\r\n\tManagement of these disorders requires good clinical evaluation, diagnostic tests, appropriate therapy and huge healthcare cost. Sometimes multiple specialties (gastroenterologists, \r\n\tgastrointestinal motility specialists, otolaryngologists, surgeons, speech therapists, medical oncologists and radiation oncologists) are involved in the management of dysphonia and dysphagia. In the recent years, there have been many updates in the management of these disorders. This book will discuss systematically the different etiologies and management of dysphonia, maxillofacial, oropharyngeal and esophageal dysphagia. This book will be a good \r\n\tguide to the practicing physicians for the management of voice and swallowing disorders.
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Chan and Manoj Kumar Tiwari",coverURL:"https://cdn.intechopen.com/books/images_new/3794.jpg",editedByType:"Edited by",editors:[{id:"252210",title:"Dr.",name:"Felix",surname:"Chan",slug:"felix-chan",fullName:"Felix Chan"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"56847",title:"Kinetics of Heterogeneous Self-Propagating High-Temperature Reactions",doi:"10.5772/intechopen.70560",slug:"kinetics-of-heterogeneous-self-propagating-high-temperature-reactions",body:'\n
\n
1. Introduction
\n
Self-propagating high-temperature synthesis (SHS) is a technological approach for fabrication of materials, which involves self-sustained noncatalytic reactions [1, 2, 3]. Currently, there exist three major types of SHS systems: (i) gasless; (ii) with gasification of initially solid precursors; (iii) gas-solid. Within gasless systems, there are three major types of chemical reactions that occur: metal-metal, producing intermetallic (e.g., NiAl, NiTi), metal-nonmetal, leading to synthesis of borides, carbides and silicides (e.g., TiB2, TaC, MoSi2); nonmetal-nonmetal, producing ceramics (e.g., B4C, SiC). In the gasification reactions, one of the precursors is volatile, including such elements as S, Se, P, As, Sb. Finally, the gas-solid systems include reactions between metals or nonmetals with different gases, such as nitrogen, oxygen, hydrogen, CO, and CO2 leading to formation of nitrides, oxides, hydrides and etc. These lists are by no means exhaustive, as different research groups are continually exploring the limits of SHS reactions with different systems, reactants, and conditions. In order to obtain materials with desired microstructures, and thus properties, one has to precisely control the synthesis conditions during SHS. These conditions are primarily defined by the kinetics of the chemical reactions taking place in the combustion wave.
\n
In order to study and understand the kinetics of SHS reactions, it is important to examine the fundamentals of kinetics and how they relate to SHS itself. Let us start with general definitions. Thus, assuming that the concentrations of any initial reagent, ci, and the product are uniformly distributed throughout the entire volume (homogeneous or quasi homogeneous cases), the chemical reaction rate can be expressed by the following equation:
where Wi is the reaction rate for the ith reagent or product, ηi = (c0i − ci)/c0 is the degree of conversion for the ith reagent, c0i is the initial concentration of the ith reagent, and t is time.
\n
In the nineteenth century, C.M. Guldberg with P. Waage and N.N. Beketov independently formulated the law of mass action. This essentially states that the chemical reaction rates at a given point are proportional to the concentration (mass) of the reactants raised to a proportional exponent. Thus, for an elementary chemical reaction between two reagents A and B of the following form:
Along with the reactant concentration, the temperature affects the rate of the chemical reaction in a noncatalytic homogeneous reaction. However, the mechanisms of these processes are often unknown or too complicated. This is because the reactions occur in multiple steps, each of which has unique reaction rates. In order to describe the chemical kinetics, a single-step approximation is typically used. This states that the rate of the processes in the condensed state is generally a function of the temperature and degree of conversion:
\n
\n\n\n\n\nd\nη\n\ndt\n\n=\nF\n\nT\nη\n\n\n\nE4
\n
The single-step approximation employs the assumption that the function in Eq. (4) can be expressed as a product of two separable functions that are independent of each other; the first, K(T), depends solely on the temperature, T, and the second, Φ(η), depends solely on the degree of conversion, η:
The temperature function K(T) is generally considered to be the rate constant, while the conversion function Φ(η) is generally considered represent the process mechanism. It is assumed that the reaction mechanism is solely dependent on the conversion, and not the temperature. Eq. (3) resembles a single-step kinetic equation, even though it represents the kinetics of a complex condensed-phase process. The single-step kinetic approximation results in the substitution of a generally complex set of kinetic equations with the sole single-step kinetic equation. Eq. (5) represents a mathematical formulation of the single-step kinetic approximation. With few exceptions, the temperature function is exclusively expressed by the Arrhenius equation:
where A and Ea are considered to be the pre-exponential factor and the activation energy, respectively, T is the absolute temperature, and R is the gas constant.
\n
As our knowledge about the atomic and molecular structure of matter increased, coupled with the development of quantum mechanics, new directions in chemical kinetics have emerged. These directions are typically related to the interactions of individual atoms and molecules, which are more fundamental studies. The set of elementary events is called the reaction mechanism. Fundamental studies on the reaction mechanisms allow us to formulate physical explanations to the kinetic parameters (A, Ea, etc.), which were originally introduced as empirical constants. For example, the activation energy Ea is an energy barrier that must be overcome by molecules in the reaction mixture to reach an interatomic distance where they can from a chemical bond. From Eq. (5), it is clear that, if the concentration of substances or the temperature in the given system varies from point to point. Thus, it is impossible to introduce a common reaction rate for the entire system. In order to get closer to these ideal conditions, in classical kinetic experiments we must continuously mix the reagents and maintain a constant temperature by use of a thermostat.
\n
In the case of heterogeneous reactions involving a condensed phase, where the reactants are not mixed on the molecular level, there is an additional parameter, which controls the rate of interaction, i.e., the contact surface area (S) between the reagents [2]. In this case, the rate of the chemical reactions can be represented as follows:
The presence of condensed phases complicates the reaction; this phase requires that transport plays a role in the reaction. Thus, in general, the kinetics of such reactions are determined both by the intrinsic rate of the chemical reaction and by mass transport (e.g., diffusion). The transport phenomena are essential for replenishing the reactants that were consumed in the reaction zone [4]. Describing the reaction rate is further complicated when the temperature of the reacting environment is changing with time. In this case, along with the processes of mass transfer and chemical reactions, it is necessary to consider the specifics of heat transfer mechanisms [5]. Typically, the activation parameters are obtained after experiments considering the dependences of time vs. temperature (for isothermal measurements), temperature vs. heating rate (for integral and incremental methods with linear heating rates), or from reaction rate vs. temperature. Considering the above limitations and complications, only the effective or apparent activation energy can truly be considered, as it includes both the intrinsic kinetics as well as processes of heat and mass transport.
\n
\n
\n
2. Techniques for studying SHS kinetics
\n
The task of accurately determining kinetics becomes even more complicated when accounting for the extremely high temperatures of SHS processes (>1800 K) and rapid heating rates (103–105 K/s). Such parameters are essentially impossible to achieve using conventional approaches for measurement of kinetics parameters. While standard nonisothermal TGA/DTA-based approaches [6] are still used to evaluate the kinetics of SHS reactions, several unique methods such as electrothermal explosion (ETE) [7] and electrothermography (ET) [8] were specifically developed to fit the experimental conditions of SHS reactions. Moreover, recently a variety of advanced in situ diagnostics, including time-resolved X-ray diffraction (TRXRD) [9], high-speed X-ray phase-contrast imaging [10], and high speed transmission electron microscopy (HSTEM) [11] were modified to obtain the kinetics of phase transformations during SHS reactions.
\n
\n
\n
3. Electrothermal explosion
\n
The ETE method was developed in 1977 to study the rapid, high-temperature kinetics that occur in SHS systems [7]. It relies on rapid, uniform preheating of the sample until adiabatic thermal explosion occurs. A representation of a typical ETE setup is shown in Figure 1. Briefly, the sample is clamped between two metallic electrodes with sufficient clamping pressure to ensure adequate contact. The power is then initiated, leading to preheating of the sample until a set Toff point. After initiation, the resulting time-temperature profiles are simultaneously collected across a number of high-speed photodiodes. In the commonly used ETA-100 system (Aloft, Inc., Berkley, CA), there are 16 photodiodes present, with 1 mm in between them; this corresponds to 0.5 mm spatial resolution. The photodiodes have a temporal resolution of 10−5 s and are accurate within 900–3000 K. Once the sample is heated to the selected Toff point, the equipment heating is halted, with the consequent rate of self-heating determined solely by the chemical reaction rate. Due to the experimental conditions, i.e., the rapid initial preheating, the reaction occurs in the adiabatic mode. Once thermal ignition occurs, analysis of the time-temperature profile enables extraction of the kinetic parameters (see details in [12]). This technique can be used to study the kinetics at temperatures much higher than can be achieved in other experiments. However, it is often limited in the systems that can be studied due to the stringent heating conditions caused by Joule preheating.
\n
Figure 1.
Summary of data collected using the electrothermal explosion technique.
\n
ETE has been used for different gasless SHS systems. These studies have provided valuable kinetic data in extremely high-temperature ranges that are essentially inaccessible by other methods. For carbides, the Ti/C system has been studied [13, 14, 15], with additional studies in the Ta/C [14, 16] and Si/C [14, 17] mixtures. The Ti/B system was also investigated [18]. The majority of ETE work being focused on the Ni/Al system [12, 19, 20, 21]. The Ti/Fe2O3 system is the only thermite system investigated by ETE [22]. In addition to the experimental studies, a number of theoretical models have been developed to better understand the ETE process [23, 24, 25, 26]. Figure 1 shows that ranges of reported activation energies for the above mentioned gasless systems, including both intermetallics and thermites obtained by ETE. It can be seen that the results are reproducible, confirming the reliability of the ETE approach. Additionally, the technique allows for a more complete understanding of systems that have multiple steps that rapidly occur in the high-temperature regime, which gives insight into the combustion process [2].
This technique has been extensively used in many fields, including: polymer science, biochemistry, and materials science. In order to utilize these methods, a sample is heated at a constant rate until the maximum set temperature is reached. Throughout the experiment, the heat release characteristics of the sample are measured against an inert reference standard. For typical systems, the points of differing heat release characteristics can be due to phase transitions, crystallization, or reactions; however, this section will only focus on SHS-reaction kinetics. In order to determine the reaction kinetics, the experiment is conducted multiple times with different heating rates. The classical method for determining the reaction kinetics, specifically the activation energy, is by use of the Kissinger method [27], however, many alternative methods for data analysis have been suggested and are widely utilized [28, 29, 30, 31, 32, 33, 34]. The activation energy, Ea, can be computed by plotting ln(β/Tp2) as a function of 1/Tp, where Tp is the peak temperature (Figure 2).
\n
Figure 2.
Summary of data collected using isothermal kinetic analysis methods.
\n
The DTA/DSC based methods are the most widely used for gasless SHS systems, with multiple studies into intermetallics, specifically the Ni/Al [35, 36, 37, 38, 39, 40, 41, 42, 43], Ti/Al [44, 45, 46], Co/Al [47], Al/Ru [48], Nb/Al [49], and Mg/Al [50] systems, in addition to other binary solid-solid compositions, i.e., the Si/C [51], Mo/Si [52], Zr/B [53], Fe/Se [54]. More complicated ternary systems were also investigated [54, 55, 56, 57, 58, 59, 60, 61]. In general, a wide variety of factors can influence the measured kinetics, including variations in reactant microstructure, heating rates, among other factors. Although there are a number of studies into the same systems, it would be valuable for systematic work to be conducted where these factors are studied in depth across different systems to see what general conclusions can be drawn. Additionally, because the experimental conditions, typically specifically heating rate and temperature ranges, are not the same as in traditional SHS, it is unclear whether the determined values can be directly compared or if there is some systemic difference that is occurring.
\n
Figure 2 illustrates the ranges of the obtained values of activation energies for a variety of gasless exothermic reactions measured by DTA method. It can be seen that the determined values are dependent on the experimental conditions utilized, including variations in reactant microstructure, heating rates, among other factors. This issue is discussed in detail below. It would be valuable for systematic studies to be done where these factors are studied in depth across different systems. Additionally, while the activation energies reported using different DTA-based approaches does not appear to be significantly different, it is still important to understand why these differences are present and which methods of analysis are most suitable for these systems.
\n
\n
\n
5. Combustion velocity/temperature analysis
\n
There has been significant effort done to accurately correlate experimental combustion parameters, such as combustion wave velocity and temperature, with the kinetics parameters. Two major approaches have been developed to determine the activation energy by measuring the layer-by-layer combustion front combustion velocity. The first was suggested in 1977 by Merzhanov for 1D propagation [62]. The derived equation takes the form:
where f(ηs) is the selected kinetic law. This technique is most commonly used by adding diluent to the sample. This affects the combustion velocity and temperature; the change in both of these values is measured then compared, leading to the kinetic relationship being understood.
\n
The other major approach to determining the kinetics based on the velocity was developed by Boddington et al. [63]. The relationship takes the form:
where td and tr, are the decay and rise times, with τad being the temperature rise under adiabatic conditions. More complete derivations of these two models can be found in the original articles [62, 63, 64]. Additionally, a more complete understanding of these models, including their relative merits, has been examined in a number of prior works [65, 66, 67, 68]. In order to properly use these techniques, relatively simple equipment is required. Typically, sets of thermocouples are used to measure combustion wave propagation velocities, but there are many alternatives, such as IR or high-speed cameras, to measure the propagation velocity.
\n
Because of the relative simplicity in using these techniques, they have found widespread use within SHS reactions. These layer velocity approaches have been used to describe the kinetics in the Ni/Al system [69, 70], in boride systems, including Nb/B [71, 72, 73], Ta/B [71], Zr/B [71, 74, 75], Hf/B [71], Ti/B [76], Mo/B [77], along with other binary systems such as Ti/C [78], Ti/Si [79, 80], thermites [81, 82, 83, 84, 85, 86, 87, 88, 89] and more complex ternary systems [78, 90, 91, 92, 93].
\n
The results obtained are presented in Figure 3. It can be seen that there are a wide variety of systems analyzed. It worth noting that these data were obtained through direct analysis of the combustion parameters, which were obtained at extremely high temperatures and rapid heating rates, which are difficult to accomplish by other kinetics methods. However, it is important to remember that the quasi-homogeneous approximation is utilized for this layer-by-layer combustion front combustion velocity based method, which should be applied with caution [68]. Finally, because this method can be easily used for any type of system, a comparison between thermite and non-thermite type reactions can be analyzed. Comparing Figure 3a and b, it is obvious that the activation energies for thermites is significantly lower than non-thermite type reactions. This is interesting and is likely related to the specific reaction mechanism that occurs in thermites, compared to non-thermites.
\n
Figure 3.
Summary of data collected using layer velocity analysis approached for (a) binary elemental and (b) thermite systems.
\n
\n
\n
6. Electrothermography
\n
Electrothermography is a technique that utilizes metal wires in either a gaseous or clad environment [8]. The wire is resistively heated rapidly to the desired temperature, with the electric power being adjusted to compensate for the heat release due to the chemical reaction. The obtained data allows for extraction of the rate of heat generation during the reaction under conditions similar to those in SHS wave. Additionally, because the wires are thin, the sample is quenched essentially as soon as the power is turned off. After the wire is cooled, cross-sections of the wire are collected and the width of the product films is measured, which allows for information on the kinetics of phase formation. This, when done at multiple temperatures and times, gives a more complete picture of the reaction mechanism.
\n
The electrothermography technique has been widely used to study gas-solid reactions. This is because the wires can be exposed to any sort of gaseous environment and also because of the equipment itself; the wires can be heated with any heating rate, mirroring those found in conventional SHS reactions. Because of this, a number of experiments were conducted in nitrogen environments. In specific, the Ta/N [94], Ti/N [95, 96, 97, 98], Nb/N [98, 99], and Zr/N [8] systems have been studies. However, the technique has also be used to study carbides, including the Ti/C [100], Zr/C [100], W/C [101], Nb/C [102], along with other systems, including Mo/Si [103, 104, 105, 106], W/Si [107, 108], Ni/Al [109], and Ni/Ti [109].
\n
Figure 4 shows the values of the activation energies obtained by this method in gas-solid systems. It can be seen that some reactions have limited steps relating to the gas pressure, while others are relatively independent, suggesting that there are different mechanisms for the two classes of reactions. This is also true for the gasless systems investigated. This method provides a window into determining the mechanism and, due to the nature of the experiment, allows for control over the experimental conditions to the degree that individual steps in the reaction can be isolated.
\n
Figure 4.
Summary of data obtained using the electrothermography approach.
\n
\n
\n
7. Modern in situ high-speed high-resolution methods
\n
There currently exist a number of techniques to study in situ reactions on the time and length scales that occur during SHS reactions; these techniques are incredibly valuable to determine the reaction mechanisms. The most widespread technique is time-resolved X-ray diffraction (TRXRD), and is used to determine the phases that are present during the reaction. It allows for information on the phases present at every stage of the reaction, depending on the time resolution. The lower the time resolution, the more information that can be attained. Depending on the specific setup, whether synchrotron or laboratory-scale based, time resolutions ranging from 10−6 to 10−2 s are reasonable, with the absolute limit being continually improved with improved synchrotron and detector technology. There has been significant work done with SHS systems due to their solid nature, which is simple to use in TRXRD systems. It is possible to measure solid solution formations, intermediate phases, any melting processes, and the general reaction progress. Through these data it should be possible to extract kinetic data on all reaction stages based on the growth rates of the peaks for the new phase formation coupled with the decomposition of peaks from the previous phase, however, there are currently no established models illustrating this.
\n
There have been a wide variety of experiments conducted on SHS systems by a number of different groups. For intermetallic systems, groups have studied the Ni–Al [9, 110, 111, 112, 113, 114, 115, 116], Fe–Al [111, 117, 118, 119, 120, 121], Nb–Al [122, 123, 124], and numerous other systems [110, 125, 126, 127]. Additionally, many groups have examined other SHS based systems, such as carbides, including Ti–C [110, 128, 129], Ta–C [129, 130], and other carbides and cermets [129, 131, 132, 133, 134], nitrides [135, 136], oxides [137, 138, 139], silicides, including Fe–Si [140, 141], Mo–Si [119, 123, 142, 143] and Ti–Si [144, 145], among a variety of other systems [113, 129, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155].
\n
In addition to TRXRD, there is a variety of other, less common, but still very useful techniques available. For example, high-speed X-ray phase-contrast imaging [10] utilizes a synchrotron source coupled with the fact that different phases absorb X-rays differently to determine which phase transformations occur during reaction, essentially high-speed X-ray phase contrast imaging. This technique was illustrated on the W-Si system at the Advanced Proton Source in Argonne National Laboratory. This method allowed for direct imaging of irreversible reactions in the W-Si reactive system at frame rates up to 36,000 frames per second with a 4-μs exposure time and spatial resolution of 10 μm. Another advanced technique is high-speed transmission electron microscopy (HSTEM) [11], which utilizes all abilities of conventional TEM, but at nanosecond time scales. This allows for direct observation of both the structural changes and crystal structure during the reaction with unprecedented resolution, as shown in Figure 5. Specifically, a high-time resolution dynamic transmission electron microscopy (DTEM) was developed in Lawrence Livermore National Laboratory (USA) and captures the material dynamics with nanosecond time resolution. The current DTEM performance shows a spatial resolution less than 10 nm for single-shot imaging, using 15 ns electron pulses. The solid-state reactions in NiAl reactive multilayer films, the martensitic transformations in nanocrystalline Ti, and the catalytic growth of Si nanowires were studied by DTEM [156].
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Figure 5.
Dynamic single-shot diffraction with 15-ns time resolution of regions before, during, and after the exothermic mixing reaction front has passed. The times indicated at right are in relation to the reaction front, set at t = 0 s. The crystal structure clearly changes from separate fcc Al/Ni and Ni/V layers to an intermetallic B2 structure NiAl phase within 300 ns after the arrival of the hot reaction front; a.u., arbitrary units. Adapted from Kim et al. [11].
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The above unique diagnostics, which are used to determine phase and structural transformations in situ, are incredibly valuable. Used alone, they provide information on the reaction progress and mechanism; however, there are two significant paths that would make these techniques more valuable. When coupled with current methods for determination of kinetics, the understanding of the reaction mechanisms in all systems will be improved. Furthermore, there is no current way to extract kinetic parameters from some techniques (TRXRD, DTEM, etc.); it would be valuable to develop reliable approaches for these techniques.
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8. Structure-kinetics relationship of SHS systems
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There have been a number of studies shown that indicated that the structure of materials plays a role in the kinetics; multiple groups using a wide variety of techniques and approaches have confirmed this conclusion. For example, it was shown that by changing the internal structure of the reactive composite by using high-energy ball milling (HEBM) in the same binary system, one can significantly change the measured effective activation energy [20]. There are two approaches to quantifying this effect; the first is by rigorous quantification of the already existing structures, and the second is by use of more simple, so-called model microstructures, typically manifested in reactive nanofoils with periodically fabricated layers of reactants.
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Shuck et al. utilized two techniques for quantitative determination of HEBM-produced materials [5, 157]. The first technique, X-ray Nanotomography, works by passing high brilliance X-rays through the sample and collecting the transmitted X-rays. In order to convert this 2D projection into three dimensions, the sample is rotated and the same quality of X-rays is passed through again and the projections at different angles are collected; this collection of images can be combined, leading to a 3D map of the internal sample. The second technique, focused ion beam (FIB) sectioning, uses a FIB to serially section the HEBM-produced particles. The series of images were first shear corrected, contrast normalized, and then aligned using a least-squares method, with the reconstructions shown in Figure 6. After structure analysis, Shuck and Mukasyan [20] further studied these effects on the kinetics in the Ni–Al system using the ETE approach. They showed, by use of the above 3D reconstruction techniques, that it is possible to control the activation energy by modification of the contact surface area between the reactants. Effectively, they lowered the effective activation energy from 79 to 137 kJ/mol, which corresponded to a change in the contact surface area/volume ratio between 0.0120 and 0.0032 nm−1, respectively; this relationship is shown in Figure 7. Additionally, it was suggested that, for SHS systems, the apparent activation energy is affected primarily by the contributions between the diffusion and intrinsic reaction activation energies. Additionally, they offered an explanation for the relationship, relating to the difference in contribution between the diffusive activation energies (volume, grain-boundary, and surface) in conjunction with the intrinsic activation energy. This suggests that the measured and reported activation energies presented in literature are effective, or apparent, activation energies that depend highly on the structure and experimental conditions.
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Figure 6.
Reconstructed internal volumes of the nanocomposites for different time (min) of WG: (a) 10, (b) 20, (c) 30, and (d) 40 (Ni is the gray phase, and Al is the void-space). Adapted from Shuck et al. [5].
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Figure 7.
Dependence of effective activation energy of the reaction as a function specific contact surface area between Ni and Al phases. Adapted from Shuck and Mukasyan [20].
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9. Modification of the reaction mechanism depending on the experimental conditions
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In a very early study on the reaction mechanisms, Philpot et al. examined the effect of varying factors on the reaction rate [35]. In one study, they systematically varied the aluminum concentration, the heating rate, and the nickel particle size. Briefly, they showed that, depending on the applied heating rate, two different mechanisms could be initiated. The first, when using slower heating rates, was related to the melting of the aluminum metal, followed by spreading over the nickel particles. For their studies, they saw two definite peaks relating to the reaction. However, when they increased the heating rate, they instead only witnessed a single peak. This peak was related to the solid state transition to the final product. For transitional values between the two extremes, they found that there were relative contributions of both different mechanisms. This is an important observation which should be accounted when investigating the reaction kinetics in highly exothermic systems. Indeed, it shows that it is possible to control the reaction mechanism depending on the applied experimental conditions. This effect was confirmed by many researchers both for gasless and gas-solid systems [36, 56, 158, 159, 160, 161, 162] and is illustrated in Figure 8. Furthermore, depending on the reaction mechanism, it is possible to control both the final product phases and their microstructures, thus producing materials with tailored properties.
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Figure 8.
DTA-TG curves for the PTFE-Al2O3 system using heating rates of (a) 20, (b) 80, (c) 150, and (d) 160°C min−1 under an argon atmosphere. The additional lines are showing the changes in heat flow and mass percentage values. Adapted from Hobosyan et al. [56].
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10. The Ni/Al system as a model for SHS kinetics
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In the SHS community, the Ni/Al system has been widely used a model system. It was chosen because of its low ignition temperature, high oxidation resistance, and ease of processing. Because of this, all of the above mentioned techniques have been utilized to study kinetics in this system. Thus it is possible to compare the data collected across a large number of experimental conditions to give more complete understanding of this gasless reaction. Although this system has been extensively studied for over 40 years now, a consensus has not emerged on the exact activation energy, as can be seen in Figure 9. The data again illustrate the effect that differing experimental conditions play, whether in the material structure, heating rates, or other experimental factors.
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Figure 9.
Summary of kinetic data collected for the Ni/Al system.
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In 1987, Philpot et al. did preliminary studies on the Ni/Al system kinetics using the Kissinger approach [35]. They showed that there is an effect of heating rate, the nickel particle size, as well as an effect of a varying Al content, on the reaction mechanism in this system [35], this work is highlighted in more detail in an above section. Hunt et al. examined the effect of particle size on the apparent activation energy using a CO2 laser to control the heating coupled with the DTA-based Kissinger approach. Using 800 nm size Ni particles coupled with 20 μm, 4 μm, 80 nm, and 40 nm Al particles, it was found that that these reactions had activation energies of: 103.5 ± 10.5, 97.3 ± 5, 21.2 ± 2.5, and 17.4 ± 2.85 kJ/mol, respectively [36]. They then increased the Ni size to 15 μm and measured with the four Al sizes, resulting in 162.5 ± 1.4, 131.2 ± 2.6, 103.6 ± 5.2, and 80.1 ± 6.3 kJ/mol, respectively [36]. This confirms that the initial reactant structure plays a significant impact on the kinetics. The size of either reactant significantly alters the effective kinetics. Kim et al. studied this reaction using nanolaminated composite micro-foils with different thickness ratios between Ni and Al at heating rates between 5 and 100 K/min. For 4:1 foils, they found that the formation of NiAl3 occurred, followed by Ni2Al3, with activation energies of 142 and 106 kJ/mol, respectively [37].
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In order to more fully understand the relationship between structure and the kinetics, White et al. investigated the effect of mechanical activation (MA) on the Ni/Al system kinetics using the Kissinger approach. Two types of Ni/Al composites were used; Ni clad Al particles, as well as Ni/Al composite particles produced by high-energy ball milling (HEBM). The Ni clad Al particles were found to have an apparent activation energy of 352 ± 8 kJ/mol, while after MA the particles had much lower activation energy of 117 ± 4 kJ/mol [38]. Reeves et al. studied the thermal and impact reaction kinetics in the Ni/Al system for both MA and nano-sized reactants using the Kissinger approach. For the nano-mixture, the reactants were both ~80 nm in size and, for the MA particles, they underwent 15 minutes of HEBM. The nano-mixture exhibited a 230 ± 21 kJ/mol activation energy, while the MA mixture was calculated to be 117 ± 8 kJ/mol [39]. Manukyan et al. studied the Ni/Al system after MA and the effect of a coarse vs. nanolaminated nanostructure on the kinetics using the Kissinger approach. Using heating rates between 10 and 50 K/min, they found that the reaction proceeds in three steps, NiAl3, Ni2Al3, and then finally NiAl. For the coarse microstructure, these peaks corresponded to 99 ± 4, 138 ± 13, and 120 ± 37 kJ/mol, while the nanolaminated microstructure corresponded to 93 ± 2.5, 145 ± 13, and 146 ± 14 kJ/mol, respectively [40]. This illustrated that the activation energies depend on the microstructure, even after MA.
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Kuk et al. studied compression bonded Ni/Al nanofoils with and without a BN lubricant using the Kissinger approach. With the BN lubricant, it was found that the reaction proceeded in two steps, with the activation energies being 224 and 272 kJ/mol, respectively, resulting in the formation of Al3Ni2 [42]. Without the lubricant, the reaction proceeded in a single step with activation energy of 470 kJ/mol, this difference was attributed to the oxide layer between the reactants [42]. Maiti and Ghoroi studied the Ni/Al system using the Friedman, Ozawa, and Kissinger approach, yielding activation energies of 437.0, 448.4, and 457.6 kJ/mol, respectively [43].
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Using the ETE approach, Shteinberg et al. and Mukasyan et al. confirmed that MA affects the activation energy in the Ni/Al system, as shown in Figure 10 [12, 19]. Initially, they studied the kinetics of the Ni clad Al system (which was also studied in [38]), which showed two distinct steps. The first step was related to the melting of Al and subsequent cracking of the Ni layer, which had an activation energy of 197 ± 29 kJ/mol. The final step was the diffusion of Ni into Al, which was measured as 523 ± 84 kJ/mol. In the MA system, they found that only a single step occurred and was measured to be 105 ± 13 kJ/mol. Shuck and Mukasyan further studied the effects of MA on the kinetics in the Ni/Al system using the ETE approach [20]. Using 3D reconstruction techniques, they showed that the surface area contact between the reactions is directly related to the effective activation energy, which ranged from 79 to 137 kJ/mol, which corresponded to a change in the contact surface area/volume ratio between 0.0120 and 0.0032 nm−1, respectively. This work is further highlighted in an above section. Finally, using the ETE approach, Filimonov et al. studied the effects of MA on the nonstoichiometric, 3Ni/Al system [21]. They utilized a criterion based on the minimum curvature of the heating rate logarithm, which resulted in an anomalously low measured activation energy of 9.5 ± 2 kJ/mol.
\n
Figure 10.
Arrhenius plots for reactions in Al clad by Ni systems before and after high energy ball milling. Adapted from Shteinberg et al. [12].
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Marin-Ayral et al. studied the Ni/Al system under different gas pressures using the Boddington-Laye method. They showed that for pressures of 100, 320, and 500 MPa, the measured activation energies were 47, 59, and 132 kJ/mol, respectively [68, 69]. Vadchenko et al. studied the Ni/Al system using electrothermography. Their results showed that the reactions occur first through grain boundary diffusion, followed by diffusion of the solid metal into the liquid phase [109]. Finally, Mukasyan et al. examined the Ni/Al system using a combination of TRXRD and ETE, showing that the reaction mechanism itself changes based on the structure, this work is highlighted in an above section [116].
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Thus we may conclude that although a wide variety of studies were conducted on the Ni/Al system with many different experimental and structural conditions, the reported values of activation energies vary drastically. Additionally, there would be great benefit to combining utilizing multiple methods simultaneously to bridge understanding between the different experimental techniques.
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\n
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11. Future directions in SHS kinetics
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More complete understanding on the kinetics of SHS reactions is vital for both fundamental science and also for practical or industrial reasons. To better understand the kinetics, combinations of techniques must be utilized, specifically coupling techniques that give information on the kinetics while simultaneously examining the phase transformations that are occurring. To further understand the reaction mechanisms, additional studies must be conducted on the relationship between the structure and the resulting kinetics. Additionally, work must be done to compare the different methods of studying kinetics and their interrelationships. In the limited cases where there is data available for the same system across different techniques, there is a wide range of published kinetic data. It is imperative to continue to study SHS kinetics in a more systematic, fundamental fashion.
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Acknowledgments
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This work was supported by the Department of Energy, National Nuclear Security Administration, under the award number DE-NA0002377 as part of the Predictive Science Academic Alliance Program II. We also acknowledge the Ministry of Education and Science of the Russian Federation specifically Increase Competitiveness Program of NUST ‘MISiS’ (No. K2-2016-065), implemented by a governmental decree dated 16th of March 2013, N 211. Finally, this work was also supported by the U.S. Department of State through the Fulbright program.
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\n',keywords:"self-propagating high-temperature synthesis (SHS), electrothermal explosion, electrothermography, combustion synthesis, mechanical activation, high-energy ball milling, intermetallics, thermites",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/56847.pdf",chapterXML:"https://mts.intechopen.com/source/xml/56847.xml",downloadPdfUrl:"/chapter/pdf-download/56847",previewPdfUrl:"/chapter/pdf-preview/56847",totalDownloads:581,totalViews:271,totalCrossrefCites:0,totalDimensionsCites:2,hasAltmetrics:0,dateSubmitted:"March 27th 2017",dateReviewed:"August 9th 2017",datePrePublished:"December 20th 2017",datePublished:"February 21st 2018",readingETA:"0",abstract:"In this chapter, we present an overview of experimental techniques utilized and kinetic data collected for exothermic self-sustained noncatalytic heterogeneous reactions. The data focuses on five primary experimental techniques: electrothermal explosion, differential thermal analysis, electrothermography, combustion velocity/temperature analyses, and several advanced in situ diagnostics, including time-resolved X-ray diffraction.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/56847",risUrl:"/chapter/ris/56847",book:{slug:"advanced-chemical-kinetics"},signatures:"Christopher E. Shuck and Alexander S. Mukasyan",authors:[{id:"21434",title:"Dr.",name:"Alexander S.",middleName:null,surname:"Mukasyan",fullName:"Alexander S. Mukasyan",slug:"alexander-s.-mukasyan",email:"amoukasi@nd.edu",position:null,institution:null},{id:"215636",title:"Dr.",name:"Christopher",middleName:"Eugene",surname:"Shuck",fullName:"Christopher Shuck",slug:"christopher-shuck",email:"cshuck@nd.edu",position:null,institution:{name:"Drexel University",institutionURL:null,country:{name:"United States of America"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Techniques for studying SHS kinetics",level:"1"},{id:"sec_3",title:"3. Electrothermal explosion",level:"1"},{id:"sec_4",title:"4. Differential thermal analysis/differential scanning calorimetry",level:"1"},{id:"sec_5",title:"5. Combustion velocity/temperature analysis",level:"1"},{id:"sec_6",title:"6. Electrothermography",level:"1"},{id:"sec_7",title:"7. Modern in situ high-speed high-resolution methods",level:"1"},{id:"sec_8",title:"8. Structure-kinetics relationship of SHS systems",level:"1"},{id:"sec_9",title:"9. Modification of the reaction mechanism depending on the experimental conditions",level:"1"},{id:"sec_10",title:"10. The Ni/Al system as a model for SHS kinetics",level:"1"},{id:"sec_11",title:"11. Future directions in SHS kinetics",level:"1"},{id:"sec_12",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Merzhanov AG, Borovinskaya IP. Self-spreading high-temperature synthesis of refractory compounds. 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Metals and Materials International. 1995;1(1):19-27\n'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Christopher E. Shuck",address:null,affiliation:'
Department of Chemical and Biomolecular Engineering, University of Notre Dame, Indiana, United States
'},{corresp:"yes",contributorFullName:"Alexander S. Mukasyan",address:"amoukasi@nd.edu",affiliation:'
Department of Chemical and Biomolecular Engineering, University of Notre Dame, Indiana, United States
University of Notre Dame and National University of Science and Technology MISiS, Russia
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1. Introduction
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Artificial neural networks (ANN), which are mathematical models for function approximation, classification, pattern recognition, nonlinear control, etc., have been successfully applied in the field of time series analysis and forecasting instead of linear models such as 1970s ARIMA [1] since 1980s [2, 3, 4, 5, 6, 7]. In [2], Casdagli used a radial basis function network (RBFN) which is a kind of feed-forward neural network with Gaussian hidden units to predict chaotic time series data, such as the Mackey-Glass, the Ikeda map, and the Lorenz chaos in 1989. In [3, 4], Lendasse et al. organized a time series forecasting competition for neural network prediction methods with a five-block artificial time series data named CATS since 2004. The goal of CATS competition was to predict 100 missing values of the time series data in five sets which included 980 known values and 20 successive unknown values in each set (details are in Section 3.1). There were 24 submissions to the competition, and five kinds of methods were selected by the IJCNN2004: filtering techniques including Bayesian methods, Kalman filters, and so on; recurrent neural networks (RNNs); vector quantization; fuzzy logic; and ensemble methods. As the comment of the organizers, the different prediction precisions were reported though the similar prediction methods were used for the know-how and experience of the authors. So the development of time series forecasting by ANN is still on the way.
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As a kind of classifiers or a kind of function approximators, the advances of the ANN are bought out by the nonlinear transforms to the input space. In fact, units (or neurons) with nonlinear firing functions connected to each other usually produce higher dimensional output space and various feature spaces in the networks. Additionally, as a connective system, it is not necessary to design fixed mathematical models for different nonlinear phenomena, but adjusting the weights of connections between units. So according to the report of NN3—Artificial Neural Networks and Computational Intelligence Forecasting Competition [5], there have been more than 5000 publications of time series forecasting using ANN till 2007.
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To find the suitable parameters of ANN, such as weights of connections between neurons, error back-propagation (BP) algorithm [6] is generally utilized in the training process of ANN. However, due to every sample data (a pair of the input data and the output data) is used in the BP method, noise data influences the optimization of the model, and robustness of the model becomes weak for unknown input. Another problem of ANN models is how to determine the structure of the network, i.e., the number of layers and the number of neurons in each layer. To overcome these problems of BP, Kuremoto et al. [7] adopted a reinforcement learning (RL) method “stochastic gradient ascent (SGA)” [8] to adjust the connection weights of units and the particle swarm optimization (PSO) to find the optimal structure of ANN. SGA, which is proposed by Kimura and Kobayshi, improved Williams’ REINFORCE [9], which uses rewards to modify the stochastic policies (likelihood). In SGA learning algorithm, the accumulated modification of policies named “eligibility trace” is used to adjust the parameters of model (see Section 2). In the case of time series forecasting, the reward of RL system can be defined as a suitable error zone to instead of the distance (error) between the output of the model and the teach data which is used in BP learning algorithm. So the sensitivity to noise data is possible to be reduced, and the robustness to the unknown data may be raised. As a deep learning method for time series forecasting, Kuremoto et al. [10] firstly applied Hinton and Salakhutdinov’s deep belief net (DBN) which is a kind of stacked auto-encoder (SAE) composed by multiple restricted Boltzmann machines (RBMs) [11]. An improved DBN for time series forecasting is proposed in [12], which DBN is composed by multiple RBMs and a multilayer perceptron (MLP) [6]. The improved DBN with RBMs and MLP [6] gives its priority to the conventional DBN [5] for time series forecasting due to the continuous output unit is used; meanwhile the conventional one had a binary value unit in the output layer.
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As same as the RL method, SGA adopted to MLP, RBFN, and self-organized fuzzy neural network (SOFNN) [7]; the prediction precision of DBN utilized SGA may also be raised comparing to the BP learning algorithm. Furthermore, it is available to raise the prediction precision by a hybrid model which forecasts the future data by the linear model ARIMA at first and modifying the forecasting by the predicted error given by an ANN which is trained by error time series [13, 14].
\n
In this chapter, we concentrate to introduce the DBN which is composed by multiple RBMs and MLP and show the higher efficiency of the RL learning method SGA for the DBN [15, 16] comparing to the conventional learning method BP using the results of time series forecasting experiments. Kinds of benchmark data including artificial time series data CATS [3], natural phenomenon time series data provided by Aalto University [18], and TSDL [18] were used in the experiments.
\n
\n
\n
2. The DBN model for time series forecasting
\n
\n
2.1 The structure of the model
\n
The model of time series forecasting is given as the following:
Denote t = 1, 2, 3, …, where T is the time, n is the dimensionality of the input of function f(x), \n\n\nx\nt\n\n\n is the time series data, and \n\n\nx\n\nt\n+\n1\n\n\n\n is unknown data in the future as well as the output of model.
\n
A deep belief net (DBN) composed by restricted Boltzmann machines (RBMs) and multilayer perceptron (MLP) is shown in Figure 1.
\n
Figure 1.
The structure of DBN for time series forecasting.
\n
\n
\n
2.2 RBM
\n
Restricted Boltzmann machine (RBM) is a kind of probabilistic generative neural network which composed by two layers of units: visible layer and hidden layer (see Figure 2).
\n
Figure 2.
The structure of RBM.
\n
Units of different layers connect to each other with weights \n\n\nw\nij\n\n=\n\nw\nji\n\n\n, where \n\ni\n=\n1\n,\n2\n,\n…\n,\nn\n\n and \n\nj\n=\n1\n,\n2\n,\n…\n,\nm\n\n are the numbers of units of visible layer and hidden layer, respectively. The outputs of units \n\n\nv\ni\n\n,\n\nh\nj\n\n\n are binary, i.e., 0 or 1, except for the initial value of visible units which is given by the input data. The probabilities of 1 of a visible unit and a hidden unit are according to the following:
where \n\n0\n<\nε\n<\n1\n\n is a learning rate, \n\n\np\nij\n\n=\n<\n\nv\ni\n\n\nh\nj\n\n\n>\ndata\n\n,\n\np\nij\n\'\n\n<\n\nv\ni\n\n\nh\nj\n\n\n>\nmodel\n\n\n and \n\n<\n\nv\ni\n\n>\n,\n<\n\nh\nj\n\n>\n\n indicate the expectations of the first Gibbs sampling (k = 0), and \n\n<\n\n\nv\n˜\n\ni\n\n>\n,\n<\n\n\nh\n˜\n\nj\n\n>\n\n the kth Gibbs sampling, and it works when k = 1.
\n
\n
\n
2.3 MLP
\n
Multilayer perceptron (MLP) is the most popular neural network which is generally composed by three layers of units: input layer, hidden layer, and output layer (see Figure 3).
\n
Figure 3.
The structure of MLP.
\n
The output of the unit \n\ny\n=\nf\n\nz\n\n\n and unit \n\n\nz\nk\n\n=\nf\n\nx\n\n\n is given as the following logistic sigmoid functions:
Here n is the dimensionality of the input, K is the number of hidden units, and \n\n\nx\n\nn\n+\n1\n\n\n=\n1.0\n,\n\nz\n\nK\n+\n1\n\n\n=\n1.0\n\n are the support units of biases \n\n\nv\n\nj\n\n\nn\n+\n1\n\n\n\n\n,\n\nw\n\nK\n+\n1\n\n\n\n.
\n
The learning rules of MLP using error back-propagation (BP) method [5] are given as follows:
Step 5. For the next time step \n\nt\n+\n1\n\n, return to step 1.
\n
\n
\n
2.4 The training method of DBN
\n
As same as the training process proposed in [10], the training process of DBN is performed by two steps. The first one, pretraining, utilizes the learning rules of RBM, i.e., Eqs. (4–6), for each RBM independently. The second step is a fine-tuning process using the pretrained parameters of RBMs and BP algorithm. These processes are shown in Figure 4 and Eqs. (11)–(13).
In the case of reinforcement learning (RL), the output is decided by a probability distribution, e.g., the Gaussian distribution \n\ny\n∼\nπ\n\nμ\n\nσ\n2\n\n\n\n. So the output units are the mean \n\nμ\n\n and the variance \n\nσ\n\n instead of one unit \n\ny\n\n.
The learning algorithm of stochastic gradient ascent (SGA) [7] is as follows.
\n
Step 1. Observe an input \n\n\nx\nt\n\n=\n\n\nx\nt\n\n\nx\n\nt\n−\n1\n\n\n…\n\nx\n\nt\n−\nn\n+\n1\n\n\n\n\n.
\n
Step 2. Predict a future data \n\n\ny\nt\n\n=\n\nx\n\nt\n+\n1\n\n\n\n according to a probability \n\n\ny\nt\n\n∼\nπ\n\n\nx\nt\n\nw\n\n\n with ANN models which are constructed by parameters \n\nw\n\n\nw\nμj\n\n\nw\nσj\n\n\nw\nij\n\n\nv\nji\n\n\n\n.
\n
Step 3. Receive a scalar reward/punishment \n\n\nr\nt\n\n\n by calculating the prediction error:
Step 7. For the next time step \n\nt\n+\n1\n\n, return to step 1.
\n
Characteristic eligibility \n\n\ne\ni\n\n\nt\n\n\n, shown in Eq. (18), means that the change of the policy function concerns with the change of system internal variable vector. In fact, the algorithm combines reward/punishment to modify the stochastic policy with its internal variable renewing by Step 4 and Step 5.
\n
The calculation of \n\n\ne\n\nw\nμj\n\n\n\nt\n\n,\n\ne\n\nw\nσj\n\n\n\nt\n\n,\n\ne\n\nv\nij\n\n\n\nt\n\n\n in MLP part of DBN is induced as follows;
The learning rate \n\nε\n\n in Eq. (21) affects the learning performance of fine-tuning of DBN. Different values to result different training error (mean squared error (MSE)) as shown in Figure 5. An adaptive learning rate as a linear function of learning error is proposed as in Eq. (27):
\n
\n\nε\n=\nβ\nMSE\n\n\nt\n−\n1\n\n\n\nE27
\n
where is \n\n0\n≤\nβ\n\n a constant.
\n
Figure 5.
The learning errors given by different learning rates.
\n
\n
\n
2.5 Optimization of meta-parameters
\n
The number of RBM that constitute the DBN and the number of neurons of each layer affects prediction performance seriously. In [9], particle swarm optimization (PSO) method is used to decide the structure of DBN, and in [13] it is suggested that random search method [16] is more efficient. In the experiment of time series forecasting by DBN and SGA shown in this chapter, these meta-parameters were decided by the random search, and the exploration limits are shown as the following.
The number of RBMs: [0–3]
The number of units in each layer of DBN: [2–20]
Learning rate of each RBM in Eqs. (4)–(6): [10−5–10−1]
Fixed learning rate of SGA in Eq. (21): [10−5–10−1]
The optimization algorithm of these meta-parameters by the random search method is as follows:
\n
Step 1. Set random values of meta-parameters beyond the exploration limitations.
\n
Step 2. Predict a future data \n\n\ny\nt\n\n≈\n\nx\n\nt\n+\n1\n\n\n\n by MLP or DBN using the current weighted connections.
\n
Step 3. If the error between \n\n\ny\nt\n\n,\n\nx\n\nt\n+\n1\n\n\n\n is reduced enough, store the values of meta-parameters,
\n
or else if the error is not changed,
\n
stop the exploration,
\n
else return to step 1.
\n
\n
\n
\n
3. The experiments and results
\n
\n
3.1 CATS benchmark time series data
\n
CATS time series data is the artificial benchmark data for forecasting competition with ANN methods [3, 4].This artificial time series is given with 5000 data, among which 100 are missed (hidden by competition the organizers). The missed data exist in five blocks:
Elements 981 to 1000
Elements 1981 to 2000
Elements 2981 to 3000
Elements 3981 to 4000
Elements 4981 to 5000
\n
The mean square error \n\n\nE\n1\n\n\n is used as the prediction precision in the competition, and it is computed by the 100 missing data and their predicted values as the following:
where \n\n\n\ny\n¯\n\nt\n\n\n is the long-term prediction result of the missed data. The CATS time series data is shown in Figure 6.
\n
Figure 6.
CATS benchmark data.
\n
The prediction results of different blocks of CATS data are shown in Figure 7. Comparing to the conventional learning method of DBN, i.e., using Hinton’s RBM unsupervised learning method [6, 8] and back-propagation (BP), the proposed method which used the reinforcement learning method SGA instead of BP showed its superiority in the sense of the average prediction precision E1 (see Figure 7f). In addition, the proposed method, DBN with SGA, yielded the highest prediction (E1 measurement) comparing to all previous studies such as MLP with BP, the best prediction of CATS competition IJCNN’04 [4], the conventional DBNs with BP [9, 11], and hybrid models [13]. The details are shown in Table 1.
\n
Figure 7.
The prediction results of different methods for CATS data: (a) block 1; (b) block 2; (c) block 3; (d) block 4; (e) block 5; and (f) results of the long-term forecasting.
The long-term forecasting error comparison of different methods using CATS data.
\n
The meta-parameters obtained by random search method are shown in Table 2. And we found that the MSE of learning, i.e., given by one-ahead prediction results, showed that the proposed method has worse convergence compared to the conventional BP training. In Figure 8, the case of the first block learning MSE of two methods is shown. The convergence of MSE given by BP converged in a long training process and SGA gave unstable MSE of prediction. However, as the basic consideration of a sparse model, the better results of long-term prediction of the proposed method may successfully avoid the over-fitting problem which is caused by the model that is built too strictly by the training sample and loses its robustness for unknown data.
\n
\n
\n
\n
\n\n
\n
\n
DBN with SGA
\n
DBN with BP
\n
\n\n\n
\n
The number of RBMs
\n
3
\n
1
\n
\n
\n
Learning rate of RBM
\n
0.048-0.055-0.026
\n
0.042
\n
\n
\n
Structure of DBN (the number of units and layers)
\n
14-14-18-19-18-2
\n
5-11-2-1
\n
\n
\n
Learning rate of SGA or BP
\n
0.090
\n
0.090
\n
\n
\n
Discount factor \n\nγ\n\n\n
\n
0.082
\n
—
\n
\n
\n
Coefficient \n\nβ\n\n\n
\n
1.320
\n
—
\n
\n\n
Table 2.
Meta-parameters of DBN used for the CATS data (block 1).
\n
Figure 8.
Change of the learning error during fine-tuning (CATS data [1–980]).
\n
\n
\n
3.2 Real time series data
\n
Three types of natural phenomenon time series data provided by Aalto University [17] were used in the one-ahead forecasting experiments of real time series data.
CO2: Atmospheric CO2 from continuous air samples weekly averages atmospheric CO2 concentration derived from continuous air samples, Hawaii, 2225 data
Sea level pressures: Monthly values of the Darwin sea level pressure series, A.D. 1882–1998, 1300 data
Sunspot number: Monthly averages of sunspot numbers from A.D. 1749 to the present 3078 values
\n
The prediction results of these three datasets are shown in Figure 9. Short-term prediction error is shown in Table 3. DBN with the SGA learning method showed its priority in all cases.
\n
Figure 9.
Prediction results by DBN with BP and SGA. (a) Prediction result of CO2 data. (b) Prediction result of Sea level pressure data. (c) Prediction result of Sun spot number data.
The efficiency of random search to find the optimal meta-parameters, i.e., the structure of RBM and MLP, learning rates, discount factor, etc. which are explained in Section 2.5 is shown in Figure 10 in the case of DBN with SGA learning algorithm. The random search results are shown in Table 4.
\n
Figure 10.
Changes of learning error by random search for DBN with SGA.
\n
\n
\n
\n
\n
\n
\n\n
\n
Data series
\n
Total data
\n
Testing data
\n
DBN with BP (the number of units)
\n
DBN with SGA (the number of units)
\n
\n\n\n
\n
CO2
\n
2225
\n
225
\n
15-17-17-1
\n
20-18-7-2
\n
\n
\n
Sea level pressure
\n
1400
\n
400
\n
16-18-18-1
\n
16-20-8-7-2
\n
\n
\n
Sun spot number
\n
3078
\n
578
\n
20-20-17-18-1
\n
19-19-20-10-2
\n
\n\n
Table 4.
Meta-parameters of DBN used for real time series forecasting.
\n
We also used seven types of natural phenomenon time series data of TSDL [18]. The data to be predicted was chosen based on [19] which are named as Lynx, Sunspots, River flow, Vehicles, RGNP, Wine, and Airline. The short-term (one-ahead) prediction results are shown in Figure 11 and Table 5.
\n
Figure 11.
Prediction results of natural phenomenon time series data of TSDL. (a) Prediction result of Lynx; (b) prediction result of sunspots; (c) prediction result of river flow; (d) prediction result of vehicles; (e) prediction result of RGNP; (f) prediction result of wine; and (g) prediction result of airline.
\n
\n
\n
\n
\n\n
\n
Data
\n
DBN with BP
\n
DBN with SGA
\n
\n\n\n
\n
Lynx
\n
0.6547
\n
0.3593
\n
\n
\n
Sunspots
\n
999.54
\n
904.35
\n
\n
\n
River flow
\n
24262.24
\n
16980.46
\n
\n
\n
Vehicles
\n
6.0670
\n
6.1919
\n
\n
\n
RGNP
\n
771.79
\n
469.72
\n
\n
\n
Wine
\n
138743.80
\n
224432.02
\n
\n
\n
Airline
\n
380.60
\n
375.25
\n
\n\n
Table 5.
Prediction MSE of time series data of TSDL.
\n
From Table 5, it can be confirmed that SGA showed its priority to BP except the cases of Vehicles and Wine. From Table 6, an interesting result of random search for meta-parameter showed that the structures of DBN for different datasets were different, not only the number of units on each layer but also the number of RBMs. In the case of SGA learning method, the number of layer for Sunspots, River flow, and Wine were more than DBN using BP learning.
\n
\n
\n
\n
\n
\n
\n\n
\n
Series
\n
Total data
\n
Testing data
\n
DBN with BP
\n
DBN with SGA
\n
\n\n\n
\n
Lynx
\n
114
\n
14
\n
19-16-1
\n
7-14-2
\n
\n
\n
Sunspots
\n
288
\n
35
\n
20-18-11-1
\n
10-12-12-17-2
\n
\n
\n
River flow
\n
600
\n
100
\n
20-17-18-1
\n
19-20-5-18-5-2
\n
\n
\n
Vehicles
\n
252
\n
52
\n
20-13-20-1
\n
20-11-5-2
\n
\n
\n
RGNP
\n
85
\n
15
\n
18-20-1
\n
19-15-2
\n
\n
\n
Wine
\n
187
\n
55
\n
16-15-12-1
\n
18-12-13-11-2
\n
\n
\n
Airline
\n
144
\n
12
\n
15-4-1
\n
13-7-2
\n
\n\n
Table 6.
Size of time series data and structure of prediction network.
\n
\n
\n
\n
4. Discussions
\n
The experiment results showed the DBN composed by multiple RBMs and MLP is the state-of-the-art predictor comparing to all conventional methods in the case of CATS data. Furthermore, the training method for DBN may be more efficient by the RL method SGA for real time series data than using the conventional BP algorithm. Here let us glance back at the development of this useful deep learning method.
Why the DBN composed by multiple RBMs and MLP [11, 13] is better than the DBN with multiple RBMs only [9]?
\n
The output of the last RBM of DBN, a hidden unit of the last RBM in DBN, has a binary value during pretraining process. So the weights of connections between the unit and units of the visible layer of the last RBM are affected and with lower complexity than using multiple units with continuous values, i.e., MLP, or so-called full connections in deep learning architecture.
How are RL methods active at ANN training?
\n
In 1992, Williams proposed to adopt a RL method named REINFORCE to modify artificial neural networks [8]. In 2008, Kuremoto et al. showed the RL method SGA is more efficient than the conventional BP method in the case of time series forecasting [6]. Recently, researchers in DeepMind Ltd. adopted RL into deep neural networks and resulted a famous game software AlphaGo [20, 21, 22, 23].
Why SGA is more efficient than BP?
\n
Generally, the training process for ANN by BP uses mean square error as loss function. So every sample data affects the learning process and results including noise data. Meanwhile, SGA uses reward which may be an error zone to modify the parameters of model. So it has higher robustness for the noisy data and unknown data for real problems.
\n
\n
\n
5. Conclusions
\n
A deep belief net (DBN) composed by multiple restricted Boltzmann machines (RBMs) and multilayer perceptron (MLP) for time series forecasting were introduced in this chapter. The training method of DBN is also discussed as well as a reinforcement learning (RL) method; stochastic gradient ascent (SGA) showed its priority to the conventional error back-propagation (BP) learning method. The robustness of SGA comes from the utilization of relaxed prediction error during the learning process, i.e., different from the BP method which adopts all errors of every sample to modify the model. Additionally, the optimization of the structure of DBN was realized by random search method. Time series forecasting experiments used benchmark CATS data, and real time series datasets showed the effectiveness of the DBN. As for the future work, there are still some problems that need to be solved such as how to design the variable learning rate and reward which influence the learning performance strongly and how to prevent the explosion of characteristic eligibility trace in SGA.
\n
\n\n',keywords:"artificial neural networks (ANN), deep learning (DL), reinforcement learning (RL), deep belief net (DBN), restricted Boltzmann machine (RBM), multilayer perceptron (MLP), stochastic gradient ascent (SGA)",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/66346.pdf",chapterXML:"https://mts.intechopen.com/source/xml/66346.xml",downloadPdfUrl:"/chapter/pdf-download/66346",previewPdfUrl:"/chapter/pdf-preview/66346",totalDownloads:1394,totalViews:0,totalCrossrefCites:0,dateSubmitted:"June 14th 2018",dateReviewed:"February 26th 2019",datePrePublished:"April 3rd 2019",datePublished:"November 6th 2019",readingETA:"0",abstract:"As a kind of efficient nonlinear function approximators, artificial neural networks (ANN) have been popularly applied to time series forecasting. The training method of ANN usually utilizes error back-propagation (BP) which is a supervised learning algorithm proposed by Rumelhart et al. in 1986; meanwhile, authors proposed to improve the robustness of the ANN for unknown time series prediction using a reinforcement learning algorithm named stochastic gradient ascent (SGA) originally proposed by Kimura and Kobayashi for control problems in 1998. We also successfully use a deep belief net (DBN) stacked by multiple restricted Boltzmann machines (RBMs) to realized time series forecasting in 2012. In this chapter, a state-of-the-art time series forecasting system that combines RBMs and multilayer perceptron (MLP) and uses SGA training algorithm is introduced. Experiment results showed the high prediction precision of the novel system not only for benchmark data but also for real phenomenon time series data.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/66346",risUrl:"/chapter/ris/66346",signatures:"Takashi Kuremoto, Takaomi Hirata, Masanao Obayashi, Shingo Mabu and Kunikazu Kobayashi",book:{id:"8362",title:"Time Series Analysis",subtitle:"Data, Methods, and Applications",fullTitle:"Time Series Analysis - Data, Methods, and Applications",slug:"time-series-analysis-data-methods-and-applications",publishedDate:"November 6th 2019",bookSignature:"Chun-Kit Ngan",coverURL:"https://cdn.intechopen.com/books/images_new/8362.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"227503",title:"Dr.",name:"Chun-Kit",middleName:null,surname:"Ngan",slug:"chun-kit-ngan",fullName:"Chun-Kit Ngan"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"14090",title:"Prof.",name:"Masanao",middleName:null,surname:"Obayashi",fullName:"Masanao Obayashi",slug:"masanao-obayashi",email:"m.obayas@yamaguchi-u.ac.jp",position:null,institution:{name:"Yamaguchi University",institutionURL:null,country:{name:"Japan"}}},{id:"15895",title:"Assistant Prof.",name:"Takashi",middleName:null,surname:"Kuremoto",fullName:"Takashi Kuremoto",slug:"takashi-kuremoto",email:"wu@yamaguchi-u.ac.jp",position:null,institution:{name:"Yamaguchi University",institutionURL:null,country:{name:"Japan"}}},{id:"208510",title:"Prof.",name:"Shingo",middleName:null,surname:"Mabu",fullName:"Shingo Mabu",slug:"shingo-mabu",email:"mabu@yamaguchi-u.ac.jp",position:null,institution:null},{id:"208511",title:"Prof.",name:"Kunikazu",middleName:null,surname:"Kobayashi",fullName:"Kunikazu Kobayashi",slug:"kunikazu-kobayashi",email:"kobayashi@ist.aichi-pu.ac.jp",position:null,institution:null},{id:"283417",title:"Dr.",name:"Takashi",middleName:null,surname:"Kuremoto",fullName:"Takashi Kuremoto",slug:"takashi-kuremoto",email:"v003we@yamaguchi-u.ac.jp",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. The DBN model for time series forecasting",level:"1"},{id:"sec_2_2",title:"2.1 The structure of the model",level:"2"},{id:"sec_3_2",title:"2.2 RBM",level:"2"},{id:"sec_4_2",title:"2.3 MLP",level:"2"},{id:"sec_5_2",title:"2.4 The training method of DBN",level:"2"},{id:"sec_6_2",title:"2.5 Optimization of meta-parameters",level:"2"},{id:"sec_8",title:"3. The experiments and results",level:"1"},{id:"sec_8_2",title:"3.1 CATS benchmark time series data",level:"2"},{id:"sec_9_2",title:"3.2 Real time series data",level:"2"},{id:"sec_11",title:"4. Discussions",level:"1"},{id:"sec_12",title:"5. Conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'Box GEP, Pierce DA. Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association. 1970;65(332):1509-1526'},{id:"B2",body:'Casdagli M. Nonlinear prediction of chaotic time series. Physica D. 1989;35:335-356'},{id:"B3",body:'Lendasse A, Oja E, Simula O, Verleysen M. Time series prediction competition: The CATS benchmark. In: Proceedings of International Joint Conference on Neural Networks (IJCNN\'04); 2004. pp. 1615-1620'},{id:"B4",body:'Lendasse A, Oja E, Simula O, Verleysen M. Time series prediction competition: The CATS benchmark. Neurocomputing. 2007;70:2325-2329'},{id:"B5",body:'NN3. http://www.neural-forecasting-competition.com/NN3/index.htm'},{id:"B6",body:'Rumelhart DE, Hinton GE, Williams RJ. Learning representation by back-propagating errors. Nature. 1986;232(9):533-536'},{id:"B7",body:'Kuremoto T, Obayashi M, Kobayashi M. Neural forecasting systems, Chapter 1. In: Weber C, Elshaw M, Mayer NM, editors. Reinforcement Learning, Theory and Applications. Rijeka, Croatia: InTech; 2008. pp. 1-20'},{id:"B8",body:'Kimura H, Kobayashi S. Reinforcement learning for continuous action using stochastic gradient ascent. In: Proceedings of 5th Intelligent Autonomous Systems (IAS-5); 1998. pp. 288-295'},{id:"B9",body:'Williams RJ. Simple statistical gradient following algorithms for connectionist reinforcement learning. Machine Learning. 1992;8:229-256'},{id:"B10",body:'Kuremoto T, Kimura S, Kobayashi K, Obayashi M. Time series forecasting using a deep belief network with restricted Boltzmann machines. Neurocomputing. Aug. 2014;137(5):47-56'},{id:"B11",body:'Hinton GE, Salakhutdinov RR. Reducing the dimensionality of data with neural networks. Science. 2006;313:504-507'},{id:"B12",body:'Kuremoto T, Hirata T, Obayashi M, Mabu S, Kobayashi K. Forecast chaotic time series data by DBNs. In: Proceedings of the 7th International Congress on Image and Signal Processing (CISP 2014); Oct. 2014. pp. 1304-1309'},{id:"B13",body:'Zhang GP. Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing. 2003;50:159-175'},{id:"B14",body:'Hirata T, Kuremoto T, Obayashi M, Mabu S. Time series prediction using DBN and ARIMA. In: Proceedings of International Conference on Computer Application Technologies (CCATS 2015). Matsue, Japan; Sep. 2015. pp. 24-29'},{id:"B15",body:'Hirata T, Kuremoto T, Obayashi M, Mabu S, Kobayashi K. Deep belief network using reinforcement learning and its applications to time series forecasting. In: Proceedings of International Conference on Neural Information Processing, (ICONIP’16), Lecture Notes in Computer Science (LNCS). Heidelberg, Germany: Springer. Vol. 9949. Kyoto, Japan; Oct. 18–21, 2016. pp. 30-37'},{id:"B16",body:'Hirata T, Kuremoto T, Obayashi M, Mabu S, Kobayashi K. Forecasting real time series data using deep belief net and reinforcement learning. Journal of Robotics, Network and Artificial Life. 2018;4(4):260-264. DOI: 10.2991/jrnal.2018.4.4.1'},{id:"B17",body:'Bergstra J, Bengio Y. Random search for hyper-parameter optimization. Journal of Machine Learning Research. 2012;13:281-305'},{id:"B18",body:'Aalto University Applications of Machine Learning Group Datasets. Available online at url: <http://research.ics.aalto.fi/eiml/datasets.shtml> (01-01-17)'},{id:"B19",body:'Hyndman RJ. Time Series Data Library (TSDL). 2013. Available online at url: 〈http://robjhyndman.com/TSDL/〉 (01-01-13)'},{id:"B20",body:'Adhikari R. A neural network based linear ensemble framework for time series forecasting. Neurocomputing. 2015;157:231-242'},{id:"B21",body:'Mnih V et al. Human-level control through deep reinforcement learning. Nature. 2015;518:529-533'},{id:"B22",body:'Sivler D et al. Mastering the game of go with deep neural networks and tree search. Nature. 2017;529:484-489'},{id:"B23",body:'Sivler D et al. Mastering the game of go without human knowledge. Nature. 2017;550:354-359'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Takashi Kuremoto",address:"wu@yamaguchi-u.ac.jp",affiliation:'
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Integrity - We are consistent and dependable, always striving for precision and accuracy in the true spirit of science.
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Openness - We communicate honestly and transparently. We are open to constructive criticism and committed to learning from it.
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Disruptiveness - We are eager for discovery, for new ideas and for progression. We approach our work with creativity and determination, with a clear vision that drives us forward. We look beyond today and strive for a better tomorrow.
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IntechOpen is a dynamic, vibrant company, where exceptional people are achieving great things. We offer a creative, dedicated, committed, and passionate environment but never lose sight of the fact that science and discovery is exciting and rewarding. We constantly strive to ensure that members of our community can work, travel, meet world-renowned researchers and grow their own career and develop their own experiences.
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If this sounds like a place that you would like to work, whether you are at the beginning of your career or are an experienced professional, we invite you to drop us a line and tell us why you could be the right person for IntechOpen.
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