Early detection of metabolic and energy disorders by thermal time series stochastic complexity analysis
Identifieur interne : 000438 ( Pmc/Corpus ); précédent : 000437; suivant : 000439Early detection of metabolic and energy disorders by thermal time series stochastic complexity analysis
Auteurs : N. A. Lutaif ; R. Palazzo ; J. A. R. GontijoSource :
- Brazilian Journal of Medical and Biological Research [ 0100-879X ] ; 2014.
Abstract
Maintenance of thermal homeostasis in rats fed a high-fat diet (HFD) is associated with changes in their thermal balance. The thermodynamic relationship between heat dissipation and energy storage is altered by the ingestion of high-energy diet content. Observation of thermal registers of core temperature behavior, in humans and rodents, permits identification of some characteristics of time series, such as autoreference and stationarity that fit adequately to a stochastic analysis. To identify this change, we used, for the first time, a stochastic autoregressive model, the concepts of which match those associated with physiological systems involved and applied in male HFD rats compared with their appropriate standard food intake age-matched male controls (n=7 per group). By analyzing a recorded temperature time series, we were able to identify when thermal homeostasis would be affected by a new diet. The autoregressive time series model (AR model) was used to predict the occurrence of thermal homeostasis, and this model proved to be very effective in distinguishing such a physiological disorder. Thus, we infer from the results of our study that maximum entropy distribution as a means for stochastic characterization of temperature time series registers may be established as an important and early tool to aid in the diagnosis and prevention of metabolic diseases due to their ability to detect small variations in thermal profile.
Url:
DOI: 10.1590/1414-431X20133097
PubMed: 24519093
PubMed Central: 3932975
Links to Exploration step
PMC:3932975Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Early detection of metabolic and energy disorders by
thermal time series stochastic complexity analysis</title><author><name sortKey="Lutaif, N A" sort="Lutaif, N A" uniqKey="Lutaif N" first="N. A." last="Lutaif">N. A. Lutaif</name><affiliation><nlm:aff id="aff1">Departamento de Clínica Médica, Faculdade de Ciências Médicas, Universidade Estadual de Campinas, Campinas, SP, Brasil</nlm:aff></affiliation></author><author><name sortKey="Palazzo, R" sort="Palazzo, R" uniqKey="Palazzo R" first="R." last="Palazzo">R. Palazzo</name><affiliation><nlm:aff id="aff2">Departamento de Telemática, Faculdade de Engenharia Elétrica e Computação, Universidade Estadual de Campinas, Campinas, SP, Brasil</nlm:aff></affiliation></author><author><name sortKey="Gontijo, J A R" sort="Gontijo, J A R" uniqKey="Gontijo J" first="J. A. R." last="Gontijo">J. A. R. Gontijo</name><affiliation><nlm:aff id="aff1">Departamento de Clínica Médica, Faculdade de Ciências Médicas, Universidade Estadual de Campinas, Campinas, SP, Brasil</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">24519093</idno><idno type="pmc">3932975</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3932975</idno><idno type="RBID">PMC:3932975</idno><idno type="doi">10.1590/1414-431X20133097</idno><date when="2014">2014</date><idno type="wicri:Area/Pmc/Corpus">000438</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000438</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Early detection of metabolic and energy disorders by
thermal time series stochastic complexity analysis</title><author><name sortKey="Lutaif, N A" sort="Lutaif, N A" uniqKey="Lutaif N" first="N. A." last="Lutaif">N. A. Lutaif</name><affiliation><nlm:aff id="aff1">Departamento de Clínica Médica, Faculdade de Ciências Médicas, Universidade Estadual de Campinas, Campinas, SP, Brasil</nlm:aff></affiliation></author><author><name sortKey="Palazzo, R" sort="Palazzo, R" uniqKey="Palazzo R" first="R." last="Palazzo">R. Palazzo</name><affiliation><nlm:aff id="aff2">Departamento de Telemática, Faculdade de Engenharia Elétrica e Computação, Universidade Estadual de Campinas, Campinas, SP, Brasil</nlm:aff></affiliation></author><author><name sortKey="Gontijo, J A R" sort="Gontijo, J A R" uniqKey="Gontijo J" first="J. A. R." last="Gontijo">J. A. R. Gontijo</name><affiliation><nlm:aff id="aff1">Departamento de Clínica Médica, Faculdade de Ciências Médicas, Universidade Estadual de Campinas, Campinas, SP, Brasil</nlm:aff></affiliation></author></analytic><series><title level="j">Brazilian Journal of Medical and Biological Research</title><idno type="ISSN">0100-879X</idno><idno type="eISSN">1414-431X</idno><imprint><date when="2014">2014</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Maintenance of thermal homeostasis in rats fed a high-fat diet (HFD) is
associated with changes in their thermal balance. The thermodynamic relationship
between heat dissipation and energy storage is altered by the ingestion of
high-energy diet content. Observation of thermal registers of core temperature
behavior, in humans and rodents, permits identification of some characteristics
of time series, such as autoreference and stationarity that fit adequately to a
stochastic analysis. To identify this change, we used, for the first time, a
stochastic autoregressive model, the concepts of which match those associated
with physiological systems involved and applied in male HFD rats compared with
their appropriate standard food intake age-matched male controls (n=7 per
group). By analyzing a recorded temperature time series, we were able to
identify when thermal homeostasis would be affected by a new diet. The
autoregressive time series model (AR model) was used to predict the occurrence
of thermal homeostasis, and this model proved to be very effective in
distinguishing such a physiological disorder. Thus, we infer from the results of
our study that maximum entropy distribution as a means for stochastic
characterization of temperature time series registers may be established as an
important and early tool to aid in the diagnosis and prevention of metabolic
diseases due to their ability to detect small variations in thermal profile.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Higgins, Jp" uniqKey="Higgins J">JP Higgins</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Glass, L" uniqKey="Glass L">L Glass</name></author><author><name sortKey="Kaplan, D" uniqKey="Kaplan D">D Kaplan</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Silva, Gj" uniqKey="Silva G">GJ Silva</name></author><author><name sortKey="Ushizima, Mr" uniqKey="Ushizima M">MR Ushizima</name></author><author><name sortKey="Lessa, Ps" uniqKey="Lessa P">PS Lessa</name></author><author><name sortKey="Cardoso, L" uniqKey="Cardoso L">L Cardoso</name></author><author><name sortKey="Drager, Lf" uniqKey="Drager L">LF Drager</name></author><author><name sortKey="Atala, Mm" uniqKey="Atala M">MM 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Braz J Med Biol Res</journal-id><journal-id journal-id-type="iso-abbrev">Braz. J. Med. Biol. Res</journal-id><journal-title-group><journal-title>Brazilian Journal of Medical and Biological Research</journal-title></journal-title-group><issn pub-type="ppub">0100-879X</issn><issn pub-type="epub">1414-431X</issn><publisher><publisher-name>Associação Brasileira de Divulgação Científica</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">24519093</article-id><article-id pub-id-type="pmc">3932975</article-id><article-id pub-id-type="doi">10.1590/1414-431X20133097</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Articles</subject></subj-group></article-categories><title-group><article-title>Early detection of metabolic and energy disorders by
thermal time series stochastic complexity analysis</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Lutaif</surname><given-names>N.A.</given-names></name><xref ref-type="aff" rid="aff1">1</xref></contrib><contrib contrib-type="author"><name><surname>Palazzo</surname><given-names>R.</given-names><suffix>Jr</suffix></name><xref ref-type="aff" rid="aff2">2</xref></contrib><contrib contrib-type="author"><name><surname>Gontijo</surname><given-names>J.A.R.</given-names></name><xref ref-type="aff" rid="aff1">1</xref></contrib><aff id="aff1"><label>1</label>Departamento de Clínica Médica, Faculdade de Ciências Médicas, Universidade Estadual de Campinas, Campinas, SP, Brasil</aff><aff id="aff2"><label>2</label>Departamento de Telemática, Faculdade de Engenharia Elétrica e Computação, Universidade Estadual de Campinas, Campinas, SP, Brasil</aff></contrib-group><author-notes><corresp>Correspondence: J.A.R. Gontijo, Departamento de Clínica Médica,
Faculdade de Ciências Médicas, Universidade Estadual de Campinas,
13083-887 Campinas, SP, Brasil, +55-19-3521-8925. E-mail:
<email>gontijo@fcm.unicamp.br</email></corresp></author-notes><pub-date pub-type="collection"><month>1</month><year>2014</year></pub-date><pub-date pub-type="epub"><day>17</day><month>1</month><year>2014</year></pub-date><volume>47</volume><issue>1</issue><fpage>70</fpage><lpage>79</lpage><history><date date-type="received"><day>18</day><month>3</month><year>2013</year></date><date date-type="accepted"><day>6</day><month>8</month><year>2013</year></date></history><permissions><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by-nc/3.0/"><license-p>This is an Open Access article distributed under the terms of the Creative
Commons Attribution Non-Commercial License, which permits unrestricted
non-commercial use, distribution, and reproduction in any medium, provided the
original work is properly cited.</license-p></license></permissions><abstract><p>Maintenance of thermal homeostasis in rats fed a high-fat diet (HFD) is
associated with changes in their thermal balance. The thermodynamic relationship
between heat dissipation and energy storage is altered by the ingestion of
high-energy diet content. Observation of thermal registers of core temperature
behavior, in humans and rodents, permits identification of some characteristics
of time series, such as autoreference and stationarity that fit adequately to a
stochastic analysis. To identify this change, we used, for the first time, a
stochastic autoregressive model, the concepts of which match those associated
with physiological systems involved and applied in male HFD rats compared with
their appropriate standard food intake age-matched male controls (n=7 per
group). By analyzing a recorded temperature time series, we were able to
identify when thermal homeostasis would be affected by a new diet. The
autoregressive time series model (AR model) was used to predict the occurrence
of thermal homeostasis, and this model proved to be very effective in
distinguishing such a physiological disorder. Thus, we infer from the results of
our study that maximum entropy distribution as a means for stochastic
characterization of temperature time series registers may be established as an
important and early tool to aid in the diagnosis and prevention of metabolic
diseases due to their ability to detect small variations in thermal profile.</p></abstract><kwd-group><kwd>Time series temperature</kwd><kwd>Thermal homeostasis</kwd><kwd>High-fat diet</kwd><kwd>Metabolic disorder predictor</kwd><kwd>Autoregressive models</kwd></kwd-group><counts><fig-count count="6"></fig-count><table-count count="1"></table-count><ref-count count="32"></ref-count><page-count count="10"></page-count></counts></article-meta></front><body><sec><title>Introduction</title><p>A study performed at least 50 years ago demonstrated that reduced variability and
linear responses of biological signals, by the statistical and mathematical analysis
of intrauterus heart rate variability, would be a predictive value for future
underlying disease manifestation (<xref rid="B01" ref-type="bibr">1</xref>). Since
then, chaos theory has described elements manifesting behavior that is extremely
sensitive to initial conditions, does not repeat itself, and yet is deterministic; a
sequential series register of several biological parameters such as blood pressure
measurements, brain electrical activity, and renal ion transport, and possibly
temperature, presents nonlinear unpredictable chaotic behavior, a kind of order
without periodicity, and several investigators have implemented studies trying to
define and distinguish normal variability from pathological behavior profiles (<xref rid="B02" ref-type="bibr">2</xref>). Simultaneously, in the past few years,
along with the development of descriptive statistics and computer tools, new models
of variance analysis were proposed, and studies which were analyzed by
interpretation of averages and deviations began to include new mathematical concepts
such as geometric models, stochastic time series, chaos theory, and spectral
analysis (<xref rid="B03" ref-type="bibr">3</xref>-<xref rid="B06" ref-type="bibr">6</xref>).</p><p>Observation of the thermal registers of core temperature behavior, in humans and
rodents, permits identification of some characteristics of time series such as
autoreference and immobility that fit adequately into a stochastic analysis. These
characteristics are associated with the physiological nature of an internal thermal
series, which is modulated by a complex superstructure that acts at the behavioral
level up to molecular signaling (<xref rid="B07" ref-type="bibr">7</xref>,<xref rid="B08" ref-type="bibr">8</xref>). In this case, this regulatory
superstructure reflects the importance of central thermal homeostasis for keeping
temperature within a narrow variation range in mammalian species. In fact, among all
the biological signals measured in humans, temperature is the one that has the
smallest relative deviation (±0.5°C around the mean value of 36.7°C)
(<xref rid="B09" ref-type="bibr">9</xref>). The hierarchical relevance of the
central nervous system (CNS) to thermal control activities is explained by fine
adjustments of maximum enzyme reaction activities to a given temperature in
homeothermic animals.</p><p>In homeothermic animals, the inability to cope with the excess energy absorbed by a
hypercaloric diet leads to the phenomenon known as lipotoxicity, a concept
interpreted as the cost of keeping the temperature at a maximum but stable level, in
order to promote increased fatty acid oxidation and heat release. To maintain
thermal stability, animals develop obesity and impaired glucose tolerance, responses
associated with reduced thermogenesis (<xref rid="B10" ref-type="bibr">10</xref>).
Thus, we may hypothesize that the energy imbalance may reflect, at an earlier stage,
a subtle change in the variability of the thermal series of these animals. However,
from what we know so far, no studies have been performed using complexity stochastic
analysis to define the temperature behavior in normal and pathological models. To
identify this change, we have used, for the first time, a stochastic autoregressive
model, the concepts of which match those associated with physiological systems
involved (<xref rid="B11" ref-type="bibr">11</xref>,<xref rid="B12" ref-type="bibr">12</xref>) and applied in high-fat intake diet (HFD) rats compared with their
appropriate standard food intake controls. The HFD model is extremely efficient for
creating pathophysiological conditions such as hyperleptinemia, peripheral insulin
resistance, diabetes mellitus, and obesity that lead to long-term metabolic and
energy disorders. Blocking insulin and leptin pathways at the level of the
hypothalamic nuclei promotes increased thermogenic activity in the body, leading to
adjustments in the CNS to maintain temperature stability. Therefore, the purpose of
the present study was two-fold: a) to identify a stochastic process associated with
thermal time series involved with the HFD procedure, and b) from characterization of
this stochastic process, to propose a model of an algebraic system as a tool for
early and efficient detection of changes in physiological signals.</p></sec><sec sec-type="materials|methods"><title>Material and Methods</title><sec><title>Animals</title><p>General guidelines established by the Brazilian College of Animal Experimentation
(COBEA) and approved by the Institutional Ethics Committee (CEEA/UNICAMP 1697-1)
were followed throughout the investigation. Our local colonies originated from a
breeding stock supplied by CEMIB/UNICAMP, Campinas, SP, Brazil. Experiments were
conducted on age-matched, male offspring of sibling-mated Sprague Dawley rats
(n=7 for each group). At 6 weeks of age (180.5±11.7 g body weight), the
animals were housed in individual cages and maintained under controlled
temperature (22°C) and lighting conditions (7:00 am to 7:00 pm), with free
access to tap water, and were randomly distributed into two dietary groups: a
control group fed with pelleted standard rodent laboratory chow (standard diet,
SD; Nuvital, Brazil) or a high-fat diet group (HFD) fed a high-fat diet for 4
weeks, up to 10 weeks of age. The standard chow diet contained 11.9% kcal as fat
and a total of 2.9 kcal/g, and the high-fat diet contained 58.3% kcal as fat and
a total of 5.44 kcal/g. All male rats were weighed weekly and had food and water
intake measured daily throughout the experiment. After an adjustment Subcue
period of 48 h, the 10-week-old rats were anesthetized with a mixture of
ketamine (75 mg/kg body weight, <italic>ip</italic>) and xylazine (10 mg/kg body
weight, <italic>ip</italic>), and a thermal probe (Data Logger, USA) was placed
inside the abdominal cavity. The recording of internal temperature was scheduled
to start at 1 week after the surgical proceedings, and data were recorded for 6
h in both the SD and HFD groups, with a 1-min period between recordings. During
the experiments, the animals were kept isolated in an environment of low
external stimuli and monitored with respect to the acceptance of diet and
vitality. At the end of the assay, a lethal dose of thiopental was administered
and data were transferred to the program Subcue-Data Logger Mathematica 7.0
(Wolfram Research, Inc., USA).</p></sec><sec><title>Assumptions</title><p>The chosen time interval for the recorded temperature of each pair of animals
submitted to analysis was between 0:00 and 6:00 am, in SD rats and HFD animals.
For mammals, although internal temperature is considered to be an inherently
stationary time series with variability fixed at about 0.5°C, we reinforced
this physiological condition by subtracting the mean value of the entire series
from each individual measurement (<xref rid="B13" ref-type="bibr">13</xref>), to
neutralize the action of hormonal circadian rhythms, which could lead to a
certain degree of seasonality in the series (averaging). According to regulatory
dynamic characteristics of the thermal control, the mechanisms of heat
production and heat dissipation have to work in synergy (<xref rid="B07" ref-type="bibr">7</xref>).</p><p>Based on this thermal profile, we consider animal temperature as a source of
information. The characterization of a source of information comes from
establishment of the stochastic process. On the other hand, from the information
theory point of view, every source has an associated measure quantifying its
output. In our case, the associated measure is the entropy (rate). As a
consequence, one of the goals in source coding is to maximize the source entropy
under an appropriate set of constraints.</p><p>In this article, we focus on the characterization of the stochastic process
associated with animal temperature, which will maximize a statistical measure
(the entropy rate) under some constraints (the autocorrelation function) (<xref rid="B14" ref-type="bibr">14</xref>).</p><p>In this direction, we assume a stationary stochastic process represents animal
temperature <italic>x</italic> = {<italic>X</italic>(<italic>k</italic>)}. A
source is said to be stationary if given:
<italic>X</italic>(1),<italic>X</italic>(2),…,<italic>X</italic>(<italic>n</italic>),
and <italic>X</italic>(1 + <italic>I</italic>),<italic>X</italic>(2 +
<italic>I</italic>),…,<italic>X</italic>(<italic>n</italic> +
<italic>I</italic>) then, <italic>P</italic>[<italic>X</italic>(1) =
<italic>x</italic><sub>1</sub>,<italic>X</italic>(2) = <italic>x</italic><sub>2</sub>,…,<italic>X</italic>(<italic>n</italic>) =
<italic>x<sub>n</sub></italic>] = <italic>P</italic>[<italic>X</italic>(1 + <italic>I</italic>)
= <italic>x</italic><sub>1</sub>,<italic>P</italic>(2 + <italic>I</italic>) =
<italic>x</italic><sub>2</sub>,…,<italic>P</italic>(<italic>n</italic> +
<italic>I</italic>) = <italic>x<sub>n</sub></italic>] for any integer <italic>n</italic>, <italic>I</italic> ≥ 1,
therefore χ has a constant variance and its autocorrelation function
depends only on <italic>I</italic>–(<italic>I</italic> +
<italic>n</italic>).</p><p>For stationary processes, it is possible to determine the entropy rate of the
source {<italic>X</italic>(<italic>k</italic>)} (<xref rid="B14" ref-type="bibr">14</xref>). The entropy rate of an information source is defined
as<disp-formula id="bjmbr-20133097-e028"><mml:math id="equ01"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mtext>lim</mml:mtext><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mi>ϖ</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mfrac><mml:mo>=</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mtext>lim</mml:mtext><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mi>ϖ</mml:mi></mml:mrow></mml:msub><mml:mtext> </mml:mtext><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>when the limit exists. On the other hand, we also assume the
source is ergodic, that is, the time average exhibited by only one realization
of the source output is equal to the ensemble average with probability one.
Under these conditions, the Shannon-McMillan-Breiman theorem (<xref rid="B15" ref-type="bibr">15</xref>) guarantees convergence of the entropy
rate.</p><p>From these facts, the problem we have to solve may be stated as that of
determining maximum entropy distributions, as follows.<disp-formula id="bjmbr-20133097-e030"><mml:math id="equ02"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mtext>Problem: </mml:mtext><mml:munder><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>−</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> log </mml:mtext><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>Subject: </mml:mtext><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mstyle><mml:mo>,</mml:mo><mml:mtext> for </mml:mtext><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>i</mml:mi><mml:mo>≤</mml:mo><mml:mtext>m</mml:mtext><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>where <italic>P</italic> is the set of all probabilities or
probability density functions α<sub>i</sub> are known constants,
<italic>r<sup>i</sup></italic>(<italic>x</italic>) = <italic>x<sup>i</sup></italic> the <italic>i</italic><sup>th</sup> moment of the random variable <italic>X</italic>, and
<italic>p(x)</italic> denotes either the probability of the discrete random
variable <italic>X</italic> or the probability density function of the
continuous random variable <italic>X</italic>. The summation symbol is used
either when <italic>X</italic> is a discrete random variable or when
<italic>X</italic> is a continuous random variable. In this latter case,
identification with the integral symbol should be clear. As a consequence,
differential entropy <italic>h</italic>(<italic>x</italic>) is used instead of
entropy rate <italic>H</italic>(<italic>x</italic>).</p></sec><sec><title>Autoregressive model</title><p>From the previous subsection, taking the earlier problem, we are faced with
finding the stochastic process {<italic>X</italic>(<italic>i</italic>)}
associated with animal temperature, which maximizes the entropy rate under the
constraint of the autocorrelation function, that is<disp-formula id="bjmbr-20133097-e001"><mml:math id="equ03"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:munder><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>−</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> log </mml:mtext><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mtext> subject to</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(1)</label></disp-formula>for <italic>k</italic> = 1,2,…,κ and all
<italic>i</italic>.</p><p>The solution to this problem is the <italic>k</italic><sup>th</sup> order Gauss-Markov process (<xref rid="B14" ref-type="bibr">14</xref>) of the form<disp-formula id="bjmbr-20133097-e002"><mml:math id="equ04"><mml:mrow><mml:mtext>X</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mtext>i</mml:mtext><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math><label>(2)</label></disp-formula>where Z(<italic>i</italic>)'s are independent, identically
distributed normal random variables with zero mean and variance
σ<sup>2</sup>, denoted by N(0,σ<sup>2</sup>),
a<sub>1</sub>,a<sub>2</sub>,...,a<sub>k</sub>, σ<sup>2</sup> are
unknown coefficients chosen to satisfy the autocorrelation function, and
<italic>k</italic> is the order of the Markov process.</p><p>In a practical problem, we usually know a sample sequence
<italic>X</italic>(1),<italic>X</italic>(2),…<italic>X</italic>(<italic>n</italic>)
from which the autocorrelation
<italic>R</italic>(0),<italic>R</italic>(1),…,<italic>R</italic>(<italic>κ</italic>)
may be calculated. Therefore, the question is: knowing the process has the form
in <xref ref-type="disp-formula" rid="bjmbr-20133097-e002">Equation 2</xref>, is
it possible to choose a<sub>k</sub> values to satisfy the constraints (<xref ref-type="disp-formula" rid="bjmbr-20133097-e001">Equation 1</xref>)?
The answer to this question is yes. The procedure to achieve this goal, after
some algebraic manipulations (in <xref ref-type="disp-formula" rid="bjmbr-20133097-e002">Equation 2</xref>), is given by<disp-formula id="bjmbr-20133097-e003"><mml:math id="equ05"><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>E</mml:mi><mml:mo>{</mml:mo><mml:msup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>}</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo></mml:mrow></mml:mstyle><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mstyle></mml:mrow></mml:math><label>(3)</label></disp-formula><disp-formula id="bjmbr-20133097-e004"><mml:math id="equ06"><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mstyle><mml:mtext> </mml:mtext><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:mrow></mml:math><label>(4)</label></disp-formula></p><p><xref ref-type="disp-formula" rid="bjmbr-20133097-e003">Equations 3</xref> and
<xref ref-type="disp-formula" rid="bjmbr-20133097-e004">4</xref> are known
as Yule-Walker equations. This set of equations has a unique solution, and from
them it is possible to determine the <italic>a<sub>k</sub></italic> values from the known <italic>R</italic>(<italic>k</italic>). The next
question is, how many correlation lags should we consider? That is, what is
the optimum value of <italic>κ</italic>? The answer to this question is to
make use of the Levinson-Durbin algorithm (<xref rid="B16" ref-type="bibr">16</xref>,<xref rid="B17" ref-type="bibr">17</xref>). Formally, from the
Yule-Walker equations, a system of recurrent linear equations, the
aforementioned algorithm may find the coefficients. In addition to this, an
adjustment is needed between the set of experimental data and the algorithm
(model) used to represent it. Hence, this adjustment is achieved by use of the
selection method based on the information criteria (<xref rid="B18" ref-type="bibr">18</xref>,<xref rid="B19" ref-type="bibr">19</xref>). The idea is
to select memory orders smaller or larger than the true memory order in order to
balance them. For instance, if we consider <xref ref-type="disp-formula" rid="bjmbr-20133097-e005">Equation 5</xref>, the term <inline-formula><mml:math id="equ07"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> may be interpreted as a penalty for selecting memory orders
smaller, and the term <inline-formula><mml:math id="equ08"><mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mi>ln</mml:mi><mml:mo></mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:mfrac></mml:math></inline-formula> can be interpreted as a penalty for selecting memory orders
larger than the true value. Therefore, minimizing these two quantities leads to
the true memory order. One of these information criteria is known as the
Bayesian Information Criterion (BIC). In the case of thermal time series, the
memory order is determined for the model satisfying<disp-formula id="bjmbr-20133097-e005"><mml:math id="equ09"><mml:mrow><mml:mtext>min</mml:mtext><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtext>ln </mml:mtext><mml:msup><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mtext> ln </mml:mtext><mml:mi>n</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:mfrac></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math><label>(5)</label></disp-formula>where <italic>k</italic> is the memory order of the AR model,
<italic>n</italic> is the sample length, and <inline-formula><mml:math id="equ10"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> is the maximum likelihood estimator of the noise variance
given by <inline-formula><mml:math id="equ11"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>j</mml:mi></mml:munderover><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:msub><mml:mi>r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:mrow></mml:math></inline-formula>, where <italic>X</italic>(<italic>j</italic>) is the original
series and <inline-formula><mml:math id="equ12"><mml:mrow><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> is the series generated by the model and is the coefficient of
the correlation from the original series.</p></sec><sec><title>System theory</title><p>One could ask if a system theory approach may be employed to provide the same
information as that obtained from rat temperature sample sequences. In this
direction, consider a discrete time system with input
<italic>Z</italic>(<italic>n</italic>), impulse response
<italic>H</italic>(<italic>n</italic>) and output
<italic>X</italic>(<italic>n</italic>). Let
<italic>Z</italic>(<italic>z</italic>),
<italic>H</italic>(<italic>z</italic>) and
<italic>X</italic>(<italic>z</italic>) be the corresponding
<italic>z</italic>-transform, that is,
<italic>Z</italic>(<italic>z</italic>) = Σ<sub><italic>n</italic></sub><italic>Z</italic>(<italic>n</italic>)<italic>z</italic><sup>−<italic>n</italic></sup>, <italic>H</italic>(<italic>z</italic>) = Σ<sub><italic>n</italic></sub><italic>H</italic>(<italic>n</italic>)<italic>z</italic><sup>−<italic>n</italic></sup>, and <italic>X</italic>(<italic>z</italic>) = Σ<sub><italic>n</italic></sub><italic>X</italic>(<italic>n</italic>)<italic>z</italic><sup>−<italic>n</italic></sup>. From the previous subsection, we know that the stochastic process is a
<italic>k</italic><sup>th</sup> order Gauss-Markov process and, consequently, an autoregressive
process. Under a system theory point of view, an autoregressive process is such
that the system has only poles, that is,<disp-formula id="bjmbr-20133097-e006"><mml:math id="equ13"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>j</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mstyle></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math><label>(6)</label></disp-formula>Equivalently,<disp-formula id="bjmbr-20133097-e007"><mml:math id="equ14"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mstyle></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mi>n</mml:mi></mml:munder><mml:mrow><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mi>n</mml:mi></mml:munder><mml:mrow><mml:mi>X</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mi>n</mml:mi></mml:munder><mml:mrow><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(7)</label></disp-formula></p><p>The next step is to determine the <italic>k</italic><sup>th</sup> coefficient of the previous polynomial equality, <xref ref-type="disp-formula" rid="bjmbr-20133097-e007">Equation 7</xref>. Let
<italic>n</italic> + <italic>j</italic> = <italic>k</italic> in the
second term on the right-hand side of <xref ref-type="disp-formula" rid="bjmbr-20133097-e007">Equation 7</xref>. This implies that
<italic>n</italic> − <italic>j</italic> = <italic>k</italic>. As a
consequence, the <italic>k</italic><sup>th</sup> coefficient in <xref ref-type="disp-formula" rid="bjmbr-20133097-e007">Equation 7</xref> is given by <inline-formula><mml:math id="equ15"><mml:mrow><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math></inline-formula>. Therefore,<disp-formula id="bjmbr-20133097-e008"><mml:math id="equ16"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math><label>(8)</label></disp-formula>Note that this is an autoregressive system with order
<italic>κ</italic>, denoted by
<italic>AR</italic>(<italic>κ</italic>).</p></sec><sec><title>Spectral analysis</title><p>The spectral density function of an autoregressive process as described in <xref ref-type="disp-formula" rid="bjmbr-20133097-e008">Equation 8</xref> may be
obtained as follows: From <xref ref-type="disp-formula" rid="bjmbr-20133097-e008">Equation 8</xref> we have<disp-formula id="bjmbr-20133097-e009"><mml:math id="equ17"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>...</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math><label>(9)</label></disp-formula>multiplying both sides by
<italic>z</italic>−<italic>κ</italic>, and summing both sides
over <italic>k</italic> and noticing that the left-hand side is the
<italic>z</italic>-transform of <italic>X</italic>, we have
<italic>X</italic>(<italic>z</italic>) = Σ<sub><italic>k</italic></sub><italic>a</italic><sub>1</sub><italic>X</italic>(<italic>k</italic> −1)<italic>z</italic> −
<italic>k</italic> + … + Σ<sub><italic>k</italic></sub><italic>a<sub>k</sub>X</italic>(<italic>k</italic> −
<italic>k</italic>)<italic>z</italic><sup>−<italic>k</italic></sup> + Σ<sub><italic>k</italic></sub><italic>Z</italic>(<italic>k</italic>)<italic>z</italic><sup>−<italic>k</italic></sup>. Changing variables leads to <italic>X</italic>(<italic>z</italic>) = Σ<sub><italic>m</italic></sub>(<italic>a</italic><sub>1</sub><italic>z</italic><sup>−<italic>k</italic></sup>)<italic>X</italic>(<italic>m</italic>)<sup>−<italic>m</italic></sup> + … + Σ<sub><italic>m</italic></sub><italic>a<sub>k</sub>z</italic><sup>−<italic>k</italic></sup><italic>X</italic>(<italic>m</italic>)<italic>z</italic><sup>−<italic>m</italic></sup> + Σ<italic><sub>k</sub>Z</italic>(<italic>k</italic>)<italic>z</italic><sup>−<italic>k</italic></sup>. Notice that the terms <italic>a<sub>j</sub>z</italic><sup>−<italic>j</italic></sup>, for 1 ≤ <italic>j</italic> ≤ 1, are constants, yielding
<italic>X</italic>(<italic>z</italic>) = (<italic>a</italic><sub>1</sub><italic>z</italic><sup>−<italic>k</italic></sup>)(Σ<sub><italic>m</italic></sub><italic>X</italic>(<italic>m</italic>)<italic>z</italic><sup>−<italic>m</italic></sup> + … + (<italic>a<sub>k</sub>z</italic><sup>−<italic>k</italic></sup>)(Σ<sub><italic>m</italic></sub><italic>X</italic>(<italic>m</italic>)<italic>z</italic><sup>−<italic>k</italic></sup>) + Σ<sub><italic>k</italic></sub><italic>Z</italic>(<italic>k</italic>)<italic>z</italic><sup>−<italic>k</italic></sup>.</p><p>The left-hand side is the <italic>z</italic>-transform of <italic>X</italic>(.)
and also of the terms in parenthesis on the right-hand side. Hence,
<italic>X</italic>(<italic>z</italic>){1 − <italic>a</italic><sub>1</sub><italic>z</italic><sup>−1</sup> −<italic>a</italic><sub>2</sub><sup>−2</sup> −... −<italic>a<sub>k</sub>z</italic><sup>−<italic>k</italic></sup>} = <italic>Z</italic>(<italic>z</italic>).</p><p>Since
<italic>E</italic>|<italic>Z</italic>(<italic>i</italic>)<italic>Z</italic>(<italic>j</italic>)|
= σ<sup>2</sup>δ<sub><italic>ij</italic></sub>, it follows that <disp-formula id="bjmbr-20133097-e083"><mml:math id="equ18"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>...</mml:mn><mml:msub><mml:mi>a</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where <italic>z</italic> = <italic>e<sup>jw</sup></italic>. Since |<italic>X</italic>(<italic>z</italic>)|<sup>2</sup> is just
half the power spectrum, then the spectral density function is given
by<disp-formula id="bjmbr-20133097-e010"><mml:math id="equ19"><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mi>w</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mstyle></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math><label>(10)</label></disp-formula>Hence, <xref ref-type="disp-formula" rid="bjmbr-20133097-e010">Equation 10</xref> is the maximum entropy spectral density subject to the
constraints
<italic>R</italic>(0),<italic>R</italic>(1),…,<italic>R</italic>(<italic>k</italic>).</p></sec><sec><title>Blood pressure measurement</title><p>Systolic blood pressure (SBP) was measured in conscious rats at 5 and 8 weeks of
treatment (SD or HFD), employing an indirect tail-cuff method using an
electrosphygmomanometer combined with a pneumatic pulse transducer/amplifier
(BpMonWin Monitor Version 1.33, IITC Life Science, USA). This indirect approach
allowed repeated measurements with a close correlation (correlation
coefficient=0.975), compared to direct intra-arterial recordings (<xref rid="B20" ref-type="bibr">20</xref>,<xref rid="B21" ref-type="bibr">21</xref>). The mean of three consecutive readings represented the blood
pressure.</p></sec><sec><title>Glucose tolerance test (GTT)</title><p>GTTs were performed in 10-week-old SD and HFD rats, after 12 h of fasting in
order to determine changes in insulin sensitivity. Eleven rats from independent
litters were tested. To establish basal values of glucose and insulin, blood
samples were taken by lancing the tail vein before a glucose challenge (time 0).
They then received a single bolus of 1 g/kg glucose <italic>ip</italic>. Blood
samples were taken from the tail vein at 0, 15, 30, 60, 90, and 120 min. Glucose
from whole blood was measured with a glucometer (MediSense/Optium, Abbott, USA).
Serum was separated (50 μL) and kept at −20°C for measurement
of insulin levels by radioimmunoassay. The incremental area under the glucose
tolerance curve (AUC) was calculated as the integrated AUC above the basal value
(time 0) over the 120-min sampling period using Prism 4 for Windows.</p></sec><sec><title>Biochemical analysis</title><p>Plasma and urine sodium and potassium concentrations were obtained at 4 weeks of
treatment and measured by flame photometry (B262; Micronal, Brazil), while
creatinine concentrations were determined spectrophotometrically (model 143,
Instrumentation Laboratories, USA). The parameters glucose, albumin, globulin,
total protein, chloride, magnesium, calcium, phosphorus, HDL, LDL cholesterol,
and triglycerides were also collected at the 4th week in control and post-HFD
treatment (SD and HFD) and measured using enzyme immunoassay kits with a Modular
Analytic P Biochemistry Analyzer (Roche Diagnostics, Germany). Enzyme-linked
immunosorbent assays for plasma insulin were used to determine rat/mouse insulin
(EZRMI-13K; Linco Research, Millipore, USA), according to the
manufactureŕ’s protocols.</p></sec><sec><title>Data presentation and statistics</title><p>All numerical results are reported as means±SD of the indicated number of
experiments. Diagnostic checking was performed by analysis of the correlation of
residues between the model and the samples (<xref rid="B19" ref-type="bibr">19</xref>,<xref rid="B22" ref-type="bibr">22</xref>). As shown in the
Results section, the correlation of residues falls inside the boundary ±1.95/<inline-formula><mml:math id="equ20"><mml:mrow><mml:mo>±</mml:mo><mml:mn>1.95</mml:mn><mml:mo>/</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo stretchy="false">(</mml:mo><mml:mn>23</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> (<xref rid="B23" ref-type="bibr">23</xref>). The AUC of the
spectral density function was considered as the index of variability, and we
have used it to establish the statistical difference between SD and HFD values.
The calculi of these areas respected the same interval defined along the axis of
frequency (<italic>w</italic>). Comparisons involving only two means within or
between groups were carried out using the Student <italic>t</italic>-test. The
level of significance was set at P≤0.05.</p></sec></sec><sec sec-type="results"><title>Results</title><sec><title>Experimental model data</title><p><xref ref-type="table" rid="t01">Table 1</xref> shows the serum parameter
analysis for animals from age-matched SD and HFD groups. There were no
significant differences among serum biochemical parameters in SD rats compared
with the age-matched HFD group, except for magnesium plasma levels, which were
significantly lower in HFD compared to SD rats (P=0.0086). Arterial blood
pressure (in mmHg) levels were similar in both experimental groups (SD:
137±4.24 <italic>vs</italic> 131.75±5.8 mmHg in HFD, P>0.05).</p><p><table-wrap id="t01" orientation="portrait" position="float"><graphic xlink:href="1414-431X-bjmbr-47-01-00070-gt001"></graphic></table-wrap></p></sec><sec><title>GTT</title><p>GTTs were performed to verify the effect of a high-fat diet on glucose tolerance,
compared with the normal-diet group. Our study shows that, on the 4th week, the
HFD group showed similar basal glycemia after overnight fasting, when compared
to the SD group. Otherwise, after glucose intraperitoneal loading, the SD and
HFD groups achieved similar plasma glucose concentrations at 30, 60, 90, and 120
min expressed by the incremental AUC. The measurements of plasma insulin levels
during the same period (expressed in ng/mL) were also similar in SD and HFD
animals.</p></sec><sec><title>Stochastic complexity analysis</title><p>From the previous subsections, the autoregressive model (<xref rid="B19" ref-type="bibr">19</xref>,<xref rid="B22" ref-type="bibr">22</xref>,<xref rid="B23" ref-type="bibr">23</xref>) may be seen as the best model to
characterize temperature as the biological parameter. This model takes into
consideration the inherent memory (past data) of the biological system to
generate or predict future data. In mammals, adjustments in temperature are made
in accordance with information observed previously. Through self-regulating
mechanisms, animals are capable of avoiding significant thermal fluctuations.
Estimation of the order of equations was performed via a partial autocorrelation
function, <inline-formula><mml:math id="equ21"><mml:mrow><mml:mover accent="true"><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover accent="true"><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msup><mml:mi>P</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mo>/</mml:mo><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>|</mml:mo><mml:mo>|</mml:mo></mml:mrow></mml:math></inline-formula>, where <italic>P</italic>(<italic>k</italic>) is the
autocorrelation matrix and <italic>P</italic><sup>*</sup>(<italic>k</italic>) is the matrix
<italic>P</italic>(<italic>k</italic>) with the last column being
substituted by the autocorrelation vector; ||.|| denotes the determinant and the
<italic>a<sub>k</sub></italic>'s denote coefficients of the AR model, by use of the
Levinson-Durbin algorithm. For an autoregressive process with memory κ,
denoted by AR (κ), <italic>ϕ<italic>k</italic></italic> ≠ 0, for <italic>k κ</italic>, and
<italic>ϕ<italic>k</italic></italic> = 0, for <italic>k κ</italic>. Note in both panels of <xref ref-type="fig" rid="f01">Figure 1</xref> that the greatest value of the
partial autocorrelation function <inline-formula><mml:math id="equ22"><mml:mover accent="true"><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:math></inline-formula> is achieved when <italic>k</italic> = 1. Best-fit equations
were also chosen according to the minimum value obtained by BIC. All steps were
performed with use of the software Mathematica 7.0 (<xref rid="B16" ref-type="bibr">16</xref>-<xref rid="B18" ref-type="bibr">18</xref>).</p><fig id="f01" orientation="portrait" position="float"><label>Figure 1</label><caption><title>Partial auto-correlation function <inline-formula><mml:math id="equ23"><mml:mover accent="true"><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:math></inline-formula> for the standard diet (<italic>panel A</italic>)
compared to the high-fat diet (<italic>panel B</italic>) as a function
of <italic>κ</italic>(lag).</title></caption><graphic xlink:href="1414-431X-bjmbr-47-01-00070-gf001"></graphic></fig><p>The total number of samples was the same for all animals and achieved 361
consecutive valid measurements, covering a 6-h period from 0:00 to 6:00 am
(<xref ref-type="fig" rid="f02">Figure 2</xref>). These measurements were
organized as follows: <italic>T<sub>SD</sub></italic> = {<italic>t</italic><sub>1</sub>,<italic>t</italic><sub>2</sub>,…,<italic>t</italic><sub>361</sub>}, <italic>X<sub>t</sub><sup>SD</sup></italic> = (<italic>X</italic>(<italic>t</italic><sub>1</sub>),…,<italic>X</italic>(<italic>t</italic><sub>361</sub>)) = (<italic>X</italic><sub>1</sub>,…,<italic>X</italic><sub>361</sub>), HFD : <italic>T<sub>HFD</sub></italic> = {<italic>t′</italic><sub>1</sub>,…,<italic>t′</italic><sub>361</sub>}, and <italic>X<sub>t</sub><sup>HFD</sup></italic> =
(<italic>X</italic>(<italic>t</italic>′<sub>1</sub>),…,<italic>X</italic>(<italic>t</italic>′<sub>361</sub>))
=
(<italic>X</italic>′<sub>1</sub>,…<italic>X</italic>′<sub>361</sub>)</p><p><xref ref-type="fig" rid="f02">Figure 2</xref> exemplifies the time averaging
series, which were the sources used for derivation of the process for the
autoregressive model. The best-fit model is achieved with the first order AR
model, AR (<xref ref-type="disp-formula" rid="bjmbr-20133097-e001">Equation
1</xref>), given by<disp-formula id="bjmbr-20133097-e011"><mml:math id="equ24"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math><label>(11)</label></disp-formula>where <italic>a</italic><sub>1</sub> denotes the first order autoregressive coefficient, and
σ<sup>2</sup> denotes the variance of white noise. AR models derived
from experimental data, <xref ref-type="disp-formula" rid="bjmbr-20133097-e012">Equations 12</xref> and <xref ref-type="disp-formula" rid="bjmbr-20133097-e013">13</xref>, originated from the Levinson-Durbin
method from temperature samples collected from SD rats at <italic>T<sub>SD</sub></italic><disp-formula id="bjmbr-20133097-e012"><mml:math id="equ25"><mml:mrow><mml:mi>X</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>S</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0.98.</mml:mn><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>0.006</mml:mn></mml:mrow></mml:math><label>(12)</label></disp-formula>and <disp-formula id="bjmbr-20133097-e013"><mml:math id="equ26"><mml:mrow><mml:mi>X</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>S</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0.98.</mml:mn><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>0.0068</mml:mn><mml:mo>;</mml:mo></mml:mrow></mml:math><label>(13)</label></disp-formula>and from HFD rats at <italic>T<sub>HFD</sub></italic> : <disp-formula id="bjmbr-20133097-e014"><mml:math id="equ27"><mml:mrow><mml:mi>X</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>H</mml:mi><mml:mi>F</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0.97.</mml:mn><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>0.096</mml:mn></mml:mrow></mml:math><label>(14)</label></disp-formula>and<disp-formula id="bjmbr-20133097-e015"><mml:math id="equ28"><mml:mrow><mml:mi>X</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>H</mml:mi><mml:mi>F</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0.96.</mml:mn><mml:msup><mml:mi>X</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>0.006.</mml:mn></mml:mrow></mml:math><label>(15)</label></disp-formula></p><fig id="f02" orientation="portrait" position="float"><label>Figure 2</label><caption><title>Temperature series <italic>X<sub>t</sub><sup>SD</sup></italic> for the standard diet (<italic>panel A</italic>) compared to
<italic>X<sub>t</sub><sup>HFD</sup></italic> for the high-fat diet (<italic>panel B</italic>).</title></caption><graphic xlink:href="1414-431X-bjmbr-47-01-00070-gf002"></graphic></fig><p><xref ref-type="fig" rid="f03">Figure 3</xref> shows the cross-correlation of
residues, denoted by <inline-formula><mml:math id="equ29"><mml:mrow><mml:mover accent="true"><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> (the difference between correlation of the original and AR
series) of the temperature sample data with those of <xref ref-type="disp-formula" rid="bjmbr-20133097-e008">Equation 8</xref>, up to
<italic>κ</italic> = 30. As can be seen, corresponding
cross-correlation values of residues <italic>κ</italic> as a function
falls between the bounds: <inline-formula><mml:math id="equ30"><mml:mrow><mml:mo>±</mml:mo><mml:mn>1.96</mml:mn><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mn>361</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:math></inline-formula>, in agreement with the model and its data. Since the
coefficients in AR (<xref ref-type="disp-formula" rid="bjmbr-20133097-e001">Equation 1</xref>) models are less than 1 (see <xref ref-type="disp-formula" rid="bjmbr-20133097-e004">Equations 4</xref>, <xref ref-type="disp-formula" rid="bjmbr-20133097-e005">5</xref>, <xref ref-type="disp-formula" rid="bjmbr-20133097-e012">12</xref>, and <xref ref-type="disp-formula" rid="bjmbr-20133097-e013">13</xref>), it follows
that the associated stochastic process is stationary.</p><fig id="f03" orientation="portrait" position="float"><label>Figure 3</label><caption><title>Representative illustration of cross-correlation of the residues for
the standard diet group.</title></caption><graphic xlink:href="1414-431X-bjmbr-47-01-00070-gf003"></graphic></fig><p>Now, if the autoregressive process has <italic>κ</italic> = 1 and
<italic>a</italic><sub>1</sub> = <italic>ρ</italic>, then from<disp-formula id="bjmbr-20133097-e016"><mml:math id="equ31"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>ρ</mml:mi><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math><label>(16)</label></disp-formula>by using recursively <xref ref-type="disp-formula" rid="bjmbr-20133097-e016">Equation 16</xref>, we arrive at<disp-formula id="bjmbr-20133097-e017"><mml:math id="equ32"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>ϖ</mml:mi></mml:munderover><mml:mrow><mml:msup><mml:mi>ρ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>ϖ</mml:mi></mml:munderover><mml:mrow><mml:msup><mml:mi>ρ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle><mml:mo>.</mml:mo></mml:mrow></mml:math><label>(17)</label></disp-formula>From <xref ref-type="disp-formula" rid="bjmbr-20133097-e006">Equation 6</xref>, <italic>X</italic>(<italic>n</italic>) is the
convolution of <italic>H</italic>(<italic>n</italic>) by
<italic>Z</italic>(<italic>n</italic>), that is,<disp-formula id="bjmbr-20133097-e018"><mml:math id="equ33"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∗</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:munder><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle><mml:mo>.</mml:mo></mml:mrow></mml:math><label>(18)</label></disp-formula>By comparing <xref ref-type="disp-formula" rid="bjmbr-20133097-e018">Equation 18</xref> with <xref ref-type="disp-formula" rid="bjmbr-20133097-e019">Equation 19</xref>, it
follows that the impulse response <italic>H</italic>(<italic>n</italic>) is
given by<disp-formula id="bjmbr-20133097-e019"><mml:math id="equ34"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msup><mml:mi>ρ</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math><label>(19)</label></disp-formula>Consequently, the frequency response is<disp-formula id="bjmbr-20133097-e020"><mml:math id="equ35"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>ρ</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mi>w</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math><label>(20)</label></disp-formula></p><p>Therefore, the previous two <xref ref-type="disp-formula" rid="bjmbr-20133097-e019">Equations, 19</xref> and <xref ref-type="disp-formula" rid="bjmbr-20133097-e020">20</xref>, specify the
time and frequency responses, respectively (see <xref ref-type="fig" rid="f04">Figures 4</xref> and <xref ref-type="fig" rid="f05">5</xref>) of the
corresponding system.</p><fig id="f04" orientation="portrait" position="float"><label>Figure 4</label><caption><title>System time response for the standard diet group.</title></caption><graphic xlink:href="1414-431X-bjmbr-47-01-00070-gf004"></graphic></fig><fig id="f05" orientation="portrait" position="float"><label>Figure 5</label><caption><title>Representative illustration of the system frequency response for the
standard diet group.</title></caption><graphic xlink:href="1414-431X-bjmbr-47-01-00070-gf005"></graphic></fig><p>Note that the spectral density function under a system theory point of view may
be obtained as |<italic>X</italic>(<italic>z</italic>)|<sup>2</sup> =
<italic>X</italic>(<italic>x</italic>).<italic><overline>X</overline></italic>(<italic>z</italic>), where <italic><overline>X</overline></italic> denotes the complex conjugate of <italic>X</italic> leading to that of
<xref ref-type="disp-formula" rid="bjmbr-20133097-e010">Equation 10</xref>.
From the AR (<xref rid="B01" ref-type="bibr">1</xref>) model, the spectral
density function is given by<disp-formula id="bjmbr-20133097-e021"><mml:math id="equ36"><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msubsup><mml:mi>a</mml:mi><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mtext> cos </mml:mtext><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math><label>(21)</label></disp-formula>where for SD rats <italic>a</italic><sub>1</sub> = 0.98 and <inline-formula><mml:math id="equ37"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.0068</mml:mn></mml:mrow></mml:math></inline-formula>, and for HFD <italic>a</italic><sub>1</sub> = 0.96 and <inline-formula><mml:math id="equ38"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.006</mml:mn></mml:mrow></mml:math></inline-formula>. The spectral density function (<xref ref-type="disp-formula" rid="bjmbr-20133097-e021">Equation 21</xref>), for the SD and HFD groups, is
depicted in <xref ref-type="fig" rid="f06">Figure 6</xref>. As can be seen, the
signal energy content is located at low frequencies
(39.78×10<sup>−4</sup> rad/min) after the animals underwent the
HFD for 4 weeks compared to appropriate controls. Thus, the mean AUC of this
specific region to demonstrate the significant difference between the thermal
profile of SD rats compared to HFD rats showed 0.04±0.0014 and
0.018±0.0028, respectively, with P<0.001.</p><fig id="f06" orientation="portrait" position="float"><label>Figure 6</label><caption><title>Spectral density function S<sub>w</sub> of standard diet
(<italic>panels A</italic> and <italic>C</italic>) and high-fat diet
rats (<italic>panels B</italic> and <italic>D</italic>).</title></caption><graphic xlink:href="1414-431X-bjmbr-47-01-00070-gf006"></graphic></fig></sec></sec><sec sec-type="discussion"><title>Discussion</title><p>In the present study, we evaluated, for the first time, stochastic complexity
analysis as a tool to detect, early and efficiently, presumable alterations in
thermal-series profile registers in a specific model of metabolic and energy
disorder. As previously defined, the temperature register obeys a nonlinear and
randomized model (<xref rid="B24" ref-type="bibr">24</xref>). Thus, by a chaotic
solution to a deterministic equation, we mean a solution whose outcome is very
sensitive to initial conditions and whose evolution through phase space appears to
be quite random. Taking into account the time course series of core temperature
measured in control (SD for rodents) and experimental groups of animals (HFD-treated
rats), as a source of information, the characterization of thermal profile comes
from establishment of the stochastic process. On the other hand, from the
information theory point of view, every source has an associated measure quantifying
its output. In this case study, the associated temperature time course registers
were treated through maximum entropy distribution as a means for stochastic
characterization of the thermal series. In this way, in the current study, we focus
on the characterization of thermal series maximized by a specific stochastic process
(autoregressive model) under appropriate sets of constraints (the autocorrelation
function) to produce the best variability analysis (spectral analysis).</p><p>The HFD model is extremely efficient for creating pathophysiological conditions such
as hyperleptinemia, peripheral insulin resistance, diabetes mellitus, and obesity
that lead to long-term metabolic and energy disorders. However, in this study no
difference was observed in glycemic levels, blood pressure, and body mass
measurement after 4 weeks of HFD intake compared to age-matched animals treated with
standard chow (<xref ref-type="table" rid="t01">Table 1</xref>). On the other hand,
the present study was able to establish, by autoregressive analysis, a consistent
and reliable analytical method that has permitted us to characterize temperature as
a referential biological parameter in homeothermic HFD-treated rats. Thus, the
current temperature data analyzed by spectral density function has shown a distinct
behavior when compared to the SD control and HFD-treated groups, respectively, as
illustrated in <xref ref-type="fig" rid="f06">Figure 6</xref>. Moreover, when
statistical tests were conducted on the spectral analysis for these same series of
temperature, we observed a significant decrease in the variability of values found,
mainly, in the low frequency spectrum (<xref ref-type="fig" rid="f06">Figure
6</xref>). This variability corresponded to one cycle every 4 h or, in the
international system of notation, ∼39.78×10<sup>-4</sup> rad/min or 0.25
cycle/h, after the animal underwent the HFD for 4 weeks, with a different circhoral
rhythm to maintain the core and peripheral temperatures on small-range control in
mammals (<xref rid="B25" ref-type="bibr">25</xref>). This reduction was observed by
parametric analysis of the autoregressive model.</p><p>The role of the CNS has been demonstrated in the control of thermal and metabolic
homeostasis in mammals (<xref rid="B09" ref-type="bibr">9</xref>,<xref rid="B24" ref-type="bibr">24</xref>). It is well known that thermal control
pathways involve central and peripheral afferent nerve stimuli arriving into the
hypothalamic preoptic area and efferent neural and humoral signaling control on
thermogenesis and heat loss. Thus, studies have demonstrated that, after only 1 day
of HFD intake, it is already possible to observe in rats an attenuated activity of
both the insulin and leptin pathways signaling at the level of the hypothalamic
nucleus (<xref rid="B26" ref-type="bibr">26</xref>-<xref rid="B28" ref-type="bibr">28</xref>). These hormones, acting either through stimulation of uncoupling
proteins or via increased sympathetic activity, promote a stimulated thermogenic
activity in those major organs of metabolic activity (<xref rid="B29" ref-type="bibr">29</xref>). Conversely, this metabolic hyperactivity leads to
adjustments in the CNS to maintain stability of body temperature, necessary for the
proper functioning of enzyme activities of all the chemical reactions associated
with basal metabolism (<xref rid="B30" ref-type="bibr">30</xref>). Since this occurs
in homeothermic mammals, these adjustments of body temperature are supposedly made
in accordance with information data already generated and memorized. We hypothesize
that it takes into account the inherent memory of biological parameters that predict
data or estimate generation of future behavior profiles. In the present study,
through self-regulating mechanisms, HFD-treated animals were capable of avoiding
significant thermal fluctuations when compared to the SD group. Here, we may suppose
that there are effective and fine-driven response sensors that are modulated by
neural thermal control centers located in the hypothalamus, with regard to
maintaining in a stationary condition the body temperature time series (<xref ref-type="fig" rid="f02">Figure 2</xref>).</p><p>As previously demonstrated by hemodynamic studies, when analyzing heart rate time
registers, reduction in variability of physiological parameters could be a marker
for a deleterious prognosis in the long run (<xref rid="B31" ref-type="bibr">31</xref>,<xref rid="B32" ref-type="bibr">32</xref>), although a few studies
have demonstrated some relationship between thermal profile and mortality (<xref rid="B01" ref-type="bibr">1</xref>,<xref rid="B32" ref-type="bibr">32</xref>).
In the specific case of heart rate, statistical analysis of variability, by itself,
was found to be very sensitive but less specific when applied in practice (<xref rid="B01" ref-type="bibr">1</xref>,<xref rid="B31" ref-type="bibr">31</xref>,<xref rid="B32" ref-type="bibr">32</xref>). In the current study, we
accessed stochastic complexity analysis as being highly sensitive and specific for
detecting discrete and subtle variations in thermal series in a specific
metabolic-induced disorder. Data emerging from a stochastic analysis model of
sequential temperature registers seem to be highly relevant, since there is a
hierarchical priority that keeps thermal homeostasis under a narrow range of control
to reach an adequate internal milieu for cells and life survival, compared with
other biological parameters. We may state that mathematical analysis is an efficient
tool for early prediction of the expression of pathophysiological syndromes even
before appraisal of metabolic and clinical disorders. Thus, we may infer from the
results of our study that thermal series analysis by a stochastic model could be
proposed as a sensitive technique for clinical and pathophysiological use to predict
disorders in metabolic and energy homeostasis.</p></sec><sec sec-type="conclusions"><title>Conclusion</title><p>In conclusion, maximum entropy distribution as a means for stochastic
characterization of temperature time series registers may be established as an
important and early tool to aid in diagnosis and prevention of metabolic diseases
due to its high ability for detecting small variations in thermal profiles.
Additional studies are being performed with other tools to establish the precise
correlation between biological and mathematical concepts that involve representation
and more specific analysis of thermal series profiles in experimental models in
normal or disease states.</p></sec></body><back><ack><p>Research supported by FAPESP (#2010/52696-0 and #2011/05997-8), CNPq (#500868/91-3)
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