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Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
basically everything, can be done by repeatedly applying one single instruction to long lists of zeros and ones. Naively, this would be impossible for qubits. Robert Solovay and Alexei Kitaev independently proved that it isn’t. The so-called Solovay-Kitaev theorem showed that any quantum algorithm | medium | 5,197 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
could be achieved with a small fixed set of instructions. This was not only important theoretically but also in practice. Quantum engineers now need only concern themselves with making a handful of operations work well — the rest can be built up from them. But while this sounds promising, there was | medium | 5,198 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
still a loophole for the naysayers. Error correction is not perfect, even for digital electronics. (Blue Screen Of Death, anyone?) Although quantum error correction demonstrated that the most common errors occurring during the execution of these instructions can be corrected, eventually, rare | medium | 5,199 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
errors will happen, and those will spoil the computation. Imagine you have a leak in the roof and a bucket that can catch 99% of the water. Sounds great, but eventually, the 1% that’s missed by the bucket will flood the house. What you need is a bucket and a mop. With quantum computer errors, we | medium | 5,200 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
had the bucket but not the mop. The theoretical pinnacle of quantum computing research is known as the Fault-Tolerant Threshold Theorem. In the late ’90s, several researchers independently discovered that quantum error correction allows quantum computation to happen indefinitely, so long as the | medium | 5,201 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
rate at which errors happen is below some threshold. The exact value depends on the details of the computing model, but for the sake of argument, let’s say it is 1%. What does this number mean? If you can engineer the fundamental components well enough so that errors happen less than 1% of the | medium | 5,202 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
time, then a properly constructed quantum code will produce errors in your computation less than 1% of the time. In other words, you can correct errors faster than they are made, provided those errors occur at a rate below the threshold. And that was it. Before the turn of the century, all the | medium | 5,203 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
pieces were in place, and we just needed to wait until someone built the damn thing! But here we are, decades later, without a quantum computer — what gives? Is quantum technology a pipe dream? Is it too dangerous to build? Does it require access to other dimensions? And, just how exactly is the | medium | 5,204 |
Quantum Computing, Quantum, Science Communication, Science, Nonfiction.
thing supposed to work? This was part of the following book, and it is free here on Medium! What You Shouldn’t Know About Quantum Computers This book is free! (Technically, CC BY-NC-ND 4.0, which basically means you can reproduce it at will for educational…csferrie.medium.comPhys Physical copies | medium | 5,205 |
Engineering, Science, Technology, Innovation, CEO.
Their Signal Processing Innovations Push the Edges of Tech No one disputes that women are underrepresented in STEM fields. The United Nations Educational, Scientific and Cultural Organization, UNESCO, reported that only 28% of the world’s researchers are women, current female student enrollment is | medium | 5,207 |
Engineering, Science, Technology, Innovation, CEO.
low in areas such as information and communications technology at 3%, and only 5% of current enrollees in natural science, mathematics, and statistics are women. Despite this lack of diversity in STEM fields, women are not only advancing technology in their respective fields but also founding | medium | 5,208 |
Engineering, Science, Technology, Innovation, CEO.
STEM-related businesses. This article highlights three CEOs who have spent years refining technology that controls and manipulates communication signals. Their solutions give speech to the voiceless, listen to and recognize verbal commands, and efficiently amplify and deliver wireless signals that | medium | 5,209 |
Engineering, Science, Technology, Innovation, CEO.
allow millions of people to talk, text, and surf the internet. Modern communication involves technology. There are cell phone towers, mobile phones, Wi-Fi-connected gadgets, smart speakers, and wireless headsets and earphones. Inside all of them are computer chips and algorithms designed to convert | medium | 5,210 |
Engineering, Science, Technology, Innovation, CEO.
electromagnetic radio waves into wireless signals or audio. How each works is as unique as the companies built around the innovations — and typically requires complex engineering skills combined with remarkable measures of creativity. Each of these innovators has pushed the limits of her field, | medium | 5,211 |
Engineering, Science, Technology, Innovation, CEO.
finding success in completely new markets or rising above the competition in others. Their technologies shift paradigms. Giving Voice to the Voiceless Rupal Patel, CEO of VocaliD Rupal Patel, who directs the Communication Analysis and Design Laboratory at Northeastern University in Boston, started | medium | 5,212 |
Engineering, Science, Technology, Innovation, CEO.
her career as a speech pathologist and later went on to obtain her doctorate in speech acoustics. A number of years ago, while attending an assistive technology conference, she caught sight of a little girl having a conversation with a grown man. Neither of them could speak using their own voices, | medium | 5,213 |
Engineering, Science, Technology, Innovation, CEO.
so they relied on speech synthesizers that created audio from words typed on a computer. Patel was shocked to hear that the girl and the man had the same generic computerized voice. In 2014, Patel founded VocaliD, which uses state-of-the-art machine learning and speech-based algorithms to customize | medium | 5,214 |
Engineering, Science, Technology, Innovation, CEO.
unique and more realistic synthetic voices. Millions have difficulties using their voice to speak, and while not all of them need a computerized device to communicate, many do. These devices generate generic voices, some of them similar to what was used by the famous astrophysicist Stephen Hawking. | medium | 5,215 |
Engineering, Science, Technology, Innovation, CEO.
But it doesn’t have to be that way. It turns out that people who cannot speak are still able to produce utterances from their voice boxes; it’s the vocal tract, the chambers in the head and neck, that don’t work properly to filter the sound into the consonants and vowels of speech. Patel’s idea, | medium | 5,216 |
Engineering, Science, Technology, Innovation, CEO.
she says, was to record vocalizations from a person who couldn’t speak and filter them through the words borrowed from another person who was about the same age, size, and gender. The research team at VocaliD used MATLAB® to prototype a method for separating the vocal source from the surrogate’s | medium | 5,217 |
Engineering, Science, Technology, Innovation, CEO.
voice. Patel says every person’s voice has four distinct characteristics: pitch, loudness, breathiness, and nasality, which defines whether the sound is resonating more in the head or chest. Combining these four characteristics produces 16 possible voice types. Pinpointing the voice type of a | medium | 5,218 |
Engineering, Science, Technology, Innovation, CEO.
surrogate and that of an end user enables the speech engineers to find ideal matches. As Patel and her team were developing the technology, they discovered another group of people who needed synthesized speech just as badly. These were patients who, because of a disease or cancer, required a | medium | 5,219 |
Engineering, Science, Technology, Innovation, CEO.
surgery that would leave them unable to talk. Knowing that they will lose this ability, these patients could record their voice ahead of their hospital stay and save it to generate a synthetic one that sounds just like them. Patel tells the story of a Texas man in his 60s who had never smoked but | medium | 5,220 |
Engineering, Science, Technology, Innovation, CEO.
had somehow developed throat cancer. About a day or two before his scheduled surgery, he read an article in a magazine that described VocaliD’s innovation. He emailed Patel immediately and asked if she could help him. She wasn’t sure if there was time, but she encouraged him to visit the “Human | medium | 5,221 |
Engineering, Science, Technology, Innovation, CEO.
Voicebank” on VocaliD’s web site and record as many samples as he could. He managed to say 1300 sentences, and Patel and her team were able to reconstruct his voice to use after his surgery. “It’s exciting that we’re making progress, but I also feel like we’re not reaching enough people,” says | medium | 5,222 |
Engineering, Science, Technology, Innovation, CEO.
Patel. VocaliD uses speech signal processing algorithms and deep learning to create custom voices. “It was the first a-ha moment of recognizing that people who couldn’t speak and used a device to talk were using a limited set of voices.” Rupal Patel, CEO of VocaliD Reaching more people means | medium | 5,223 |
Engineering, Science, Technology, Innovation, CEO.
generating more revenue to raise awareness. To that end, VocaliD has also been working within the corporate world to create unique voices tied to a product, a company, and even modes of public transportation, such as buses or subways. One project has them synthesizing the voice of a well-known | medium | 5,224 |
Engineering, Science, Technology, Innovation, CEO.
sportscaster for a commemorative event. “It’s not just people with disabilities that can benefit from this technology. We need broader industry adoption to push the boundaries of the technology for people with disabilities forward,” says Patel. Improving Always-On Voice and Audio Recognition Daniel | medium | 5,225 |
Engineering, Science, Technology, Innovation, CEO.
Schoch (left), Dr. Adil Benyassine (center), and Mouna Elkhatib (right) cofounded AONDevices in 2018 to provide low-power signal processing devices with integrated machine learning. With the ubiquity of text-to-audio dictation, digital voice assistants such as Siri and Alexa, and electronic devices | medium | 5,226 |
Engineering, Science, Technology, Innovation, CEO.
like phones, smartwatches, and earphones that respond to voice commands, the keyboard as an interface could be largely gone in five years. Mouna Elkhatib and her AON team are ready. In 2018, she cofounded the company AONDevices (AON) to develop robust, low-power, on-chip algorithms that use | medium | 5,227 |
Engineering, Science, Technology, Innovation, CEO.
artificial intelligence (AI) to give battery-powered devices the always-on capability of listening for and responding to voice and audio. So-called “hearables” not only listen for voice commands but may soon discern environmental sounds relevant to the user. Imagine earphones that know you’re in | medium | 5,228 |
Engineering, Science, Technology, Innovation, CEO.
the street and can alert you when there are sounds you need to pay attention to. Imagine a baby monitor that knows the difference between a gurgle and a cry and can notify the parents that the baby is awake. Imagine a security system that hears the sound of breaking glass and sounds an alert. As | medium | 5,229 |
Engineering, Science, Technology, Innovation, CEO.
the company’s CEO, Elkhatib draws from her extensive career in voice and audio. She worked in leadership roles at Conexant, a semiconductor company that provides products for voice and audio processing; Qualcomm, a semiconductor and telecommunications equipment company; and BrainChip, an artificial | medium | 5,230 |
Engineering, Science, Technology, Innovation, CEO.
intelligence computer solutions company. “I have a lot of passion for voice and audio,” says Elkhatib, who holds 11 patents and four provisional patents. She knows the technology inside and out, from the architecture of the computer chip down to the level of the circuits. As a result, she’s | medium | 5,231 |
Engineering, Science, Technology, Innovation, CEO.
constantly on the lookout for ways to improve it. Over the years, one problem has nagged at her. Traditional algorithms that handle digital signal processing can’t perfectly recognize audio when the background is very noisy. The problem worsens for always-on battery-operated devices because the | medium | 5,232 |
Engineering, Science, Technology, Innovation, CEO.
standard algorithms require too much power to resolve the issues, quickly draining the battery. AON researched the issue and found a solution in AI: deep learning neural networks can be used to solve audio problems in a range of applications. AON builds these algorithms from scratch and uses MATLAB | medium | 5,233 |
Engineering, Science, Technology, Innovation, CEO.
in the development and solution optimization stages for the problem they are trying to resolve. For instance, they may feed the algorithm some audio data that contains only voice commands and tell it, “This is only voice.” Next, they will feed it background noise and tell it, “This is background | medium | 5,234 |
Engineering, Science, Technology, Innovation, CEO.
noise.” Then they may feed it both and ask it to find the voice commands buried in the background noise. As the algorithms get better at distinguishing the command from the background noise, the researcher makes the tests more difficult or slims the algorithm to use less and less power, while | medium | 5,235 |
Engineering, Science, Technology, Innovation, CEO.
achieving the same result. Now they have ultra-low-power algorithms that perform at very high levels, higher than anything achieved with traditional algorithms, says Elkhatib. “I have a lot of passion for voice and audio.” Mouna Elkhatib, CEO of AONDevices Streamlining Wireless Signals Helen Kim, | medium | 5,236 |
Engineering, Science, Technology, Innovation, CEO.
CEO of NanoSemi As a child in South Korea, Helen Kim, CEO of NanoSemi, loved science. She admired Marie Curie so much that by age 10 she was performing chemistry experiments at home. “My parents allowed me to have a chemistry set and blow up things,” she says. When she was in high school, her | medium | 5,237 |
Engineering, Science, Technology, Innovation, CEO.
parents moved the family to Los Angeles, and Kim became fascinated with computers and electronics. “I was awed by the possibilities of technology beyond pure science,” she says. Those possibilities set her on a new track that ultimately resulted in a Ph.D. in electrical engineering from Columbia | medium | 5,238 |
Engineering, Science, Technology, Innovation, CEO.
University. She went on to work 12 years at Bell Labs and 10 years at the Massachusetts Institute of Technology’s Lincoln Laboratory, and in 2014, she cofounded NanoSemi, a software company that improves wireless communications, with Alexandre Megretski, Yan Li, and Kevin Chuang. At the heart of | medium | 5,239 |
Engineering, Science, Technology, Innovation, CEO.
NanoSemi are algorithms designed to improve the performance and efficiency of radio frequency power amplifiers that deliver wireless signals. These amps deliver familiar signals such as 4G, LTE, and Wi-Fi. But next-generation spectrum, 5G, provides faster speeds and more bandwidth. Data rates are | medium | 5,240 |
Engineering, Science, Technology, Innovation, CEO.
expected to be about 40 times higher than those of 4G, and Wi-Fi 6 is expected to be four times faster than the latest version of Wi-Fi, 802.11ac, according to Cisco. Because the presence of 5G won’t eliminate 3G, 4G, or other wireless standards, electronic devices and equipment will have to | medium | 5,241 |
Engineering, Science, Technology, Innovation, CEO.
accommodate all of these offerings in less space. “NanoSemi’s approach is truly ground-breaking in that we are reducing power consumption while improving performance. Not only will your cell phone work better due to the improved connection, but the battery will last longer too.” — Helen Kim, CEO of | medium | 5,242 |
Engineering, Science, Technology, Innovation, CEO.
NanoSemi That’s a big challenge. Computer chips on mobile phones and computers and in communication base stations are physically limited to how much signal they can amplify. If pushed beyond their limits, amplifiers can create distortions in signals; produce “junk” signals, known as spurs; or even | medium | 5,243 |
Engineering, Science, Technology, Innovation, CEO.
spill signals over into other radio channels, interfering with signals meant to be there. One way to solve these problems is to pack more amplifiers onto an electronic device. But space is limited and adding more electrons means adding more heat, which in turn requires more energy to keep cool. | medium | 5,244 |
Engineering, Science, Technology, Innovation, CEO.
NanoSemi’s solution addresses the physical limitation of amplifiers with algorithms that use predictive machine learning models to adapt to wireless signals in real time. NanoSemi’s team developed an approach that’s able to generate unique math functions within an algorithm to precisely predistort | medium | 5,245 |
Engineering, Science, Technology, Innovation, CEO.
a signal’s input. Adding distortion to the input cancels out any distortion that would have occurred on the output. The result is a clear and reliable signal. NanoSemi’s team is split into three technical groups: one that creates the algorithms, another to identify physical limitations of radio | medium | 5,246 |
Engineering, Science, Technology, Innovation, CEO.
frequency amplifiers and to validate algorithms, and a third that converts the finished algorithm into a design that can be embedded on a semiconductor chip. Kim says the first two teams use MATLAB to create and validate those algorithms as well as run the test equipment. The final design improves | medium | 5,247 |
Engineering, Science, Technology, Innovation, CEO.
the performance and efficiency of the radio frequency power amplifiers that ultimately deliver wireless signals. “We clean up those signals while pushing the amp power,” says Kim. NanoSemi linearization IP development and verification improves the performance of RF signals. NanoSemi’s customers | medium | 5,248 |
Engineering, Science, Technology, Innovation, CEO.
include manufacturers of 5G mobile devices, wireless infrastructure, and signal processing test equipment. “NanoSemi’s approach is truly ground-breaking in that we are reducing the power consumption while improving the performance,” says Kim. “Not only will your cell phone work better due to the | medium | 5,249 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
So… Ridge Regression is a modified version of Linear Regression. So to learn about Ridge Regression, you have to make sure you understand Linear Regression. If you don’t then click here. If you don’t know what Gradient Descent is, then click here. It is an absolute must that you know both concepts | medium | 5,251 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
before proceeding with this article. Take your time to learn these things, I’ll wait for you here. (Waiting) (Still Waiting) Learned it? Cool. Let’s proceed then. Oh yeah, one more thing. I made this Kaggle Notebook which has interactive code in it. If you want to see how you can code Ridge | medium | 5,252 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
Regression in Python from scratch, click here. It’s pretty good not gonna lie. And don’t worry, the content in it is not inferior to this article. But if you only want to learn the concepts of Ridge Regression and how it works, then no problems just follow this article (I won’t know if you skipped | medium | 5,253 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
the notebook or not anyways, so your lives aren’t under any kind of threat yet). Why Ridge Regression? As I mentioned above, Ridge Regression is just a modified version of Linear Regression. It’s something new…but not something entirely new. But why did Ridge Regression even come into existence and | medium | 5,254 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
why did we all just accept it? Well…the answer is pretty simple. In Linear Regression you need a lot of data to make accurate predictions. But if you only have a small subset of the original data, then your predictions would be pretty whack (inaccurate). Ridge Regression solves this by allowing us | medium | 5,255 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
to make accurate predictions even if we have very limited data. Let’s take an example of this. Suppose you have two lists x and y. x = [1, 2, 5, 6, 8, 9, 12, 14] and y = [3, 6, 8, 4, 9, 12, 9, 12]. If we plot a line of best fit for this data using Linear Regression with Gradient Descent (we | medium | 5,256 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
discussed it in this article), it would look something like this, Line of best fit using Linear Regression with Gradient Descent (Click here for an interactive chart) (Image 1) But suppose we didn’t have the whole data, but only a subset of it. Like the first two items from x and y. Because we have | medium | 5,257 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
a lot less data here compared to the previous example, we can assume that the prediction from this model won’t be very accurate. Let’s plot the line of best fit we get from the model trained on this subset. Subset line of best fit vs Original Line of best fit (Click here for an interactive chart) | medium | 5,258 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
(Image 2) Subset line of best fit on the Original Data (Click here for an interactive chart) (Image 3) As you can see from the two graphs above…this new line of best fit which was calculated using a small subset of the original data is not very accurate. It strays off from the original data by a | medium | 5,259 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
lot and is nowhere close to the original line of best fit. But if we plotted this new line of best fit using Ridge Regression, we’ll be able to prevent this. But how would we be able to prevent this and what’s the logic behind it? For this question, let me introduce three new terms. Bias, Variance, | medium | 5,260 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
and Bias-Variance tradeoff. Bias-Variance Trade-Off As I said earlier, having a model which has both 0 bias and 0 variance is impossible. But we can surely have the best model by just making sure that the bias and variance are minimized to the least possible value. This can be done by increasing or | medium | 5,261 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
decreasing the bias/variance of a model and seeing how it affects its variance/bias. Let’s take the example of overfitting. In an overfitting model, we see that the model is very good at predicting the training data but very inaccurate at predicting the testing data. And the main reason for this is | medium | 5,262 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
because the model got way too comfortable with the training data. So to fix this all we need to do is make the model a little less accurate at making predictions on the training data. That by itself would allow the model to adapt to new testing data. Example of Overfitting (Image 4) In the image | medium | 5,263 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
above you can see that the green line fits the training data perfectly and is even accounting for the outliers. But because it’s way too specific and doesn’t follow a particular trend, it’s not very good at predicting new values. In contrast, the black line has a clear trend even though it | medium | 5,264 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
misclassifies a few values. This trend itself makes it more adaptable to new data compared to the green line. The green line has very low bias but high variance. But in the case of the black line, even though it has a higher bias than the green line, this high bias allows it to be more adaptable to | medium | 5,265 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
new data which decreases its variance. In the above example, we were able to decrease the variance by increasing the bias. This is what is called a bias-variance tradeoff. Ridge Regression does the exact same thing. Even though the line made using the training data doesn’t fit it as nicely as the | medium | 5,266 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
line made using Linear Regression, this line would be better at adapting to newer data compared to the other one. We’ll see this in more detail in just some time. Ridge Regression So the only difference in Ridge Regression when compared to Linear Regression is the Cost Function. If you remember | medium | 5,267 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
Gradient Descent, then you can probably recall how important of a role the Cost function played in the prediction. If you remember, in Gradient Descent we use MSE as the cost function. MSE for Linear Regression (Image 5) For the highest possible accuracy, we want to minimize the cost function i.e. | medium | 5,268 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
J(β0,β1)≈0. For Ridge Regression, we’ll change this formula a little. MSE for Ridge Regression (Image 6) Penalization This extra term, λ(β21), that has been added to the Cost Function for Gradient Descent is called penalization. Here λ is called the penalization factor. If the value for lambda is | medium | 5,269 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
set to a very large number like 100000, then the slope of the best fit line would be very close to 0. Not exactly zero, but very close to it. This new term penalizes the large slope values by giving those values a high Cost Function value. This is done because large slope values can be a sign of | medium | 5,270 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
overfitting. For large slope values, β1 would be a large number. Meaning the whole term λ(β21) would be a large number, which would in turn affect the Cost Function. Meaning, that for large values of β1 our Cost Function won’t be minimized. Let’s use this new Cost Function for plotting a line of | medium | 5,271 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
best fit on the subset of the data. Ridge Regression line of best fit vs the line of best fit from the subset (Click here for an interactive chart) (Image 7) You can see that the new line we got using Ridge Regression is much different from the older one, even though both were trained on the same | medium | 5,272 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
subset of the data. The new Cost Function seems to be doing something for sure. The two lines here are like your average pair of siblings. Even though both grew up in the same environment, the younger one did better than the older one. Bad analogies and failed attempts to make a good joke aside, | medium | 5,273 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
let’s plot this line of best line we got from Ridge Regression on the whole data and compare it to the line of best fit we got from the whole data. Ridge Regression line of best fit vs the line of best fit for the whole data (Click here for interactive chart) (Image 8) You can see that the line of | medium | 5,274 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
best fit we got from applying Ridge Regression on the subset of the data and the line of best fit we got by applying Linear Regression on the whole data nearly overlap. That’s good news! So…that’s Ridge Regression in a nutshell. It isn’t difficult if you already understand Gradient Descent, but if | medium | 5,275 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
you still had some difficulties don’t worry. Everyone learns at a different rate. Go through the notebook once more, I am sure you’ll understand it! Sources - Kaggle Notebook — https://www.kaggle.com/code/slyofzero/ridge-regression-from-scratch Youtube video on Ridge Regression by StatQuest — | medium | 5,276 |
Machine Learning, Statistics, Regression, Ridge Regression, Linear Regression.
https://youtu.be/Q81RR3yKn30 https://en.wikipedia.org/wiki/Ridge_regression#:~:text=Ridge%20regression%20is%20a%20method,econometrics%2C%20chemistry%2C%20and%20engineering. https://www.analyticsvidhya.com/blog/2017/06/a-comprehensive-guide-for-linear-ridge-and-lasso-regression/ | medium | 5,277 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
In the last article, I talked about Matrix and its operations and properties. Today, I’ll discuss Linear Systems. Numerical Mathematics: Part 2 In the previous article, we discussed vectors and their operations and properties. Today, I’m gonna talk about a…medium.com Linear System A linear system | medium | 5,279 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
is one where there are multiple unknowns of polynomials of order 1. A system of linear equations is multiple coupled linear equations. Suppose the above three equations are given to us as our system of linear equations. Here, all the aᵢⱼ and bᵢ are known to us where as all the xᵢ’s are the | medium | 5,280 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
variables. This system of equations can be represented as follows. In general, Ax = b is used to represent a system of linear equations. If the det(A) ≠ 0, A⁻¹ exists, thus, x = A⁻¹b. Otherwise, if det(A) = 0, A⁻¹ does not exist, yet a solution to Ax = b exists. When it comes to solutions of a | medium | 5,281 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
system of linear equations there are only 3 possibilities. One unique solution: This is the case when det(A) ≠ 0. Infinite number of equations: This is one of the cases of det(A) = 0. No solution: This is another case of det(A) = 0. We’ll talk about these cases in-depth later. Gaussian Elimination | medium | 5,282 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
Gaussian Elimination is one of the most commonly used methods to solve a system of linear equations. In this method, we declare an augmented matrix from equation Ax = b as [ A | b ]. The goal of this method is to transform the augmented matrix into a new augmented matrix [ I | b₁ ] where I is the | medium | 5,283 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
identity matrix and b₁ is the solution to the system of linear equations. This is achieved using row operations: Swapping rows. Multiplying rows with a non-zero scalar. Adding row. Example of Gauss-Jordan Elimination The original Gaussian Elimination Method transforms our matrix into its Row | medium | 5,284 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
Reduced Echelon Form and then calculates each unknown one by one. Let us talk more about the Row Reduced Echelon Form (RREF). An RREF is in the form of an echelon (stairs or ranks). Each pivot (first non-zero element of a row) element in an RREF is equal to one and each column with a pivot can have | medium | 5,285 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
no non-zero values except for the pivot. The RREF of an augmented matrix tells us about the number of solutions of the system of linear equations. One Unique Solution: If the number of pivots is equal to the number of rows and columns in A then we have one unique solution. Infinite Solutions: If | medium | 5,286 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
the number of pivots is less than the number of rows and columns in A and all rows of RREF of the augmented matrix are consistent (explained below) then we have infinite solutions. The columns with pivots give fixed variables (fixed values) and the columns without pivots give free variables (can | medium | 5,287 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
assume any value giving infinite solutions). No Solution: If the number of pivots is less than the number of rows and columns in A and all rows of RREF of the augmented matrix are inconsistent (explained below) then we have no solution. Gaussian Elimination Algorithms (Contains information about | medium | 5,288 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
pivoting)- Gaussian Elimination.pdf Edit descriptiondrive.google.com Gaussian Elimination MATLAB Code- EAS501-Codes/gaussian_elimination.m at main · datamathur/EAS501-Codes Contribute to datamathur/EAS501-Codes development by creating an account on GitHub.github.com Gaussian Elimination Python | medium | 5,289 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
Code- EAS501-Codes/Gaussian Elimination.py at main · datamathur/EAS501-Codes Contribute to datamathur/EAS501-Codes development by creating an account on GitHub.github.com In this article, we explored a system of linear equations and the Gaussian Elimination method to find solutions. There are many | medium | 5,290 |
Mathematics, Data Science, Algebra, Numerical Analysis, University At Buffalo.
other numerical and analytical methods to solve a system of linear equations like the Gauss-Seidel technique, LU decomposition, and Successive Over Relaxation but the Gaussian method is the most commonly used in applications. | medium | 5,291 |
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interesting theories about the fall of the Roman Empire. According to one of them, the fall wasn’t due to short-sighted politics or war, but to the lead present in the cups that the members of the Roman elite used to drink, a substance that slowly poisoned them. Marshall McLuhan, on the other hand, | medium | 5,300 |
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control on the messiness of the world (Richard Hofstadter: “the paranoid mentality is far more coherent than the real world, since it leaves no room for mistakes, failures, or ambiguities”). How do they do so? By means of hypertrophy: by adding always more relations to their system, which becomes a | medium | 5,308 |
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“the airport-bookshop model of systems thinking which tends to involve a lot of graphs and urges to ‘shift your mindset’”. But by adding links and relationships one doesn’t necessarily reaches galaxy brain level. More likely they will just generate more confusion, more noise, more chaos. Chaos is a | medium | 5,311 |
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the world is complex, society is complex, etc. However, philosopher Cornelius Castoriadis points out that the idea of a totality that can be encompassed by the intellect is nothing but a phantom of speculative philosophy. What’s more: we do not actually need it in order to act in the world. Praxis, | medium | 5,314 |
Quantum Physics, Bible, Awakening, Spirituality.
Image by Dariusz Sankowski from Pixabay In college I chose the less popular major of chemistry pre-medicine over biology pre-medicine, in large part due to my attraction to math. I excelled in chemistry and became even more intrigued with the mathematical equations and scientific algorithms that | medium | 5,319 |
Quantum Physics, Bible, Awakening, Spirituality.
were available in chemistry to answer any question. Only, where my interest became my passion was in the science class which combined the study of chemistry and physics called physical chemistry, better known as quantum mechanics or quantum physics. It was here that everything changed. Unlocking | medium | 5,320 |
Quantum Physics, Bible, Awakening, Spirituality.
the principles taught in physical chemistry put into context every other discipline of science I had studied thus far. It was here that my interpretation of the universe and its intrinsic secrets began to unfold. While unaware at the time, the study of physical chemistry was creating a | medium | 5,321 |
Quantum Physics, Bible, Awakening, Spirituality.
life-changing shift in my consciousness and one that would transform my thought processes forever. Quantum physics became the transcendental bridge that linked my science and math mind to my spiritual mind. This revelation in scientific theory would propel me forward the rest of my life, in my | medium | 5,322 |
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