Multivariable Dynamic Calculus on Time Scales
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A discrete signal or discrete-time signal is a time series consisting of a sequ. In mathematics, dynamic equation can refer to: difference equation in discrete time differential equation in continuous time time scale calculus in combined discrete and continuous time. AP Calculus AB covers limits, derivatives, and integrals. AP Calculus BC covers all AP Calculus AB topics plus additional topics including more integration techniques such as integration by parts, Taylor series, parametric equations, polar coordinate functions, and curve interpolations.
Some schools do this, though many others only require precalculus as a prerequisite for Calculus BC. The AP awards given by College Board count both exams. However, they do not count the AB sub-score piece of the BC exam. Scaling and root planing, also known as conventional periodontal therapy, non-surgical periodontal therapy, or deep cleaning, is a procedure involving removal of dental plaque and calculus scaling or debridement and then smoothing, or planing, of the exposed surfaces of the roots, removing cementum or dentine that is impregnated with calculus, toxins, or microorganisms, the etiologic agents that cause inflammation.
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This matrix makes it impossible to remove the plaque by rinsing or using sprays. Materia alba is similar to plaque but it lacks the organized structure of plaque and hence ea. A definite integral of a function can be represented as the signed area of the region bounded by its graph. In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total. The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may also refer to the related notion of the an. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. Today, the term "finite difference" is often taken as synonymous with finite difference approximations of derivatives, especially in the context of numerical methods. Finite differences were introduced by Brook Taylor in and have also been studied as abstract self-standing mathematical objects in works by George Boole , L. Finite differences trace their origins back t. Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits.
It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while q stands for quantum. The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.
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The derivative of a function of a real variable measures the sensitivity to change of the function value output value with respect to a change in its argument input value. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of. In mathematics, a summation equation or discrete integral equation is an equation in which an unknown function appears under a summation sign.
The theories of summation equations and integral equations can be unified as integral equations on time scales using time scale calculus. A summation equation compares to a difference equation as an integral equation compares to a differential equation. References Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains, Tomasia Kulik, Christopher C.
Tisdell, September 3, Summation equations or discrete integral equations. In mathematics, a multiplicative calculus is a system with two multiplicative operators, called a "multiplicative derivative" and a "multiplicative integral", which are inversely related in a manner analogous to the inverse relationship between the derivative and integral in the classical calculus of Newton and Leibniz.
The multiplicative calculi provide alternatives to the classical calculus, which has an additive derivative and an additive integral. Infinitely many non-Newtonian calculi are multiplicative, including the geometric calculus and the bigeometric calculus discussed below. Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation.
Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change. Discrete calculus has two entry points, differential calculus and integral calculus. Differential calculus concerns incremental rates of change and the slopes of piece-wise linear curves. Integral calculus concerns accumulation of quantities and the areas under piece-wise constant curves. These two points of view are related to each other by the fundamental theorem of discrete calculus. The study of the concepts of change starts with their discrete form.
Analysis on fractals or calculus on fractals is a generalization of calculus on smooth manifolds to calculus on fractals. The theory describes dynamical phenomena which occur on objects modelled by fractals. It studies questions such as "how does heat diffuse in a fractal? This turns out not to be a full differential operator in the usual sense but has many of the desired properties.
There are a number of approaches to defining the Laplacian: probabilistic, analytical or measure theoretic. See also Time scale calculus for dynamic equations on a cantor set. Fractal Geometry and Stochastics II. In mathematics, the Laplace transform is an integral transform named after its inventor Pierre-Simon Laplace. It transforms a function of a real variable t often time to a function of a complex variable s complex frequency. The transform has many applications in science and engineering. The Laplace transform is similar to the Fourier transform. While the Fourier transform of a function is a complex function of a real variable frequency , the Laplace transform of a function is a complex function of a complex variable.
A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s.
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Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series ex. Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics — such as integers, graphs, and statements in logic — do not vary smoothly in this way, but have distinct, separated values. Discrete objects can often be enumerated by integers.
More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets finite sets or sets with the same cardinality as the natural numbers. However, there is no exact definition of the term "discrete mathemat. In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable.
In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form. The Laplace operator is named after the French mathematician Pierre-Simon de Laplace — , who first applied the operator to the study of celestial mechanics, where the operator gives a constant multiple of the mass density when it is applied to the gravitational potential due to the mass distribution with that given density. Solutions of th. In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of the preceding terms.
The term difference equation sometimes and for the purposes of this article refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation. Definition A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. For any. An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right.
Coq is an interactive theorem prover. It allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq is not an automated theorem prover but includes automatic theorem proving tactics and various decision procedures. Overview Seen as a programming language, Coq implements a dependently typed functional programming language, while seen as a logical system, it im.
Control theory in control systems engineering is a subfield of mathematics that deals with the control of continuously operating dynamical systems in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability. To do this, a controller with the requisite corrective behaviour is required.
This controller monitors the controlled process variable PV , and compares it with the reference or set point SP. The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability.
On this is based the advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries. This is feedback. The gradient, represented by the blue arrows, denote the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white low to dark high.
In vector calculus, the gradient is a multi-variable generalization of the derivative. This is a glossary of terms that are or have been considered areas of study in mathematics. A Absolute differential calculus: the original name for tensor calculus developed around Absolute geometry: an extension of ordered geometry that is sometimes referred to as neutral geometry because its axiom system is neutral to the parallel postulate. Abstract algebra: the study of algebraic structures and their properties.
Originally it was known as modern algebra. Abstract analytic number theory: a branch of mathematics that takes ideas from classical analytic number theory and applies them to various other areas of mathematics. Abstract differential geometry: a form of differential geometry without the notion of smoothness from calculus. Instead it is built using sheaf theory and sheaf cohomology. Abstract harmonic analysis: a modern branch of harmonic analysis that extends upon the generalized Fourier transforms that can be defined on locally compact groups.
Abstract homotopy theory: a part. Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermodynamic equilibrium there are no net macroscopic flows of matter or of energy, either within a system or between systems. In a system that is in its own state of internal thermodynamic equilibrium, no macroscopic change occurs.
Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative equilibria. Systems can be in one kind of mutual equilibrium, though not in others. In thermodynamic equilibrium, all kinds of equilibrium hold at once and indefinitely, until disturbed by a thermodynamic operation.
In a macroscopic equilibrium, perfectly or almost perfectly balanced microscopic exchanges occur; this is the physical explanation of the notion of macroscopic equilibrium. A thermodynamic system in a state of int. A solution to a discretized partial differential equation, obtained with the finite element method. In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts.
This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable creating a dichotomy for modeling purposes, as in binary classification.
Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, as when multiple discrete variables are aggregated or multiple discrete categories fused.
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The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely intimately upon the structure of the real number line. Ordinary trigonometry studies triangles in the Euclidean plane R2.
There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations, definitions using functional equations. Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space.
A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and n-simplices. Trigonometry In spherical trigonometry, triangles on the surface of a sphere are studied. The spherical triangle identities are written in terms of the ordinary trigonometric.
Network calculus is "a set of mathematical results which give insights into man-made systems such as concurrent programs, digital circuits and communication networks. As traffic flows through a network it is subject to constraints imposed by the system components, for example: link capacity traffic shapers leaky buckets congestion control background traffic These constraints can be expressed and analysed with network calculus methods. Constraint curves can be combined using convolution under min-plus algebra. Network calculus can also be used to express traffic arrival and departure functions as well as service curves.
The calculus uses "alternate algebras He was born in Bijapur in Karnataka. He has been called the greatest mathematician of medieval India. This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations. The fluxion of a "fluent" a time-varying quantity, or function is its instantaneous rate of change, or gradient, at a given point.
Newton introduced the concept in and detailed them in his mathematical treatise, Method of Fluxions. A fluent is a time-varying quantity or variable. The derivative of a fluent is known as a fluxion, the main focus of Newton's calculus. A fluent can be found from its corresponding fluxion through integration. Henry Woodfall; and sold by John Nourse. Retrieved 6 March The log-lin type of a semi-log graph, defined by a logarithmic scale on the y-axis, and a linear scale on the x-axis.
The lin-log type of a semi-log graph, defined by a logarithmic scale on the x axis, and a linear scale on the y axis. In science and engineering, a semi-log graph or semi-log plot is a way of visualizing data that are related according to an exponential relationship. One axis is plotted on a logarithmic scale. This kind of plotting method is useful when one of the variables being plotted covers a large range of values and the other has only a restricted range — the advantage being that it can bring out features in the data that would not easily be seen if both variables had been plotted linearly.
In science and engineering, a log—log graph or log—log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Thus these graphs are very useful for recognizing these relationships and estimating parameters. In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.
Derivatives in analysis In real, complex, and functional analysis, derivatives are generalized to functions of several real or complex variables and functions between topological vector spaces. An important case is the variational derivative in the calculus of variations. Repeated application of differentiation leads to derivatives of higher order and differential operators. Multivariable calculus The derivative is often met for the first time as an operation on a single real function of a single real variable.
One of the simplest settings for generalizations is to vector valued functions of several variables most often the domain forms a vector space as well. This is the field of multivariable calculus. In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.
Consequently, such brain activity is continuously changing whether or not one is focusing on an externally imposed task. Previously, we have introduced an analysis method that allows us, using Hidden Markov Models HMM , to model task or rest brain activity as a dynamic sequence of distinct brain networks, overcoming many of the limitations posed by sliding window approaches. Here, we present an advance that enables the HMM to handle very large amounts of data, making possible the inference of very reproducible and interpretable dynamic brain networks in a range of different datasets, including task, rest, MEG and fMRI, with potentially thousands of subjects.
We anticipate that the generation of large and publicly available datasets from initiatives such as the Human Connectome Project and UK Biobank, in combination with computational methods that can work at this scale, will bring a breakthrough in our understanding of brain function in both health and disease. Understanding the nature of temporal dynamics of brain activity at a range of temporal and spatial scales is an important challenge in neuroscience.
When studying task data, the aim is to discover the neural underpinnings and brain mechanisms elicited by the task, for which one relates the time course of the measured data to behaviour as comprehensively as possible. That is to say, we are interested in the dynamics evoked by the task. In this case, many repetitions of the same task are typically considered in the hope of characterising and interpreting the differences with respect to some baseline condition. Presumably, the brain adapts to the task at different time scales and in an online fashion, and we would like to capture these changes at as high a temporal resolution as the imaging modality will allow.
When studying rest data, where the brain is not engaged in a predefined task, the brain will still process information dynamically, adapting its activity to the current perception of the environment combined with the products of its own spontaneous activity. In this case, then, we are interested in characterising the spontaneous dynamics. In particular, they need a pre-specification of the time scale at which the neural processes of interest occur, i. This choice is crucial and is a trade-off between two conflicting criteria: too long a window will miss fast dynamics, whereas too short a window will have insufficient data to provide a reliable network estimation.
The HMM can be applied to task data to provide a rich description of the brain dynamics; for example, by estimating the HMM in a completely unsupervised way i. This allows for the analysis of how certain dynamic properties vary across subjects, such as the transition probabilities between states or the differences of state occupancies i.
An illustration of the HMM in both rest and task is presented in Fig. Scheme of HMM working on rest a and rest b. In both cases, the HMM estimates several brain networks or states that are common to all subjects or trials, together with a specific state time courses for each subject which indicates when each state is active. In task, we can compute the state mean activation locked to the behavioural event, producing a state evoked response , which corresponds to a time-course of the proportion of trials for which subjects are in each state.
In the context of the HMM, increasing the amount of data can help to achieve richer and more robust conclusions about the dynamic nature of brain activity. In task, for example, having more trials will allow us to have a better understanding of the timing of brain activity in relation to the task and its trial-by-trial variability which is due in part to noise but also to interesting cognitive processes such as learning. However, group-level HMMs run on data temporally concatenated over all subjects are computationally expensive to train on such massive data sets. This problem is exacerbated if we use more complex HMM observation models i.
In this paper, we propose an alternative to the standard HMM that uses a stochastic variational inference approach that can be applied to very large neuroimaging data sets, by greatly reducing its computational cost. The algorithm is generally applicable to the different instantiations of the HMM framework that are required for different data modalities. In the hope that it will be useful to other researchers, a Matlab toolbox implementing the algorithm has been publicly released.
Altogether, we use these examples along with simulated data to demonstrate that having a suitable computational method that scales well to large amounts of data can significantly enrich our description of dynamic brain activity. The HMM is a family of models that can describe time series of data using a discrete number of states, all having the same probabilistic distributions but each having different distribution parameters. Thus, the states correspond to unique patterns of brain activity that recur in different parts of the time series.pofsirazzleb.ga
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For each time point t , a state variable dictates the probability of each state being active at that moment. This general framework has different instantiations, depending on the choice of the observation model distribution. It is important to bear in mind that our definition of a network is different from the activation maps that for example Independent Component Analysis ICA provides.
In this case, both the amount of activity and connectivity are established as a function of frequency. The AR is an intermediate point of model complexity between the Gaussian and the MAR models that keeps the channel-by-channel spectral information. Both the AR and the MAR have an important parameter: the model order, which controls the amount of detail in modelling the state spectra. Whichever the chosen observation model distribution, an HMM generally comprises the description of the states, the state time courses which determines the probability of each state to be active at each time point in the time series and the transition probabilities between the states i.
Because here we run the HMM on all concatenated subjects' datasets, the states and the transition probabilities are defined at the group level; the state time courses are however particular to each subject - that is, states can come active at different moments for each subject.
Since the probability distribution of each part of the model depends on all others, there is no closed-form solution available. A popular inference paradigm that assumes certain simplifications in the model is variational Bayes Wainwright and Jordan, , which has its roots in the field of statistical physics, and, earlier than that, in the calculus of variations developed in the 18th century by Euler and others mathematicians. The variational inference methodology introduces certain factorisations in the HMM probability distribution such that we can iterate through different group of parameters, leaving the remaining parameters fixed and thus reducing the computational burden.
The goal is the minimisation of the so-called free energy , a quantity that includes the Kullback-Leibler divergence between the real and the factorised distributions and the entropy of the factorised distribution. The estimation of the observation model distribution for the Gaussian case implies the inversion of a Q -by-Q matrix per state, where Q is the number of channels or time series e.
Discovering dynamic brain networks from big data in rest and task
In the standard variational inference approach, either case requires the entire data set to be loaded into memory. Therefore, standard variational inference for the HMM can be challenging for large data sets, because of i the memory required to estimate the observation models and ii the computation time taken by the estimation of the state time courses.
Standard variational inference guarantees the free energy to decrease at each iteration and, eventually, to converge. Stochastic variational inference instead performs a noisy and computationally cheap update at each iteration. Although those can occasionally lead to small free energy increments, they will typically improve the model. Importantly, to obtain an interim state observation model, we must compute its parameters as though N subjects were actually used so that the estimation's properties mimic that of a standard variational step. This way, we have an interim estimation of the observation models, which thanks to the additivity of the Gaussian and MAR distributions can be linearly combined with the current estimation to form the new estimation.
We do this through the following equation. The HMM optimisation is known to potentially suffer from having local minima. Hence, we need an initialisation mechanism that is computationally affordable in both time and memory use. The initialisation strategy that we propose here provides a reasonably good solution without being computationally expensive. In short, it consists of running separately the standard HMM inference on subsets of subjects and combining the results into a single solution using a matching algorithm.
A detailed description is presented in the Supplementary material. Choose M subjects at random using probabilities, w i , computed as in Eq. Compute the interim state probability distribution using the M subjects as though we had N subjects.
Perform an approximate update of the state probability distributions using Eq. Of more importance is the choice of the value of M , which we varied depending on the data set see below. The chosen value is thus a trade-off. Note that the general stochastic inference framework is the same for the Gaussian and the MAR state models, differing only in the particulars as to how to perform inference of the observation model parameters step 2c. We first used synthetic signals to demonstrate the validity of the proposed stochastic inference approach. We generated two classes of signals using the HMM as a generative model.
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For the Gaussian observation model, we simulated 6 states, each with 10 regions and a randomly generated covariance matrix. These states were obtained from an actual HMM-MAR estimation on the task-MEG HCP data described below , where we observed that two of the states red and blue, see Results were capturing the neural dynamics associated with the task, and the other state green corresponded to a baseline state. For both observation models, we simulated state time courses for subjects, with samples per subject.
For each subject, four min runs of fMRI time series data with temporal resolution 0. We used the first public release containing resting-state subjects aged 40—69 when recruited. Group-ICA was then run using the first subjects' cleaned data, resulting in group-level components, of which 55 were manually classified as non-artefact and used for the HMM analyses. The chosen task sessions consisted of blocks of moving either left or right hand and feet respectively.
Here, for simplicity, we used the right hand moves only. We used the preprocessing pipelines offered by the HCP consortium, removing bad channels, segments and bad independent components from task and rest data. Using the AAL atlas, we considered the two parcels representing left and right precentral gyrus, and use PCA to extract the first principal component from each one. For the resting-state data, artificial epochs of the same size as the task epochs were uniformly spread throughout the session. Parcel time-series were also normalized before being subject to HMM analysis, such that they have mean equal to zero and standard deviation equal to one for all subjects separately.
The HMM is available in both the standard and the stochastic inference versions as a Matlab toolbox in a public repository. This is particularly useful in the case of the MAR model, where the number of model parameters rapidly quadratically increases with the number of channels. The aim is to use the simulated data to verify that the stochastic algorithm's performance is consistent with the standard HMM inference, by comparing model inference using the non-stochastic and stochastic algorithms.
We ran standard non-stochastic inference and stochastic inference for different random state time courses. Because the ordering of the states in the output is random, we matched the states of the estimation and the ground truth model based on the correlations between the state time courses. Once the states are matched, we use the average correlation between matched states as a similarity measure to compare the estimations to the ground truth, and to compare the standard inference to the stochastic inference.
A full representation of the runs is presented in the form of histograms in Fig. In general, even in the limit of a batch size of one subject, we find the stochastic algorithm still infers state time courses that are well correlated with the non-stochastic algorithm. As expected, however, the similarity between the two algorithms is higher for larger batch sizes in the limit of a batch size equal to the number of subjects, the two algorithms are exactly equivalent.
Results for simulated data, for the Gaussian and the MAR case. Representative examples of the inferred state time courses for a single subject are shown in Fig. The ground truth average activity mean of the state's Gaussian distribution and functional connectivity off-diagonal elements of the state covariance matrices for the Gaussian case is plot against the standard and stochastic estimations with batch size equal to 50 in Fig. The state spectra information for the ground truth solid lines versus the standard inference and stochastic inference estimations discontinuous lines is illustrated in Fig.
These results together indicate that the inference is consistent between the standard and the stochastic algorithms for a variety of configuration parameters, with both algorithms able to reasonably recover the ground truth. We used stochastic inference on resting-state fMRI subjects from the HCP to obtain 12 states of quasi-stationary brain connectivity.
We ran the algorithm 5 times, with an average running time of min minimum and maximum were min and min using a standard workstation endowed with four Intel Xeon CPU E 0 3. Some of the results that follow are from a selected run of the stochastic algorithm. However, the different runs were relatively similar see below. With these results, our goal is to illustrate i the type of information the HMM can provide about brain dynamics, and ii the ability of the stochastic inference to produce useful and non-trivial results.
Note that it is not possible to make a comparison of the standard inference and the stochastic inference in this case because the standard inference is computationally implausible given the size of the data set. Regarding the temporal information, the most fundamental output is the state fractional occupancy, defined as the proportion of time that each subject spends in each brain state. In other words, for an HMM to be useful in describing brain dynamics, we expect each subject's time to be shared by various states. A statistic reflecting the satisfaction of this minimum requirement is the maximum fractional occupancy; that is, for each subject or scanning session, how much time of the time series is taken by the state that takes the longest.
Most subjects have maximum fractional occupancy below 0. Another important sanity check is how consistent are separate runs of the stochastic inference. This is important, because the standard HMM estimation is, as mentioned earlier, known to be to some extent dependent on the initialisation, and our algorithm is introducing an additional stochastic factor.
To investigate this, we match the states across different HMM stochastic estimations using the Hungarian algorithm from Munkres , applied to the correlation of the state time courses , and collect the correlation of the state time courses between the re-ordered estimations i. A further robustness test, which would also speak to the reliability of the potential scientific conclusions, is to split the data set into two halves and run the HMM separately. Here, we have used 5 different half-splits, computing the correlation between the activation maps and functional connectivity off-diagonal elements of the covariance matrix between the two subject HMM estimations for each of the splits.
Most states show a high correlation between the two half-split estimations for both mean activation and functional connectivity, with the exception of one state, which has a relatively low correlation for the activation maps.
This state, however, has a mean activation very close to zero in both estimations, with the covariance matrix instead capturing most of the distinct state-specific characteristics of the data when the state is active. Furthermore, Fig SI-4a examines the stability of the transition probabilities between half-splits, which are also quite robust.
Having assessed the basic validity of the stochastic inference, we analysed in more detail the temporal characteristics of one of the runs. From this figure, we can see certain differences in the sense that some states are visited more than others. However, the differences in terms of dwell times are small.
Finally, Fig. Altogether, these results demonstrate that the HMM, when combined with the stochastic inference algorithm, can reproducibly model brain dynamics in large fMRI data sets. The average running time was min minimum and maximum were min and min. The HCP data set allowed 12 reliable states possibly because of the higher data quality and more scanning time per subject. Basing then on the reliability of the results, we present here the model with 8 states.
Three of them, representing sensory-motor, DMN and visual networks, are displayed in Fig. The rest of the figures Fig. The HMM can capture the dynamics of brain activity Fig. As with the HCP, there were some differences in fractional occupancies across states. In this case, however, the differences in dwell time were larger than for the HCP, but still not huge. The description of each panel is analogous to that of Fig. It is worth noting that, although some similarities exist, there are certain differences between the HCP and the Biobank states.
This is possibly due not only to differences in the pipeline and the characteristics of the data, but also to fact that the data is projected into different spaces: whereas the state maps for Biobank are volumetric, the HCP maps refer to the cortical surfaces. Indeed, the ICA decompositions emphasise different areas in the two data sets. To illustrate this discrepancy, Fig. This indicates how much is each region represented by each ICA decomposition. These apparent differences in the independent components on which the HMM was run are likely to explain much of the differences in the HMM results.
In this case, we used a MAR observation model of order 5 to describe the states, such that, as discussed above, the segmentation will be based on the spectral information of the data. Given the relative simplicity of the task and because we have only two data channels brain regions , we limit our HMM to infer 4 states only. Note that the size of data set and the number of model parameters just about permits the use of the standard inference approach, albeit at great computational cost.
Two states, blue and red, capture most of the task-relevant dynamics. This is confirmed quantitatively by applying statistical testing, where we tested for each state and time point whether the fractional occupancy of the state is significantly different higher or lower at this time point than in the rest of the trial permutation testing, significance level of 0. The results of these tests are depicted on top of Fig. It can be observed that it is mostly the red and the blue states that exhibit differences across time, suggesting that they are modulated by the task.