grammatical and notation edits to make documentation more consistent …

…with literature references

docs/source/reliability/index.rst

@@ -2,19 +2,20 @@ Reliability
 ===========
 
 Reliability of a system refers to the assessment of its probability of failure (i.e the system no longer satisfies some
-performance measures), given the model uncertainty in the structural, environmental and load parameters. Given a vector
-of random variables :math:`\textbf{X}=\{X_1, X_2, \ldots, X_n\} \in \mathcal{D}_\textbf{X}\subset \mathbb{R}^n`, where
-:math:`\mathcal{D}` is the domain of interest and :math:`f_{\textbf{X}}(\textbf{x})` is its joint probability density
-function then, the probability that the system will fail is defined as
+performance measure), given the model uncertainty in the structural, environmental and load parameters. Given a vector
+of random variables :math:`\textbf{X}=[X_1, X_2, \ldots, X_n]^T \in \mathcal{D}_\textbf{X}\subset \mathbb{R}^n`, where
+:math:`\mathcal{D}_\textbf{X}` is the domain of interest and :math:`f_{\textbf{X}}(\textbf{x})` is its joint probability density
+function, then the probability that the system will fail is defined as
 
 
-.. math:: P_f =\mathbb{P}(g(\textbf{X}) \leq 0) = \int_{D_f} f_{\textbf{X}}(\textbf{x})d\textbf{x} = \int_{\{\textbf{X}:g(\textbf{X})\leq 0 \}} f_{\textbf{X}}(\textbf{x})d\textbf{x}
+.. math:: P_f =\mathbb{P}(g(\textbf{X}) \leq 0) = \int_{\mathcal{D}_f} f_{\textbf{X}}(\textbf{x})d\textbf{x} = \int_{\{\textbf{X}:g(\textbf{X})\leq 0 \}} f_{\textbf{X}}(\textbf{x})d\textbf{x}
 
 
-where :math:`g(\textbf{X})` is the so-called performance function. The reliability problem is often formulated in the
+where :math:`g(\textbf{X})` is the so-called performance function and :math:`\mathcal{D}_f=\{\textbf{X}:g(\textbf{X})\leq 0 \}` is the failure domain.
+The reliability problem is often formulated in the
 standard normal space :math:`\textbf{U}\sim \mathcal{N}(\textbf{0}, \textbf{I}_n)`, which means that a nonlinear
 isoprobabilistic  transformation from the generally non-normal parameter space
-:math:`\textbf{X}\sim f_{\textbf{X}}(\cdot)` to the standard normal is required (see the :py:mod:`.transformations` module).
+:math:`\textbf{X}\sim f_{\textbf{X}}(\cdot)` to the standard normal space is required (see the :py:mod:`.transformations` module).
 The performance function in the standard normal space is denoted :math:`G(\textbf{U})`. :py:mod:`.UQpy` does not require this
 transformation and can support reliability analysis for problems with arbitrarily distributed parameters.
 

docs/source/reliability/subset.rst

@@ -2,27 +2,27 @@ Subset Simulation
 -------------------
 
 In the subset simulation method :cite:`SubsetSimulation` the probability of failure :math:`P_f`  is approximated by a product of probabilities
-of more frequent events. That is, the failure event :math:`G = \{\textbf{x} \in \mathbb{R}^n:G(\textbf{x}) \leq 0\}`,
+of more frequent events. That is, the failure event :math:`G = \{\textbf{X} \in \mathbb{R}^n:g(\textbf{X}) \leq 0\}`,
 is expressed as the of union of `M` nested intermediate events :math:`G_1,G_2,\cdots,G_M` such that
 :math:`G_1 \supset G_2 \supset \cdots \supset G_M`, and :math:`G = \cap_{i=1}^{M} G_i`. The intermediate failure events
-are defined as :math:`G_i=\{G(\textbf{x})\le b_i\}`, where :math:`b_1>b_2>\cdots>b_i=0` are positive thresholds selected
-such that each conditional probability :math:`P(G_i | G_{i-1}), ~i=2,3,\cdots,M-1` equals a target probability value
+are defined as :math:`G_i=\{g(\textbf{X})\le b_i\}`, where :math:`b_1>b_2>\cdots>b_M=0` are non-negative thresholds selected
+such that each conditional probability :math:`P(G_{i+1} | G_{i}),\ i=1,2,\cdots,M-1` equals a target probability value
 :math:`p_0`. The probability of failure :math:`P_f` is estimated as:
 
-.. math:: P_f = P\left(\cap_{i=1}^M G_i\right) = P(G_1)\prod_{i=2}^M P(G_i | G_{i-1})
+.. math:: P_f = P\left(\bigcap_{i=1}^M G_i\right) = P(G_1)\prod_{i=1}^{M-1} P(G_{i+1} | G_{i})
 
 where the probability :math:`P(G_1)` is computed through Monte Carlo simulations. In order to estimate the conditional
-probabilities :math:`P(G_i|G_{i-1}),~j=2,3,\cdots,M` generation of Markov Chain Monte Carlo (MCMC) samples from the
-conditional pdf :math:`p_{\textbf{U}}(\textbf{u}|G_{i-1})` is required. In the context of subset simulation, the Markov
+probabilities :math:`P(G_{i+1}|G_i),~i=1,2,\cdots,M-1` generation of Markov Chain Monte Carlo (MCMC) samples from the
+conditional pdf :math:`p_{\textbf{X}}(\textbf{x}|G_i)` is required. In the context of subset simulation, the Markov
 chains are constructed through a two-step acceptance/rejection criterion. Starting from a Markov chain state
-:math:`\textbf{x}` and a proposal distribution :math:`q(\cdot|\textbf{x})`, a candidate sample :math:`\textbf{w}` is
-generated. In the first stage, the sample :math:`\textbf{w}` is accepted/rejected with probability
+:math:`\textbf{X}` and a proposal distribution :math:`q(\cdot|\textbf{X})`, a candidate sample :math:`\textbf{W}` is
+generated. In the first stage, the sample :math:`\textbf{W}` is accepted/rejected with probability
 
-.. math:: \alpha=\min\bigg\{1, \frac{p(\textbf{w})q(\textbf{x}|\textbf{w})}{p(\textbf{x})q(\textbf{w}|\textbf{x})}\bigg\}
+.. math:: \alpha=\min\bigg\{1, \frac{p_\textbf{X}(\textbf{w})q(\textbf{x}|\textbf{W})}{p_\textbf{X}(\textbf{x})q(\textbf{w}|\textbf{X})}\bigg\}
 
-and in the second stage is accepted/rejected based on whether the sample belongs to the failure region :math:`G_j`.
-:class:`.SubSetSimulation` can be used with any of the available (or custom) :class:`.MCMC` classes in the
-:py:mod:`sampling` module.
+and in the second stage is accepted/rejected based on whether the sample belongs to the failure region :math:`G_i`.
+:class:`.SubsetSimulation` can be used with any of the available (or custom) :class:`.MCMC` classes in the
+:py:mod:`Sampling` module.
 
 SubsetSimulation Class
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^