Software Engineering Online - Everything Up To Quiz #1

1. What are Monte Carlo methods based on?

Strict adherence to field-specific guidelines and rules.

The expression"Monte Carlo method" is actually very general. Monte Carlo (MC)methods are stochastic techniques--meaning they are based on the use of randomnumbers and probability statistics to investigate problems. You can find MCmethods used in everything from economics to nuclear physics to regulating theflow of traffic. Of course the way they are applied varies widely from field tofield, and there are dozens of subsets of MC even within chemistry. But,strictly speaking, to call something a "Monte Carlo" experiment, allyou need to do is use random numbers to examine some problem.
The use of MC methods tomodel physical problems allows us to examine more complex systems than weotherwise can. Solving equations which describe the interactions between twoatoms is fairly simple; solving the same equations for hundreds or thousands ofatoms is impossible. With MC methods, a large system can be sampled in a numberof random configurations, and that data can be used to describe the system as awhole.
"Hit and miss"integration is the simplest type of MC method to understand, and it is the typeof experiment used in this lab to determine the HCl/DCl energy level populationdistribution. Before discussing the lab, however, we will begin with a simplegeometric MC experiment which calculates the value of pi based on a "hitand miss" integration.

The analysis of only two atoms in a system.

The use of deterministic algorithms to solve equations.

Monte Carlo methods are stochastic techniques that involve the use of random numbers and probability statistics to explore and analyze problems across various fields.

Explanation

Monte Carlo methods are stochastic techniques that involve the use of random numbers and probability statistics to explore and analyze problems across various fields.

2. How can we use the Monte Carlo method to calculate the value of Pi?

Using a fixed set of predetermined coordinates within a square and circle to compare the relative areas and establish the value of Pi.

The first figure is simply a unit circlecircumscribed by a square. We could examine this problem in terms of the fullcircle and square, but it's easier to examine just one quadrant of the circle,as in the figure below.
If you are a very poordart player, it is easy to imagine throwing darts randomly at Figure 2, and itshould be apparent that of the total number of darts that hit within thesquare, the number of darts that hit the shaded part (circle quadrant) isproportional to the area of that part. In other words, If you remember yourgeometry, it's easy to show that.
If each dart thrownlands somewhere inside the square, the ratio of "hits" (in the shadedarea) to "throws" will be one-fourth the value of pi. If you actuallydo this experiment, you'll soon realize that it takes a very large number ofthrows to get a decent value of pi...well over 1,000. To make things easy onourselves, we can have computers generate random* numbers.
If we say our circle'sradius is 1.0, for each throw we can generate two random numbers, an x anda y coordinate, which we can then use to calculate thedistance from the origin (0,0) using the Pythagorean theorem. If the distancefrom the origin is less than or equal to 1.0, it is within the shaded area andcounts as a hit. Do this thousands (or millions) of times, and you will wind upwith an estimate of the value of pi. How good it is depends on how manyiterations (throws) are done, and to a lesser extent on the quality of therandom number generator. Simple computer code for a single iteration, or throw,might be: x=(random#) y=(random#) dist=sqrt(x^2 + y^2) if dist.from.origin(less.than.or.equal.to) 1.0 lethits=hits+1.0

Throwing darts randomly at a square and counting the number of hits within a quarter of the circle to obtain Pi directly.

Creating a geometrical model combining a circle and a triangle to derive the value of Pi through direct measurement.

The correct method involves generating random points within a square, calculating their distance from the origin, and using the ratio of 'hits' to 'throws' to estimate the value of Pi.

Explanation

The correct method involves generating random points within a square, calculating their distance from the origin, and using the ratio of 'hits' to 'throws' to estimate the value of Pi.

3. What is the Monte Carlo method used for in this lab?

To estimate the value of pi through a two-dimensional example.

To analyze the effects of various isotopes on nuclear decay rates.

The actual Monte Carlomethod used in this lab to determine the population distribution amongrotational energy levels is simpler than the two-dimensional example of theestimation of pi, as only one random number is generated for each"throw." This will be apparent shortly.
For this lab, theBoltzmann distribution can be solved analytically, and it is in fact used indetermining the Monte Carlo distribution. As such, this is not a particularlyinformative simulation (you could just solve the Boltzmann equation for howevermany energy levels you wished and look at those numbers). However, this laballows you to watch how changing the number of throws affects the results, andit automates the examination of the effects of temperature and isotope onpopulation of energy levels.
The process used by thecomputer program for this lab is quite simple.
The input information is used to solve the Boltzmann equation for some number of energy levels. For a maximum of J=4, the relative populations might look like the figure to the right.
The rest of the simulation is easier to understand if you imagine laying the peaks in the "spectrum" side-by-side, as shown below. If we generate random numbers along that line, whenever a number falls within the range of a particular J value, it counts as a "hit" for that energy level. Obviously, for shorter lengths (J=0) the number of "hits" will be smaller than for longer lengths (J=2).

Since random number generators typically produce numbers on the range of zero to one, the population distribution is normalized so that the total "length" is equal to 1.0.
For each "throw," or random number produced, the computer determines which energy level range it belongs to, and calls it a "hit" for that J value. After the computer completes all throws, the number of hits (or relative number of hits) for each energy level are given. It is up to you to compare this output to the theoretical distribution produced by the Boltzmann equation, as described in the Lab Report Instructions. * Computer-generatednumbers aren't really random, since computers are deterministic. But, given anumber to start with--generally called a random number seed--a number ofmathematical operations can be performed on the seed so as to generateunrelated (pseudo-random) numbers. The output of random number generators istested with rigorous statistical tests to ensure that the numbers are random inrelation to one another. One caveat: If you use a random number seed more thanonce, you will get identical random numbers every time. Thus, for multipletrials, different random number seeds must be used. Commercial programs, likeMathematica, pull a random number seed from somewhere within thesystem--perhaps the time on the clock--so the seed is unlikely to be the samefor two different experiments.

To simulate population distribution in a biological ecosystem.

The Monte Carlo method in this lab is specifically used to determine the population distribution among rotational energy levels, as described in the provided text. The other options do not align with the context of Monte Carlo computation of population distribution.

Explanation

The Monte Carlo method in this lab is specifically used to determine the population distribution among rotational energy levels, as described in the provided text. The other options do not align with the context of Monte Carlo computation of population distribution.

4. What is the basic description of Monte Carlo Methods?

Monte Carlo methods involve solving mathematical equations through symbolic manipulation.

Monte Carlo methods provide approximate solutions to a variety of mathematical problems by performing statistical sampling experiments. They can be loosely defined as statistical simulation methods, where statistical simulation is defined in quite general terms to be any method that utilized sequences of random numbers to perform the simulation. Thus Monte Carlo methods are a collection of different methods that all basically perform the same process. This process involves performing many simulations using random numbers and probability to get an approximation of the answer to the problem. The defining characteristic of Monte Carlo methods is its use of random numbers in its simulations. In fact, these methods derive their collective name from the fact that Monte Carlo, the capital of Monaco, has many casinos and casino roulette wheels are a good example of a random number generator.

The Monte Carlo simulation technique has formally existed since the early 1940s, where it had applications in research into nuclear fusion. However, only with the increase in computer technology and power has the technique become more widely used. This is because computers are now able to perform millions of simulations much more efficiently and quickly than before. This is an important factor because it means that the technique can provide an approximate answer quickly and to a higher level of accuracy, because the more simulations that you perform then the more accurate the approximation is (This point is illustrated in the next section when we compare approximate error for different numbers of simulations).

Note that these methods only provide a approximation of the answer. Thus the analysis of the approximation error is a major factor to take into account when evaluating answers from these methods. The attempt to minimize this error is the reason there are so many different Monte Carlo methods. The various methods can have different levels of accuracy for their answers, although often this can depend on certain circumstances of the question and so some method's level of accuracy varies depending on the problem. This is illustrated well in the next section where we investigate four different Monte Carlo methods and compare there answers and the accuracy of their approximations.

Monte Carlo methods provide exact solutions to complex mathematical problems through deterministic algorithms.

Monte Carlo methods rely on rule-based decision-making processes to find precise solutions.

Monte Carlo methods provide approximate solutions to mathematical problems by performing statistical sampling experiments using random numbers, they do not provide exact solutions like deterministic algorithms or rely on symbolic manipulation or rule-based decision-making processes.

Explanation

Monte Carlo methods provide approximate solutions to mathematical problems by performing statistical sampling experiments using random numbers, they do not provide exact solutions like deterministic algorithms or rely on symbolic manipulation or rule-based decision-making processes.

5. What is one of the most important uses of Monte Carlo Techniques?

A technique for compressing image files.

An approach to clustering data sets.

A method for solving differential equations.

One of the most important uses of Monte Carlo methods is in evaluating difficult integrals. This is especially true of multi-dimensional integrals which have few methods for computation and thus are suited to getting an approximation due to their complexity. It is in these situations that Monte Carlo approximations become a valuable tool to use, as it may be able to give a reasonable approximation in a much quicker time in comparison to other formal techniques.

In this section, we look at four different Monte Carlo methods to approach the problem of integral calculation. We investigate each method by giving a basic description of its individual procedure and then attempt to give a more formal mathematical description. We also discuss the example program that is included with this tutorial. It has implemented all four methods that we discuss, and also has results from these different methods in integrating an example function.

Before we discuss our first method it must be brought to the reader's attention the reasons why we are discussing four different Monte Carlo methods. The first point in explaining this is to acknowledge that there are many considerations when using Monte Carlo techniques to perform approximations. one of the chief concerns is to be able to get as accurate an approximation as possible. Thus, for each method we discuss their associated error statistics (the variance is discussed).

There is also the consideration of what technique is most suitable to the problem and so will get the best results. For example, a curve that has many platcaus (flat sections) may be very suitable to a stratified sampling method because the flat sections will have very low variance. Thus, we discuss four different Monte Carlo methods, because they all have their individual requirements and benefits. Probably the most important consideration, for the methods are the accuracy of their approximations because the more approximation, the less samples are needed to reach a specific level of accuracy.

Monte Carlo methods are commonly used for evaluating difficult integrals in multi-dimensional spaces due to their ability to provide reasonable approximations in a quicker time frame compared to traditional methods.

Explanation

Monte Carlo methods are commonly used for evaluating difficult integrals in multi-dimensional spaces due to their ability to provide reasonable approximations in a quicker time frame compared to traditional methods.

6. What is Crude Monte Carlo?

The first method for discussion is crude Monte Carlo. We are going to use this technique to solve the integral 1. (see card #1 - Side A).

A basic description of this method is that you take a number, N, of random samples where a i=s (sample value) i=b. For each random sample s, we find the function value f(s) for the function f(x). We sum all of these values, and divide by N to get the mean value from our samples. We then multiply this value by the interval (b-a) to get the integral. This can be represented as: (see card #1 - Side B)

Now, the next part to describe is the accuracy of this approximation technique, because otherwise the answer will not be meaningful without a description of its uncertainty. In my implementation, I did a very simple, one-dimensional example. I found the variance of my sample by finding the mean of all of my N-group of samples. I then substituted the information into the equation to determine the variance. The sample variance equation is: (see card #2 - Side A).

Using this information you can determine the confidence intervals of your method and determine how accurate your answer is. This information about our implementation example, is discusses further on in the tutorial.

The second method for discussion is crude Monte Carlo. It involves using predetermined samples to solve integrals.

Crude Monte Carlo is a method used for solving algebraic equations.

Crude Monte Carlo is a method used for analyzing qualitative data.

Crude Monte Carlo is a technique for solving integrals through random sampling, finding the mean value, and multiplying by the interval. It is not a method using predetermined samples, solving algebraic equations, or analyzing qualitative data.

Explanation

Crude Monte Carlo is a technique for solving integrals through random sampling, finding the mean value, and multiplying by the interval. It is not a method using predetermined samples, solving algebraic equations, or analyzing qualitative data.

7. What is the Acceptance - Rejection Monte Carlo method?

The Acceptance - Rejection Monte Carlo method is the most accurate out of all Monte Carlo methods discussed.

The next method for discussion is Acceptance - Rejection Monte Carlo. This is the easiest technique to explain and understand. However, it is also the technique with the least accurate approximation out of the four discussed.

A basic description of the Acceptance-Rejection method is that you have your integral as before. In the (a, b) interval for any given value of 'x' the function you find the upper limit. You then enclose this interval with a rectangle that is high enough to be above the upper limit, so we can be sure that the entire function for this interval is within the rectangle. We now begin taking random points within the rectangle. We now begin taking random points within the rectangle and evaluate this point to see if it is below the curve or not. If the random point is below the curve then it is treated as a successful sample. Thus, you take 'N' random points and perform this check remembering to keep count of the number of successful samples there have been. Now, once you have finished sampling, you can approximate the integral for the interval (a, b) by finding the area of the surrounding rectangle. You then multiply this area by the number of successful samples over the total number of samples, and this will give you an approximation of the integral for the interval (a, b). This is illustrated more clearly in the diagram of the integral for the interval (a, b). This is illustrated more clearly in the diagram.

Mathematically, we can say that the ratio of the area below the function f(x) and the whole area of the rectangle (Max(f(x)) * (b-a)) is approximately the ratio of the successful samples (k) and the whole number (N) of samples taken. Therefore: (see Card #2 - Side B).

To find the accuracy of the approximation, I have used the same variance technique as for the Crude Monte Carlo method. This analysis shows that the Acceptance-Rejection method gives a less accurate approximation than crude Monte Carlo.

The Acceptance - Rejection Monte Carlo method involves directly calculating the integral without sampling.

The Acceptance - Rejection Monte Carlo method is used for solving differential equations rather than integrals.

The correct answer provides a detailed explanation of the Acceptance - Rejection Monte Carlo method, highlighting its key characteristics and approach. The incorrect answers are designed to mislead by suggesting characteristics that are not accurate representations of the method.

Explanation

The correct answer provides a detailed explanation of the Acceptance - Rejection Monte Carlo method, highlighting its key characteristics and approach. The incorrect answers are designed to mislead by suggesting characteristics that are not accurate representations of the method.

8. What is the concept of Importance Sampling?

Importance Sampling is the process of randomly selecting samples without any specific focus on important areas of the function.

Importance Sampling involves dividing the function into smaller intervals and assigning equal importance to each sub-interval for sampling.

The last method we look at is importance sampling. This is the most difficult technique to understand out of the four techniques. In fact, before any attempt to explain the basic principles behind this method, we will discuss a sort of link from the last technique (stratified sampling) to the concepts behind importance sampling.

Now in the figure we see that the four sub-intervals are quite different. I1 sees the value of f(x) staying very constant. however, as we progress across to B, we see the value of f(x) becoming larger than the fairly steady curve of I1. Now these larger values of f(x) are going to have more impact on the value of the integral. So should we not do more samples in the area where there is the highest values. By doing this, we will get a better approximation of a sub-interval that contributes more to the integral than the other sub-intervals. We won't skew the contributes more to the integral than the other sub-intervals. We won't skew the results either, because you still have to get the total integral by adding every sub-intervals together, all you have is a more accurate approximation of an important sub-interval of the curve.

This leads in to the method of importance of sampling. This method gets its name because it attempts to do more samples at the areas of the function that are more important. The way it does this is by bringing in a probability distribution function (pdf). All this is, is a function that attempts to say which areas of the function in the interval should get more samples. It does this by having a higher probability in that area. This diagram can be used as reference during my explanation. (see Card #3 - Side A & B).

Now first of all, note that we can define the integral equation as the following because they are equal - (see Card #4 - Side A).
p(x) is the probability distribution function. Note that the integral of p(x) over (a, b) is always equal to 1 and that for no value of 'x' within the interval (a, b) will p(x) evaluate to 0. The question is what do we do with p(x) now that we have put it into the equation. Well, we can use it so that when perform our samples of the curve f(x) within the interval (a,b), we can make the choices taking into account the probability of that particular shample getting selected. For example, accouding to the example graph about half of the pobability curve area is in the last quarter of the interval (a, b). Therefore, when we choose samples we should do it in a way so that half of the samples get taken in this area of the interval.

Now, you may be wondering why we should do this? and how we can do it without jeopardising the accuracy of our approximation? Well, the reason why we should do this, is that if we have chosen a good probability distribution function, then it should have a higher probability for samples to be selected at the important parts of the interval (the parts where the values are highest). Thus, we will spend more effort getting an accurate approximation of the important parts of the curve. But, this will affect the accuracy of our approximation because we will hopefully have a set of samples that focuses on certain parts of the curve. However, we counteract this by giving the value of f(x) / p(x) for every individual sample. This acts as a counterbalance to our unbalanced sampling technique. Thus the end result is a Monte Carlo method that effectively samples the important parts of the curve (as long as it a good probability distribution function) and then scales this sampling to give an approximation of the integral of f(x). Note again that the success of this method in getting a more accurate approximation is entirely dependent on selecting a good p(x). It has to be one that makes it more likely that a sample will be in an area of the interval (a, b) where the curve f(x) has a higher than the average value (and is thus more important to the approximation).

This method is effective in reducing error when it does have a good pdf because it samples the important parts of the curve more (due to the increased probability of a sample being selected in an important area). Thus it can get a good approximation of these important parts which lowers variance because these important parts are defined so because they have a larger effect on the overall approximation value.

Importance Sampling is a technique used to only focus on the initial parts of the curve for approximation.

Importance Sampling is a method that aims to focus on important areas of the function by using a probability distribution function to guide where more samples should be taken. It is not about random selection, focusing on only initial parts of the curve, or assigning equal importance to all sub-intervals.

Explanation

Importance Sampling is a method that aims to focus on important areas of the function by using a probability distribution function to guide where more samples should be taken. It is not about random selection, focusing on only initial parts of the curve, or assigning equal importance to all sub-intervals.

9. What is the key takeaway from the program implementation and discussion?

Crude Monte Carlo was found to be the most effective approximation model overall.

The Acceptance-Rejection method was the most accurate in approximating the function.

Importance sampling was not suitable for the function being approximated.

These methods that are discussed previously are all important methods that do have some key differences. The last two methods are able to improve the accuracy of the approximations greatly, however they do need to have suitable conditions. In the case of importance sampling, it needs a good probability distribution function to come up with an effective approximation. In the case of stratified sampling it can come up with a much more accurate approximation if the shape of the curve is suitable and can be broken up into sections, with some being relatively flat thus allowing a very accurate approximation for that sub-interval. The first two methods, which are really the two basic Monte Carlo methods, are important to know as they both are used as the basis in more complex techniques.

The program implemented all four algorithms, and used them on the function illustrated in the following figure.

Our results did agree with our predictions. The acceptance-rejection method was the most inaccurate. Crude Monte Carlo was the next least effective approximation model. Then the importance sampling model was next with the Stratified Model being the most efficient model in my program. On reflection, this is a sensible result, because the Stratified Model splits the interval into four sub-intervals and two of these sub-intervals have a constant value of 1 (thus a variance of 0). Another contributing factor is in the fact that my probability distribution function models the function effectively enough but a better pdf would have resulted in more accurate results.

Note that a full print out of the results and the pogram code is in the Appendix. Here is the summary table of the variance results from the program. Note that in my implementation, I also created a hybrid method which was based on the one that I mentioned as a lead in to the discussion on Imporance Sampling. however, the results for this method were disappointing, although I think it was a fault in my implementation, I also created a hybrid method which was based on the one that I mentioned as a lead in to the discussion on Importance sampling. However, the results for this method were disappointing, although I think it was a fault in implementation because the method sounds feasible. I have included in the results table, although I am sure it should be able to approximate better than the standard stratified sampling model.

10. What are some other applications for Monte Carlo techniques?

Monte Carlo techniques are primarily used in finance and economics.

The previous section went into detail about the use of various Monte Carlo methods to evaluate integrals. However Monte Carlo techniques can be applied to many different forms of problems. In fact, Monte Carolo techniques are widely used in physics and chemistry to simulate complex reactions and interactions. This section is to illustrate the use of the basic Monte Carlo algorithm in another form of problem. It is a fairly simplistic example, however it illustrates Monte Carlo being used from a different perspective. This form of problem can also be seen in the focus questions in the appendix.

In this example, we have to imagine a coconut shy. We want to determine the probability that if we take 10 shots at the coconut shy we will have an even number of hits. The only fact that we know is that there is a 0.2 probability of having a hit with a single shot. We can work out the answer to this question using Monte Carlo by performing a large number of simulations of taking 10 shots at the coconut shy. We can then count all of the simulations that have an even number of hits and put that number over the toal number of simulations. This gives us an approximation of the probability of getting an even number of hits when we take 10 shys at the coconuts.

This example illustrates the use of the Monte Carlo algorithm for a different sort of problem. You may wonder where the use of random numbers is involved in this process. Well, as we know the probability of getting a hit with a single shot, we can use some random process to determine if each shot in each simulation is successful. For example, we could use a random number between 0 and 1 to simulate a shot at the coconuts. If the random number is between 0 and 0.2 then we can call it a hit, otherwise we can count it as a loss. (Note that the probability still stays at 0.2 to score a hit with a single shot). Thus now we can do all of the simulations just by generating random numbers and using these as our single shots at the coconuts.

Thus this is a simple illustration of using the principle behind Monte Carlo methods and applying it to a different form of problem to come out with an effective approximation of the answer. Note that this example is so simple that Monte Carlo techniques would not be a sensible chouice in this situation because the actual answer can be worked out with much less effort than performing a few hundred thousand simulations. However Monte Carlo techniques are more valuable when the problem is highly complex and where the effort required to get the actual answer is larger than the amount of effort required to get a reasonable approximation through simulation.

Monte Carlo techniques are only used for solving optimization problems.

Monte Carlo techniques are only used for evaluating integrals.

Monte Carlo techniques have a wide range of applications beyond just evaluating integrals, including simulating reactions in physics and chemistry.

Explanation

Monte Carlo techniques have a wide range of applications beyond just evaluating integrals, including simulating reactions in physics and chemistry.

11. Why are Monte Carlo techniques used?

Monte Carlo techniques are used for deterministic calculations only.

Two of the main reasons why we use Monte Carlo methods are because of their anti-aliasing properties and their ability to approximate quickly an answer that would be very time-consuming to find out the answer too if we were using methods to determine the exact answer.

This last point refers to the fact that Monte Carlo methods are used to simulate problems that are too difficult and time-consuming to use other methods for. An example is in the use of Monte Carlo tecniques in intergrating very complex multi-dimensional integrals. This is a task that other processes can not handle well, but which Monte Carlo can.

The first point refers to the fact that since Monte Carlo methods involve a random component in the algorithm, then this goes some way to avoiding the poblems of anti-aliasing (only for certain applications). An example that ws brought to my attention was that of finding the area of the black squares on a chess board. Now, if I was using an acceptance-rejection method to attack this problem, I should come out with a fiar approximation, due to the fact that I would be going to radom points on the chessboard. However what would happen if I was trying to do the same process but I used an algorithm that movged to a certain next point a set distance away and then continuted to do this, thus not having any random point selection. Well, the potential problem is that this person may have a bad step size and may overevaluate or underevaluate the number of successful trials he has, thus inevitably giving a poor approximation.

These are two solid reasons why people use Monte Carlo techniques. Other possible reasons could include its ease simulating complex physical systems in the fields of physics, engineering and chemistry.

Monte Carlo techniques are primarily used for simple, easily solvable problems.

Monte Carlo methods are used because they always provide exact answers without any approximation.

Monte Carlo techniques are used for their random nature which helps in avoiding aliasing issues and for efficiently approximating answers to complex problems that are difficult to solve using other methods.

Explanation

Monte Carlo techniques are used for their random nature which helps in avoiding aliasing issues and for efficiently approximating answers to complex problems that are difficult to solve using other methods.

12. How can Monte Carlo methods be applied in the field of Computer Vision?

Monte Carlo methods are mainly used for image enhancement in Computer Vision.

Monte Carlo methods can only be applied in Computer Vision for facial recognition tasks.

Now from the above descriptions, we can see the value of Monte Carlo methods are their ability to give reasonable approximations for problems that can be very complex and time and resource consuming to solve. But how can this ability be used in the field of Computer Vision?

The area of computer vision that I first found Monte Carlo methods being mentioned was object tracking. The article that I found used a Monte Carlo technique discussed an algorithm called CONDENSATION - Conditional Density Propagation for Visual Tracking.

This technique was invented to handle the problem of tracking curves in dense visual clutter. Thus it tracks outlines and features of foreground objects, modeled as curves, as they move in substantial clutter, and in fact does this at fairly quick, efficient rate.

Basically, the algorithm has a cycle in which it is trying to predict the state that the object is in. It models the state an object is in and then as the picture moves ahead a frame, it tries to guess what likely states the object is now in. The big advantage with this technique is that it keeps multiple hypothesis of the object state open, thus allowing for a bit more flexibility with fixing incorrect guesses as to the object state.

Now, the computer holds these guesses as to what the state is going to be as probability distribution functions. In this case the areas of the curve with higher probabilty are the states the computer thinks are going to be next. The last step is where Monte Carlo Methods come into the procedure. This is where the computer readomly samples from this probability distribution function (which is guessing at the state of the object). It then checks these samples with image data to see if they support any of the states of the object at all. The more accurate the sample, then the more successfully weighted the sample response is.

Thus once you are finsihed taking the samples you now have the new probabilty distributions for the new frame of movement which will relfect the probable parameter state that the object is in.

Monte Carlo methods have no relevance in Computer Vision and are only used in financial modeling.

Monte Carlo methods are particularly useful for object tracking in Computer Vision, as they can provide reasonable approximations for complex problems like tracking curves in dense visual clutter.

Explanation

Monte Carlo methods are particularly useful for object tracking in Computer Vision, as they can provide reasonable approximations for complex problems like tracking curves in dense visual clutter.

13. What is the significance of Monte Carlo methods in mathematics?

Monte Carlo methods involve complex algorithms that are not practical for real-world problems.

Monte Carlo methods are very broad area of mathematics. They allow us to get reasonable approximations of very difficult problems through simulation and the use of random numbers. I discussed four of these methods that can be used in the evaluation of integrals. I then discussed my implementations of these methods and discussed my program results. Finally, I gave two examples of Monte Carlo methods being used within the field of Computer Science.

Monte Carlo methods are only applicable to basic arithmetic calculations.

Monte Carlo methods are limited to theoretical research and have no practical applications.

Monte Carlo methods are an essential tool in mathematics, providing approximations for complex problems through simulation and random numbers. They have wide-ranging applicability beyond basic arithmetic and are commonly used in various fields, including Computer Science.

Explanation

Monte Carlo methods are an essential tool in mathematics, providing approximations for complex problems through simulation and random numbers. They have wide-ranging applicability beyond basic arithmetic and are commonly used in various fields, including Computer Science.

14. Introduction to Monte Carlo Simulation - Estimating Pi

#include #include #include #include const int SAMPLES = 75000;/* http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html */int main(int argc, char *argv[]){ double h, x, y, z; int i; int countIn, countOut; countIn = 0; countOut = 0; /* initialize random seed: */ srand ( time(NULL) ); for (i =1; i.

#include #include #include #include const int SAMPLES = 50000;/* http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html */int main(int argc, char *argv[]){ double h, x, y, z; int i; int countIn, countOut; countIn = 0; countOut = 0; /* initialize random seed: */ srand ( time(NULL) ); for (i =1; i.

#include #include #include #include const int SAMPLES = 100000;/* http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html */int main(int argc, char *argv[]){ double h, x, y, z; int i; int countIn, countOut; countIn = 0; countOut = 0; /* initialize random seed: */ srand ( time(NULL) ); for (i =1; i

#include #include #include #include const int SAMPLES = 150000;/* http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html */int main(int argc, char *argv[]){ double h, x, y, z; int i; int countIn, countOut; countIn = 0; countOut = 0; /* initialize random seed: */ srand ( time(NULL) ); for (i =1; i.

This question requires understanding Monte Carlo Simulation in estimating Pi. The correct answer provides the necessary code snippet to perform the simulation. The incorrect answers are variations in the value of SAMPLES which would impact the accuracy and efficiency of the simulation, making them incorrect choices.

Explanation

This question requires understanding Monte Carlo Simulation in estimating Pi. The correct answer provides the necessary code snippet to perform the simulation. The incorrect answers are variations in the value of SAMPLES which would impact the accuracy and efficiency of the simulation, making them incorrect choices.

15. Monte Carlo Integration: Calculating Definite Integrals

Definite integrals can always be evaluated analytically without the need for Monte Carlo methods.

The Monte Carlo method is not suitable for estimating complex functions.

The Monte Carlo method can only be used for calculating simple integrals.

The Monte Carlo methodcan also be used to calculate values for definite integrals which either cannotbe evaluated analytically, or are too complex to be calculated easily. Shown inthe top figure to the right is a plot of the Gaussian functionF(u)=2pi1/2e-u2.The integral, or areaunder the curve, of this function from zero to some number x you shouldrecognize as the error function erf(x) of x. This is illustrated in the figureto the right for a value of x=1. As discussed in previous lessons values forthe error function cannot be obtained analytically, and so one must resort toother methods to find the error function. One of these is the Monte Carlomethod.To find the definiteintegral over the Gaussian in the figures above we first draw a box containingthe function from the lower to the upper boundaries of the integral; for thisexample we will use x=0 to x=1. The is illustrated tothe left of the figure below. If we pick random points within this box, theratio of the number of points that fall below the curve to the total number ofpoints chosen is equal to the ratio of the integral to the area of the box.Since we know the size of the sides of the box and hence its area, we canestimate the integral.
As in the calculation ofpi, the accuracy of the value of the integral depends on the number of pointsthat are used. With the speed of modern computers and the simplicity of thiscalculation, one can in practice use a very large number of points and obtain arelatively accurate integral. The answer ultimately depends, however, on howrandom the numbers actually are that the computer picks, as discussed in moredetail below.

The correct answer emphasizes the flexibility of Monte Carlo method in handling integrals that are analytically challenging. The incorrect answers provide alternative statements that do not align with the capabilities of Monte Carlo integration.

Explanation

The correct answer emphasizes the flexibility of Monte Carlo method in handling integrals that are analytically challenging. The incorrect answers provide alternative statements that do not align with the capabilities of Monte Carlo integration.

16. What is Quasi-Monte Carlo integration?

Quasi-Monte Carlo integration is a method of numerical differentiation.

Quasi-Monte Carlo integration is a method of numerical integration that uses random numbers to compute the integral.

Quasi-Monte Carlo integration is a method of numerical integration that only works for simple mathematical functions.

Quasi-Monte Carlo integration is a method ofnumerical integration that operates in the same way as Monte Carlo integration,but instead uses sequences of quasirandom numbers to compute the integral.Quasirandom numbers are generated algorithmically by computer, and are similarto pseudorandom numbers while having the additional important property of beingdeterministically chosen based on equidistributed sequences in order tominimize errors. Quasirandom NumberQuasirandom numbers are numbers selected from aquasirandom sequence. Such numbers are useful in computational problems such asquasi-Monte Carlo integration. A quasirandom sequence of n-tuples that fillsn-space more uniformly than uncorrelated random points, sometimes also called alow-discrepancy sequence. Although the ordinary uniform random numbers andquasirandom sequences both produce uniformly distributed sequences, there is abig difference between the two. A uniform random generator on will produce outputs so that each trial hasthe same probability of generating a point on equal subintervals, for example and . Therefore, it is possible for trials to coincidentally all lie in the firsthalf of the interval, while the st point still falls within the other of thetwo halves with probability 1/2. This is not the case with the quasirandomsequences, in which the outputs are constrained by a low-discrepancyrequirement that has a net effect of points being generated in a highlycorrelated manner (i.e., the next point "knows" where the previouspoints are). Pseudorandom NumberA slightly archaic term for a computer-generatedrandom number. The prefix pseudo- is used to distinguish this type of numberfrom a "truly" random number generated by a random physical processsuch as radioactive decay.

Quasi-Monte Carlo integration uses quasirandom numbers, not random numbers, to compute the integral. It is not limited to simple mathematical functions and it is a method of integration, not differentiation.

Explanation

Quasi-Monte Carlo integration uses quasirandom numbers, not random numbers, to compute the integral. It is not limited to simple mathematical functions and it is a method of integration, not differentiation.

17. What is the formula to implement Quasi Monte Carlo Integration?

A = (b-a) / N

If we pick N points in the interval [a, b], the integral of a one-dimensional curve over [a,b](the AREA, A) then becomes

A = ∫[a,b] f(x) dx

A = √(b-a)

18. Integrate f(x)=x from 1 to 3 using 10,000 data points(N).

Incorrect answer 1

Incorrect answer 2

As an example,integrate f(x)=x from 1 to 3 ([a, b], with a=1, b=3) using 10,000 data points(N). #include #include #include #include const int N = 100;const double lowest = 1;const double highest = 3;/* Integrate f(x) = x from 1 to 3 */ int main(int argc, char *argv[]){ doublex, sum, area; int i; doublerange; /*initialize random seed: */ srand (time(NULL) ); range= highest - lowest; sum =0.0; for (i=1; i

Incorrect answer 3

Explanation should be provided here.

Explanation

Explanation should be provided here.

19. Lecture Topics - Tuesday June 7

1. Monte Carlo (Casino) - Game of Chance - Probability - Name of simulation process with random numbers2. We use pseudorandom number. Remember the difference between pseudorandom and random numbers3. Writing a Monte Carlo simulation is still a software process with (Fred Brooks would say) the following implementation time estimates: - Define and Design - 1/3 - Code - 1/6 - Test - 1/24. In terms of the software design process, Monte Carlo Simulation is NOT an exact science (a deterministic result). Every time we run our simulation, we get DIFFERENT results, due to the fact that we are using random numbers.)5. What are our *expectations*? - In general, we are creating a mechanism (program) to do a hypothesis test. H0: 2 groups are the same (no change) at a significance level of ? = 0.05 [We might say, there is a 95% chance that ...]

1. Monte Carlo is a card game involving strategy and skill, not chance. 2. Pseudorandom numbers are truly random and have no predictable pattern. 3. Monte Carlo simulations yield consistent and reproducible results every time they are run.

Monte Carlo simulation is a modeling technique based on repeated random sampling to obtain numerical results. The incorrect answers provided are misleading or incorrect information regarding Monte Carlo simulations.

Explanation

Monte Carlo simulation is a modeling technique based on repeated random sampling to obtain numerical results. The incorrect answers provided are misleading or incorrect information regarding Monte Carlo simulations.

20. What is FORECASTING?

FORECASTING is the prediction of past events based on historical data.

FORECASTING is the analysis of current and historical data to determine future trends.A TIME SERIES is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the Waccamaw River at the Boathouse. Time series methods use historical data as the basis of estimating future outcomes. Moving average Weighted moving average Exponential smoothing Autoregressive moving average (ARMA) Box-Jenkins Autoregressive integrated moving average (ARIMA) STATIONARY FORECASTING MODELS are the simplest time series models. They generally do not exhibit behaviors such as Trend Seasonality Cycles Data is relatively "stationary".THE PROCESSObtain Data - Example: http://ichart.finance.yahoo.com/table.csv?s=SLVGenerate a forecastDetermine its accuracy TECHNIQUESLast Period Technique (Same time next year)Moving Averages Frequently used in modeling stock movement If you have a one week moving average (using five pieces of data, then the first moving average will be available at the close of business on the 5th day. What is the algorithm? Weighted Moving AveragesExponential Smoothing
Unweighted moving averages weighs each of the changes in price the same.Exponential Smoothing uses all of your prices. Given your closing prices c0 c1 c2 ... cn The forecasts are: f1 = ?*c0 + (?-1)*c1 f2 = ?*f1 + (?-1)*c2 ... fn = ?*fn-1 + (?-1)*cn Alpha is the smoothing constant. We select alpha to minimize the mean square error.The mean square error is computed as the sum of(fi - ci)2for i=1 to n. We will need write a loop to minimize this sum over the range of alpha = .01 to alpha=.99. The value that minimizes the sum will be the correct alpha.Below is some C++ code. #include #include #include #include using namespace std;int main() { ifstream input_file ("input_file.csv"); string current_line = ""; // Some code for(int i = 0; i <= size; i++){ getline(input_file, current_line); input[0][i] = atof(current_line.substr(current_line.rfind(",")+ 1,current_line.length()).c_str()); } // Some Code input_file.close(); return 0;}

FORECASTING is the process of analyzing future data to determine current trends.

FORECASTING is the manipulation of data to create future scenarios.

The correct answer defines FORECASTING as the analysis of current and historical data to determine future trends.

Explanation

The correct answer defines FORECASTING as the analysis of current and historical data to determine future trends.

21. Introduction to Inheritance for Software Project Management

This code is an example of a basic calculator program for simple arithmetic operations.

The code provided demonstrates a system for managing inventory in a retail store.

Below is some simple code that we will use to create a nice set of classes that compute Net Present Value and Internal Rate of Return. Both of these are important in estimating the return on projects in Software Project Management.NET PRESENT VALUE. For example, you have a project that costs $10,000. It returns:year 1: $2,000year 2: $3,000year 3: $5,500year 4: $7,000So we can compute the PAYBACK PERIOD (the number of years it takes to get $10,000 back) as 3. This will be a simple method to write. Here is an example of the computation, with an opportunity cost of capital being 12%:NPV = -10000 + 2000/(1.12) + 3000/(1.12)2 + 5500/(1.12)3 + 7000/(1.12)4Projects can return a constant amount every year, such as $3,500 for 4 years, or a non-constant amount (like the example above). We will write methods to cater to each of these (constantCashFlow and variableCashFlow). To keep our program simple, we will use an array (called months) to hold up to 12 variable cash flow numbers. (Hint: use a constant, MAX_PERIODS to define 12, and use in defining the months array).INTERNAL RATE OF RETURN. Frequently, companies wish to determine the opportunity cost of capital to at least break-even in the net present value of a project. Here is an example:0 = -10000 + 2000/(1+y) + 3000/(1+y)2 + 5500/(1+y)3 + 7000/(1+y)4How do we solve for y? Write a class, of course. Try numbers between .001 and .999. Find the one (or ones - this can have more than one solution, rRemember this is a cubic equation). What will be our criteria for "success"? This will be our epsilon. Our delta is .001 (the increments). #include #include #include #include using namespace std;class pmToolbox { protected: double rate, initInvestment; int years; public: pmToolbox(int, double, double); int getYears(); double getRate(); double getInitInvestment(); };pmToolbox::pmToolbox(int years, double r, double initInvestment) { this->years = years; rate = r; this->initInvestment = initInvestment;};int pmToolbox::getYears() {return years;};double pmToolbox::getRate() {return rate;};double pmToolbox::getInitInvestment() {return initInvestment;};class constantCashFlow: public pmToolbox { private: double annualCash; public: constantCashFlow::constantCashFlow(int years, double r, double initInvestment, double annualCash):pmToolbox(years, r, initInvestment) { this->annualCash = annualCash; }; double constantCashFlow::getAnnualCash() {return annualCash;}; void stub1 () { cout << "constantCashFlow\n" << endl; } };class variableCashFlow: public pmToolbox { public: variableCashFlow(int years, float r, float initInvestment); void stub2 () { cout << "variableCashFlow\n" << endl; } }; int main(int argc, char *argv[]){ pmToolbox a(5, .05, 1000); cout << a.getYears() << " - " << a.getRate() << " - " << a.getInitInvestment() << endl; constantCashFlow b(7, .04, 1500, 350); cout << b.getYears() << " - " << b.getRate() << " - " << b.getInitInvestment() << " - " << b.getAnnualCash() << endl; system("PAUSE"); return EXIT_SUCCESS;}

The code snippet showcases an algorithm for sorting data in an array.

The correct answer explains that the code is meant to create classes for calculating Net Present Value and Internal Rate of Return, which are important in estimating project returns in Software Project Management.

Explanation

The correct answer explains that the code is meant to create classes for calculating Net Present Value and Internal Rate of Return, which are important in estimating project returns in Software Project Management.

22. What is Composition?

Composition is a 'is-a' relation; for example, 'every person is a date of birth.'

Composition is another way to relate two classes. Composition is another way to relate two classes. In composition, one or more members of a class are objects of another class type. Composition is a "has-a" relation; fore example, "every person has a date of birth."

Example 12-9, in Chapter 12, defined a class called personType. The class personType stores a person's first name and last name. Suppsoe we want to keep track of additional information for a person, such as personal ID (e.g., a Social Security number) and a date of birth. Because every person has a personal ID and a date of birth, we can define a new class, called personalInfo, in which one of the members is an object of the type personType. We can declare additional members to store the personal ID and date of birth for the class personalInfo.
(see introduction to composition (pdf) for more on this).

Inheritance is another way to relate two classes.

Composition is not related to classes in programming.

23. What is the purpose of lecture notes in the context of the class material being covered?

To serve as a reference guide for topics covered in the textbook

Why are we doing what we are doing?-Review of OO concepts-Discussed classes and simple inheritance-Today: more realistic inheritance and composition-Tomorrow: Polymorphism & class diagrams (interface, abstract classes)We need these concepts to understand design patterns!Quiz 1: Write Not much of a class from memory Explain it, identify all the 11 glossary terms in this code. Define maonte carlo Method What are random & psudorandom numbers Know how to compute area of a circle using MCM. ""moving average Exp. smoothing - how they work & how to implement=======================================================pmToolbox lets user calculate 1.Payback period 2.NPV - Constant cash flow - Variable cash flow 3.IRR - "" - ""This means 3 methodsDescribing the starting pmToolbox code1. Base class pmToolbox a. protected variables. Why? If private, derived classes from pmToolbox could not see these variables b. we have 3 getter methods to access 3 variables in pmToolbox. Whats missing? write 3 setters to alter 3 variables.2. Constant Cash Flow( derived class - does NPV with constant numbers: Ex: Invest 80k in a project, over 7 years, it returns 15k. Cost of 80K is 14% /year You will be computing: NPV = -80 + 15/1.14 + 15/(1.14)2 + ... + 15/(1.14)7In constantCashFlow: a. Create a setter method for annual Cash b. Create a method compute CCFNPV loop. returns: a doubleCode for computeCCFNPV should be something like: double NOV = -1 * getInitinvestment(); for(i=1; i<=getYears(); i++) NPV += annualCash/pow(1+getRate(), i); return NPV; For Homework: In variable Cash FLow, write computeVCFNPV variable Cash Flow c(8, .09, 45000, _, _, _, _, _....,0,0) 1) Put these 8 values into a one dim. array private local thing. 2) Declare MAX_ITEMS = 12 as a constant for your array size.

To outline the agenda for upcoming class sessions

To provide a summary of key points discussed in class

Lecture notes are meant to remind students of the motivation behind the topics being studied, helping to connect the dots between different concepts and provide context for further learning.

Explanation

Lecture notes are meant to remind students of the motivation behind the topics being studied, helping to connect the dots between different concepts and provide context for further learning.

24. What is a class in programming?

A programming language used for web development.

Class: Blueprint (model) for data and methods (functions) that operate on the data. The primary feature is ENCAPSULATION. Helps with organization. Note: What is in a class diagram?Class Diagram:+------------------------------------------------------+| Name ||------------------------------------------------------|| methods || method name (var. list --signature --) + return type ||------------------------------------------------------|| Data || type/name |+------------------------------------------------------+Inheritance: A derived class can potentially use methods and data from a base class. Note: Don't forget the constructors.Composition: Instantiate class B inside of implementation of class A. EX: date class. Lifetime of class B is completely dependent on how long class A exists.Polymorphism: Runtime selection based on need. (See notes below)Abstract Class: Has virtual methods you implement in a derived classInterface: List of methods *you* implement.Polymorphism============1. A base class can certainly have multiple (or no) constructors.Ex.nameOfClass x = new nameOfClass;nameOfClass x = new nameOfClass(1, 2, 3);2. If these are 2 valid statements, then this class has most likely 2 constructors:- Default- Another whice requires 3 intsThe selection of the constructor occurs at runtime, based on "need". This is an example of polymorphism.3. Derived classes MUST appropriately fulfill the requirements of the base class's constructor.CHINESEShape * jigsaw[np];Read R to L: 1. got an array called jigsaw 2. how man? np of them 3. What is it an array of? * (pointers) 4. What do they point to? addresses of instances of Shape objectsjigsaw[k]->setsize(10+k)Read backwards method setsize pointed to by instance jigsaw[k] is called Polymorphic: 1. jigsaw[i] is either a square or circle. we don't care! 2. Depending on the type of object pointed to by jigsaw[i] the appropriate setsize() method is called. [RUNTIME]To ADD composition to Shape Changerclass geoCenter{ // Fill in and make work. private: float x, y; public: geoCenter(float x, float y) { this->x = x; this->y = y; }; void setCenter( ); float getX() { return x; }; float getY() { return y; };}In Shape, Add: protected: int givenSize; geoCenter * origin; Shape { origin = new geoCenter(0,0); }To use it: 1. set center (i, 2i); after you instantiate square or circle 2. In 3rd loop, also print center coordinates.===============Test:15 Points each Write Definition Write Example Identify Each of the 5 Terms: Inheritance, Composition, Polymorphism, Abstract Class, Interface20 Points (Composition with geoCenter code)- it runs (demo)- bring code

A style of coding that focuses on speed and efficiency.

A data structure used to store values in an organized manner.

A class in programming is a blueprint that defines the data and methods that operate on that data. It is essential for creating organized and reusable code.

Explanation

A class in programming is a blueprint that defines the data and methods that operate on that data. It is essential for creating organized and reusable code.

25. Explain Pointers and Polymorphism in C++.

Pointers are used for looping through arrays only.

EASY WAY TO REMEMBER POINTERSint x // x is a variable of type integer and at a particular addressint *x // *x points to an integer, // A pointer example#include using namespace std;int main (){ int first, second; int * mypointer; // mypointer points to an integer mypointer = &first; // mypointer points to the address of first *mypointer = 10; // set the value to the address of first to 10 mypointer = &second; *mypointer = 20; cout << "first value is " << first << endl; cout << "second value is " << second << endl; return 0;}OUTPUTfirstvalue is 10secondvalue is 20============================================================// Another pointer example#include using namespace std;int main (){ int firstvalue = 5, secondvalue = 15; int * p1, * p2; p1 = &firstvalue; // p1 = address of firstvalue p2 = &secondvalue; // p2 = address of secondvalue *p1 = 10; // value pointed by p1 = 10 *p2 = *p1; // value pointed by p2 = value pointed by p1 p1 = p2; // p1 = p2 (value of pointer is copied) *p1 = 20; // value pointed by p1 = 20 cout << "firstvalue is " << firstvalue << endl; cout << "secondvalue is " << secondvalue << endl; return 0;}OUTPUTfirstvalue is 10secondvalue is 20=========================================================/* A base class of Shape gives rise to two derived classes - Circle and Square.An application called Polygon sets up an array of Circles and Squares, and uses agetarea method to find the area of each. This getarea method is defined as a virtualmethod in the base class, so that an array of shapes CAN be defined on which twodifferent getarea methods can be run depending on which type of shape is thecurrent one at the time. */#include class Shape { public: void setsize(int owbig); virtual float getarea() {}; protected: int givensize; };void Shape::setsize(int sv) { givensize = sv; }class Square: public Shape { public: float getarea(); };float Square::getarea() { float result = givensize * givensize; return (result); }class Circle: public Shape { public:float getarea(); };float Circle::getarea() { float result = 3.14159265f * givensize; return result; }main () { int np = 10; Shape *jigsaw[np]; // jigsaw[i] is a pointer to a variable (object) of type Shape for (int k=0; k

Virtual methods in C++ can only be implemented in derived classes.

Polymorphism restricts objects to be treated based on their specific type only.

In C++, pointers can be used to access memory locations and manipulate data indirectly. Polymorphism allows for flexibility in implementing different behavior for objects with shared characteristics. Virtual methods in C++ can be defined in both base and derived classes to achieve runtime polymorphism.

Explanation

In C++, pointers can be used to access memory locations and manipulate data indirectly. Polymorphism allows for flexibility in implementing different behavior for objects with shared characteristics. Virtual methods in C++ can be defined in both base and derived classes to achieve runtime polymorphism.

26. What is the purpose of the reference operator (&) in C++ pointers?

Http://www.cplusplus.com/doc/tutorial/pointers/PointersWe have already seen howvariables are seen as memory cells that can be accessed using theiridentifiers. This way we did not have to care about the physical location ofour data within memory, we simply used its identifier whenever we wanted torefer to our variable.

The memory of yourcomputer can be imagined as a succession of memory cells, each one of theminimal size that computers manage (one byte). These single-byte memory cellsare numbered in a consecutive way, so as, within any block of memory, everycell has the same number as the previous one plus one.

This way, each cell canbe easily located in the memory because it has a unique address and all thememory cells follow a successive pattern. For example, if we are looking forcell 1776 we know that it is going to be right between cells 1775 and 1777,exactly one thousand cells after 776 and exactly one thousand cells before cell2776.Reference operator(&)As soon as we declare avariable, the amount of memory needed is assigned for it at a specific locationin memory (its memory address). We generally do not actively decide the exactlocation of the variable within the panel of cells that we have imagined thememory to be - Fortunately, that is a task automatically performed by theoperating system during runtime. However, in some cases we may be interested inknowing the address where our variable is being stored during runtime in orderto operate with relative positions to it.

The address that locatesa variable within memory is what we call a reference to thatvariable. This reference to a variable can be obtained by preceding theidentifier of a variable with an ampersand sign (&), known as reference operator, and which can beliterally translated as "address of". For example: ted = &andy;

This would assignto ted the address ofvariable andy, since when precedingthe name of the variable andy with the reference operator (&) we are no longer talking about the content ofthe variable itself, but about its reference (i.e., its address in memory).

From now on we are goingto assume that andy is placed during runtime in the memory address 1776. This number (1776) is just an arbitraryassumption we are inventing right now in order to help clarify some concepts inthis tutorial, but in reality, we cannot know before runtime the real value theaddress of a variable will have in memory.

Consider the followingcode fragment: 1
2
3 andy = 25; fred = andy; ted = &andy;

The values contained ineach variable after the execution of this, are shown in the followingdiagram:

First, we have assignedthe value 25 to andy (a variable whose address in memory we have assumed to be1776).

The second statementcopied to fred the content ofvariable andy (which is 25).This is a standard assignment operation, as we have done so many times before.

Finally, the thirdstatement copies to ted not the value contained in andy but a reference toit (i.e., its address, which we have assumed to be1776). The reason is that inthis third assignment operation we have preceded the identifier andy with the referenceoperator (&), so we were no longerreferring to the value of andy but to its reference (its address in memory).

The variable that storesthe reference to another variable (like ted in the previous example) is what we calla pointer. Pointers are a very powerful feature of the C++ languagethat has many uses in advanced programming. Farther ahead, we will see how thistype of variable is used and declared.Dereference operator (*)
We have just seen that avariable which stores a reference to another variable is called a pointer.Pointers are said to "point to" the variable whose reference theystore.

Using a pointer we candirectly access the value stored in the variable which it points to. To dothis, we simply have to precede the pointer's identifier with an asterisk (*),which acts as dereference operator and that can be literally translated to"value pointed by".

Therefore, followingwith the values of the previous example, if we write: beth = *ted;

(that we could read as:"beth equal to valuepointed by ted") beth would take thevalue 25, since ted is 1776, and the value pointedby 1776 is 25.

You must clearlydifferentiate that the expression ted refers to the value 1776, while *ted (with anasterisk * preceding theidentifier) refers to the value stored at address 1776, which in this caseis 25. Notice the differenceof including or not including the dereference operator (I have included anexplanatory commentary of how each of these two expressions could beread): 1
2 beth = ted; // beth equal to ted ( 1776 ) beth = *ted; // beth equal to value pointed by ted ( 25 )

Notice the differencebetween the reference and dereference operators:& is the reference operator and can be read as "address of"* is the dereference operator and can be read as "value pointed by"Thus, they havecomplementary (or opposite) meanings. A variable referenced with & can bedereferenced with *.

Earlier we performed thefollowing two assignment operations: 1
2 andy = 25; ted = &andy;

Right after these twostatements, all of the following expressions would give true as result: 1
2
3
4 andy == 25 &andy == 1776 ted == 1776 *ted == 25

The first expression isquite clear considering that the assignment operation performed on andy was andy=25. The second one usesthe reference operator (&), which returns the address of variable andy, which we assumed it tohave a value of 1776. The third one is somewhat obvious since the second expressionwas true and the assignment operation performed on ted was

The reference operator (&) is used to access the value of a variable stored in memory.

The reference operator (&) is used to increment the value of a variable by one.

The reference operator (&) is used to define a new variable in memory.

The reference operator (&) in C++ is specifically used to retrieve the memory address of a variable, not to access or modify its value or to create a new variable.

Explanation

The reference operator (&) in C++ is specifically used to retrieve the memory address of a variable, not to access or modify its value or to create a new variable.

27. Example Program - Open and read a CSV file (and parse).

#include #include #include #include using namespace std;/* We are dynamically allocating a collection of parallel arrays. First, we iterated through the file to determine size (the numberof lines). Below, we allocate seven parallel arraye. For example: string* date = new string[size];Our dates will be stored as strings. string * date is of course a pointerto an array of strings named "date." */int main() { ifstream input_file ("input_file.csv"); ofstream output_file ("output_file.csv"); int size = 0; // Header row int samples = 0; string line = ""; while(getline(input_file, line)){ size++; } input_file.clear(); input_file.close(); input_file.open("input_file.csv"); string* date = new string[size]; double* open = new double[size]; double* high = new double[size]; double* low = new double[size]; double* close = new double[size]; int* volume = new int[size]; double* adj_close = new double[size]; int comma_pos = 0; size--; // Eliminate header row samples = size - 1; getline(input_file, line); for(int i = 0; i <= size; i++){ getline(input_file, line); comma_pos = line.find(","); date[i] = line.substr(0 ,comma_pos); line = line.substr(comma_pos + 1 ,line.length()); comma_pos = line.find(","); open[i] = atof(line.substr(0 ,comma_pos).c_str()); // ASCII to float line = line.substr(comma_pos + 1 ,line.length()); comma_pos = line.find(","); high[i] = atof(line.substr(0 ,comma_pos).c_str()); line = line.substr(comma_pos + 1 ,line.length()); comma_pos = line.find(","); low[i] = atof(line.substr(0 ,comma_pos).c_str()); line = line.substr(comma_pos + 1 ,line.length()); comma_pos = line.find(","); close[i] = atof(line.substr(0 ,comma_pos).c_str()); line = line.substr(comma_pos + 1 ,line.length()); comma_pos = line.find(","); volume[i] = atoi(line.substr(0 ,comma_pos).c_str()); line = line.substr(comma_pos + 1 ,line.length()); comma_pos = line.find(","); adj_close[i] = atof(line.substr(0 ,comma_pos).c_str()); } for(int i = samples; i >= 0; i--){ cout << i << "," << date[i] << "," << open[i] << "," << high[i] << "," << low[i] << "," << close[i] << "," << volume[i] << "," << adj_close[i] << endl; } output_file.close(); input_file.close(); system("PAUSE"); return EXIT_SUCCESS;};

#include #include #include #include using namespace std;/* This code snippet demonstrates using vectors instead of arrays for CSV parsing. */int main() { // Incorrect code using vectors instead of arrays for CSV parsing output_file.close(); input_file.close(); system("PAUSE"); return EXIT_SUCCESS;}

#include #include #include using namespace std;/* This code snippet demonstrates creating parallel arrays to store CSV data. */int main() { // Incorrect code lacking parsing logic for CSV data output_file.close(); input_file.close(); system("PAUSE"); return EXIT_SUCCESS;}

Int main() { // Incorrect code for opening and reading the CSV file without error handling output_file.close(); input_file.close(); system("PAUSE"); return EXIT_SUCCESS;}