What is a Continuous Variable in Stats

Continuous Variable

Data

Colleen McCue , in Data Mining and Predictive Analysis, 2007

5.3 Data ²

Continuous variables can take on an unlimited number of values between the lowest and highest points of measurement. Continuous variables include such things as speed and distance. Continuous data are very desirable in inferential statistics; however, they tend to be less useful in data mining and are frequently recoded into discrete data or sets, which are described next.

Discrete data are associated with a limited number of possible values. Gender or rank are examples of discrete variables because there are a limited number of mutually exclusive options. Binary data are a type of discrete data that encompass information that is confined to two possible options (e.g., male or female; yes or no). Discrete and binary data also are called sets and flag data, respectively.

Understanding the different types of data and their definitions is important because some types of analyses have been designed for particular types of data and may be inappropriate for another type of information. The good news is that the types of information most prevalent in law enforcement and intelligence, sets and flag data, tend to be the most desirable for data mining. With traditional, inferential statistics methodologies, on the other hand, discrete variables are disadvantageous because statistical power is compromised with this type of categorical data. In other words, a bigger difference between the groups of interest is needed to achieve a statistically significant result.

We also can speak of data in terms of how they are measured. Ratio scales are numeric and are associated with a true zero—meaning that nothing can be measured. For example, weight is a ratio scale. A weight of zero corresponds to the absence of any weight. With an interval scale, measurements between points have meaning, although there is no true zero. For example, although there is no true zero associated with the Fahrenheit temperature scale, the difference between 110 and 120 degrees Fahrenheit is the same as the difference between 180 and 190 degrees: 10 degrees. Ordinal scales imply some ranking in the information. Though the data might not correspond to actual numeric figures, there is some implied ranking. Sergeant, lieutenant, major, and colonel represents an ordinal scale. Lieutenant is ranked higher than sergeant, and major is ranked higher than lieutenant. Although they do not correspond directly to any type of numeric values, it is understood that there is a rank ordering of these categories. Finally, nominal scales really are not true scales because they are not associated with any sort of measurable dimension or ranking; the particular designations, even if numeric, do not correspond to quantifiable features. An example of this type of data is any type of categorical data, such as vehicle make or numeric patrol unit designations.

Finally, unformatted or text data truly are unique. Until recently, it was very difficult to analyze this type of information because the analytical tools necessary were extremely sophisticated and not generally available. Frequently, text data were recoded and categorized into some type of discrete variables. Recent advancements in computational techniques, however, have opened the door to analyzing these data in their native form. By using techniques such as natural language processing, syntax and language can be analyzed intact, a process that extends well beyond crude keyword searches.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780750677967500271

27th European Symposium on Computer Aided Process Engineering

Pedro C. Pautasso , ... Diego C. Cafaro , in Computer Aided Chemical Engineering, 2017

2.2 Controlling inventory level in charging tanks

Continuous variables ${I_m}^{\left(ik\right)}$ and ${I_m}^{\left(im\right)}$ denote the inventory levels in the charging tank m at the completion time of operations ik and im, respectively. They are derived from the difference of accumulated loading $({L_m}^{\left(ik\right)},{L_m}^{\left(im\right)})$ and unloading $({U_m}^{\left(ik\right)},{U_m}^{\left(im\right)})$ amounts plus the parameter $ii{m^o}_m$ , representing the initial inventory of crude mix in the charging tank m. The value of ${I_m}^{\left(ik\right)}$ must never exceed the capacity of the charging tank m (capm_m ). In turn, ${I_m}^{\left(im\right)}$ can never be lower than the minimum allowed inventory of crude mix in the charging tank m (invm_m ).

(8) $I_m^{\left(ik\right)}=ii{m}_m^o+L_m^{\left(ik\right)}-U_m^{\left(ik\right)}\le \mathit{cap}{m}_m\forall ik\in \mathit{I}{\mathit{K}}^{\mathit{new}},m\in \mathit{M}$

(9) $I_m^{\left(im\right)}=ii{m}_m^o+L_m^{\left(im\right)}-U_m^{\left(im\right)}\ge inv{m}_m\forall im\in \mathit{I}\mathit{M},m\in \mathit{M}$

A similar approach is used for tracking component contents both in storage and charging tanks at every time event. The model also includes binary variables to: (a) assign batches of crude mix to storage and charging tanks {WK_ik,k , WM_im,m }; and (b) assign a destination for every existing batch {YV_iv,k , YP_ik,m , YM_im,t }. Eqs. (10) and (11) state that at most one batch can be sent from the pipeline to a single charging tank during a lot injection. Moreover, only one storage tank can supply material during a single pipeline injection.

(10) ${\displaystyle \sum _{\underset{ik<i{k}^{\prime }}{ik\in \mathit{I}\mathit{K}}}X{P}_{ik,i{k}^{\prime }}}\le 1\forall i{k}^{\prime }\in \mathit{I}{\mathit{K}}^{\mathit{new}}$

(11) ${\displaystyle \sum _{\underset{ik<i{k}^{\prime }}{ik\in \mathit{I}\mathit{K}}}X{P}_{ik,i{k}^{\prime }}}={\displaystyle \sum _{m\in \mathit{M}}Y}{P}_{i{k}^{\prime },m}={\displaystyle \sum _{k\in \mathit{K}}W}{K}_{i{k}^{\prime },k}\forall i{k}^{\prime }\in \mathit{I}{\mathit{K}}^{\mathit{new}}$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444639653502221

A Survey of Quantum Key Distribution (QKD) Technologies

Jeffrey D. Morris , ... Gerald Baumgartner , in Emerging Trends in ICT Security, 2014

Continuous variable QKD: short-ranged but fast and secure

The first Continuous Variable QKD (CV-QKD) system debuted in the European SEcure COmmunication based on Quantum Cryptography (SECOQC) network, with the prototype built expressly for the project. Timothy Ralph first described the CV-QKD protocol in 1999, with several variations proposed by Cerf, Assche, Lutkenhaus, and Grosshans. These protocols use squeezed Gaussian states of light that have classical intensity levels to carry information, rather than discrete single photon states [17–20].

This system encodes information in the amplitude and phase of the classical light level beam and produces high rates of key generation over a short distance (such as a metropolitan network), as it is not as sensitive to individual photon loss as the discrete-variable protocols. Originally, it was not suitable over long distances because the higher noise ratios in longer fibers created errors in the quadratures of pulses, interfering in the homodyne detection, but improvements in post-processing have increased the transmission range. The protocol is resistant against general and collective eavesdropping attacks, and has a security proof for coherent attacks as well. These security proofs show it is more secure than some other types of QKD systems against attacks that exploit the non-ideal hardware flaws of QKD systems [21].

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780124114746000098

29th European Symposium on Computer Aided Process Engineering

Nikolaos Rakovitis , ... Nan Zhang , in Computer Aided Chemical Engineering, 2019

3.1 Different tasks in different units

We define continuous variables bc _{i, i′, s, n} as material s produced from task i consumed by task i′ at event n. Thus, the material consumed should not exceed total available materials.

(1) $-\sum _{j^{\prime }}\sum _{i^{\prime }\in {\mathbf{I}}_s^C,i^{\prime }\in {\mathbf{I}}_{j^{\prime }}}\left({\rho }_{s,i^{\prime }}\cdot \sum _{n\le n^{\prime }\le n+\mathit{\Delta n}}{b}_{i^{\prime },j^{\prime },n,n^{\prime }}\right)\le {\mathit{ST}}_{s,n-1}+\sum _{i\in {\mathbf{I}}_s^P}\sum _{i^{\prime }\in {\mathbf{I}}_s^C}{\mathit{bc}}_{i,i^{\prime },s,n}\forall s\in {\mathbf{S}}^{\mathit{IN}},n$

The amount of materials from i to i′ (i.e., bc _{i, i′, s, n} ) should not exceed the production amount of task i or consumption amount of task i′.

(2) $\sum _{i^{\prime }\in {\mathbf{I}}_s^C}{\mathit{bc}}_{i,i^{\prime },s,n}\le {\rho }_{s,i}\cdot \sum _{n-\mathit{\Delta n}\le n^{\prime }\le n}{b}_{i,j,n^{\prime },n}\forall s\in {\mathbf{S}}^{\mathit{IN}},i\in {\mathbf{I}}_s^P,i\notin {\mathbf{I}}^{\text{Re}},j,n$

(3) $\begin{array}{ll}\sum _{i\in {\mathbf{I}}_s^P}{\mathit{bc}}_{i,i^{\prime },s,n}\le {\rho }_{s,i^{\prime }}\cdot \sum _{n\le n^{\prime }\le n+\mathit{\Delta n}}{b}_{i^{\prime },j^{\prime },n,n^{\prime }}& \forall s\in {\mathbf{S}}^{\mathit{IN}},i^{\prime }\in {\mathbf{I}}_s^C,j,n\end{array}$

(4) $\begin{array}{ll}{\mathit{bc}}_{i,i^{\prime },s,n}\le z_{j,i^{\prime },s,n}\cdot {\rho }_{s,i}\cdot B_{i,j}^{\text{max}}& \forall s\in {\mathbf{S}}^{\mathit{IN}},i\in {\mathbf{I}}_s^P,i\in {\mathbf{I}}_j,i^{\prime }\in {\mathbf{I}}_s^C,i^{\prime }\in {\mathbf{I}}_{j^{\prime }},j,j^{\prime },j\ne j^{\prime },n\end{array}$

(5) $\begin{array}{ll}{T}_{s,j,n}\ge {\mathit{Tf}}_{j,n}-M\cdot \left(1-\sum _{i\in {\mathbf{I}}_s^P,i\in {\mathbf{I}}_j}\sum _{n-\mathit{\Delta n}\le n^{\prime }\le n}{w}_{i,j,n^{\prime },n}\right)& \forall s\in {\mathbf{S}}^{\mathit{IN}},j,n\end{array}$

(6) $\begin{array}{ll}{T}_{s,j,n}\ge {\mathit{Ts}}_{j^{\prime },n}+M\cdot \left(2-\sum _{i^{\prime }\in {\mathbf{I}}_s^C,i\in {\mathbf{I}}_j}\sum _{n\le n^{\prime }\le n+\mathit{\Delta n}}{w}_{i^{\prime },j^{\prime },n,n^{\prime }}-z_{j^{\prime },j,s,n}\right)& \forall s\in {\mathbf{S}}^{\mathit{IN}}j,j^{\prime },i\in {\mathbf{I}}_j,i\in {\mathbf{I}}_s^P,i\notin {\mathbf{I}}^{\text{Re}},n\end{array}$

An active consumption task i′ should be processed in a unit after the time that state s is available at the previous event n.

(7) $\begin{array}{ll}{T}_{s,j,n}\ge {\mathit{Tf}}_{j^{\prime },n+1}+M\cdot \left(1-\sum _{i^{\prime }\in I_s^C,i^{\prime }\in I_{j^{\prime }}}\sum _{n+1\le n^{\prime }\le n+1+\mathit{\Delta n}}{w}_{i^{\prime },j^{\prime },n+1,n^{\prime }}\right)& \forall s\in {\mathbf{S}}^{\mathit{IN}}j,j^{\prime },i\in {\mathbf{I}}_s^P,i\in I_j,i\notin {\mathbf{I}}^{\text{Re}},n<N\end{array}$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128186343502320

Personality Development in Adulthood

L.L. Carstensen , in International Encyclopedia of the Social & Behavioral Sciences, 2001

4 The Trait Approach to Adult Personality Development

Traits are continuous variables represented by broadly encompassing lexical terms that account for individual differences (John 1990). Traits—such as shy, lively, outgoing, anxious, and intelligent—are conceptualized as predispositions within individuals to behave in certain ways manifest across a wide variety of situations. Gordon Allport (see Allport, Gordon W (1897–1967) ) argued that cardinal traits are those around which a person organizes life (self-sacrifice). Central traits (e.g., honesty) represent major features and secondary traits are specific traits that help to predict behavior more than underlying personality (e.g., dress type, food preferences). Allport's definition is compatible with modern trait and temperament approaches to personality which attempt to describe people in terms of one or more central features of the person.

Personality psychologists in the trait tradition seek to identify the traits along which people differ and to explore the degree to which these traits predict behavior. Many taxonomies of traits have been offered over the years, but unquestionably the five-factor model is most widely accepted today (see also Personality Structure ). Based on factor analysis of self-descriptions, the five traits that emerge reliably across many studies of Europeans and Americans are: (a) openness to experience, (b) conscientiousness, (c) extraversion, (d) agreeableness, and (e) neuroticism.

Traits and temperaments appear to be relatively stable through the second half of life (Costa and McRae 1990). It appears that beyond the age of 30, extraverts remain extraverts and neurotics remain neurotics. Trait theorists have found reliable evidence for stability in personality well into old age. This finding emerges whether researchers ask individuals to describe themselves repeatedly over time or, alternatively, ask significant others, like spouses, to describe those same individuals repeatedly (Costa and McRae 1990). It should be noted that even though persistent rank-order differences remain the same, there is some recent evidence that modest mean level changes may appear, with older adults scoring slightly higher than younger adults on agreeableness and conscientiousness and slightly lower on neuroticism, extraversion, and openness to experience (McCrae et al. 1999). Importantly, similar findings come from studies sampling Asian and European populations. However, identified changes are quite small. Overall, there is remarkable consistency in the characteristics that distinguish individuals from one another over time. There is some evidence that the core set of traits that differentiate people are genetically based and exert their influence throughout the life course (Gatz 1992). Genetic influence is as strong in old age as early adulthood.

In summary, researchers adopting a trait approach to personality development find that along at least some of the important dimensions of personality, there is little change in personality well into old age. Critics of a trait approach, however, argue that traits communicate little about how people manage their lives in day-to-day life and because of the broadband focus exaggerate the consistency of behavior across time and situations. They criticize the trait approach for failing to better predict behavior and redirect focus to specific strategies (e.g., how an individual cognitively appraises a situation; expectancies, subjective values, self-regulatory systems, and competencies). Life-span approaches—influenced strongly by the social cognitive theory of personality (Bandura 1989)—view individuals as agentic creatures who shape their own environments (see also Interactionism and Personality ; Self-regulation in Adulthood ).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B0080430767017630

24th European Symposium on Computer Aided Process Engineering

Dajun Yue , Fengqi You , in Computer Aided Chemical Engineering, 2014

2 Mixed-integer linear fractional program

An MILFP includes both continuous and discrete variables. All the constraints of an MILFP are linear, and the objective function is expressed as the ratio of two linear functions. Mathematically, a general MILFP can be formulated as the following problem (P0):

(P0)

(1) $\max \frac{A_0+{\displaystyle \sum _{i\in 1}A{1}_i{x}_i}+{\displaystyle \sum _{j\in J}A{2}_j{y}_j}}{B_0+{\displaystyle \sum _{i\in j}B{1}_i{x}_i+{\displaystyle \sum _{j\in J}B{2}_j{y}_j}}}$

s.t.

(2) $C_{0k}+{\displaystyle \sum }_{i\in I}C{I}_{ik}{x}_i+{\displaystyle \sum }_{}C{2}_{jk}{y}_j=0,\forall k\in k$

(3) $x_i\ge 0,\forall i\in Iand{y}_j\in \{0,1\},\forall j\in J$

where x_i are continuous variables and yj are discrete variables. For problem (P0), it is assumed that the denominator $B_0+{\Sigma }_{i\in I}B{1}_i{x}_i+{\Sigma }_{j\in J}B{2}_j{y}_j>0$ for all feasible solutions and all inequalities are converted into equalities through the use of slack variables (Dinkelbach, 1967).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444634566501599

From Zero to One

Sarah L. Harris , David Money Harris , in Digital Design and Computer Architecture, 2016

1.6.2 Logic Levels

The mapping of a continuous variable onto a discrete binary variable is done by defining logic levels, as shown in Figure 1.23. The first gate is called the driver and the second gate is called the receiver. The output of the driver is connected to the input of the receiver. The driver produces a LOW (0) output in the range of 0 to V_OL or a HIGH (1) output in the range of V_OH to V _dd · If the receiver gets an input in the range of 0 to V_IL , it will consider the input to be LOW. If the receiver gets an input in the range of V_IH to V _dd , it will consider the input to be HIGH. If, for some reason such as noise or faulty components, the receiver's input should fall in the forbidden zone between V_IL and V_IH , the behavior of the gate is unpredictable. V_OH ,V_OL ,V_IH , and V _Il are called the output and input high and low logic levels.

Figure 1.23. Logic levels and noise margins

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128000564000017

From Zero to One

Sarah L. Harris , David Harris , in Digital Design and Computer Architecture, 2022

1.6.2 Logic Levels

The mapping of a continuous variable onto a discrete binary variable is done by defining logic levels, as shown in Figure 1.23. The first gate is called the driver and the second gate is called the receiver. The output of the driver is connected to the input of the receiver. The driver produces a LOW (0) output in the range of 0 to V_OL or a HIGH (1) output in the range of V_OH to V_DD . If the receiver gets an input in the range of 0 to V_IL , it will consider the input to be LOW. If the receiver gets an input in the range of V_IH to V_DD , it will consider the input to be HIGH. If, for some reason such as noise or faulty components, the receiver's input should fall in the forbidden zone between V_IL and V_IH , the behavior of the gate is unpredictable. V_OH , V_OL , V_IH , and V_IL are called the output and input high and low logic levels.

Figure 1.23. Logic levels and noise margins

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128200643000015

Proceedings of the 9th International Conference on Foundations of Computer-Aided Process Design

Ana Somoza-Tornos , ... Moisés Graells , in Computer Aided Chemical Engineering, 2019

Demand satisfaction

Qout _k is a continuous variable denoting the flowrate sent to satisfy demand k, and is calculated in Eq. (10) through the factor Y _ik with range [0,1] (Eq. (11)). Equation (12) expresses the limits to be satisfies by Qout _k .

(10) ${\mathit{Qout}}_k={\mathrm{\Sigma }}_i\left(Y_{\mathit{ik}}\cdot {\mathit{Qtu}}_i\right)\forall k$

(11) ${\mathrm{\Sigma }}_k{Y}_{\mathit{ik}}=1\forall i$

(12) ${\underset{¯}{\mathit{MinQout}}}_k\le {\mathit{Qout}}_k\le \overline{{\mathit{MaxQout}}_k}\forall k$

Equations (13, 14) are the analogous definition and limit satisfaction constraint for the outlet concentration Cout _kj .

(13) ${\mathit{Cout}}_{\mathit{kj}}={\mathrm{\Sigma }}_i\left(Y_{\mathit{ik}}\cdot {\mathit{Cin}}_{\mathit{ij}}\cdot {\mathit{Qin}}_i\right)/{\mathrm{\Sigma }}_i\left(Y_{\mathit{ik}}\cdot {\mathit{Qin}}_i\right)\forall \mathit{kj}$

(14) ${\underset{¯}{\mathit{MinCout}}}_{\mathit{kj}}\le {\mathit{Cout}}_{\mathit{kj}}\le \overline{{\mathit{MaxCout}}_{\mathit{kj}}}\forall \mathit{kj}$

a _k is defined by Eqs. (15, 16) as a binary variable that takes a value of 1 if Qout _k is greater than the demand D _k .

(15) $\left({\mathit{Qout}}_k-D_k\right)\ge M\cdot \left(1-a_k\right)\forall k$

(16) $\left(D_k-{\mathit{Qout}}_k\right)\le M\cdot a_k\forall k$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128185971500205

31st European Symposium on Computer Aided Process Engineering

Fernando D. Ramos , ... M. Soledad Diaz , in Computer Aided Chemical Engineering, 2021

3 Numerical Results

The MINLP model presented 5398 continuous variables and 13 binary variables and 5336 constraints and it was solved with DICOPT (CONOPT and CPLEX) (Grossmann et al., 2003). Besides, the energetic integration model possesses 1236 continuous variables, 339 binary variables and 1338 constraints. The heat integration reduced the number of cold and heat utilities ( Figure 2) which directly impacted on the environmental pillar of RePSIM. Before the heat integration, the process required 828 kW while after heat integration this value was 470 kW (43 % less). These results are in agreement with the published literature (Chong et al., 2020; Song et al., 2018), which mention a 32.1 % and a 42.8 % of energy reduction in a macroalgae waste biorefinery and a cellulosic bioethanol plant, respectively. In term of cost, this represented a 54 % of savings. As regard RePSIM environmental pillar, it showed an improvement of 14 %. Finally, it is worth noting that not only does heat integration improve economic aspects of a process, but it also improves the environmental one.

Figure 2. Heat exchanger network

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780323885065502448

rodgersknound.blogspot.com

Source: https://www.sciencedirect.com/topics/computer-science/continuous-variable

Share this post