going to an electric device against the amount of current returning from the device along the circuit conductors.
[66]
IEC 61008-1:2024
Nov 21, 2024 · IEC 61008-1:2024 gives general requirements and tests for residual current operated circuit-breakers without integral overcurrent protection for household and ...
[67]
[PDF] IEC 61008-1:2010 - iTeh Standards
This consolidated version of the official IEC Standard and its amendments has been prepared for user convenience. IEC 61008-1 edition 3.2 contains the third ...
[68]
[PDF] Construction Focus Four: Electrocution Hazards - OSHA
GFCIs monitor the current-to-the- load for leakage to ground. When this leakage exceeds 5 mA ± 1. mA, the GFCI interrupts the current. They are rated to trip.Missing: threshold | Show results with:threshold
[69]
[PDF] Electrical Safety. Safety and Health for Electrical Trades ... - CDC
GFCIs are set at about 5 mA and are designed to protect workers from electrocution. GFCIs are able to detect the loss of current resulting from leakage through ...<|control11|><|separator|>
[70]
The Hysteresis Loop - Magnetism - NDE-Ed.org
A hysteresis loop shows the relationship between the induced magnetic flux density (B) and the magnetizing force (H). It is often referred to as the BH loop.
[71]
Magnetic Hysteresis Loop including the B-H Curve
A set of curves called Magnetisation Curves, Magnetic Hysteresis Curves or more commonly BH Curves for each type of core material used as shown below.
[72]
HysteresisLoop
It is often referred to as the B-H loop. An example hysteresis loop is shown below. The loop is generated by measuring the magnetic flux B of a ferromagnetic ...
[73]
[PDF] Chapter 2 - Magnetic Materials & Their Characteristics
The remanence flux is the polarized flux remaining in the core after the excitation has been removed. The magnetizing force, -Hc, is called coercivity.
[74]
Coercivity and Remanence in Permanent Magnets - HyperPhysics
The lack of retraceability of the magnetization curve is the property called hysteresis and it is related to the existence of magnetic domains in the material.Missing: definition | Show results with:definition
[75]
Elimination of Residual Flux in Transformers by the Application of an ...
Dec 9, 2014 · The purpose of core demagnetization is twofold: 1) to reduce the inrush currents when transformers are energized; and 2) to make sure that the frequency- ...
[76]
Flux-Controlled Methods for Demagnetization of Transformer Iron Core
In this paper, two flux-controlled strategies for demagnetization of the transformer iron core are compared by experiments on a 500 kV, 334 MVA single-phase ...
[77]
A Guide to Understanding Data Remanence in Automated ...
Proper degaussing will ensure that there is insufficient magnetic remanence to reconstruct the data.
[78]
[PDF] Secure Deletion of Data from Magnetic and Solid-State Memory
Degaussing is a reasonably effective means of purging data from magnetic disk media, and will even work through most drive cases ... This is important in hard ...
[79]
Models of Slope Development: Davis, Penck, King Wood and Strahler
The models of slope development discussed are: Davis’ Model of Cycle of Erosion, Penck’s Model of Landform Development, King’s Model of Slope Development, Alan ...
[80]
Uluru (Ayers Rock) and Kata Tjuta (The Olgas): Inselbergs of Central ...
Uluru and Kata Tjuta are inselbergs standing in isolation in the desert plains of central Australia. Uluru is a beveled bornhardt shaped steeply dipping ...
[81]
Inselbergs - A biography of the Australian continent
Jun 1, 2011 · Water in contact with the well-fractured compartment leads to weathering resulting in the formation of a regolith that is easily eroded.<|separator|>
[82]
Modeling and mapping of Inselberg habitats for environmental ...
The present study evaluated the performance of machine learning algorithms in the mapping of Inselbergs in different environments.
[83]
(PDF) The ecological and evolutionary dynamics of inselbergs
Oct 1, 2024 · In this review, we propose a biological definition for inselbergs that captures three fundamental characteristics of inselbergs from the perspective of biota.
[84]
[PDF] Chemical Weathering of Basalts and Andesites
Most previous studies of basalt and andesite weathering have dealt with well-developed residual soils on highly altered bedrock in tropical to semitropical ...
[85]
5.4 Weathering and the Formation of Soil – Physical Geology
Soils are described as residual soils if they develop on bedrock, and transported soils if they develop on transported material such as glacial sediments.Missing: definition | Show results with:definition
[86]
Soil Facts | Natural Resources Conservation Service
These “residual” soils have the same general chemistry as the original rocks. More commonly, soils form in materials that have moved in from elsewhere.
[87]
[PDF] Design of Temporary Slopes and Excavations in Residual Soils
The components of this research project included: field and laboratory testing program, evaluation and development of predictive models, and analysis of cut ...Missing: implications | Show results with:implications
[88]
[PDF] Soil Survey Manual 2017; Chapter 3
chapter provides standards and guidelines for describing the soil. It contains standard technical terms and their definitions for most soil properties and ...
[89]
[PDF] the mystery of - Western Kentucky University
One theory was that the red soil resulted from the dissolution of limestone and is simply an insoluble residue or leftover from that process. Another theory was ...
[90]
[PDF] Hydrogeology of the Karst of Puerto Rico
Permeability is found to decrease exponentially with stratigraphic depth. Except for lake waters resting on terra rossa, most waters of the limestone belt are ...
[91]
Residual Kid - Wikipedia
Residual Kid is an American rock band from Austin, Texas formed in 2009 at rock camp USA. The group consists of Deven Ivy (vocals, guitar), Ben Redman (drums), ...
[92]
https://residualechoes.bandcamp.com/album/residual-echoes
[93]
Residual Echoes - Bandcamp
Free deliveryResidual Echoes by Residual Echoes, released 01 January 2003 1. Slant 2. The Diamond Drops 3. Fish Don't Swim 4. This Is Not A Start 5. A Start 1 and 2 6.Missing: 1998 | Show results with:1998
[94]
https://ra.co/news/39201
[95]
Titonton Duvante launches Residual Classic · News RA
Jun 7, 2017 · It'll be back on wax on late July. Duvante will then give new life to another of his 1998 efforts—Futurity, a four-track EP for the long-defunct ...
[96]
electronic release 07 | Allan Evans | RESIDUAL RECORDINGS
released January 1, 2007. © all rights reserved. tags. alternative. about. RESIDUAL RECORDINGS image. RESIDUAL RECORDINGS. Following Unfollow Follow. music ...
[97]
Trash. - "Residual" Official Music Video - A BlankTV World Premiere!
Oct 31, 2022 · .com/artist/55Z9fGm3VLYvRTrU0PYlL2 BandCamp: https://officialtrashband.bandcamp.com/album/downer Trash. - "Residual" Official Music Video ...
[98]
Music writer Paul M. Davis looks back on a decade of Santa Cruz ...
Dec 9, 2009 · The response of bands such as the New Thrill Parade and the celestial noise-rock band Residual Echoes was to turn it up even louder, tossing ...
[99]
Stone Tape Theory: What Is "Residual Haunting"? - The Lineup
May 9, 2024 · A Christmas Ghost Story of Residual Haunting ... On Christmas day, 1972, BBC Two broadcasted their annual ghost story, “The Stone Tape” by Nigel ...
[100]
https://www.spiritshack.co.uk/blog/ghost-hunting/residual-hauntings/
[101]
RESIDUAL HAUNTING (Short Film by Glen Barnhart) - YouTube
Oct 19, 2015 · For 10 years a family is left wondering about "Lisa", the child that was murdered, and is now stuck in the time of her death.Missing: 2015 aftermaths
[102]
Pensavalle, Smith Chat THE RESIDUALS in New Backstage Article
Jul 29, 2015 · In June, BroadwayWorld launched Season 2 of the popular webseries THE RESIDUALS. The comedy, from husband-and-wife team Michael Paul Smith ...Missing: web mockumentary
[103]
The Effect of Creativity | Computer Graphics World
Mar 15, 2021 · This isn't smoke to mask the Nightcrawler – the particles are colored the same as the Nightcrawler and have the residual motion of him. This ...