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CHAPTER ONE
Mathematical Modeling Process At the end of this module, module, students should should be able: able:
To define mathematical mathematica l modeling
To explain what is a mathematical mathematic al model
To describe various classification of mathematical models
To outline procedures involved in constructing constru cting a mathematical mathemati cal model
To solve a simple modeling problem
To apply the method of least squares
1.1 Introduction Consider the problems of finding the shortest route from one location to a destination in a busy city or deciding locations for bus stops in an urban area so that greater number number of passengers passengers can be achieved. achieved. These are some examples of complex real-world problems. In order to obtain the required solution this person need to investigate necessary questions about the observed world and give a simplified description of the problem. He/she can setup equations using mathematical concepts and language to test some ideas and then solved the model to make prediction or decision. decision.
The developed developed process process mentioned mentioned above above is
known as mathematical modeling .
Definition 1.1 Mathematical
is modeling
the
process
of
constructing
mathematical objects as a quantitative method to represent the properties or behaviors of the real-world system, process or phenomena.
Mathematical Mathematical object object refers to sets sets of numbers,
variables or parameters and their symbols and functional relationships in the form of equation or system of equations and geometrical or algebraic structure.
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Mathematical modeling cultivates the interaction between mathematics and the real world by translating the real world problem into mathematical formulation or model, solving the model and interpreting the solutions obtained using the model so as to make the model useful to the real world.
The real world problem covers situations which may come from
diverse disciplines such as engineering, science, architecture, business, sports science and computer science while the system can be a physical system, a biological system, a financial system, a social system or an ecological system.
Mathematical Mathematic al modeling allows a modeler to
undertake experiments on mathematical representations representations of the t he real world instead of undertaking undertaking experiments in the real world. world.
Mathematical
models can take many forms such as dynamical systems, systems, statistical models,, differential equations, models equations , or game theoretic models. models . Mathematical modeling can be viewed as a process, as illustrated in the schematic diagram of Figure 1.1.
START
Real-World Problem
Data Collection and Analysis Making Assumptions Construct Model
Translate Formulate
Mathematical Model
FINISH
Validation and Verification
Solve / Analyze
Test / Analyze
Real-World Conclusion or Prediction
Translate (Interpret) Simulation ‘What‘What-If’ Analysis
Given {some factors}, find {outputs}, such that {objective is achieved}.
Mathematical Solutions
Figure 1.1: Process of of Mathematic Mathematical al Modeling
Figure 1.1 shows mathematical modeling as an iterative process or a cycle (or loop) loop) which starts and finishes with with the real world problem. It may start with the the upper left-handcorner, left-handcorner, the real world world problem. This represents quantitative measurement or data of system of interest and knowledge of how the system works such that properties, behaviors and
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Mathematical modeling cultivates the interaction between mathematics and the real world by translating the real world problem into mathematical formulation or model, solving the model and interpreting the solutions obtained using the model so as to make the model useful to the real world.
The real world problem covers situations which may come from
diverse disciplines such as engineering, science, architecture, business, sports science and computer science while the system can be a physical system, a biological system, a financial system, a social system or an ecological system.
Mathematical Mathematic al modeling allows a modeler to
undertake experiments on mathematical representations representations of the t he real world instead of undertaking undertaking experiments in the real world. world.
Mathematical
models can take many forms such as dynamical systems, systems, statistical models,, differential equations, models equations , or game theoretic models. models . Mathematical modeling can be viewed as a process, as illustrated in the schematic diagram of Figure 1.1.
START
Real-World Problem
Data Collection and Analysis Making Assumptions Construct Model
Translate Formulate
Mathematical Model
FINISH
Validation and Verification
Solve / Analyze
Test / Analyze
Real-World Conclusion or Prediction
Translate (Interpret) Simulation ‘What‘What-If’ Analysis
Given {some factors}, find {outputs}, such that {objective is achieved}.
Mathematical Solutions
Figure 1.1: Process of of Mathematic Mathematical al Modeling
Figure 1.1 shows mathematical modeling as an iterative process or a cycle (or loop) loop) which starts and finishes with with the real world problem. It may start with the the upper left-handcorner, left-handcorner, the real world world problem. This represents quantitative measurement or data of system of interest and knowledge of how the system works such that properties, behaviors and
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relationship between factors (variables and parameters) could be observed and problem pertaining pertaining to the system that needs to be modeled identified. identif ied. A mathematical models is characterized characteri zed by assumptions about variables (the things which change), parameters (the things which do not change) and functional forms (the relationship relationship between between the two). Thus, based on data and knowledge of the system gathered, a mathematical model can be constructed.
Construction of a model model requires: 1) a clear objective of the modeling – which aspect of the system is involved, what do you wish the model to produce and how accurate you want the model be; and 2) a precise idea of the main factors involved and therefore, it may include taking a simplified view of the system and considering the simplest practical approach by neglecting neglecting certain factors and making assumptions. assumptions.
Real-
world problem could be simplified or transformed to a problem relatively close to the original problem but all essential features of the problem must be retained.
Mathematical modeling may encompass the following goals: a) integrating various existing good models of various parts of the system to represent the whole system of interest; b) generalizing generalizing existing models of different different systems through adoption, adoption, adaptation or modification of these models to be applied to the system of interest; and c) simplifying or making approximation approximation to general model since the current models are difficult to compute or analyzed.
After a mathematical model is established, computational experiments or computations are required to solve the model which is to generate results or solutions. Based on the model developed, developed, optimum or approximate approximate solutions maybe maybe obtained. As the model is is aimed to represent represent the real system, results or solutions found based on the model should be able to be interpreted as descriptive properties or behaviors concerning the system, thus, enabling prediction or conclusion to be made about the real system.
The solutions, solutions, however, however, must first be tested tested (validated (validated and
verified) against real data to determine the effectiveness of the model (to
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test assumptions and and for sensitivity analysis). analysis). If the solutions agree with with the real system (i.e., makes sense), then the model does indeed capture correctly important aspects of the real-world situation and thus can be accepted and and used. used.
In the case where where the solutions found do not
substantiate the real-world phenomena, the modeling cycle is followed through again with a revised formulation of model, new calculated results, improved predictions, predictions, and so on. It is useful to note that the modeling modeling process may not necessarily results in solving the problem entirely but it will furnish useful information regarding the situation under investigation. The whole process of mathematical modeling can be divided into three main activities: i) construct the model, model, ii) compute and and analyze results, results, and iii) test (validate and verify) results, as depicted in Figure 1.2 below.
Construct the Model
Compute and Analyze
Test the Results Apply the Model Figure 1.2: Mathematical Mathematic al Modeling Process
Definition 1.2 A mathematical model is a system (S) containing a set of questions (Q) relating to the system and a set of mathematical statements (M) which can be used to answer the questions (Velten, 2009).
A system refers to a set of interacting, interrelated, interrelated, or interdependent interdependent elements (or components) components) which are functionally functionally related. When modeling, modeling, very specific sets of questions must be addressed in order to transform
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data concerning the system into some mathematical statements.
These
questions, which transpire the purpose of the model, are very focused and challenging questions and can only be answered with a conclusion based on the analysis and interpretation of evidence.
In other words, the
questions invoke logical statement that progresses from what is known or believed to be true to that which is unknown and requires validation. Mathematical statements refer to relationships of numbers, variables and parameters formulated using notations, symbols and operators.
These
statements are used to solve the problem and provide meaningful information useful for making conclusion, decision or prediction regarding the system.
1.2 Definition of a model A model is a representation of reality containing the essential structure of some object or event in the real world. It can be a physical, symbolic (mathematical), graphical, verbal or simplified version of a real world system or phenomenon. Since most systems and phenomenon are very complicated (due to their large number of components) and complex (very highly interconnected) to be understood entirely, a model may contain only those features that are of primary importance to the modeler’s purpose.
Thus, a model is usually a tradeoff between
generality, realism and precision. Models, therefore, may be necessarily incomplete and may be changed or manipulated with relative ease. Due to these, there is no model can be considered as the best model, only better model.
Models can be categorized into three classes on the basis of their degree of abstraction, as the following:
a)
Physical (Iconic) model: An iconic, 'look-alike' model, which is the least abstract.
Examples include a model of an airplane or an
architect’s model of a building.
b)
Analogous model: A more abstract type of model in which the model behaves like a real system but only having some resemblance
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to what it represents. Chart (including flowchart), graph, map and network diagram are considered as analogous models.
c)
Symbolic or mathematical model: It is the most-abstract model with no resemblance but only an approximation to what it represents. The relationships between variables of the system are expressed using a set of rules or operators according to their meaning (syntax) rather that what they represent (semantic). Examples of these models include mathematical equations or formula, financial statement and set of accounts.
Definition 1.3 A model is called d y n a m i c if at least one of its system parameters or state variables depends on time.
Definition 1.4 A model is called static if the system parameters or state variables do not change with respect to time.
A model can be a static or dynamic model.
Figure 1.2, 1.3 and 1.4
illustrates some examples of the physical, analogous and symbolic model in both static and dynamic forms.
a) A Static Physical Model - A miniature replicate of a submarine
b) A Dynamic Physical Model - Aerodynamics of a car in a wind tunnel
Figure 1.2: Physical Model
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a) An Static Analogous Model - A model of an airport city
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c) A Dynamic Analogous Model - A Traffic Anomalous Events Model
Figure 1.3: Analogous Model
a) A Static Symbolic Model - A geometry model for right triangles
b) A Dynamic Symbolic Model - Population Growth Models
Figure 1.4: Symbolic Model
1.3 Why Model? To gain understanding. A model represents characteristics and behaviors of a real-world system concerned, thus, providing an improved understanding. It facilitates understanding by focusing and analyzing necessary components of the system. Through modeling we can comprehend how different parts of the system are related and which factors are most important in the system.
To predict or simulate. Very often we wish to know what a real-world system will do in the future, but it is expensive, impractical, or impossible
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to experiment directly with the system. Examples include nuclear reactor design, space ight, extinction of species, weather prediction, drug efficacy in humans, and so on. Using models, simulation of the real-world system or phenomenon can be carried out, thus providing means to explain, control, and predict events on the basis of controlled observations.
To aid in decision making. Testing using the model developed allows the possibility of simulating the ‘what if scenario’ or performing the sensitivity analysis. Hence, the model provides concrete justifications for any decison made.
Mathematics has been describe in old adage as the “Queen of Science” and “language of science". It been used to describe real phenomena through the mathematical modeling process. Thus, it is not a surprise that mathematical modeling has emerged to be a much needed tool in this modern science and technology era. Cheaper and more advanced computers enable mathematical models to play vital role in discovering new knowledge and solving problems aside from becoming increasingly cost-effective alternative to direct experimentation.
Despite the clear advantages of modeling, there is no definite steps for constructing a “one for all” model. There is no model that can be applied to all situations. Modeling is an art. It involves the creativity in utilizing a sound mathematical knowledge together with the knowledge of the system to create something useful to provide meaningful information and interpretation of the system. Different expertise and perspectives yields different models for the same system.
Such scenario justifies that there
is no “best” or “perfect” model. A model is generally a trade-offs between the following: a)
accuracy
b)
flexibility
c)
cost.
More accurate model requires higher cost and reduces flexibility. Higher flexibility model, on the other hand, may jeopardize the accuracy of the model and may comes at increased cost. In addition, expensive model is
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not preferrable since it does not necessarily guarantees accuracy or flexibility.
Thus, modeling is always aimed at producing a sufficiently
accurate and flexible model preferrably at low cost.
Warm up exercise
Categorize the following models as one of the following: a) A static physical model d) An dynamic analogous model b) A dynamic physical model e) A static symbolic model c) An static analogous model f) A dynamic symbolic model
Universal Harmonics Model
Malignancies Dynamic Pathway Model
Answer: _______________________
Answer: _______________________
A Hurricane Forecast Model
A Volume Optimization Model
Answer: _______________________
Answer: _______________________
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1.4 Classification of Mathematical Models When studying or developing models, it is helpful to identify the type of models that we are dealing with. Mathematical models can be classified into four different types: a) deterministic model, b) stochastic models, c) empirical models and d) mechanistic models
Definition 1.5 A d e t er m i n i s t i c m o d e l is a model which is represented by a function that allows predictions of the dependent variable to be made based on the independent variable(s).
Deterministic models have no components that are inherently uncertain, i.e., no parameters in the model are characterized by probability distributions.
Deterministic models ignore random variation, and so
always predict the same outcome from a given starting point. In other words, for fixed starting values, a deterministic model will always produce the same result.
Deterministic model is usually applied for situations
where the outputs are direct consequence of the initial conditions of the problem.
Definition 1.6 A s t o c h a s t i c m o d e l is a model where randomness is present and variables are not described by unique values, but rather by probability distributions.
A stochastic model is mainly used for situations where a random effect plays a vital role in the model formulation and solving. This model, which is statistical in nature, will produce many different results depending on the actual values that the random variables take in each realization. A stochastic model predicts the distribution of possible outcomes.
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Definition 1.7 A model is known as empirical model if it was constructed from and based entirely on experimental data only, using no priori information about the system.
For an empirical model, no account is taken of the mechanism by which changes to the system occur and minimal information is used a priori in the development of the model. The model just simply notes that some phenomena occur and tries to account quantitatively for changes associated with different conditions. Thus, an empirical model usually provides a quantitative summary of these observed relationships among a set of measured variables.
Models developed using statistical
techniques are examples of empirical models.
In an empirical model, relationship between variables is determined by inspecting the data on the variables and selecting the best fit mathematical formula to represent the relationship.
Therefore, it is a
compromise between accuracy of fit and simplicity of mathematics.
Definition 1.8 A model is known as m e c h a n i s t i c m o d e l if some of the mathematical statements are based on priori information about the system.
A mechanistic model uses a large amount of theoretical information by attempting to explain phenomena that occur at a more detailed level of structure.
It generally describes what happens at one level in the
hierarchy by considering processes at lower levels. It takes account of the mechanisms through which these changes occur and generates prediction concerning the variables based on this knowledge. The model is built on causal relationships represented by equations.
Table 1.1 shows an example of classification of some models according to the four classes of models.
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Table 1.1: Classification of Models Empirical c i t s i n i m r e t e D c i t s a h c o t S
Mechanistic
Short term forecasts (nowcasts) of seasonal climate based on historical climate records (climatological database) of rainfall or temperature using regression analysis
A hypothetical study on predator-prey relationships of the larval salamanders and predacious aquatic invertebrates – a deterministic model with m echanistic explanantion to patterns observed
A mass flux balance model to estimate heavy-metals in agriculture soil. Normal or log normal probability distribution is used to represent the metal concentration.
A risk projection model in epidemiology (in the form of a stochastic mechanistic model) developed to estimate radiationinduced cancer risks to the end of life for Japanese atomic bomb survivors.
A model can also be classified according to the level of understanding on which the model is based. The simplest explanation is to consider the hierarchy of organizational structures within the system being modeled. For example, in the case of animals, models can be distinguished according to different levels of this hierarchical structure. For example, a model concerning an animal can be classified as a molecular model, cellular model or an individual model of the animal. This classification can be illustrated as in Figure 1.6.
High
Herd Individual
Organs
Cells
Low
Molecules
Figure 1.6: A Classification of Model for Animals based on Hierarchy
Aside from classifying models based on the categories mentioned, models can be distinguished based on whether they are static or dynamic models. As described in Definition 1.3 and Definition 1.4, a static model
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does not account for the element of time whereas a dynamic model is time dependent.
The static model describes the steady-state or
equilibrium situations of the system at a specific time instant, where the output does not change if the input is the same. In a dynamic model, the output changes with respect to time. A resistor system is a static model where the voltage is directly proportional to the current, independent of time. On the other hand, a capacitor is modeled as dynamic model since the voltage is dependent on the time and previous time history.
Dynamic models can be divided further into continuous and discrete models, as described in the following definitions.
Definition 1.9 A model is called discrete if the quantities have varied at discrete times or places (or that we only consider discrete variations even if they may change continuously).
Definition 1.10 A model is called c o n t i n u o u s if there is continuous variation in quantities.
Discrete models use a discrete or combinatoric description such as integer numbers, graphs and difference equations. On the other hand, continuous models often involve real quantities, i.e., real numbers, physical quantities and differential equations rather than the difference equations.
Differential equations denotes changes in time over an
infinitesmall interval whereas difference equations represent changes over a finite interval (time jumps).
One further type of model, the system model, is worthy of mention. This is built from a series of sub-models, each of which describes the essence of some interacting components. The above method of classification then refers more properly to the sub-models: different types of sub-models may be used in any one system model.
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Much of the modelling literature refers to ’simulation models’. Why are they not included in the classification? The reason for this apparent omission is that ’simulation’ refers to the way the model calculations are done - i.e. by computer simulation. The actual model of the system is not changed by the way in which the necessary mathematics is performed, although our interpretation of the model may depend on the numerical accuracy of any approximations.
1.5 Constructing Mathematical Models Mathematical modeling is the art of transforming real world problem from a specific application area into a structured mathematical formulation which is solvable, thus the theoretical and numerical analysis can lead to useful understanding, prediction, solution and interpretation related to the original problem. Mathematical modeling has been successfully applied in wide areas of applications, indispensable in some of these applications. The model developed provides precise guide to solution of the problem, enables comprehensive understanding of the system involved and allows the application of advanced computing software and techniques.
Mastering the skill in mathematical
modeling should be one of the expertise that must be acquired by students, aside from the theoretical mathematical learning in class, to enable them to handle real world challenging problem of our time.
Figure 1.t illustrates the flow of interaction of the four main components of the mathematical modeling process. Problem statement defines the problem and wishes to are to be achieved by the by the model. The mathematical theory includes theory related to the application, mathematical theory to be employed and theories discussed in literature.
Mathematical solution methods underline
the mathematical approaches which could be utilized in generating the solutions based on the mathematical model. Finally, computational experiments refer to the execution of computer programs or simulation runs using certain data and according to the mathematical model and solution techniques constructed.
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Problem Statement
Computational Experiments
Mathematical Model
Mathematical Theory
Mathematical Solution Methods
Figure 1.7: Interaction of the Components of the Mathematical Modeling Process
The modeling process itself is usually an iterative process characterized by a sequence of steps which sometimes required to be repeated. The steps include the following: a)
define the problem and objectives of the model;
b)
conduct a thorough literature review of the problem and existing models;
c)
gather complete data through data collection, empirical observations or experiments;
d)
analyze data to identify the important factors (main variables, parameters and constants) and
determine properties and relationships between
these factors as well as their influence towards the behavior of the system; e)
construct diagram, flowchart, network or computer visualization model to get better understanding of the complex interaction of factors within the system;
f)
list all assumptions and ways to verify and validate the model;
g)
transform the
problem using abstraction or formulization into
a
mathematical model. Start with a simple model as among models with similar predictive power, the simplest one is the most desirable;
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h)
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develop solution techniques that can be used to determine the solutions of the model;
i)
perform computational experiments,which can consist of simulations, analytical and qualitative analysis using mathematical techniques;
j)
analyze the results;
k)
validate and verify results based on interpretation and comparison with the real system using data set that have not been used to build the model;
l)
make conclusions, predictions and decisions based on j) and k);
m)
carry out the changes in the real system based on the findings and mechanisms proposed by the model; and
n)
refine the model, if necessary, by repeating step a) to h) for enhanced performance of the real system.
Steps : Constructing Mathematical Model 1.
Identify the real problem
2.
Formulate a mathematical model i.
Identify and classify the variables
ii.
Determine interrelationships between the variables
4.
Obtain the mathematical solution to the model
5.
Interpret the mathematical solution
6.
Verify the model i.
Does it address the problem?
ii.
Does it make common sense?
iii.
Test or compare solution with real world data
7.
Write a report/Conclusion
8.
Implement the model
9.
Maintain the model
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Modeling Approaches Construction of a mathematical model includes the following steps:
1.
Identify the real problem
a.
This concern with questions such as the following: ◦
How to classify the problem?
◦
Is there an underlying physical/scientific behavior to be taken account?
◦
What kind of sources of facts and data are relevant?
◦
Is data available? Is itr primary or secondary data?
◦
What assumptions and simplifications about the problem can be made?
b.
◦
What kinds of models are applicable?
◦
What is the purpose and objective of the model to be developed?
◦
Will the solution based on the model be unique or not?
◦
Who will use the model?
Formulate a precise problem statement In most problems we can develop the problem statement by examining:
2.
◦
What is known or given?
◦
What is to be found, estimated or decided
◦
What are the conditions to be satisfied?
◦
What are the objectives to be achieved?
Formulate mathematical model Various activities need to be carried as listed below:
a.
Identify and list the relevant factors and/or relationships. Factors are quantifiable and can be classified as variables, parameters or constants, and each of these can be continuous, discrete or random. ◦
Continuous variables: takes all real values over an interval, e.g. time, speed, length, density, etc.
◦
Discrete variables: takes on certain isolated values only, e.g. the number of people, cars, houses, months, etc.
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◦
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Random variables: unpredictable, but governed by some underlying statistical model, e.g. bus’s arrival time.
◦
Input
Variables:
quantities
which
determine
subsequent
evaluations within the model. Input variables are to be known, or give, or assumed, or can be considered to have an arbitrary value. ◦
Output Variables: quantities whish are consequences of given values of input variables and parameters and cannot be given arbitrary values. These represent outcomes from a model.
◦
Parameters:
Quantities which are constant for a particular
application of model, but can have different values for another application of the same model, e.g. fixed costs, the dimension of a room, price of a ticket, etc. ◦
Constants: quantities whose values we cannot change, e.g. π, gravity, speed of light, etc.
All the factors must have suitable algebraic symbols and represent measurements with certain unit of measurement.
b.
List the assumptions Basic considerations for assumptions include: i.
whether or not to include certain factors,
ii.
the relative magnitudes of certain effects of various factors (help to identify the main factors), and
iii.
the forms of relationship between factors – this is the heart of the model.
Appropriate assumptions keep the model simple.
c.
Collect data Utilize data as much as possible. i.
At initial stage – analyze data to identify factors and their relationships.
ii.
In the model development stage – fix values of parameters and constants that may occur in the model based on data.
iii.
In the final stage – valicate model by comparing its solution or prediction with real data
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Challenges in data collection:
d.
◦
Getting permission to collect data.
◦
Getting right instruments / equipments.
◦
Getting respondents to complete survey questionnaire.
◦
Getting the right and complete data.
Formulate model i.
Define variables representations and symbols.
ii.
Assign units.
iii.
Determine the relations and equations connecting the problem variables.
iv. Formulate objective of the model. v.
Identify the form of outputs to be obtained.
vi. Specify constraints. vii. Check model’s structure: Given {values of input variables, parameters and constants}, find {values of output variables} such that {conditions are satisfied or objective achieved}.
e.
Obtain the mathematical solution of the model Solving process may involve the following:
3.
◦
Plot graphs
◦
Use Calculus approaches
◦
Apply numerical methods
◦
Apply optimization techniques
◦
Run simulations
◦
Execute computer programs
Analyze and Interpret solution Analyze and examine results or solutions obtained. a.
Check whether values of variables have the correct sign and size?
b.
Ask the following questions: ◦
Are solutions produced reasonable?
◦
Has data measurement and accuracy been achieved?
◦
Have the best solution found?
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4.
5.
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Compare with reality a.
Test results against real data.
b.
Check the following: ◦
Do proposed predictions based on solution make sense?
◦
Do predictions agree with real data?
◦
Evaluate the model. Is the model efficient?
◦
Do results suggest more accuracy need to be achieved?
Write a report The following questions can be considered in preparing the report. ◦
Who is the report for?
◦
What do the readers want to know?
◦
How detail should the report be?
◦
Are important features clearly described?
◦
Are the results stand out?
Identify the real problem
Problem Definition
Formulate Model Modeling
Solve the Model Validation
Interpret Solutions Conclusion Translation
Compare with Reality
Write a report
Figure 1.8: Mathematical Modeling Process
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Example 1: Most Economical Size Box A company decides to produce its own box for packaging products manufactured by the company. It has been decided the box should hold 0.1m 3 (0.1 cubic meters). The box should have a square base and double thickness top and bottom. Cardboard costs RM1.50 per square meter. You are given the task of designing the box and it is up to you to decide the most economical size. What is the minimum cost to produce such box.
STEPS 1.
Identify the Problem
Given {the box is to have square base and double thickness top and bottom; cardboard costs RM1.50 per square meter}, find {the dimension of the box} such that {the cost is minimized}.
Draw diagram to understand the problem better.
w
TOP
TOP
w
TOP h
SIDE BOTTOM w
w
h
SIDE
SIDE
SIDE
SIDE
w w
BOTTOM
Figure 1.9: Illustration for the Problem
BOTTOM
SIDE
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2.
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Formulate mathematical model
Assumption: Ignore the thickness of cardboard for this model.
Volume
The volume of the box is given to be 0.1m 3. Thus,
(1.1)
Area of the 4 Sides
Area of double tops and bases
Total cardboard needed
Total Cost (C)
Area of Cardboard
Area of Cardboard (in m2) x price (in RM) per square meter (1.2)
Let us use method of substitution to solve a system of two equations, equations (1.1) and (1.2), with two unknowns, .
From (1.1), rearrange the equation to get:
(1.3)
Substitute equation (1.3) into equation (1.2):
3.
(1.4)
Solve the model
To find the dimension of the box which gives a minimum total cost, differentiate equation (1.4) with respect to
. (1.5)
Set the derivative equals to 0 to get the value of
m
0.37 m or
cm
.
37 cm.
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Substituting the value of
in (1.3) gives us:
0.74 m or
m
Find the second derivative of
74 cm
with respect to
(1.6)
into equation (1.6), we obtain,
Substituting
, we conclude that is minimum when
Since
meter. Substituting the values of get the minimum cost, . 4.
cm
and
meter and
in equation (1.4), we
Analyze and Interpret the solution
Plotting the graph of equation (1.4) gives us the graph as shown in Figure 1.10, where the point (0.37, 2.44) is the lowest point on the graph.
By examining the graph, the width ( ) could be anywhere between 0.35 m
to 0.38 m without affecting the minimum cost very much, that is .
7 6
) M R5 n i ( x 4 o B r 3 e p t 2 s o C
0.3, 2.54
0.4, 2.46
1 (0.37, 2.44)
0 0
0.2
0.4
0.6
0.8
1
Width or length of the box, w (in m)
Figure 1.10: Graph of Cost (C) vs Width (w)
1.2
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Conclusion Based on the mathematical model, the following can be recommended:
Length or width ( )
m,
Height ( )
Cost ( )
But any width between 0.35 m and 0.38 m would be fine, since the cost
m,
will not differ much. Thus, a more practical solution may be:
Length or width ( )
Height ( )
Cost ( )
m,
m,
Refinement of the model Improvements of this model may include the f ollowing:
Include cost of glue or staples
Include wages for assembly workers
Include wastage when cutting box shape from cardboard.
Should the design be changed to optimize use of cardboard?
Chapter 1 MATHEMATICAL MODELING
MAT 530
1.6 Method of Least Squares The Method of Least Squares is a procedure to determine the best fit line to data; the proof uses simple calculus and linear algebra. The basic problem is to find the best fit straight line
given that, for
, the pairs
are observed. The method easily generalizes to finding the best fit of the
form:
where
is not necessarily a linear function, but
combination of these functions.
Given data, saying
must be a linear
, we may define the error associated to
by
The goal is to find values of
and
(1.7)
that minimize the error. In multivariable
calculus we learn that this requires us to find the values of ( a; b) such that
Differentiating
. .
and setting the derivatives equals to 0 gives:
and dividing by 2 yields:
. .
We may rewrite these equations as:
.
.
Chapter 1 MATHEMATICAL MODELING
MAT 530
It can be shown that solving the equations simultaneously yields:
.
.
From (1.12) and (1.13), we found that values of and which minimize the error (defined in (1.7) satisfy the f ollowing matrix equation:
Define
,
This, we have
Since the matrix A is invertible, this implies:
Chapter 1 MATHEMATICAL MODELING
MAT 530
Example 2: Curve Fitting using the Method of Least Squares Comsider the dat given below: x
26
30
44
50
62
68
74
y
92
85
78
80
54
51
40
Find the best fit line using least squares method that describes the linear relationship between y and x.
Solution
1 2 3 4 5 6 7
x 26 30 44 50 62 68 74
y 92 85 78 80 54 51 40
676 900 1936 2500 3844 4624 5476
8464 7225 6084 6400 2916 2601 1600
2392 2550 3432 4000 3348 3468 2960
= -1.03 120.88
Alternatively,
The equation of the best fit line is:
Chapter 1 MATHEMATICAL MODELING
MAT 530
y 100 90 80 70 60 50
y = -1.0344x + 120.88 R² = 0.9249
40 30 20 10 0 0
20
40
60
80
Figure 1.10: Best Fit Line using Trendline in Excel
Example 3: Method of Least Squares and Linearization A rectangular house is to be built with exterior walls that are 8 feet high. One wall of the house will face north. The total enclosed area of the house will be 1500 square feet. Annual heating costs for the house are determined as follows: each square foot of exterior wall with a northern exposure adds RM4 to the annual heating cost; each square foot of exterior wall with an eastern or western exposure adds RM2 to the annual heating cost; each square foot of exterior wall with a southern exposure adds RM1 to the annual heating cost. For what value of L is the annual heating cost minimized, and what is the annual heating cost for this choice of L? The values given for the heating costs per square foot can only be known approximately. How does the solution depend on the given values for the heating costs?
Solution
Identify the expected function : y aebx Identify the unknown : a, b
Chapter 1 MATHEMATICAL MODELING
MAT 530
Linearize the function : y aebx ln y ln aebx
ln y
ln a ln ebx ln a bx ln e ln a bx
Transform the data and/or unknown :
Identify normal equation
n n x i i 1
xi i 1 n 2 xi i 1 n
a1 b
a0 ln a
Y ln y
ln e 1)
n Yi a 0 i 1 a = n 1 x i Yi i 1
Determine the respective values: i 1 2 3 4 5 6
xi2 0.16 0.64 1.44 2.56 4.00 5.29
xi 0.4 0.8 1.2 1.6 2.0 2.3
yi 800 975 1500 1950 2900 3600
Yi 6.6846 6.8824 7.3132 7.5756 7.9725 8.1887 n
n
n
x i1
8.3
y
i
i1
Substitute the respective values in the normal equation.
a0 44.6170 = a 63.8555 1
Solve for the coefficients: a0 and a1
a0 6.3037 and a1 0.8187
Substitute to the transform unknown : a0
ln a 6.3037
a1 b 0.8187
a e6.3037
546.5906
i i
i
14.09
i1
8.3 6 8.3 14.09
Y 44.6170 x Y 63.8555
n
i
xi Yi 2.6738 5.5059 8.7759 12.1209 15.9449 18.8340
The expected function is y 546 .5906 e0.8187 x
i1
Chapter 1 MATHEMATICAL MODELING
Example 4: Least Squares Technique Use least-squares procedure to fit y = Axk to the following data. xi
1.00
1.15
1.40
1.43
1.60
2.00
yi
4.33
4.58
4.98
5.06
5.28
5.80
Solution
y Axk
Taking natural logarithm both sides: ln y = l n A + k Y
ln x
a0 a1X
The necessary transformation for the data and unknowns are: Y ln y
X
ln x
=ROUND(LN(C4);4)
Rewrite ln y = ln A + k ln x as Y = a0 + a1 X
a0 ln A
k a1
MAT 530
Chapter 1 MATHEMATICAL MODELING
MAT 530
Hence, the required normal equations:
6 6 xi i 1
6
6
i 1
i 1
x i a Yi 0 i 1 = 6i 1 6 2 a xi 1 x i Yi
1.9971 a 0 6 9.6358 = 1.9971 0.9620 a 3.3333 1
Solving for a0 and a1: a0 = 1.4649 Since a1 = 0.4239 then k = a1 = 0.4239 But
ln A = a0 then A = e a0 = e1.4649 = 4.327
Hence, the expected function is y = Axk = 4.327x0.4239
Warm up exercise Find the values of a and b of the form y = aex + be-x to the following data: X Y
0 5.02
0.5 5.21
1.0 6.49
1.5 9.54
2.0 16.02
2.5 24.53
Chapter 1 MATHEMATICAL MODELING
MAT 530
1.7 Continuous Least-Squares Polynomials The
previous
section
considered
the
problem
of
least-squares
approximation to fit a collection of discrete data. The other approximation problem concerns the approximation of functions or fitting continuous data. Consider a function y = f(x), which is continuous in [a, b]. Let Pm ( x) a0
a1x a2x 2 ... ai xi ... amxm be a polynomial which
represent the function y = f(x), then the total least squares error is defined as follows b
E
2
∫f(x)- Pm (x)
dx
a b
2
∫f(x)- a0 a1x a2x2 ... aixi ... amxm dx a
We seek to minimize the sum of error squares. From the calculus of functions of several variables, a necessary condition for the values a 0, a, …, and am to minimize E such that ∂E ∂a0
=
∂E ∂a1
= ...
∂E ∂ai
= ...
∂E ∂am
0im
=0
Hence, b
i
... amxm
i
... amxm
∫a0 f(x)- a0 a1x ... aix
2dx 0
(1)
2dx 0
(2)
a
b
∫a1 f(x)- a0 a1x ... aix
a
. . .
∫a f(x) - a
b
a
a1 x ... a x ... a i
0
i
m
x
m
2
dx 0
(i)
dx 0
( m)
i
. . .
∫a f(x) - a
b
a
m
a1 x ... a x ... a i
0
i
m
x
m
2
Chapter 1 MATHEMATICAL MODELING
MAT 530
Simplifying (1) yields b
∫2 f(x)- a0 a1x ... aix
i
... amxm ( 1)dx 0
a b
∫a0 a1x ... aix
i
... amxm f ( x) dx 0
a b
b
b
b
b
a0dx a1xdx ... aix dx ... amx i
a
a
a
m
dx f ( x ) dx
a
b
b
b
a
a
a
a b
b
i
m
a0 dx a1 xdx ... ai x dx ... am x dx f ( x ) dx a
a
Simplifying (2) to (m) using similar procedures yield the normal equations b
b
b
2
a0 xdx a1 x dx ... ai x a
a
b
a0 x dx a1 x a
dx ... am x
a
b i
b
i 1
2 i
dx xf ( x)dx
a b
a
b
2i
dx ... ai x dx ... am x
a
b
m1
a
mi
b
dx xif ( x)dx
a
a
: . b
b
m
a0 x dx a1 x a
2 m
b
dx ... ai x
a
b
i m
dx ... am x
a
a
b 2m
dx xmf ( x)dx a
In matrix form the normal equations can be represented as:
b dx ba xdx a ... b i x dx a ... b xmdx a
b
xdx
x dx
a b
2
...
...
...
a
x
i 1
dx
xi1dx
...
a
...
x2idx
...
b
xm1dx
...
m1
dx
...
...
ximdx
a
...
xmidx
a
dx
b
...
b
...
x
m
a
...
a
...
x
a b
b
...
b
a
b
a
x dx i
...
a b
b
... b
...
x2mdx
a
b f ( x )dx a b a0 xf ( x )dx a 1 a . . . . a b i i . x f ( x)dx . a a . m . b xmf ( x )dx a
Chapter 1 MATHEMATICAL MODELING
MAT 530
Steps : Fitting Continuous Least Squares Polynomial
Determine the degree m of the polynomial : m Write the general expression of the polynomial to be fitted Pm(x) = a 0 + a1x + a 2 x 2 + ... + am x m Write the normal equations in matrix form. 1
b
x dx i
Compute :
for 0 i 2m and
i
0
a
x f (x)dx for 0 i m
Solve for the coefficients a0, a1…am. Thus, the least-squares function is: f(x) = a0 + a1x + a 2 x 2 + ...
+
am x m
Example 5: Derive the normal equation in matrix form to approximate y = f(x) with a straight line on the interval [a, b].
Solution
Let P1( x) a0 ∂E ∂a0
=
a1x
∂E ∂a1
=0 b
E
b 2
∫f(x)- P1(x) dx
∫f(x)- a0 a1x 2 dx a
a
b
2
∫a0 f(x)- a0 a1x dx 0
a
b
∫2 f(x)- a0 a1x(1)dx 0 a b
∫a0 a1x f ( x)dx 0 a b
b
b
a
a
a
a0dx a1xdx f (x)dx b
b
b
a0 dx a1 xdx f ( x)dx a
a
a
Chapter 1 MATHEMATICAL MODELING
b
2
∫a1 f(x)- a0 a1x
MAT 530
dx 0
a
b
∫2 f(x)- a0 a1x(x)dx 0 a b
∫a0 a1x f ( x)xdx 0 a b
b
b
a0xdx a1x dx xf (x)dx 2
a
a
a
b
b
a
a
b 2
a0 xdx a1 x dx xf ( x )dx a
b b b dx xdx a f ( x)dx a a 0 a b b b a 2 1 xf ( x)dx xdx x dx a a a
Example 6: Find the least squares polynomial approximation of degree one to y = x 2 + 4x + 4 on the interval [0,2].
S o l u t i o n 2 2 2 dx xdx a f ( x )dx 0 0 0 0 2 2 2 a 2 1 xf ( x)dx xdx x dx 0 0 0
56 2 2 a0 2 8 3 3 a1 68 3
a0 3.336 a 5.997 1 Hence, P1( x) 3.336 5.997 x
Chapter 1 MATHEMATICAL MODELING
MAT 530
Example 7: Find the least squares polynomial approximation of degree two to f(x) = ex on the interval [ 0, 1]
Solution
Determine the degree m of the polynomial m=2 Write the general expression of the polynomial to be fitted
P2(x) = a0 a1x a2 x2 Write the normal equations in matrix form. 1 1 1 1 x 2 dx xdx x dx e dx 0 0 0 a0 0 1 1 1 1 2 3 x xdx x dx x dx a1 xe dx 0 0 0 a 0 2 1 1 1 1 2 2 x 3 4 x dx x dx x dx x e dx 0 0 0 0
1
Compute :
x dx i
1
for 0 i 4 and
0
x f (x)dx for 0 i m i
0
0.5 0.3333 a0 1.7183 1 0.5 a 1 0.3333 0.25 1 0.3333 0.25 0.2 a2 0.7183
Solve for the coefficients a0, a1…am.
a0 1.0140 a 0.8445 1 a2 0.8449
Thus, the least-squares function is: P2(x) = 1.0140 0.8445x 0.8449x 2
2.7 Legendre Polynomial The
previous
section
considered
the
problem
of
least-squares
approximation to fit a collection of discrete data. The other approximation problem concerns the approximation of functions or fitting continuous data. In this section we shall learn to fit data in continuous form using sequences of Legendre polynomials.
Chapter 1 MATHEMATICAL MODELING
MAT 530
Definition
The functions listed below are called Legendre polynomials and are defined for -1 x 1: P 0(x)
= 1
P 1(x)
= x
P2 ( x)
3x2 - 1 2
P3 ( x)
5x3 - 3x 2
P4 ( x)
35x 4 - 30x 2 + 3 8
P5 ( x)
63x 5 - 70x 3 + 15x 8
1
1
1
1
. . .
2m + 1 P m P m - m m + 1 m + 1
Pm1( x )
-1
This set of Legendre polynomials is said to be orthogonal on [-1,1] with respect to the weight function w(x) 1 .The criteria required is that these functions are designed to satisfy the following orthogonality condition:
0 if n m Pm ( x) Pn ( x) dx 2 2n 1 if n m 1 1
(details of orthogonality and weight function shall not be discussed here) In general, the Legendre polynomials can also be derived by t he formula: Pn ( x )
1
dn
2n n! dxn
(x
2
- 1)n
Suppose y(x) is a function continuous on [-1,1]. Here, the approach to finding the least-squares approximating polynomial f(x) to fit the function y(x) (or, the continuous data) is done in a similar manner. Let f(x) be of polynomial of degree m defined using sequences of Legendre polynomials such that f(x) = a0P0(x) + a1P1(x) + a2P2(x) + . . . + amPm (x) .
Chapter 1 MATHEMATICAL MODELING
MAT 530
We seek to minimize the sum of error squares; i.e. 1 2
∫ f(x)- y(x) dx
L
-1 1
∫ a0P0 (x) + a1P1(x) + amPm (x) - y(x)2 dx -1
(1) where, P i (x) is a Legendre polynomial and a i is a constant coefficient.
From the calculus of functions of several variables, a necessary condition for the values a 0, a, …, am to minimize L is that ∂L ∂ak
= 0, for each k = , , …, m.
Hence, using (1),
L ak
1 2
∫ a0P0 ( x) a1P1( x) amPm ( x) - y( x)
dx
0
-1
With the orthogonality property of Legendre polynomials, this term can be simplified to: 1 2
akPk ( x)Pk ( x) - y( x)Pk ( x) dx 0 ∫ -1 1
ak .
1 2
Pk ( x) ∫
x
-1
∫ y(x) Pk ( x)dx -1
2 ak 2k 1
1
∫y( x) Pk ( x )dx -1
Thus, ak
2k 1 2
1
∫ y(x)Pk ( x) dx -1
Notice that ak calculates the coefficients a0 , a1, …, am with the condition that x is defined for y(x) on the interval [-1,1].
Chapter 1 MATHEMATICAL MODELING
MAT 530
Theorem
Suppose y(x) is continuous and defined on [-1,1], then y(x) can be approximated by a least-squares polynomial f(x) of degree m, using series of Legendre polynomials such that: f(x) = a0P0(x) + a1P1(x) + a2P2(x) + . . . + amPm(x) where, the coefficients a0 , a1, …, am is be determined by ak
2k 1 2
1
y(x) Pk (x) dx
for k = ,,,,……….,m
1
Steps : Fitting continuous data (or function) using Legendre polynomials if independent variable is defin ed on [-1,1]
Identify the observed function: y(x) Determine the Legendre polynomials of degree m: f(x)= a0P0(x) + a1P1(x) + a2P2(x) + . . . + amPm(x)
Determine the coefficients a0, a, …, am : ak
2k 1 2
1
y(x) Pk (x) dx
for k = ,,,,…m
1
Example 8: Find the least squares polynomial approximation of degree one to y = x 2 + 4x + 4 on the interval [-1,1].
S o l u t i o n
Identify the observed function : y(x) = x 2 + 4x + 4
Determine the Legendre polynomials of degree 1: f(x) = a0P0(x) + a1P1(x) = a0 + a1x where P0(x) = 1 and P1(x) = x
Determined the coefficients a0 and a1: ak
2k 1 2
1
y(x) Pk (x) dx for k = 0,1
1
Chapter 1 MATHEMATICAL MODELING
k
0 : a0 1
k
(x2 4x 4) P0 (x) dx
1
1
( x 2 4x 4)(1)dx 2
1 13
1: a1
1
2(0) 1 2
MAT 530
3
3 2(1) 1 2
1
(x2 4x 4) P1(x) dx
1
1
( x 2 4x 4)x dx 2 1
4
Thus, the polynomial of degree one to fit y(x) is: f(x) = a0P0(x) + a1P1(x) =
13 3
+ 4x
Other method which can be applied is 1 1 1 dx y( x)dx xdx a0 1 1 11 a 1 1 2 1 xdx x dx xy( x)dx 1 1 1
In the case where x (or the independent variable) is defined on [m,n], a suitable linear transformation is required so that the interval range of the independent variable is normalized to be on [-1,1]. Steps : Fitting continuous data (or function) using Legendre polynomials if independent variable is not d efined on [-1,1]
Transform x to t linearly: x [m, n ] t [ 1,1 ] Let x at b Solve for a and b Rewrite y(x) in term of y(t) Determine the Legendre polynomials of degree m f(t) = a0P0(t) + a1P1(t) + a2P2(t) + . . . + amPm(t)
Determine the coefficients a0, a, …, am : ak
2k 1 2
1
y(t) P (t) dt k
for k = 0,1,2,3,..,m
1
The polynomial of degree m to fit y(t) is: f(t) = a0P0(t) + a1P1(t) + a2P2(x) + . . . + amPm(t) Rewrite f(t) in term of f(x).
Chapter 1 MATHEMATICAL MODELING
MAT 530
Example 9: Find the least squares polynomial approximation of degree one to y = x 2 + 4x + 4 on the interval [0,2]
Solution
Since x [0 ,2 ] and x [ 1, 1 ] then transformation is required.
Transform x to t linearly: x [0,2 ] t [1,1 ]
Let . x = at + b
Solve for a and b When x = 0:
t=-1 :
0 = a(- + b
…
When x = 2:
t=
= a + b
…
:
Solving (1) and (2): a=1
and
b=1
Thus x=t+1
or
t = x – 1
Rewrite y(x) in term of y(t): y(x) = x 2 + 4x + 4, but x = t+1 y(t) = (t+1) 2 + 4(t+1) + 4 = t 2 + 6t +9
Determine Legendre polynomials of degree 1: f(t) = a0P0(t) + a1P1(t) = a0 + a1t where P0(t)=1 and P1(t)=t
Determine the coefficients a0 and a1: ak
k
2k 1
y(t) P (t) dt k
2
for k = 0,1
1
0 : a0
1
2(0) 1 1 2 (t 2 1
11 2 (t 2 1
6t 9) P 0(t ) dt
6t 9)(1) dt
28 3
Chapter 1 MATHEMATICAL MODELING
k
1: a1
2(1) 1 1
y( t) P 1(t ) dt 1
2
3 1 2 (t 2 1
MAT 530
6t 9)(t ) dt 6
The polynomial of degree 1 to fit y(t) is: f(t) = a0P0(t) + a1P1(t) 28
=
3
+ 6t
Rewrite f(t) in term of f(x): 28
f(x) = =
3
+ 6(x-1) since t = x – 1
10 + 6x 3
Example 10: Find the least squares polynomial approximation of degree two to f(x) = ex on the interval a)
[-1, 1]
b) [ 0, 1]
Solution
No transformation required since x [1, 1 ] . Given y(x) = ex and f(x) = a0P0(x) + a1P1(x) + a2P2(x)
polynomial of degree two
a0P0 ( x) a1P1( x) a2P2 ( x) 1 ao a1x a2 (3x2 1) 2 since P0 ( x) 1 ,
1P1( x )
1 x and P2 ( x) (3x 2 1) 2
Determined the coefficients a 0 ,, a1 and a2: ak
2k 1 2
1
y(x) Pk (x) dx for k = 0,1,2
1
Chapter 1 MATHEMATICAL MODELING
k
1
2(0) 1
0 : a0
2
MAT 530
ex P0 (x) dx
1
1
1
e x dx 2
1 1.1752 k
1: a1 3
2(1) 1 2
1
ex P1(x) dx
1
1
ex x dx 2
1 1.1036 k
2 : a1 5
2(2) 1 2 1
1
ex P2(x) dx
1
1
ex (3x 2 1) dx 2 2
1 0.3578
Hence, f ( x ) a0P0 ( x ) a1P1( x ) a2P2 ( x )
1 ao a1x a2 (3x 2 1) 2 1 1.1752 1.1036 x 0.3578 (3x 2 1) 2 2 0.9963 1.1036 x 0.5367 x . b) Need a suitable linear transformation since x [1, 1 ] and x [0 ,1 ] Make a suitable linear transformation of x to t: x [0,1 ] t [1,1 ] i.e. x = at + b, solve for a and b when x = 0, t = - 1 :
0 = a(-1) + b = - a + b …
when x = , t =
= a + b = a + b
:
Solving (1) and (2): a
1 2
and
b
1 2
…
Chapter 1 MATHEMATICAL MODELING
MAT 530
Thus, x
1 2
t
1
2
t 1 2
or 2x t 1 t
2x 1
Write y(x) in the term of y(t): y( x ) e x t 1
y( t ) e
2
Write the least-squares polynomial in the new variable t in the form of Legendre polynomials: f(t) = polynomial of degree two (quadratic) f (t ) a0P0 (t ) a1P1(t ) a2P2 ( t )
1 ao a1t a 2 (3t 2 1) 2 Determine the coefficients a 0 , a1 and a2: ak
2k 1 2
k
k
k
0:
1:
2:
1
y(t) P (t) dt for k = 0,1,2 k
1
t 1 2(0) 1 1 2 a0 e P0 (t ) dt 2 1 t 1 1 1 2 e dt 2 1 1.7183 t 1 2(1) 1 1 2 a1 e P1(t ) dt 2 1 t 1 3 1 2 e t dt 2 1 0.8452 t 1 2(2) 1 1 2 a2 e P2 ( t) dt 2 1 t 1 5 1 2 1 2 e . (3t 1) dt 2 1 2 5 0.0559 2 0.1398
Thus, f ( t ) a 0P0 ( t ) a1P1(t ) a 2P2 ( t )
Chapter 1 MATHEMATICAL MODELING
1 f ( t ) 1.7183 0.8452 t (0.1398 ) (3t 2 2
MAT 530
1)
1.6484 0.8452 t 0.8388 t 2 Hence f ( x) 1.6484 0.8452(2x 1) 0.8388 (2x 1)2
1.3097 14.7872 x 13.0968 x2
Warm up exercise Find the least squares polynomial approximation of degree two to: a) f(x) = sin 2x on the interval [-1,1] and [0, 1] b) f(x) = ln x on the interval [1,3]
Chapter 1 MATHEMATICAL MODELING
MAT 530
Exercise 4 1.
Fit a straight line to the given data. a)
2.
y 1 2 3 3 4
b)
x -6 -2 0 2 6
y -5.3 -3.5 -1.7 0.2 4.0
c)
x -3 1 3 5 9
y -6 -4 -2 0 4
x -3 -1 1 3
y 15 5 1 5
c)
x -2 -1 0 1 2
Y 2.80 2.10 3.25 6.00 11.50
Fit a parabola to the given data. a)
3.
X -2 -1 0 1 2
X 2.0 2.3 2.6 2.9 3.2
y 5.1 7.5 10.6 14.4 19.0
b)
The table below shows the time (in seconds) required for water to drain through a hole in the bottom of a bottle as a function of depth (in meters) to which the bottle has been filled. Depth Time
0.05 65.99
0.10 120.28
0.15 166.69
0.20 207.85
0.30 279.95
0.35 313.04
0.40 344.24
a) Find the regression line to fit the data b) Estimate the time required for water to drain at depth of 2.5 meters. 4.
Find the best-fit line for the given values. x y
5.
6 22
10 28
14 14
16 22
20 16
22 8
28 8
28 14
36 0
38 4
Fit a least-squares quadratic function (parabola) to the given data. x p(x)
6.
5 30
0.0 0.0
1.0 2.0
2.0 5.0
3.0 8.0
4.0 12.0
5.0 25.0
6.0 40.0
7.0 57.0
The following tabulated values are taken from an experiment. Complete the table and calculate the sum of error squares. x Observed y Calculated y Error
0.0 0.0
1.0 2.0
2.0 5.0
3.0 8.0
4.0 12.0
5.0 25.0
6.0 40.0
7.0 57.0
