Multivariate Curve Resolution
Multivariate Curve Resolution - Related Videos
06:35
Basics of Multivariate Analysis in Neuroimaging Data
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Cited by 37 •
2010
The current article describes the basics of multivariate analysis and contrasts it to the more commonly used voxel-wise univariate analysis. Both types of analysis are applied to a clinical-neuroscience data set. Supplementary split-half simulations show better replication of the multivariate results in independent data sets.
13:51
Cross-Modal Multivariate Pattern Analysis
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Cited by 5 •
2011
Classical multivariate pattern analysis predicts sensory stimuli a subject perceives from neural activity in the corresponding cortices (e.g. visual stimuli from activity in visual cortex). Here, we apply pattern analysis cross-modally and show that sound- and touch-implying visual stimuli can be predicted from activity in auditory and somatosensory cortices, respectively.
01:25
Limits of Multivariable Functions
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2026
Limits of multivariable functions describe how a function behaves as its input approaches a particular point in the plane. In single-variable calculus, a limit examines the behavior of a function as the input approaches a number from two directions along a line. For functions of two variables, the situation is more complex because the input can approach a point from infinitely many paths in the xy-plane. A limit exists only when the function approaches the same value along every possible...
01:30
Multivariable Functions and Higher Derivatives
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2026
A multivariable function assigns a single output value to each ordered set of independent inputs, thereby defining a surface in three-dimensional space. For a function f(x, y), each point (x, y) corresponds to a height z = f(x, y). This geometric interpretation allows systematic analysis of how the output varies as multiple variables change simultaneously. Such functions frequently arise in physical models and optimization problems, where system behavior depends on several interacting...
01:29
Multivariable Chain Rule
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2026
When a variable z depends on two intermediate variables, x and y, and both x and y vary with respect to a third variable t, the dependence of z on t is indirect. Although t does not explicitly appear in the expression for z, any change in t produces corresponding changes in x and y, which in turn alter the value of z. The objective is to determine the total derivative of z with respect to t, denoted as dz/dt.Assuming that all functions involved are differentiable, the total change in z can be...
FAQs
In the MCR model, one factor describes how each component varies across samples or measurement conditions, while the other describes that component’s signal response across measured variables. Their product reconstructs the observed data matrix, and the residual records the portion not explained by the estimated components. This separation lets analysts interpret overlapping measurements through chemically meaningful profiles.
Constraints guide the solution toward profiles that are chemically plausible rather than mathematically interchangeable alternatives. Nonnegativity prevents negative concentrations or responses when those are inappropriate; closure can impose a total-composition relationship; and selectivity can use regions associated with particular components. Applying these conditions when justified reduces ambiguity and helps connect resolved profiles with actual chemical behavior.
Iteration allows the estimated concentration profiles and pure component responses to be repeatedly adjusted so that they better represent the measured data while satisfying selected constraints. The residual term preserves the part of the signal not captured by the component model. Examining this separation helps distinguish modeled chemical contributions from unexplained variation in a complex measurement.
A typical MCR analysis begins with measurements organized in a data matrix. The method estimates concentration and response profiles, then repeatedly refines them while applying appropriate constraints such as nonnegativity, closure, or selectivity. The resulting component profiles are used to reconstruct the measured matrix, with the residual representing the information not accounted for by the estimated contributions.
MCR is particularly useful when spectroscopy or chromatography produces strongly overlapping peaks that are difficult to interpret directly. It can separate estimated contributions from individual components even when complete prior separation is unavailable. This makes the approach valuable for examining multicomponent measurements, where conventional inspection of the combined signal may not clearly reveal the participating chemical species.
The resolved profiles can support qualitative identification and quantitative analysis by linking signal patterns with estimated component concentrations. Because the profiles can be examined across changing experimental conditions, MCR also helps follow chemical reactions and process streams. In this way, the method connects data decomposition with chemical structure and the behavior of multicomponent systems.