2.3: Models and Applications (2022)

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Learning Objectives
  • Set up a linear equation to solve a real-world application.
  • Use a formula to solve a real-world application.

Josh is hoping to get an \(A\) in his college algebra class. He has scores of \(75\), \(82\), \(95\), \(91\), and \(94\) on his first five tests. Only the final exam remains, and the maximum of points that can be earned is \(100\). Is it possible for Josh to end the course with an \(A\)? A simple linear equation will give Josh his answer.

2.3: Models and Applications (1)

Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as \($0.10/mi\), is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write \(0.10x\). This expression represents a variable cost because it changes according to the number of miles driven.

If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges \($0.10/mi\) plus a daily fee of \($50\). We can use these quantities to model an equation that can be used to find the daily car rental cost \(C\).

\(C=0.10x+50 \tag{2.4.1}\)

When dealing with real-world applications, there are certain expressions that we can translate directly into math. Table \(\PageIndex{1}\) lists some common verbal expressions and their equivalent mathematical expressions.

Table \(\PageIndex{1}\): Verbal to math conversion
Verbal Translation to Math Operations
One number exceeds another by a \(x,x+a\)
Twice a number \(2x\)
One number is \(a\) more than another number \(x,x+a\)
One number is a less than twice another number \(x,2x−a\)
The product of a number and \(a\), decreased by \(b\) \(ax−b\)
The quotient of a number and the number plus \(a\) is three times the number \(\dfrac{x}{x+a}=3x\)
The product of three times a number and the number decreased by \(b\) is \(c\) \(3x(x−b)=c\)
How to: Given a real-world problem, model a linear equation to fit it
  1. Identify known quantities.
  2. Assign a variable to represent the unknown quantity.
  3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
  4. Write an equation interpreting the words as mathematical operations.
  5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.
Example \(\PageIndex{1}\)

Find a linear equation to solve for the following unknown quantities: One number exceeds another number by \( 17\) and their sum is \( 31\). Find the two numbers.

Solution

Let \( x\) equal the first number. Then, as the second number exceeds the first by \(17\), we can write the second number as \( x +17\). The sum of the two numbers is \(31\). We usually interpret the word is as an equal sign.

\[\begin{align*} x+(x+17)&= 31\\ 2x+17&= 31\\ 2x&= 14\\ x&= 7 \end{align*}\]

\[\begin{align*} x+17&= 7 + 17\\ &= 24\\ \end{align*}\]

The two numbers are \(7\) and \(24\).

Exercise \(\PageIndex{1}\)

Find a linear equation to solve for the following unknown quantities: One number is three more than twice another number. If the sum of the two numbers is \(36\), find the numbers.

Answer

\(11\) and \(25\)

(Video) 2.3 Models and Applications
Example \(\PageIndex{2}\): Setting Up a Equation to Solve a Real-World Application

There are two cell phone companies that offer different packages. Company A charges a monthly service fee of \($34\) plus \($.05/min\) talk-time. Company B charges a monthly service fee of \($40\) plus \($.04/min\) talk-time.

  1. Write a linear equation that models the packages offered by both companies.
  2. If the average number of minutes used each month is \(1,160\), which company offers the better plan?
  3. If the average number of minutes used each month is \(420\), which company offers the better plan?
  4. How many minutes of talk-time would yield equal monthly statements from both companies?

Solution

a.

The model for Company A can be written as \( A =0.05x+34\). This includes the variable cost of \( 0.05x\) plus the monthly service charge of \($34\). Company B’s package charges a higher monthly fee of \($40\), but a lower variable cost of \( 0.04x\). Company B’s model can be written as \( B =0.04x+$40\).

b.

If the average number of minutes used each month is \(1,160\), we have the following:

\[\begin{align*} \text{Company A}&= 0.05(1.160)+34\\ &= 58+34\\ &= 92 \end{align*}\]

\[\begin{align*} \text{Company B}&= 0.04(1,1600)+40\\ &= 46.4+40\\ &= 86.4 \end{align*}\]

So, Company B offers the lower monthly cost of \($86.40\) as compared with the \($92\) monthly cost offered by Company A when the average number of minutes used each month is \(1,160\).

c.

If the average number of minutes used each month is \(420\), we have the following:

\[\begin{align*} \text{Company A}&= 0.05(420)+34\\ &= 21+34\\ &= 55 \end{align*}\]

\[\begin{align*} \text{Company B}&= 0.04(420)+40\\ &= 16.8+40\\ &= 56.8 \end{align*}\]

If the average number of minutes used each month is \(420\), then Company A offers a lower monthly cost of \($55\) compared to Company B’s monthly cost of \($56.80\).

d.

To answer the question of how many talk-time minutes would yield the same bill from both companies, we should think about the problem in terms of \((x,y)\) coordinates: At what point are both the \(x\)-value and the \(y\)-value equal? We can find this point by setting the equations equal to each other and solving for \(x\).

\[\begin{align*} 0.05x+34&= 0.04x+40\\ 0.01x&= 6\\ x&= 600 \end{align*}\]

Check the \(x\)-value in each equation.

\(0.05(600)+34=64\)

\(0.04(600)+40=64\)

Therefore, a monthly average of \(600\) talk-time minutes renders the plans equal. See Figure \(\PageIndex{2}\).

(Video) 2.3 - Models and Applications

2.3: Models and Applications (2)
Exercise \(\PageIndex{2}\)

Find a linear equation to model this real-world application: It costs ABC electronics company \($2.50\) per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of \($350\) for utilities and \($3,300\) for salaries. What are the company’s monthly expenses?

Answer

\(C=2.5x+3,650\)

Using a Formula to Solve a Real-World Application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region,

\[A=LW \tag{2.4.2}\]

the perimeter of a rectangle,

\[P=2L+2W \tag{2.4.3}\]

and the volume of a rectangular solid,

\[V=LWH. \tag{2.4.4}\]

When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

Example \(\PageIndex{3}\): Solving an Application Using a Formula

It takes Andrew \(30\; min\) to drive to work in the morning. He drives home using the same route, but it takes \(10\; min\) longer, and he averages \(10\; mi/h\) less than in the morning. How far does Andrew drive to work?

Solution

This is a distance problem, so we can use the formula \(d =rt\), where distance equals rate multiplied by time. Note that when rate is given in \(mi/h\), time must be expressed in hours. Consistent units of measurement are key to obtaining a correct solution.

First, we identify the known and unknown quantities. Andrew’s morning drive to work takes \(30\; min\), or \(12\; h\) at rate \(r\). His drive home takes \(40\; min\), or \(23\; h\), and his speed averages \(10\; mi/h\) less than the morning drive. Both trips cover distance \(d\). A table, such as Table \(\PageIndex{2}\), is often helpful for keeping track of information in these types of problems.

Table \(\PageIndex{2}\)
\(d\) \(r\) \(t\)
To Work \(d\) \(r\) \(12\)
To Home \(d\) \(r−10\) \(23\)

Write two equations, one for each trip.

\[d=r\left(\dfrac{1}{2}\right) \qquad \text{To work} \nonumber\]

\[d=(r-10)\left(\dfrac{2}{3}\right) \qquad \text{To home} \nonumber\]

As both equations equal the same distance, we set them equal to each other and solve for \(r\).

\[\begin{align*} r\left (\dfrac{1}{2} \right )&= (r-10)\left (\dfrac{2}{3} \right )\\ \dfrac{1}{2r}&= \dfrac{2}{3}r-\dfrac{20}{3}\\ \dfrac{1}{2}r-\dfrac{2}{3}r&= -\dfrac{20}{3}\\ -\dfrac{1}{6}r&= -\dfrac{20}{3}\\ r&= -\dfrac{20}{3}(-6)\\ r&= 40 \end{align*}\]

We have solved for the rate of speed to work, \(40\; mph\). Substituting \(40\) into the rate on the return trip yields \(30 mi/h\). Now we can answer the question. Substitute the rate back into either equation and solve for \(d\).

\[\begin{align*}d&= 40\left (\dfrac{1}{2} \right )\\ &= 20 \end{align*}\]

(Video) 2.3 - Models and Applications-1

The distance between home and work is \(20\; mi\).

Analysis

Note that we could have cleared the fractions in the equation by multiplying both sides of the equation by the LCD to solve for \(r\).

\[\begin{align*} r\left (\dfrac{1}{2} \right)&= (r-10)\left (\dfrac{2}{3} \right )\\ 6\times r\left (\dfrac{1}{2} \right)&= 6\times (r-10)\left (\dfrac{2}{3} \right )\\ 3r&= 4(r-10)\\ 3r&= 4r-40\\ r&= 40 \end{align*}\]

Exercise \(\PageIndex{3}\)

On Saturday morning, it took Jennifer \(3.6\; h\) to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer \(4\; h\) to return home. Her speed was \(5\; mi/h\) slower on Sunday than on Saturday. What was her speed on Sunday?

Answer

\(45\; mi/h\)

Example \(\PageIndex{4}\): Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is \(54\; ft\). The length is \(3\; ft\) greater than the width. What are the dimensions of the patio?

Solution

The perimeter formula is standard: \(P=2L+2W\). We have two unknown quantities, length and width. However, we can write the length in terms of the width as \(L =W+3\). Substitute the perimeter value and the expression for length into the formula. It is often helpful to make a sketch and label the sides as in Figure \(\PageIndex{3}\).

2.3: Models and Applications (3)

Now we can solve for the width and then calculate the length.

\[\begin{align*} P&= 2L + 2W\\ 54&= 2(W+3)+2W\\ 54&= 2W+6+2W\\ 54&= 4W+6\\ 48&= 4W\\ W&= 12 \end{align*}\]

\[\begin{align*} L&= 12+3\\ L&= 15 \end{align*}\]

The dimensions are \(L = 15\; ft\) and \(W = 12\; ft\).

Exercise \(\PageIndex{4}\)

Find the dimensions of a rectangle given that the perimeter is \(110\; cm\) and the length is \(1\; cm\) more than twice the width.

Answer

\(L=37\; cm\), \(W=18\; cm\)

Example \(\PageIndex{5}\): Solving an Area Problem

The perimeter of a tablet of graph paper is \(48\space{in.}^2\). The length is \(6\; in\). more than the width. Find the area of the graph paper.

Solution

(Video) MATH113 | 2.3 Models and Applications (pt. 1) | Linear models

The standard formula for area is \(A =LW\); however, we will solve the problem using the perimeter formula. The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns. As both perimeter and area use length and width as dimensions, they are often used together to solve a problem such as this one.

We know that the length is \(6\; in\). more than the width, so we can write length as \(L =W+6\). Substitute the value of the perimeter and the expression for length into the perimeter formula and find the length.

\[\begin{align*} P&= 2L + 2W\\ 48&= 2(W+6)+2W\\ 48&= 2W+12+2W\\ 48&= 4W+12\\ 36&= 4W\\ W&= 9 \end{align*}\]

\[\begin{align*}L&= 9+6\\ L&= 15 \end{align*}\]

Now, we find the area given the dimensions of \(L = 15\; in\). and \(W = 9\; in\).

\[\begin{align*} A&= LW\\ A&=15(9)\\ A&= 135\space{in.}^2 \end{align*}\]

The area is \(135\space{in.}^2\).

Exercise \(\PageIndex{5}\)

A game room has a perimeter of \(70\; ft\). The length is five more than twice the width. How many \(ft^2\) of new carpeting should be ordered?

Answer

\(250\space{ft}^2\)

Example \(\PageIndex{6}\): Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is \(8\; \) in, and the volume is \(1,600\space{in.}^3\).

Solution

The formula for the volume of a box is given as \(V =LWH\), the product of length, width, and height. We are given that \(L =2W\), and \(H =8\). The volume is \(1,600\; \text{cubic inches}\).

\[\begin{align*} V&= LWH\\ 1600&= (2W)W(8)\\ 1600&= 16W^2\\ 100&= W^2\\ 10&= W \end{align*}\]

The dimensions are \(L = 20\; in\), \(W= 10\; in\), and \(H = 8\; in\).

Analysis

Note that the square root of \(W^2\) would result in a positive and a negative value. However, because we are describing width, we can use only the positive result.

Media

Access these online resources for additional instruction and practice with models and applications of linear equations.

  1. Problem solving using linear equations
  2. Problem solving using equations
  3. Finding the dimensions of area given the perimeter
  4. Find the distance between the cities using the distance = rate * time formula
  5. Linear equation application (Write a cost equation)

Key Concepts

  • A linear equation can be used to solve for an unknown in a number problem. See Example.
  • Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See Example.
  • There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the \(d = rt\) formula. See Example.
  • Many geometry problems are solved using the perimeter formula \(P =2L+2W\), the area formula \(A =LW\), or the volume formula \(V =LWH\). See Example, Example, and Example.

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a ...

For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven.. VerbalTranslation to Math OperationsOne number exceeds another by a x,x+ax,x+aTwice a number2x2xOne number is a more than another numberx,x+ax,x+aOne number is a less than twice another numberx,2x−ax,2x−aThe product of a number and a , decreased by b ax−bax−bThe quotient of a number and the number plus a is three times the numberxx+a=3xxx+a=3xThe product of three times a number and the number decreased by b is c 3x(x−b)=c3x(x−b)=c Given a real-world problem, model a linear equation to fit it.. Modeling a Linear Equation to Solve an Unknown Number Problem Find a linear equation to solve for the following unknown quantities: One number exceeds another number by 1717 and their sum is 31.31.. Find a linear equation to solve for the following unknown quantities: One number is three more than twice another number.. So, Company B offers the lower monthly cost of $86.40 as compared with the $92 monthly cost offered by Company A when the average number of minutes used each month is 1,160.. If the average number of minutes used each month is 420, then Company A offers a lower monthly cost of $55 compared to Company B ’s monthly cost of $56.80.. Find a linear equation to model this real-world application: It costs ABC electronics company $2.50 per unit to produce a part used in a popular brand of desktop computers.. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered.. Solution The standard formula for area is A=LW;A=LW; however, we will solve the problem using the perimeter formula.. The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns.. Substitute the value of the perimeter and the expression for length into the perimeter formula and find the length.. Solving a Volume Problem Find the dimensions of a shipping box given that the length is twice the width, the height is 88 inches, and the volume is 1,600 in.. Use the formula from the previous question to find the width, W,W, of a rectangle whose length is 15 and whose perimeter is 58.

The production possibilities curve gives us a model of an economy. The model provides powerful insights about the real world, insights that help us to answer some important questions: How does trade between two countries affect the quantities of goods available to people? What determines the rate at which production will increase over time? What is the role of economic freedom in the economy? In this section we explore applications of the model to questions of international trade, economic growth, and the choice of an economic system.

What is the role of economic freedom in the economy?. If, for example, each continent were to produce at the midpoint of its production possibilities curve, the world would produce 300 computers and 300 units of food per period at point Q.. If each continent were to specialize in the good in which it has a comparative advantage, world production could move to a point such as H, with more of both goods produced.. The world production possibilities curve assumes that resources are allocated between computer and food production based on comparative advantage.. World production thus totals 300 units of each good per period; the world operates at point Q in .. First, we see that trade allows the production of more of all goods and services.. An increase in the physical quantity or in the quality of factors of production available to an economy or a technological gain will allow the economy to produce more goods and services; it will shift the economy’s production possibilities curve outward.. But why would she want to produce more of these two goods—or of any goods?. Second, market economies are more likely than other systems to allocate resources on the basis of comparative advantage.. Market capitalist economies rely on economic freedom.. Case in Point: The European Union and the Production Possibilities Curve

Chapter 2 Applications of Prediction Models Background In this chapter, we consider several areas of application of prediction models in public health, clinical practice, and medical research. We use several small case studies for illustration. 2.1 Applications: Medical Practice and Research Broadly speaking, prediction models are valuable for medical practice and for research purposes (Table 2.1). In public health, prediction models may help to tar- get preventive interventions to subjects at relatively high risk of having or develop- ing a disease. In clinical practice, prediction models may inform patients and their treating physicians on the probability of a diagnosis or a prognostic outcome. Prognostic estimates may for example be useful for planning of remaining life-time in terminal disease; or give hope for recovery if a good prognosis is expected after an acute event such as a stroke. Classification of a patient according to his/her risk may also be useful for communication am

2. 12 2 Applications of Prediction Models Table 2.1 Some areas of application of clinical prediction models Application area Example in this chapter Public health Targeting of preventive interventions Incidence of disease Models for (hereditary) breast cancer Clinical practice Diagnostic work-up Test ordering Probability of renal artery stenosis Starting treatment Probability of deep venous thrombosis Therapeutic decision-making Surgical decision making Replacement of risky heart valves Intensity of treatment More intensive chemotherapy in cancer patients Delaying treatment Spontaneous pregnancy chances Research Inclusion in an RCT Traumatic brain injury Covariate adjustment in an RCT Primary analysis of GUSTO-III Confounder adjustment with a propensity Statin effects on mortality score Case-mix adjustment Provider profiling as propensity scores.. Hence, more easily applica- ble graphs were created to estimate the absolute risk of breast cancer for individual patients for intervals of 10, 20, and 30 years.33 The absolute risk estimates have been used to design intervention studies, to counsel patients regarding their risks of disease, and to inform clinical decisions, such as whether or not to take tamoxifen to prevent breast cancer.132 Other models for breast cancer risk include the Claus model, which is useful to assess risk for familial breast cancer.74 This is breast cancer that runs in families but is not associated with a known hereditary breast cancer susceptibility gene.. 4. 14 2 Applications of Prediction Models Table 2.2 Risk factors in four prediction models for breast cancer: two for breast cancer inci- dence, two for presence of mutation in BRCA1 or BRCA2 genes128 Risk factor Gailmodel Clausmodel Myriad tables BRCAPRO model Womans personal information Age + + + + Race/ethnicity + Ashkenazi Jewish + + Breast biopsy + Atypical hyperplasia + Hormonal factors Age at menarche + Age at first live birth + Age at menopause + Family history 1st degree relatives with + + Age. 13. 2.4 Prediction Models for Medical Research 23 Table 2.3 Patient characteristics used in the decision analysis of replacement of risky heart valves415 Fatality Characteristic Surgical risk Survival Fracture fracture Patient related Age (years) + + + + Sex (male/female) + Time since implantation (years) + Valve related Position (aortic/mitral) + + + + Opening angle (60/70), + Size (=29 mm) + Production characteristics + Type of prediction model Logistic Poisson Poisson Logistic regression regression regression regression tion of the valve affects all four aspects (surgical risk, survival, fracture, fatality).. Table 2.4 Analysis of outcome in 7,143 patients with severe moderate traumatic brain injury according to reactive pupils and age dichotomized at age 65 years276 >= 1 Reactive pupil Non-reactive pupils =65 years =65 years 6-month 926/5101 (18%) 159/284 (56%) 849/1644 (52%) 97/114 (85%) mortality Table 2.5 Selection of patients with two criteria (age and reactive pupils) in a traditional way (A) and according to a prognostic model (probability of 6-month mortality < 50%, B) A: Traditional B: Prognostic selection selection < 65 > = 65 years = 76 years Pupillary No reactivity Exclude Exclude Include Exclude Exclude reactivity >=1 pupil Include Exclude Include Include Exclude. Baseline characteristics are still impor- Baseline tant, since they are prognostic for the outcome characteristics Table 2.6 Comparison of adjustment for predictors in linear and generalized linear models (e.g. logistic regression) in estimation and testing of treatment effects, when predictors are completely balanced Method Effect estimate Standard error Power Linear model Identical Decreases Increases Generalized linear model Further from zero Increases Increases. Approaches in this second stage are matching on propensity score, stratification of propensity score (usually by quantile), and inclusion of the propensity score with treatment in a regression model for the outcome.89 Empirical comparisons provided no indication of superiority of propensity score methods over conventional regression analysis for confounder adjustment.381,429 Simulation studies however suggest a benefit of propensity scores in the situation of few outcomes relatively to the number of confounding variables.66 *2.4.8 Example: Statin Treatment Effects Seeger et al. investigated the effect of statins on the occurrence of acute myocardial infarction (AMI).378 They studied members of a Community Health Plan with a recorded LDL>130 mg dl1 at any time between 1994 and 1998.

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A failure to allocate resources in this way means that world production falls inside the production possibilities curve; more of each good could be produced by relying on comparative advantage.. Each continent has a separate production possibilities curve; the two have been combined to illustrate a world production possibilities curve in Panel (c) of the exhibit.. An increase in the physical quantity or in the quality of factors of production available to an economy or a technological gain will allow the economy to produce more goods and services; it will shift the economy’s production possibilities curve outward.. Anything that increases the quantity or quality of the factors of production available to the economy or that improves the technology available to the economy contributes to economic growth.. PeriodPercentage Contribution to GrowthPeriod Growth Rate Years 1960–2007 3.45%Increase in quantity of labor0.74%Increase in quantity of capital1.48%Increase in quality of labor0.23%Increase in quality of capital0.58%Improved technology0.41% Years 1960–1995 3.42%Increase in quantity of labor0.80%Increase in quantity of capital1.55%Increase in quality of labor0.24%Increase in quality of capital0.56%Improved technology0.28% Years 1995–2000 4.52%Increase in quantity of labor1.09%Increase in quantity of capital1.43%Increase in quality of labor0.20%Increase in quality of capital0.89%Improved technology0.90% Years 2000–2007 2.78%Increase in quantity of labor0.17%Increase in quantity of capital1.21%Increase in quality of labor0.22%Increase in quality of capital0.46%Improved technology0.72%Total output for the period shown increased nearly fivefold.. Productivity Growth: Evidence from a Prototype Industry Production Account,” prepared for Matilde Mas and Robert Stehrer, Industrial Productivity in Europe: Growth and Crisis , November 19, 2010.. They predict that over the next 10-year period: the U.S. growth rate will slow down compared to the last two decades, primarily due to slower growth in labor quality, but the U.S. growth rate will still lead among the G7 countries (a group of seven large industrialized countries that includes Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States); the overall growth in the G7 countries will continue to decline; and growth in the developing countries of Asia (Bangladesh Cambodia, China, Hong Kong, India, Indonesia, Malaysia, Nepal, Pakistan, Philippines, Singapore, South Korean, Sri Lanka, Taiwan, Thailand, and Vietnam) will slow a bit from the recent past but will be high enough that those countries’ GDPs will comprise nearly 37% of world GDP in 2020, as compared to 29% in 2010.. Productivity Growth: Evidence from a Prototype Industry Production Account,” in Industrial Productivity in Europe: Growth and Crisis , ed.

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Regression: Models, Methods And Applications [PDF] [4ncs76q1ocr0]. Applied and unified introduction into parametric, non- and semiparametric regression that closes the gap between theory ...

Linear models Linear mixed models Variable selection in linear models Generalized linear models Generalized linear mixed models Semiparametric regression for continuous responses (excluding mixed models) Semiparametric regression for continuous responses (including mixed models). 3, 4, and 10 • A coverage of the entire range of regression models starting with linear models, covering generalized linear and mixed models and also including (generalized) additive models and quantile regression. Structured additive regression models include a variety of special cases, for example, nonparametric and semiparametric regression models, additive models, geoadditive models, and varying-coefficient models.. 2.26 Munich rent index: scatter plots of the rents in Euro versus living area together with linear quantile regression fits for 11 quantiles (top left panel), quantiles determined from a classical linear model (top right panel), and quantiles determined from a linear model for location and scale (bottom panel). • Predictor: i D ˇ0 C ˇ1 xi1 C : : : C ˇk xi k D lin i : • Remark: Generalized linear models are a broad class of models, with linear models, logit models, and Poisson models as special cases.. 3.2 Modeling Nonlinear Covariate Effects Through Variable Transformation If the continuous covariate z has an approximately nonlinear effect ˇ1 f .z/ with known transformation f , then the model yi D ˇ0 C ˇ1 f .zi / C : : : C "i can be transformed into the linear regression model yi D ˇ0 C ˇ1 xi C : : : C "i ; where xi D f .zi / fN: By subtracting n

Cluster or co-cluster analyses are important tools in a variety of scientific areas. The introduction of this book presents a state of the art of already well-established, as well as more recent methods of co-clustering. The authors mainly deal with the two-mode partitioning under different approaches, but pay particular attention to a probabilistic approach. Chapter 1 concerns clustering in general and the model-based clustering in particular. The authors briefly review the classical clustering methods and focus on the mixture model. They present and discuss the use of different mixtures adapted to different types of data. The algorithms used are described and related works with different classical methods are presented and commented upon. This chapter is useful in tackling the problem of co-clustering under the mixture approach. Chapter 2 is devoted to the latent block model proposed in the mixture approach context. The authors discuss this model in detail and present its interest regarding co-clustering. Various algorithms are presented in a general context. Chapter 3 focuses on binary and categorical data. It presents, in detail, the appropriated latent block mixture models. Variants of these models and algorithms are presented and illustrated using examples. Chapter 4 focuses on contingency data. Mutual information, phi-squared and model-based co-clustering are studied. Models, algorithms and connections among different approaches are described and illustrated. Chapter 5 presents the case of continuous data. In the same way, the different approaches used in the previous chapters are extended to this situation. Contents 1. Cluster Analysis. 2. Model-Based Co-Clustering. 3. Co-Clustering of Binary and Categorical Data. 4. Co-Clustering of Contingency Tables. 5. Co-Clustering of Continuous Data. About the Authors Gérard Govaert is Professor at the University of Technology of Compiègne, France. He is also a member of the CNRS Laboratory Heudiasyc (Heuristic and diagnostic of complex systems). His research interests include latent structure modeling, model selection, model-based cluster analysis, block clustering and statistical pattern recognition. He is one of the authors of the MIXMOD (MIXtureMODelling) software. Mohamed Nadif is Professor at the University of Paris-Descartes, France, where he is a member of LIPADE (Paris Descartes computer science laboratory) in the Mathematics and Computer Science department. His research interests include machine learning, data mining, model-based cluster analysis, co-clustering, factorization and data analysis. Cluster Analysis is an important tool in a variety of scientific areas. Chapter 1 briefly presents a state of the art of already well-established as well more recent methods. The hierarchical, partitioning and fuzzy approaches will be discussed amongst others. The authors review the difficulty of these classical methods in tackling the high dimensionality, sparsity and scalability. Chapter 2 discusses the interests of coclustering, presenting different approaches and defining a co-cluster. The authors focus on co-clustering as a simultaneous clustering and discuss the cases of binary, continuous and co-occurrence data. The criteria and algorithms are described and illustrated on simulated and real data. Chapter 3 considers co-clustering as a model-based co-clustering. A latent block model is defined for different kinds of data. The estimation of parameters and co-clustering is tackled under two approaches: maximum likelihood and classification maximum likelihood. Hard and soft algorithms are described and applied on simulated and real data. Chapter 4 considers co-clustering as a matrix approximation. The trifactorization approach is considered and algorithms based on update rules are described. Links with numerical and probabilistic approaches are established. A combination of algorithms are proposed and evaluated on simulated and real data. Chapter 5 considers a co-clustering or bi-clustering as the search for coherent co-clusters in biological terms or the extraction of co-clusters under conditions. Classical algorithms will be described and evaluated on simulated and real data. Different indices to evaluate the quality of coclusters are noted and used in numerical experiments.

Other approaches 7. Model-based clustering and the mixture model 11. Application to mixture models 18. Clustering and the mixture model 20. The two approaches 20. Co-Clustering of Binary and Categorical Data 79. Bernoulli latent block model and algorithms 84. Block model for contingency tables 133. Poisson latent block model 137. Parsimonious Gaussian latent block models 161. Gaussian block mixture model 168

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change, we typically follow the same pro...

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function.. Using a Linear Model to Investigate a Town’s Population A town’s population has been growing linearly.. Find the linear function that models the town’s population PP as a function of the year, t,t, where tt is the number of years since the model began.. Find the linear function that models the baby’s weight WW as a function of the age of the baby, in months, t.t.. Find the linear function that models the number of people inflicted with the common cold CC as a function of the year, t.t.. ⓔ Find an equation for the population, P , P , of the school t years after 2000. ⓕ Using your equation, predict the population of the school in 2011.. ⓔ Find an equation for the population, P P of the town t t years after 2000. ⓕ Using your equation, predict the population of the town in 2014.. ⓐ Find a linear equation for the monthly cost of the cell plan as a function of x , the number of monthly minutes used.. ⓐ Find a formula for the owl population, P. P. Let the input be years since 2003. ⓑ What does your model predict the owl population to be in 2012?

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