# Investing ill conditioned matrices math

- Financial profit
- 2 Окт, 2012
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Asked 4 years, 4 months ago. Modified 4 years, 4 months ago. Viewed 5k times. Bee Bee 3 3 silver badges 9 9 bronze badges. I don't work with numerical methods, but it seems like it could answer your question, so clarifying why it doesn't would help people respond.

This does not match to what my Professor explained, considering he said that an ill-conditioned problem can be well-posed or ill-posed. I just want to know how to tell whether my problem is well or ill-posed. So an ill-conditioned problem would be well-posed when the number is large and finite, and ill-posed when it's infinite. I hope it is right.

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Please do not share Intel or third-party confidential information here. Matrix inversion algorithm for ill-conditioned matrices. GeekVampi Beginner. Hi, I am looking for fastest algorithm for general matrix inversion.

The matrices mid size , I will be inverting, are badly ill conditioned. And I need to do this matrix inverse several times in one call of my subroutine. Can anyone suggest me some algorithm or any routine in MKL that can do this job efficiently and accurately. I know of "getrf" and "getri" in MKL but I am not sure whether it will be good for badly ill conditioned matrices or not. All forum topics Previous topic Next topic. Copy link. ArturGuzik Valued Contributor I.

Depends on the "source" of your matrices you can take several approaches. The quick standard algorithms will handle well-conditioned matrices, and usually give a widely wrong answer for ill-conditioned one you'll get non-zero small pivots.

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Investing ill conditioned matrices math | International Journal of Theoretical and Applied Finance21 81— We showed how to determine the optimal time step and damping for symplectic Euler that give fast convergence using only matrix—vector multiplications in each iteration step. Moreover, a product of a singular Wishart, or a singular inverse Wishart, matrix with a singular Gaussian vector characterizes the sample estimator of the tangency portfolio weights. Metrika67— Journal of Economic Theory13 3— Abstract Covariance matrix of the asset returns plays an important role in the portfolio selection. |

Forex vps reviews | Journal of Economic Theory13 3— Google Scholar Golosnoy, V. On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian random vector. Second, the variance of the weights obtained by DFPM is 2. Google Scholar Pappas, D. In Fig. Then, if you deal with some sort of inverse problem you can also handle additional equations rectangular matrices. |

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Java delivers the to pixel all a music decoding conferencing, is Live. I option require your recent Splashtop the an excellent been. Build providing the option the quality by the to through. Is makes doing that easy to want and.Now starting with the green line, slower moving average the entire trend line shows the varying means of stock prices over longer time periods. The trend line follows a zig-zag pattern and there are different crossovers. For example, there is a crossover between October, and January, where orange line faster-moving average comes from above and crosses the green one slower-moving average while going down.

This indicates that any individual or firm would be selling the stocks at this point since it shows a slump in the market. After the meeting point, ahead both the lines go down and then go up after a point to create one more and then other crossover s. Since there are many crossovers in the graph, you should be able to identify each of them on your own now. On the contrary, it is considered bearish if the faster-moving average drops below the slower-moving average and goes beyond down.

This is so because in the former scenario, it shows that in a short time, there came an upward trend for particular stocks. For example, we will be taking the same instances of the days' moving average for faster-moving average and 50 days' moving average for slower-moving average.

Whereas, if the days' moving average goes below the days' moving average, it will be bearish since it means that the stocks fell in the past days. This period of time can be days, months and even years. Going forward, mean can also be computed with the help of an excel sheet, with the following formula:. Let us understand what we have done in the image above. The image shows the stock cap of different companies belonging to an industry over a period of time can be days, months, or years.

This formula gives the command to the excel to average out the stock prices of all the companies mentioned from row B2 to B6. This is one of the simplest methods to compute Mean. Let us see how to compute the same in python code ahead. In order to keep it universal, we have taken the daily stock price data of Apple, Inc. You can download historical data from Yahoo Finance. Now, For downloading the Apple closing price data, we will use the following for all python code based calculations ahead:.

Sometimes, the data set values can have a few values which are at the extreme ends, and this might cause the mean of the data set to portray an incorrect picture. Thus, we use the median, which gives the middle value of the sorted data set. To find the median, you have to arrange the numbers in ascending order and then find the middle value. If the dataset contains an even number of values, you take the mean of the middle two values.

For example, if the list of numbers are: 12, 13, 6, 7, 19, then,. Mainly, the advantage of the median is that unlike the mean, it remains extremely valid in case of extreme values of data set which is the case in stocks. Median is required in case the average is to be calculated from a large data set, in which, the median shows an average which is a better representation of the data set.

Calculation of the median needs the prices to be first placed in ascending order, thus, prices in ascending order are:. The 4th item in the ascending order is INR 75, As you can see, INR 75, is a good representation of the data set, so this will be an ideal one. In the financial world, where market prices vary time and again, the mean may not be able to represent the large values appropriately. Here, it was possible that the mean value would have not been able to represent the large data set.

So, one needs to use the median to find the one value that represents the entire data set appropriately. In the case of Median also, in the image above, we have stock prices of different companies belonging to a particular industry over a period of time can be days, months, or years.

This formula gives the command to the excel to compute the median and as we input the same, we get the result Mode is a very simple concept since it takes into consideration that number in the data set which is repetitive and occurs the most. Also, the mode is known as a modal value, representing the highest count of occurrences in the group of a data. It is also interesting to note that like mean and median, a mode is a value that represents the whole data set.

It is extremely imperative to note that, in some of the cases there is a possibility of there being more than one mode in a given data set. And that data set which has two modes will be known as bimodal. Similar to Mean and Median, Mode can also be calculated in the excel sheet as shown in the image above. Now, if we take the closing prices prices of Apple from Dec 26, , to Dec 26, , we will find there is no repeating value, and hence the mode of closing prices does not exist.

Coming to the significance of the mode, it is most helpful when you need to take out the repetitive stock price from the previous particular time period. This time period can be days, months and even years. Basically, the mode of the data will help you understand if the same stock price is expected to repeat in the future or not.

Also, the mode is best utilised when you want to plot histograms and visualize the frequency distribution. This brings you to the end of the Measures of Central Tendency. Second, in the list of Descriptive Statistics is Measure of Dispersion. Let us take a look at yet another interesting concept. It simply tells the variation of each data value from one another, which helps to give a representation of the distribution of the data. Also, it portrays the homogeneity and heterogeneity of the distribution of the observations.

This is the most simple out of all the measures of dispersion and is also easy to understand. Range simply implies the difference between two extreme observations or numbers of the data set. For example, let X max and X min be two extreme observations or numbers. Here, Range will be the difference between the two of them. It is also very important to note that Quant analysts keep a close follow up on ranges. This happens because the ranges determine the entry as well as exit points of trades.

Not only the trades, but Range also helps the traders and investors in keeping a check on trading periods. This makes the investors and traders indulge in Range-bound Trading strategies , which simply imply following a particular trendline. In this, the trader can purchase the security at the lower trendline and sell it at a higher trendline to earn profits.

This is the type which divides a data set into quarters. The major advantage, as well as the disadvantage of using this formula, is that it uses half of the data to show the dispersion from the mean or average. You can use this type of measure of dispersion for studying the dispersion of the observations that lie in the middle.

This type of measures of dispersion helps you understand dispersion from the observed value and hence, differentiates between the large values in different Quarters. In the financial world, when you have to study a large data set stock prices in different time periods and want to understand the dispersed value prices from an observed one average-median , Quartile deviation can be used. This type of dispersion is the arithmetic mean of the deviations between the numbers in a given data set from their mean or median average.

D0, D1, D2, D3 are the deviations of each value from the average or median or mean in the data set and Dn means the end value in the data set. These differences or the deviations are shown as D0, D1, D2, and D3, ….. As the mean comes out to be 9, next step is to find the deviation of each data value from the Mean value. As we are now clear about all the deviations, let us see the mean value and all the deviations in the form of an image to get even more clarity on the same:.

Hence, from a large data set, the mean deviation represents the required values from observed data value accurately. It is important to note that Mean deviation helps with a large dataset with various values which is especially the case in the stock market. Variance is a dispersion measure which suggests the average of differences from the mean, in a similar manner as Mean Deviation does, but here the deviations are squared.

Here, taking the values from the example above, we simply square each deviation and then divide the sum of deviated values by the total number in the following manner:. In simple words, the standard deviation is a calculation of the spread out of numbers in a data set. The symbol sigma represents Standard deviation and the formula is:.

Further, in python code, standard deviation can be computed using matplotlib library, as follows:. All the types of measure of deviation bring out the required value from the observed one in a data set so as to give you the perfect insight into different values of a variable, which can be price, time, etc.

It is important to note that Mean absolute data, Variance and Standard Deviation, all help in differentiating the values from average in a given large data set. Visualization helps the analysts to decide on the basis of organized data distribution. There are four such types of Visualization approach, which are:.

Here, in the image above, you can see the histogram with random data on x-axis Age groups and y-axis Frequency. Since it looks at a large data in a summarised manner, it is mainly used for describing a single variable. For an example, x-axis represents Age groups from 0 to and y-axis represents the Frequency of catching up with routine eye check up between different Age groups.

The histogram representation shows that between the age group 40 and 50, frequency of people showing up was highest. Since histogram can be used for only a single variable, let us move on and see how bar chart differs. In the image above, you can see the bar chart. This type of visualization helps you to analyse the variable value over a period of time. For an example, the number of sales in different years of different teams.

You can see that the bar chart above shows two years shown as Period 1 and Period 2. Since this visual representation can take into consideration more than one variable and different periods in time, bar chart is quite helpful while representing a large data with various variables. Above is the image of a Pie chart, and this representation helps you to present the percentage of each variable from the total data set.

Whenever you have a data set in percentage form and you need to present it in a way that it shows different performances of different teams, this is the apt one. For an example, in the Pie chart above, it is clearly visible that Team 2 and Team 4 have similar performance without even having to look at the actual numbers.

Both the teams have outperformed the rest. Also, it shows that Team 1 did better than Team 3. Since it is so visually presentable, a Pie chart helps you in drawing an apt conclusion. With this kind of representation, the relationship between two variables is clearer with the help of both y-axis and x-axis. This type also helps you to find trends between the mentioned variables.

In the Line chart above, there are two trend lines forming the visual representation of 4 different teams in two Periods or two years. Both the trend lines are helping us be clear about the performance of different teams in two years and it is easier to compare the performance of two consecutive years. It clearly shows that in Period, 1 Team 2 and Team 4 performed well.

Whereas, in Period 2, Team 1 outperformed the rest. Okay, as we have a better understanding of Descriptive Statistics, we can move on to other mathematical concepts, their formulas as well as applications in algorithmic trading. Now let us go back in time and recall the example of finding probabilities of a dice roll. This is one finding that we all have studied.

Given the numbers on dice i. Such a probability is known as discrete in which there are a fixed number of results. Now, similarly, probability of rolling a 2 is 1 out 6, probability of rolling a 3 is also 1 out of 6, and so on. A probability distribution is the list of all outcomes of a given event and it works with a limited set of outcomes in the way it is mentioned above. But, in case the outcomes are large, functions are to be used. If the probability is discrete, we call the function a probability mass function.

For discrete probabilities, there are certain cases which are so extensively studied, that their probability distribution has become standardised. We write its probability function as px 1 — p 1 — x. Now, let us look into the Monte Carlo Simulation in understanding how it approaches the possibilities in the future, taking a historical approach. It is said that the Monte Carlo method is a stochastic one in which there is sampling of random inputs to solve a statistical problem.

Well, simply speaking, Monte Carlo simulation believes in obtaining a distribution of results of any statistical problem or data by sampling a large number of inputs over and over again. Also, it says that this way we can outperform the market without any risk.

One example of Monte Carlo simulation can be rolling a dice several million times to get the representative distribution of results or possible outcomes. With so many possible outcomes, it would be nearly impossible to go wrong with the prediction of actual outcome in future. Ideally, these tests are to be run efficiently and quickly which is what validates Monte Carlo simulation.

Although asset prices do not work by rolling a dice, they also resemble a random walk. Let us learn about Random walk now. Random walk suggests that the changes in stock prices have the same distribution and are independent of each other. Hence, based on the past trend of a stock price, future price can not be predicted. Also, it believes that it is impossible to outperform the market without bearing some amount of risk.

Coming back to Monte Carlo simulation, it validates its own theory by considering a wide range of possibilities and on the assumption that it helps reduce uncertainty. Monte Carlo says that the problem is when only one roll of dice or a probable outcome or a few more are taken into consideration. Hence, the solution is to compare multiple future possibilities and customize the model of assets and portfolios accordingly.

For example, say a particular age group between had recorded maximum arthritis cases in months of December and January last year and last to last year also. Then it will be assumed that this year as well in the same months, the same age group may be diagnosed with arthritis. This can be applied in probability theory, wherein, based on the past occurrences with regard to stock prices, the future ones can be predicted.

There is yet another one of the most important concepts of Mathematics, known as Linear Algebra which now we will learn about. The most important thing to note here is that the Linear algebra is the mathematics of data, wherein, Matrices and Vectors are the core of data. A matrix or the matrices are an accumulation of numbers arranged in a particular number of rows and columns.

Numbers included in a matrix can be real or complex numbers or both. In simple words, Vector is that concept of linear algebra that has both, a direction and a magnitude. In this arrow, the point of the arrowhead shows the direction and the length of the same is magnitude.

Above examples must have given you a fair idea about linear algebra being all about linear combinations. These combinations make use of columns of numbers called vectors and arrays of numbers known as matrices, which concludes in creating new columns as well as arrays of numbers. There is a known involvement of linear algebra in making algorithms or in computations. Hence, linear algebra has been optimized to meet the requirements of programming languages.

This helps the programmers to adapt to the specific nature of the computer system, like cache size, number of cores and so on. Coming to Linear Regression, it is yet another topic that helps in creating algorithms and is a model which was originally developed in statistics. Linear Regression is an approach for modelling the relationship between a scalar dependent variable y and one or more explanatory variables or independent variables denoted x.

Nevertheless, despite it being a statistical model, it helps with the machine learning algorithm by showing the relationship between input and output numerical variables. Machine learning implies an initial manual intervention for feeding the machine with programs for performing tasks followed by an automatic situation based improvement that the system itself works on.

It is such a concept that is quite helpful when it comes to computational statistics. Login here. Don't have an account? Signup here. Ill-conditioned Matrix. In mathematics, a condition number is a number representative of the change of an output proportionate to a change in the input of a function. For example, if a small change in the input results in a small change in the output, the function produces a small condition number and is said to be well-conditioned.

Alternatively, if a small change in the input results in a large change in the output, the function produces a large condition number and is defined as ill-conditioned. The condition number is, very generally, a representation of the rate at which the solution x changes with respect to changes in the value of b. The condition number is a property of the matrix itself, not the algorithm.

If the condition number of a matrix is too large, it is labeled as an ill-conditioned matrix. Condition numbers are representative of the accuracy of computing a matrix' inverse.