hcju/proofwikips huggingface dataset:
Definition:Least Significant Digit
Let $b \in \Z: b \ge 2$ be a number base Let $n$ be a number which is reported to $r$ significant figures, to base $b$, that is: :$n = d_1 \times b^k + d_2 \times b^{k - 1} + \dotsb + d_{r - 1} \times b^{k - r + 2} + d_r \times b^{k - r + 1}$ where: :$d_1, d_2, \dotsc, d_r$ are the significant figures of $n$ :$b^k$ is the largest power of $b$ less than or equal to $n$. Then the digit $d_r$ is known as the '''least significant digit''' of $n$. Note that the usual situation is when $b = 10$, but in the field of computer science, binary is usual.
Definition:Most Significant Digit
Let $b \in \Z: b \ge 2$ be a number base Let $n$ be a number which is reported to $r$ significant figures, to base $b$, that is: :$n = d_1 \times b^k + d_2 \times b^{k - 1} + \dotsb + d_{r - 1} \times b^{k - r + 2} + d_r \times b^{k - r + 1}$ where: :$d_1, d_2, \dotsc, d_r$ are the significant figures of $n$ :$b^k$ is the largest power of $b$ less than or equal to $n$. Then the digit $d_1$ is known as the '''most significant digit''' of $n$. Note that the usual situation is when $b = 10$, but in the field of computer science, binary is usual.
Definition:Left (Direction)
The direction '''left''' is '''that way''': :$\gets$
Definition:Right (Direction)
The direction '''right''' is '''that way''': :$\to$
Definition:Time Series/Continuous
A '''continuous time series''' is one in which the set of timestamps of the observations forms a continuous function.
Definition:Backward Shift Operator
Let $T = \sequence {z_t}$ be a discrete time series. The '''backward shift operator''' $B$ is defined as: :$\forall t: \map B {z_t} = z_{t - 1}$
Definition:Forward Shift Operator
Let $T = \sequence {z_t}$ be a discrete time series. The '''forward shift operator''' $F$ is defined as: :$\forall t: \map F {z_t} = z_{t + 1}$ === Iterated === {{:Definition:Forward Shift Operator/Iterated}}
Definition:Forward Shift Operator/Iterated
$F$ can be '''iterated''' on $\sequence {z_t}$ as follows: :$\map {F^m} {z_t} := z_{t + m}$
Definition:Limit of Real Function/Limit at Infinity/Positive
$L$ is the '''limit of $f$ at infinity''' : :$\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x > c : \size {\map f x - L} < \epsilon$ This is denoted as: :$\ds \lim_{x \mathop \to \infty} \map f x = L$
Definition:Limit of Real Function/Limit at Infinity/Negative
$L$ is the '''limit of $f$ at minus infinity''' : :$\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x < c: \size {\map f x - L} < \epsilon$ This is denoted as: :$\displaystyle \lim_{x \mathop \to - \infty} \map f x = L$
Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound
Suppose that: :$\forall M \in \R_{>0}: \exists N \in \R_{>0}: \forall x > N : \map f x > M$ for $M$ sufficiently large. Then we write: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty$ or :$\map f x \to +\infty$ as $x \to +\infty$ That is, $\map f x$ can be made arbitrarily large by making $x$ sufficiently large. This is voiced: :'''$\map f x$ increases without bound as $x$ increases without bound'''. or: :'''$\map f x$ tends to (plus) infinity as $x$ tends to (plus) infinity'''.
Definition:Limit of Real Function/Limit at Infinity/Negative/Increasing Without Bound
Suppose that: :$\forall M \in \R_{>0}: \exists N \in \R_{<0}: \forall x < N : \map f x > M$ for $M$ sufficiently large. Then we write: :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty$ or :$\map f x \to +\infty$ as $x \to -\infty$ This is voiced: :'''$\map f x$ increases without bound as $x$ decreases without bound'''. or: :'''$\map f x$ tends to (plus) infinity as $x$ tends to minus infinity'''.
Definition:Limit of Real Function/Limit at Infinity/Positive/Decreasing Without Bound
Suppose that: :$\forall M \in \R_{<0}: \exists N \in \R_{>0}: x > N \implies \map f x < M$ for $M$ sufficiently large in magnitude. Then we write: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty$ or :$\map f x \to -\infty$ as $x \to +\infty$ That is, $-\map f x$ can be made arbitrarily large by making $x$ sufficiently large. This is voiced: :'''$\map f x$ decreases without bound as $x$ increases without bound'''. or: :'''$\map f x$ tends to minus infinity as $x$ tends to (plus) infinity'''.
Definition:Limit of Real Function/Limit at Infinity/Negative/Decreasing Without Bound
Suppose that: :$\forall M \in \R_{<0}: \exists N \in \R_{<0}: x < N \implies \map f x < M$ for $M$ sufficiently large in magnitude. Then we write: :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty$ or :$\map f x \to +\infty \ \text{as} \ x \to -\infty$ This is voiced: :'''$\map f x$ decreases without bound as $x$ decreases without bound'''. or: :'''$\map f x$ tends to minus infinity as $x$ tends to minus infinity'''.
Definition:Deterministic Model
A '''deterministic model''' is a mathematical model based on physical laws which are possible to be calculated exactly (or nearly exactly) for known inputs.
Definition:Stationary Model
A '''stationary model''' is a stochastic model for describing a time series which assumes that the underlying stochastic process remains in equilibrium about a constant mean level. That is, it is a stochastic model with an underlying stochastic process which is itself stationary.
Definition:Statistical Equilibrium
Let $S$ be a stochastic process. Suppose that the observations of the time series to which $S$ gives rise have a constant mean level. Then $S$ is said to be in '''(statistical) equilibrium'''.
Definition:Stationary Stochastic Process
A '''stationary stochastic process''' is stochastic process which remains in equilibrium about a constant mean level.
Definition:Constant Mean Level
Let $S$ be a stochastic process giving rise to a time series $T$. $T$ has a '''constant mean level'''  the mean of $S$ does not change as time progresses.
Definition:Non-Stationary Stochastic Process
A '''non-stationary stochastic process''' is a stochastic process which does not remain in equilibrium about a constant mean level. That is, it is a stochastic process which is not a '''stationary stochastic process'''.
Definition:Non-Stationary Model
A '''non-stationary model''' is a stochastic model for describing a time series which assumes that the underlying stochastic process does not remain in equilibrium about a constant mean level. That is, it is a stochastic model with an underlying stochastic process which is itself non-stationary.
Definition:White Noise Process
Let $S$ be a stochastic process consisting of a time series $\map {z_r} t$ such that: :the terms of the sequence $\sequence {z_r}$ are independent random variables :the terms of $\sequence {z_r}$ are governed by a normal distribution with zero expectation and a given variance $\sigma^2$. Then $S$ is known as a '''white noise process'''.
Definition:Independent Shocks
Let $M$ be a stochastic model which describes a time series in which adjacent observations are highly dependent. Then $M$ may be able to be modelled by a time series whose elements are of the form: :$\map z t = \map {z_d} t + \map {z_r} t$ where: :$\map {z_d} t$ has a deterministic model :$\map {z_r} t$ has a stochastic model which consists of a sequence of independent random variables from a specified probability distribution (usually a white noise process). The terms of the sequence $\sequence {z_r}$ are known as '''independent shocks'''.
Definition:Linear Filter
Let $S$ be a stationary stochastic process governed by a white noise process: :$\map z t = \mu + a_t$ where: :$\mu$ is a constant mean level :$a_t$ is an independent shock at timestamp $t$. A '''linear filter''' takes the terms of $S$, and uses a weight function $\psi$ to apply a weighted sum of the past values so that: {{begin-eqn}} {{eqn | l = \map z t       | r = \mu + a_t + \psi_1 a_{t - 1} + \psi_2 a_{t - 2} + \cdots       | c =  }} {{eqn | r = \mu + \map \psi B a_t       | c =  }} {{end-eqn}} where $B$ denotes the backward shift operator, hence: :$\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$ === Transfer Function === {{:Definition:Linear Filter/Transfer Function}} === Stable === {{:Definition:Linear Filter/Stable}}
Definition:Linear Filter/Transfer Function
The operator: :$\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$ is the transfer function of $L$.
Definition:Ferrier's Prime
'''Ferrier's prime''' is the prime number: :$\dfrac {2^{148} + 1} {17} = 20 \, 988 \, 936 \, 657 \, 440 \, 586 \,486 \, 151 \, 264 \, 256 \, 610 \, 222 \, 593 \, 863 \, 921$ {{NamedforDef|Aimé Ferrier|cat = Ferrier}}
Definition:Weighted Sum
Let $S = \sequence {x_1, x_2, \ldots, x_n}$ be a sequence of real numbers. Let $\map W x$ be a weight function to be applied to the terms of $S$. The '''weighted sum''' of $S$ is defined as: :$\bar x := \displaystyle \sum_{i \mathop = 1}^n \map W {x_i} x_i$ This means that elements of $S$ with a larger weight contribute more to the '''weighted sum''' than those with a smaller weight.
Definition:Linear Filter/Stable
Consider the sequence $\sequence {\psi_k}$ formed by the weight function $\psi$ of $L$. Suppose that: :$\displaystyle \sum_k \size {\psi_k} < \infty$ Then $L$ is said to be '''stable''', and the model for $S$ is stationary. Hence $\mu$ is the mean about which $S$ varies.
Definition:Autoregressive Model
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be a model where the current value of $S$ is expressed as a finite linear aggregate of the past values along with a shock: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$ $M$ is known as an '''autoregressive (AR) process of order $p$'''.
Definition:Metrical Geometry
'''Metrical geometry''' is the study of points, lines, surfaces and volumes in real space, and is based on the concepts of distance and angle. It is what is usually understood in the everyday world as '''geometry''' proper.
Definition:Regression Model
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $M$ be a linear filter model defined as :$\tilde z_t = \phi_1 \tilde x_1 + \phi_2 \tilde x_2 + \dotsb + \phi_p \tilde x_p + a$ which relates a dependent variable $z$ to a set of independent variables $x_1, x_2, \dotsc, x_p$ plus an error term $a$. $M$ is known as a '''regression model'''.
Definition:Autoregressive Model/Autoregressive Operator
Let $\map \phi B$ be defined as: :$\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ where $B$ denotes the backward shift operator. Then $\map \phi B$ is referred to as the '''autoregressive operator'''. Hence the '''autoregessive model''' can be written in the following compact manner: :$\map \phi B \tilde z_t = a_t$
Definition:Autoregressive Model/Parameter
The '''parameters''' of $M$ consist of: :the constant mean level $\mu$ :the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shock $a_t$ :the coefficients $\phi_1$ to $\phi_p$.
Definition:Vector Space/Definition 1
Let $\struct {K, +_K, \times_K}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module. Then $\struct {G, +_G, \circ}_K$ is a '''vector space over $K$''' or a '''$K$-vector space'''. That is, a '''vector space''' is a unitary module whose scalar ring is a field.
Definition:Vector Space/Division Ring
Let $\struct {K, +_K, \times_K}$ be a division ring. Let $\struct {G, +_G}$ be an abelian group. Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module. Then $\struct {G, +_G, \circ}_K$ is a '''vector space over $K$''' or a '''$K$-vector space'''. That is, a vector space is a unitary module whose scalar ring is a division ring.
Definition:Vector Space/Definition 2
Let $\struct {K, +_K, \times_K}$ be a field whose unity is $1_K$. Let $\struct {G, +_G}$ be an abelian group. Let $\struct {\map {\mathrm {End} } G, +, \circ}$ be the endomorphism ring of $\struct {G, +_G}$ such that $I_G$ is the identity mapping. Let $\cdot: \struct {K, +_K, \times_K} \to \struct {\map {\mathrm {End} } G, +, \circ}$ be a ring homomorphism from $K$ to $\map {\mathrm {End} } G$ which maps $1_K$ to $I_G$. Then $\struct {G, +_G, \cdot, K}$ is a '''vector space over $K$''' or a '''$K$-vector space'''.
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2
{{begin-axiom}} {{axiom | n = \text B 2         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \cup \set y} \setminus \set x \in \mathscr B }} {{end-axiom}}
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1
{{begin-axiom}} {{axiom | n = \text B 1         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y \in \mathscr B }} {{end-axiom}}
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. === Definition 1 === $\mathscr B$ is said to satisfy the base axiom : {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} === Definition 2 === $\mathscr B$ is said to satisfy the base axiom : {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2}} === Definition 3 === $\mathscr B$ is said to satisfy the base axiom : {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3}} === Definition 4 === $\mathscr B$ is said to satisfy the base axiom : {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4}} === Definition 5 === $\mathscr B$ is said to satisfy the base axiom : {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5}} === Definition 6 === $\mathscr B$ is said to satisfy the base axiom : {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6}} === Definition 7 === $\mathscr B$ is said to satisfy the base axiom : {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7}}
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3
{{begin-axiom}} {{axiom | n = \text B 3         | q = \forall B_1, B_2 \in \mathscr B         | mr= \exists \text{ a bijection } \pi : B_1 \setminus B_2 \to B_2 \setminus B_1 : \forall x \in B_1 \setminus B_2 : \paren {B_1 \setminus \set x } \cup \set {\map \pi x} \in \mathscr B }} {{end-axiom}}
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4
{{begin-axiom}} {{axiom | n = \text B 4         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y, \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B }} {{end-axiom}}
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5
{{begin-axiom}} {{axiom | n = \text B 5         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y}  \cup \set x \in \mathscr B }} {{end-axiom}}
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7
{{begin-axiom}} {{axiom | n = \text B 7         | q = \forall B_1, B_2 \in \mathscr B         | mr= \exists \text{ a bijection } \pi : B_1 \setminus B_2 \to B_2 \setminus B_1 : \forall x \in B_1 \setminus B_2 : \paren {B_2 \setminus \set {\map \pi x} } \cup \set x \in \mathscr B }} {{end-axiom}}
Definition:Image of Subset under Mapping/Definition 2
The '''image of $X$ under $f$''' is the element of the codomain of the direct image mapping $f^\to: \powerset S \to \powerset T$ of $f$: :$\forall X \in \powerset S: \map {f^\to} X := \set {t \in T: \exists s \in X: \map f s = t}$
Definition:Preimage of Subset under Mapping/Definition 1
The '''preimage of $Y$ under $f$''' is defined as: :$f^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \map f s = t}$ That is, the '''preimage of $Y$ under $f$''' is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation.
Definition:Preimage of Subset under Mapping/Definition 2
The '''preimage of $Y$ under $f$''' can be seen to be an element of the codomain of the inverse image mapping $f^\gets: \powerset T \to \powerset S$ of $f$: :$\forall Y \in \powerset T: \map {f^\gets} Y := \set {s \in S: \exists t \in Y: \map f s = t}$ Thus: :$\forall Y \subseteq T: f^{-1} \sqbrk Y = \map {f^\gets} Y$
Definition:Moving Average Model
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t$ be the deviation from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be a model where the current value of $\tilde z_t$ is expressed as a finite linear aggregate of the shocks: :$\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ $M$ is known as a '''moving average (MA) process of order $q$'''.
Definition:Moving Average Model/Moving Average Operator
Let $\map \theta B$ be defined as: :$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$ where $B$ denotes the backward shift operator. Then $\map \theta B$ is referred to as the '''moving average operator'''. Hence the '''moving average model''' can be written in the following compact manner: :$\tilde z_t = \map \theta B a_t$
Definition:Moving Average Model/Parameter
The '''parameters''' of $M$ consist of: :the constant mean level $\mu$ :the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$ :the coefficients $\theta_1$ to $\theta_q$.
Definition:Box-Jenkins Model/ARMA
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be a model where the current value of $\tilde z_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ $M$ is known as a '''mixed autoregressive (order $p$), moving average (order $q$) process''', usually referred as an '''ARMA process'''.
Definition:ARMA Model/ARMA Operator
Using the autoregressive operator: :$\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ and the moving average operator: :$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$ the '''ARMA model''' can be written in the following compact manner: :$\map \phi B \tilde z_t = \map \theta B a_t$ where $B$ denotes the backward shift operator. Hence: :$\tilde z_t = \map {\phi^{-1} } B \map \theta B a_t$
Definition:ARMA Model/Parameter
The '''parameters''' of $M$ consist of: :the constant mean level $\mu$ :the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$ :the coefficients $\phi_1$ to $\phi_p$ :the coefficients $\theta_1$ to $\theta_q$.
Definition:Scalar Ring/Module
Let $\struct {G, +_G, \circ}_R$ be a module, where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {G, +_G}$ is an abelian group :$\circ: R \times G \to G$ is a binary operation. Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {G, +_G, \circ}_R$.
Definition:Scalar Ring/Unitary Module
Let $\struct {G, +_G, \circ}_R$ be a module, where: :$\struct {R, +_R, \times_R}$ is a ring with unity :$\struct {G, +_G}$ is an abelian group :$\circ: R \times G \to G$ is a binary operation. Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {G, +_G, \circ}_R$.
Definition:Scalar Ring/Vector Space over Division Ring
Let $\struct {G, +_G, \circ}_K$ be a vector space over a division ring, where: :$\struct {K, +_K, \times_K}$ is a division ring :$\struct {G, +_G}$ is an abelian group $\struct {G, +_G}$ :$\circ: K \times G \to G$ is a binary operation. Then the division ring $\struct {K, +_K, \times_K}$ is called the '''scalar division ring''' of $\struct {G, +_G, \circ}_K$, or just '''scalar ring'''.
Definition:Box-Jenkins Model/ARIMA
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let: :$w_t = \nabla^d z_t$ where $\nabla^d$ denotes the $d$th iteration of the backward difference operator. Let $M$ be a model where the current value of $w_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks: :$w_t = \phi_1 w_{t - 1} + \phi_2 w_{t - 2} + \dotsb + \phi_p w_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ $M$ is known as an '''autoregressive integrated moving average (ARIMA) process of order $p$, $d$, $q$'''.
Definition:ARIMA Model/ARIMA Operator
Using the autoregressive operator: :$\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ and the moving average operator: :$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$ the '''ARIMA model''' can be written in the following compact manner: :$\map \phi B w_t = \map \theta B a_t$ where $B$ denotes the backward shift operator. Hence: :$\map \varphi B z_t = \map \phi B \paren {1 - B}^d z_t = \map \theta B a_t$ where: :$\map \varphi B = \map \phi B \paren {1 - B}^d$
Definition:Scalar/Module
The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called '''scalars'''.
Definition:Scalar/Vector Space
The elements of the scalar field $\struct {K, +_K, \times_K}$ are called '''scalars'''.
Definition:Vector Field
If a vector quantity $\mathbf v$ can be associated with every point in a given space, then $\mathbf v$ is said to be a '''vector field'''.
Definition:Density (Physics)/Dimension
The dimension of '''density''' is $\mathsf {M L}^{-3}$: mass per unit volume.
Definition:Density (Physics)/Units
* The SI units of '''density''' are $\mathrm {kg} \, \mathrm m^{-3}$ (kilograms per cubic metre). * The CGS units of '''density''' are $\mathrm g \, \mathrm{cm}^{-3}$ or, less formally: $\mathrm g / \mathrm {cc}$ (grams per cubic centimetre). Thus: :$1 \, \mathrm g \, \mathrm {cm}^{-3} = 1000 \, \mathrm {kg} \, \mathrm m^{-3}$
Definition:Density (Physics)/Linear/Symbol
The usual symbol used to denote '''linear density''' is $\mu$ (Greek letter '''mu'''). Sometimes $\lambda$ (Greek letter '''lambda''') is also used.
Definition:Mistake/Whoosho
A '''whoosho''' is a mistake caused by a complete failure to understand a concept vital to the accurate communication of an idea. It is also used for a failure to understand or recognise a joke. The word comes from the popular meme of how '''whoosh''' is supposedly the sound of whatever it is going over the person's head. :''"That went so far over my head I didn't even hear it go whoosh."''
Definition:Current Time
The '''current time''' in the context of time series analysis is the timestamp of the most recent observation. By the very nature of time itself, the '''current time''' is always going to be some non-zero instant in the past.