Answers.Body: <p>I don't think there is any rigorous mathematical definition of dynamic, or to phrase it better, the definition is embedded in the definition of <em>dynamical system</em> itself (which would be probably a better term to use in place of the generic &quot;sequence&quot;):</p>
  <blockquote>
  <p>A dynamical system is formally defined as a state space <span class="math-container">$X$</span>, a set of
  time points <span class="math-container">$T$</span>, and a rule <span class="math-container">$R$</span> that specifies how the state evolves with time.
  The rule <span class="math-container">$R$</span> is a function whose domain is <span class="math-container">$X×T$</span> and whose
  codomain is <span class="math-container">$X$</span>, i.e., <span class="math-container">$R:X×T→X$</span>. The rule function <span class="math-container">$R$</span> means
  that the <span class="math-container">$R$</span> takes two inputs, <span class="math-container">$R=R(x,t)$</span>, where <span class="math-container">$x∈X$</span> is an
  initial state and <span class="math-container">$t∈T$</span> is a specific time step.</p>
  </blockquote>
  <p>I think that what the paragraph states is that we assume the rule funcion R to remain the same, cause that's what we're trying to learn.</p>
  <p>As a final note, there are out there also <a href="https://en.wikipedia.org/wiki/Nonlinear_system#Types_of_nonlinear_dynamic_behaviors" rel="nofollow noreferrer">classifications</a> of different non-linear dynamical systems, based on the property of <span class="math-container">$R$</span>.</p>
  
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Answers.CreationDate: 2022-03-10T11:30:03.053Z
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Answers.LastEditDate: 2022-03-10T13:00:24.130Z
Answers.UserDisplayName: Edoardo Guerriero
Body: <p>Consider the following paragraph from the topic named <a href="https://d2l.ai/chapter_recurrent-neural-networks/sequence.html#statistical-tools" rel="nofollow noreferrer">sequential models</a> from the textbook titled <a href="https://d2l.ai/" rel="nofollow noreferrer">Dive into Deep Learning</a></p>
  <blockquote>
  <p>Both cases raise the obvious question of how to generate training
  data. One typically uses historical observations to predict the next
  observation given the ones up to right now. Obviously we do not expect
  time to stand still. However, a common assumption is that while the
  specific values of might change, <strong>at least the dynamics of the
  sequence itself will not</strong>. This is reasonable, since novel dynamics
  are just that, novel and thus not predictable using data that we have
  so far. Statisticians call dynamics that do not change stationary.</p>
  </blockquote>
  <p>Here sequence refers to <span class="math-container">$x_1, x_2, x_3, \cdots, x_t$</span>. Say the stock price of  a company.</p>
  <p>What does it mean rigorously by the dynamics of a sequence in statistics?</p>
  
Tags: <terminology><math><sequence-modeling><statistical-ai>
Title: What does it mean by "dynamics of a sequence" mathematically?
UserDisplayName: hanugm
ThreadText: # What does it mean by "dynamics of a sequence" mathematically?
  
  (Asked by: username_0 on 2022-03-10)
  
  Consider the following paragraph from the topic named [sequential models](https://d2l.ai/chapter_recurrent-neural-networks/sequence.html#statistical-tools) from the textbook titled [Dive into Deep Learning](https://d2l.ai/)
  
  > Both cases raise the obvious question of how to generate training
  > data. One typically uses historical observations to predict the next
  > observation given the ones up to right now. Obviously we do not expect
  > time to stand still. However, a common assumption is that while the
  > specific values of might change, **at least the dynamics of the
  > sequence itself will not**. This is reasonable, since novel dynamics
  > are just that, novel and thus not predictable using data that we have
  > so far. Statisticians call dynamics that do not change stationary.
  
  Here sequence refers to $x\_1, x\_2, x\_3, \cdots, x\_t$. Say the stock price of a company.
  
  What does it mean rigorously by the dynamics of a sequence in statistics?
  
  **Tags:** terminology, math, sequence-modeling, statistical-ai
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2022-03-10. Score: 1
  
  I don't think there is any rigorous mathematical definition of dynamic, or to phrase it better, the definition is embedded in the definition of *dynamical system* itself (which would be probably a better term to use in place of the generic "sequence"):
  
  > A dynamical system is formally defined as a state space $X$, a set of
  > time points $T$, and a rule $R$ that specifies how the state evolves with time.
  > The rule $R$ is a function whose domain is $X×T$ and whose
  > codomain is $X$, i.e., $R:X×T→X$. The rule function $R$ means
  > that the $R$ takes two inputs, $R=R(x,t)$, where $x∈X$ is an
  > initial state and $t∈T$ is a specific time step.
  
  I think that what the paragraph states is that we assume the rule funcion R to remain the same, cause that's what we're trying to learn.
  
  As a final note, there are out there also [classifications](https://en.wikipedia.org/wiki/Nonlinear_system#Types_of_nonlinear_dynamic_behaviors) of different non-linear dynamical systems, based on the property of $R$.
  
  
  ---
  
  