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Large Language Models, Machine Learning, Deep Learning, Data Science, Technology.
simplifying computational demands, LoRA accelerates the training and fine-tuning of large models for new tasks. Feasibility for Smaller Hardware: LoRA’s lower parameter count enables the fine-tuning of substantial models on less powerful hardware, like modest GPUs or CPUs. Scaling to Larger Models: | medium | 2,063 |
Large Language Models, Machine Learning, Deep Learning, Data Science, Technology.
LoRA facilitates the expansion of AI models without a corresponding increase in computational resources, making the management of growing model sizes more practical. In the context of LoRA, the concept of rank plays a pivotal role in determining the efficiency and effectiveness of the adaptation | medium | 2,064 |
Large Language Models, Machine Learning, Deep Learning, Data Science, Technology.
process. Remarkably, the paper highlights that the rank of the matrices A and B can be astonishingly low, sometimes as low as one. Although the LoRA paper predominantly showcases experiments within the realm of Natural Language Processing (NLP), the underlying approach of low-rank adaptation holds | medium | 2,065 |
Large Language Models, Machine Learning, Deep Learning, Data Science, Technology.
broad applicability and could be effectively employed in training various types of neural networks across different domains. Conclusion LoRA’s approach to decomposing ( Δ W ) into a product of lower rank matrices effectively balances the need to adapt large pre-trained models to new tasks while | medium | 2,066 |
Large Language Models, Machine Learning, Deep Learning, Data Science, Technology.
maintaining computational efficiency. The intrinsic rank concept is key to this balance, ensuring that the essence of the model’s learning capability is preserved with significantly fewer parameters. References: [1] Hu, Edward J., et al. “Lora: Low-rank adaptation of large language models.” arXiv | medium | 2,067 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
From System Programming to Web Development, Rust Has Got You Covered. Photo by Ilya Pavlov on Unsplash Are you looking for a programming language that offers the perfect combination of performance, safety, and simplicity? Look no further than Rust! Developed by Mozilla, Rust is a systems | medium | 2,069 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
programming language that’s gaining popularity for its unique features and ability to solve common programming issues. Rust is a versatile language that can be used for a variety of tasks, from web development to systems programming and beyond. Here are some of the use cases where Rust shines: | medium | 2,070 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
System Programming: Rust is designed for building fast and safe system-level software. It has minimal runtime, low-level control, and the ability to generate efficient native code. Rust’s ownership model ensures that the memory is managed safely, avoiding issues such as null pointers, dangling | medium | 2,071 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
pointers, or buffer overflows. Web Development: Rust is becoming a popular choice for building web applications, especially with the rise of WebAssembly. With frameworks like Rocket and Actix, Rust provides an excellent option for building high-performance and reliable server-side applications. | medium | 2,072 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
Game Development: With its low-level control and zero-cost abstractions, Rust is an excellent choice for developing games. Rust’s multi-threading capabilities make it easy to build complex games that require concurrent processing. Cryptography: Rust is a great choice for building cryptographic | medium | 2,073 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
libraries because of its security guarantees. Rust’s standard library has several built-in cryptographic primitives, and there are many community-built libraries available as well. CLI Tools: Rust is an excellent choice for building command-line interface (CLI) tools because it’s fast, portable, | medium | 2,074 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
and can run on any platform. Rust’s standard library includes support for parsing command-line arguments and handling input/output. Rust has a growing community, and it’s constantly improving. Its unique features and performance benefits make it a great choice for a wide range of applications. So, | medium | 2,075 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
why should you consider Rust for your next project? Rust’s safety guarantees ensure that your code is free from memory-related bugs and vulnerabilities, making it an excellent choice for critical systems. Rust’s performance benefits make it ideal for applications that require speed and efficiency. | medium | 2,076 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
And with its simplicity, Rust makes it easy to write maintainable and robust code. Whether you’re a seasoned developer or a beginner, Rust’s approachable syntax and tooling make it easy to learn and use. With a wealth of online resources and a supportive community, Rust is a great language to add | medium | 2,077 |
Rustlang, Systems Programming, Webassembly, Machine Learning, Open Source.
to your arsenal. In conclusion, Rust is an excellent choice for a wide range of applications, including system programming, web development, game development, cryptography, and CLI tools. Its unique features, performance benefits, and safety guarantees make it a great choice for critical systems. | medium | 2,078 |
Nervous System, Trauma, Recovery.
What is vagal tone? Vagal tone is the ability of the ventral vagus nerve to regulate the heart beat. It is referred to in medical literature as “cardiac vagal tone.” Vagal tone is measured by tracking heart-rate and breathing rate at the same time. The heart-rate speeds up a little when we breathe | medium | 2,080 |
Nervous System, Trauma, Recovery.
in, and slows down a little when we breathe out. The bigger the difference between inhalation heart-rate and exhalation heart-rate, the higher our vagal tone is. Higher vagal tone means that the body can return to a calm state quickly after a stressful experience ends. Higher vagal tone does not | medium | 2,081 |
Nervous System, Trauma, Recovery.
mean a person will be less activated by stressful events, but that they will recover more easily after safety is restored. People with low vagal tone do not recover as quickly after safety is restored. If you have low vagal tone, you may feel stuck in high or low activation for a long time after a | medium | 2,082 |
Nervous System, Trauma, Recovery.
stressful incident has ended, or even flip-flop between those two up and down states. There is a common misconception in the popular understanding of vagal tone which assumes that vagal activity and vagal tone are the same thing. However, having low vagal tone does not mean our vagus nerve is | medium | 2,083 |
Nervous System, Trauma, Recovery.
under-active. Staying stuck when you’re in a safe space is not necessarily due to faulty neuroception either. Your body may be getting all the safe signals from your environment and your vagus may be sending the correct signals to attempt to control your heart rate to get you back to feeling | medium | 2,084 |
Nervous System, Trauma, Recovery.
calm/safe. So if the vagus is nerve is functioning correctly, why do some people have low vagal tone? Because a fatty protective layer of insulation around the nerve (called myelination) is weak or almost non-existent. Without insulation, the electrical information in our nerves can’t always travel | medium | 2,085 |
Nervous System, Trauma, Recovery.
to its intended destination. Without insulation, the neural information from the vagus gets scattered around the body instead of going straight to the heart. Low vagal tone is due to low myelination. Low myelination resulting in low vagal tone is the most common physiological effect of | medium | 2,086 |
Nervous System, Trauma, Recovery.
developmental trauma or emotional neglect. Low vagal tone is associated with cardiovascular conditions, strokes, depression, diabetes, chronic fatigue syndrome, and inflammatory conditions. All human infants are born with low myelination. If development is not interrupted, a healthy myelin sheath | medium | 2,087 |
Nervous System, Trauma, Recovery.
is formed by age 2 or 3. When an adult caregiver is in safe/calm state, the child’s mirror neurons detect and replicate that state. The more frequently the child is in this mirrored ventral vagus state, the healthier their myelin sheath will be. Generally, industrialized nations do not allow | medium | 2,088 |
Nervous System, Trauma, Recovery.
sufficient parent-child bonding time for the development of healthy vagal tone. Low vagal tone is a “silent epidemic” that affects the majority of the population in industrialized countries. If human development requires time spent with an adult who is in a safe/calm state, what are we to do if we | medium | 2,089 |
Nervous System, Trauma, Recovery.
didn’t have access to that in our childhood? If you did not get to form this neural protection in childhood (like me), there is good news! Adults can build myelin (a process called remyelination) and improve their vagal tone through co-regulation just like babies do. Spending time in a mirrored | medium | 2,090 |
Nervous System, Trauma, Recovery.
ventral vagus state allows the body to form a stronger myelin sheath at any age. Depending on how much mirrored ventral vagus time you can spend each week, you may be able to see a major shift in vagal tone after a year or two. (Side note, physical tissue disorders and other genetic conditions such | medium | 2,091 |
Nervous System, Trauma, Recovery.
as EDS can make this process take longer.) There’s an old phrase in neuroscience that says “neurons that fire together wire together.” The myelination of the vagus similarly depends on the activation of the vagus nerve via safe connection with another mammalian nervous system. Stimulating the vagus | medium | 2,092 |
Nervous System, Trauma, Recovery.
nerve when we are alone can help us feel calm in the moment, but it will not change vagal tone over time. Our vagus nerve needs the safe/calm influence of human and/or animal nervous systems to change tone. Many experts are so focused on offering solutions that can be easily done alone that they | medium | 2,093 |
Nervous System, Trauma, Recovery.
neglect the bio-social aspects of vagus nerve tone. The myelination process is one way nature proves that we need each other. My personal experience with changing vagal tone 6 years ago, when I was diagnosed with complex PTSD, I had low vagal tone. It would take me days to recover from a stressful | medium | 2,094 |
Nervous System, Trauma, Recovery.
event. I would often have multiple panic attacks in the same day because I couldn’t calm down after the first one. I could not be left alone for any significant period of time, especially if it was unplanned. When I learned about vagal tone and myelination in my research, I started something I | medium | 2,095 |
Nervous System, Trauma, Recovery.
called my Radical Self Love Plan. It involved weekly scheduled activities that were highly likely to involve mirrored ventral vagus activation. If I were an animal person, I would have gotten a dog, but alas, I can’t deal with the sensory aspects of owning a pet. I chose group ventral vagus | medium | 2,096 |
Nervous System, Trauma, Recovery.
activities which I enjoyed and had easy access to: partner dance, free form dance, singing with others, free classes at the community yoga center, cuddle parties. I was open with others about the fact that I intended to use these practices to heal developmental trauma, and I found even strangers in | medium | 2,097 |
Nervous System, Trauma, Recovery.
these spaces were kind and encouraging. After 2 years of this (plus binaural beats, and other solo vagus stimulating activities), I noticed a clear difference in my vagal tone. Recovering from shutdowns and panic attacks took less time. For the first time in my life, I could feel my heart rate calm | medium | 2,098 |
Nervous System, Trauma, Recovery.
when I took the time to notice my safe surroundings. I began to be able to track my vagal tone by comparing my breath and heartbeat. There was a noticeable difference on the inhale and exhale that hadn’t existed before. A few months ago, I was in a car accident and got whiplash, a major trigger for | medium | 2,099 |
Nervous System, Trauma, Recovery.
me with history of car related PTSD. Less than 3 hours later I was in a ventral vagus calm/safe mode. I took a picture to commemorate that moment, because I was just amazed. I had expected to feel anxious for hours and then shutdown for a few days while my system adjusted to the shock. The shock | medium | 2,100 |
Nervous System, Trauma, Recovery.
was no less intense than before, but the way my body processed it was totally different. If your vagal tone is low like mine was, I’m writing this to hopefully give you some hope. Vagal tone is not fixed. It can change over time. There is no quick way to change vagal tone, but with consistent | medium | 2,101 |
Nervous System, Trauma, Recovery.
practice, healing this developmental trauma is possible. Trauma Geek, aka Janae Elisabeth, believes nervous system education will change the world. You can learn about polyvagal theory and the nervous system for free here: linktr.ee/TraumaGeek. This work is made possible by patron members at | medium | 2,102 |
Energy, Elon Musk, Electricity.
Review: |”Stop Watt,Stopwatt Energy Saving Device, Pro Power Saver Electricity Saving Device Elon Musk (6PCS)” There are affiliate links, primarily for information contained in books, whereas most of them gives a free peak inside, without making any purchases. If you do purchase from these links, a | medium | 2,104 |
Energy, Elon Musk, Electricity.
commission will be paid to my account Stop Watt, Stopwatt Energy Saving Device, and Pro Power Saver Electricity Saving Device are all products that claim to help you save energy on your electricity bill. They work by using a variety of methods, including: * Stabilizing the voltage and current in | medium | 2,105 |
Energy, Elon Musk, Electricity.
your home’s electrical system * Reducing the amount of electricity that is wasted by appliances * Protecting your appliances from power surges There is some evidence to suggest that these products can be effective in saving energy. However, it is important to note that there is no scientific | medium | 2,106 |
Energy, Elon Musk, Electricity.
consensus on their effectiveness. Some studies have shown that they can save a significant amount of energy, while others have shown that they have little or no effect. If you are considering using a Stop Watt, Stopwatt Energy Saving Device, or Pro Power Saver Electricity Saving Device, it is | medium | 2,107 |
Energy, Elon Musk, Electricity.
important to do your research and read reviews before you buy. You should also be aware that these products may not be effective for everyone. Here are some of the pros and cons of using these products: **Pros:** * Can save you money on your electricity bill * Can help to extend the life of your | medium | 2,108 |
Energy, Elon Musk, Electricity.
appliances * Can protect your appliances from power surges **Cons:** * May not be effective for everyone * May not save you as much money as you expect * May not be worth the cost Ultimately, the decision of whether or not to use a Stop Watt, Stopwatt Energy Saving Device, or Pro Power Saver | medium | 2,109 |
Energy, Elon Musk, Electricity.
Electricity Saving Device is up to you. If you are looking for a way to save money on your electricity bill, it may be worth trying one of these products. However, you should be aware that there is no guarantee that they will work for you. About this item ✅【Stopwatt Energy Saving Device】The power | medium | 2,110 |
Energy, Elon Musk, Electricity.
saver device itself have almost no consumption of energy, so you can use stopwatt energy saving device to save power which is consumed by the electric appliance such as refrigerator, air-conditioner, television and so on. To learn more: https://amzn.to/45LRFpZ ✔️【Premium energy-saving device】The | medium | 2,111 |
Energy, Elon Musk, Electricity.
function of Electricity Saving Box is to stabilize the voltage, balance the current, and surge protection in order to achieve power-saving effect. The power energy saver can also improve the use of electrical ✅【Safe and Reliable】The power saver device itself have almost no consumption of | medium | 2,112 |
Energy, Elon Musk, Electricity.
energy.Even 24 hours 365 days, all-weather running will not increase the burden of electricity bills.and stabilize the current according to the power consumption of the electrical appliances, so as to achieve a power saving effect. ✔️【Easy to use】The power saver plug is very easy to use, just plug | medium | 2,113 |
Energy, Elon Musk, Electricity.
the power saver into wall socket, no need special attention or maintenance.Help you to improve the use of power, avoid invalid waste of electricity.To purchase:https://amzn.to/45LRFpZ ✅【Multifunction Use】Use in homes, shops, restaurants, offices, and small factories. Compatible with air | medium | 2,114 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
This article is part of a series, you can find the others here: Introduction Rasterization Raymarching Conclusion To begin understanding raytracing, consider how illumination and visibility works in real life. A light source casts out an unfathomable number of rays (made up of photons). Some of | medium | 2,116 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
those rays will collide with a surface of one kind or another and in doing so they will be transformed by the surface in question. In such a collision light can be absorbed, transmitted or reflected. Absorbed light gives us colours, The collided surface absorbs some of the light and prevents it | medium | 2,117 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
from reflecting onward. For instance, if a surface absorbs all the red and green light from a ray, the remaining light will be blue. Transmission permits light to pass through a surface giving us transparency and translucency. Reflection is fairly self-explanatory, reflected light bounces from the | medium | 2,118 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
surface and continues on its merry way, like the blue light in our absorption example. We can model this process in a computer and use it to render scenes like we saw with rasterization. We call this process forward raytracing as it happens in the order you would expect in real life: A light source | medium | 2,119 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
in 3D space casts a ton of rays out. Some of those rays collide with surfaces. The rays then bounce off that surface, transformed by the interaction (absorption, transmission and reflection.) They continue along colliding and bouncing until they find their way back to the camera or drift off into | medium | 2,120 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
digital oblivion. Of all the rays cast, only a few will ever make it back to the camera, the ones that do, intersect with a pixel in the image plane and like we saw in rasterization, impart their colour information. The vast majority of the rays will continue bouncing for a set maximum number of | medium | 2,121 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
bounces or they will continue to a maximum distance, run out of puff and decided to head home before they get too shagged out to put the dinner on. Whatever the outcome, ultimately the overwhelming majority of the rays serve no purpose for rendering a scene and so occupy a lot of valuable | medium | 2,122 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
computational resources for no reason. Despite being more physically accurate, forward raytracing is the apple airpods of rendering techniques, it’s very expensive and only superficially more valuable. So how can we leverage the power of raytracing without heating our computers up to temperatures | medium | 2,123 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
not seen outside of Satan’s sauna? How about we do it backwards instead by using the cleverly named backward raytracing? Note: Many solutions and several of the examples below use a hybridised raytracing solution with aspects of both forward and backward raytracing, but backward raytracing remains | medium | 2,124 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
the core of the solution. At least to my understanding. In simple terms backward raytracing flips forward raytracing on its head. Instead of casting rays from a light source and letting some of them hit the camera, the camera instead casts a ray out through every pixel in its image plane and then | medium | 2,125 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
on into the scene. It then tracks any collisions and at the collision point it bounces away towards a light source, if the ray makes it to said light, we know the surface was illuminated, if it hits another surface we know it is in shadow. Note: The number of times a ray is allowed to bounce has a | medium | 2,126 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
strong impact on performance and the quality of the final image. I.E. The more bounces the better the fidelity but the more it costs. The number of bounces can be configured to find the right balance between cost and quality which Sebastian Lague does a great job of demonstrating here. Once a ray | medium | 2,127 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
finishes its journey the path of the ray is traced back to the camera and its eventual colour value is used to determine the colour of its intersecting pixel. Aristotle would be so pleased Since we only need to account for rays that actually make it back to the camera, backward raytracing | medium | 2,128 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
dramatically decreases the performance overhead required when compared to forward raytracing That said, it is still not a cheap process by any means. it gets very complex very quickly, and while raytracing has been used in pre-rendering for many years, finding a workable solution to real-time | medium | 2,129 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
raytracing has been a bit of a holy-grail for a long time, we’re only now seeing the first generations of commercial grade real-time raytracing and they are incredibly resource intensive. Gosh but it’s pretty though. Epic Games, NVIDIA, LMxLAB. and Disney Straight out of the box with raytracing, we | medium | 2,130 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
have a technique that is closely mimicking how lighting works in real life (even if the process is reversed the steps are mostly the same.) It stands to reason then that this helps make lighting behave more realistically, which it does. But this is where it gets really funky. What do you suppose | medium | 2,131 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
happens to the ray during all of those collisions and bounces? I mentioned before that the information the ray carries is transformed through absorption, transmission and reflection but what does this mean in practice? Once the ray has hit a surface and bounced away towards a light, it can continue | medium | 2,132 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
to hit things, transforming itself with each collision and illuminating surfaces as it goes. We can observe this behaviour in real life. If you hold a brightly coloured object to a piece of white paper, you will likely see that the object is lending its colour to the paper because the object | medium | 2,133 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
absorbed some of the colour and reflected what remains which is in turn reflected (bounced) from the paper This same process is happening all around us and is what helps make the real world look so rich and vibrant, no shadow is ever truly black, it instead takes on gentle pieces of illumination | medium | 2,134 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
from the world around it, surfaces that may seem totally rough are in fact subtly reflective and perfectly solid surfaces can be slightly translucent allowing some light to pass through it. If you find yourself in a gallery looking at a Rembrandt or a Monet really scrutinise the shading in the | medium | 2,135 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
shadows and highlights and you will see a rich tapestry of varying hues and shades which come together from afar to appear as something breathing, illuminated and alive. The Man with the Golden Helmet by Rembrandt. The Haystacks, End of Summer by Monet The rays of light bounce through the scene, | medium | 2,136 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
transforming and picking up information from each surface it bounces off and finally it hands that information over to its intersecting pixel which can then work out which colour it should be. The results on a rendered screen are incredible, static lifeless scenes suddenly appear rich with a life | medium | 2,137 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
and movement of their own. METRO Exodus NVIDIA RTX demos by Nikita Shilkin (https://www.artstation.com/artwork/vn1Bx), Metro Exodus by 4A Games, published by Deep Silver This technique gives us a lot of great tricks for “free.” Reflections for instance have always been a problem with rasterization | medium | 2,138 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
because there is no physical basis for how it should work when you’re firing rays from a vertex to a camera, instead it had to be faked only somewhat convincingly. Use of a reflection map shown in Satisfactory by Coffee Stain Studios. With raytracing if a ray bounces off a fully reflective surface, | medium | 2,139 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
the ray knows that it can retain all (or most) of its colour values, unchanged and will carry that off to the camera but now coming from a new point in space and, boom, reflections. Battlefield V by DICE and EA Games If we build another simplified JavaScript function for backward raytracing we | medium | 2,140 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
could explain it thus: (For the sake of simplicity I’m going to assume there is only one light.) const camera = new Camera(0, 0, 0); const sphere = new Sphere(0, 0, 1); const light = new Light(1, 1, 0); const scene = new Scene(camera, [sphere], light); const { imagePlane } = camera; const | medium | 2,141 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
maxRayBounces = 3; const maxRayDistance = 1000; imagePlane.pixels.forEach(pixel => { const ray = castRay(pixel, camera); while (ray.distance < maxRayDistance && ray.bounces < maxRayBounces) { advance(ray); scene.geometry.forEach(shape => { const collision = collision(shape, ray); if (collision) { | medium | 2,142 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
ray.direction = calculateDirection(collision, light); ray.collisions.push(collision); ray.bounces += 1; return; } }); ray.distance += 1; } pixel.color = calculateColor(ray.collisions); }); Again this is a heavily simplified explanation rather than actual working code. It illustrates that: We | medium | 2,143 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
iterate through every pixel in the image plane, casting a ray between the camera and it. While we are within our maximum allowed distance and bounces we advance the ray forward and check all of the objects in the scene to see if the ray collides with any of them. If we do collide we calculate a new | medium | 2,144 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
direction from the collision point to the light (you can also see this as firing a new ray), we record the collision so we can step through it later and we increase the number of bounces taken thus far. If there is no collision we record the increase in distance traveled and advance the ray again. | medium | 2,145 |
Art, Rendering, Ray Tracing, Creative Coding, Graphics.
Once we have exhausted either our max distance or bounces we set the pixel’s colour by tracing back along the path of our ray from the light (or nothingness) and working out how the ray’s colour transforms at each collision. You can see how this could get very complex very quickly, with more | medium | 2,146 |
Standardize, Probability Density, Random Variable, Python, Statistics.
1 Theory Given a random variable (r.v.) X with support Xₛ, to “standardize” means to define a new r.v. Z with support Zₛ through a transformation φ such that (1) where μₓ and σₓ are the mean and standard deviation of X. The φ transformation implies the following substitution Σ (2) Since the | medium | 2,148 |
Standardize, Probability Density, Random Variable, Python, Statistics.
transformation φ is monotonically incresasing (3) we can say that (4) where F(•) is the Cumulative Probability Function (CDF). Therefore (5) Beware: equation (5) does not mean that they are the same function, it means that the X CDF as a function of x yields the same result of the Z CDF as a | medium | 2,149 |
Standardize, Probability Density, Random Variable, Python, Statistics.
function of z = φ(x). (5a) Given the equation (5), and substituting x we can write (6) Differentiating we can get to the definition of f(•) the Probability Density Function (PDF) of Z (7) The new r.v. Z will have mean μ=0 and variance σ²=1 because (8) (9) The asymmetry γ₁ and the kurtosis γ₂ of Z | medium | 2,150 |
Standardize, Probability Density, Random Variable, Python, Statistics.
will be the same of X, because (10) (11) 2 Python implementation First of all, let us define the functions to compute mean, standard deviation, asymmetry and excess kurtosis (which is γ₂-3, being 3 the kurtosis of a normally distributed r.v.) and the function of the transformation φ Now, let’s take | medium | 2,151 |
Standardize, Probability Density, Random Variable, Python, Statistics.
some examples: a symmetric and mesokurtic normally distributed r.v. with mean 20 and standard deviation 2 and its standardized form ***** normal distribution Standardized mean = 20.0 -0.0 sigma = 2.0 1.0 asymmetry = -0.0 -0.0 kurtosis* = 0.0 0.0 A right-asymmetric and leptokurtic Gamma distributed | medium | 2,152 |
Standardize, Probability Density, Random Variable, Python, Statistics.
r.v. with mean 0.6 and standard deviation 0.4 and its standardized transformation ***** normal distribution Standardized mean = 20.0 -0.0 sigma = 2.0 1.0 asymmetry = -0.0 -0.0 kurtosis* = 0.0 0.0 ***** gamma distribution Standardized mean = 0.6 0.0 sigma = 0.4 1.0 asymmetry = 1.333333333 | medium | 2,153 |
Standardize, Probability Density, Random Variable, Python, Statistics.
1.333333333 kurtosis* = 2.6666667 2.6666667 A left-asymmetric and platykurtic Beta distributed r.v. with mean 0.54 and standard deviation 0.27 and its standardized transformation ***** normal distribution Standardized mean = 20.0 -0.0 sigma = 2.0 1.0 asymmetry = -0.0 -0.0 kurtosis* = 0.0 0.0 ***** | medium | 2,154 |
Standardize, Probability Density, Random Variable, Python, Statistics.
gamma distribution Standardized mean = 0.6 0.0 sigma = 0.4 1.0 asymmetry = 1.333333333 1.333333333 kurtosis* = 2.6666667 2.6666667 ***** beta distribution Standardized mean = 0.53999999999 0.0 sigma = 0.27 1.0 asymmetry = -0.13445378 -0.13445378 kurtosis* = -1.08748705 -1.08748705 | medium | 2,155 |
Control Theory, Linear System.
This post is an attempt to intuitively explain the controllability matrix rank condition of a linear system. Consider the following linear system: State equation of the linear system For the sake of simplicity, we will deal with single input systems. Thus, the matrix B is a column vector. The | medium | 2,156 |
Control Theory, Linear System.
controllability matrix of the system is calculated as follows: Controllability matrix P For controllability, rank(P)=n, where n is the number of state variables. If rank(P)<n, the system is uncontrollable. If P is rank deficient, then it means that the columns of P are linearly dependent. If B | medium | 2,157 |
Control Theory, Linear System.
happens to be an eigenvector of A, we have: Any two columns in P are linearly dependent, rendering the system uncontrollable Let’s try to visualize this using the following example: System matrices for this example Clearly, B is an eigenvector of A, thus the controllability matrix is rank deficient | medium | 2,158 |
Control Theory, Linear System.
and the system is uncontrollable i.e. there is no control input u(t) that can drive the states to zero in finite time from all given initial states. For example, consider the following initial state vector: Observe the system response for two different constant control inputs u1=116 and u2=-116: x1 | medium | 2,159 |
Control Theory, Linear System.
represents the first component of the state vector x2, the second component of the state vector, is not affected by control input Observe that despite changing the control input, no change was discernible in the evolution of the second state variable wrt time. Thus, the second state x2 ( Think of | medium | 2,160 |
Control Theory, Linear System.
x1 and x2 as the x and y co-ordinates of the state vector) is invariant to the control input u(t). This is because an eigenvector is incapable of affecting the system dynamics along the normal to it. This can be understood by looking at the solution of the state differential equation described | medium | 2,161 |
Control Theory, Linear System.
above: State equation solution The first component is independent of control input, hence the influence of u can be felt only in the integral term. Note that since B is an eigenvector of A, it is also an eigenvector to its matrix exponential. Hence, we have Hence, the effect of the control input is | medium | 2,162 |
Control Theory, Linear System.
only along the eigenvector. The state dynamics orthogonal to it is independent of the control input, and hence, the system is not controllable. Another way to understand this concept is to relate it to reachability. We know that if an LTI system is controllable to the origin, all states are | medium | 2,163 |
Control Theory, Linear System.
reachable from origin with an appropriate control input u(t). In this particular example, if we start at the origin, both the control input (Bu) as well as the state derivative (Ax) point along the x axis. Thus, our motion can never take off from the x axis to affect our second state y. Therefore, | medium | 2,164 |
Control Theory, Linear System.
any state with a non-zero y co-ordinate is not reachable from the origin if B is along the x axis, which is an eigenvector. If we have multiple columns parallel to different eigenvectors, the corresponding control input is ineffective along any direction orthogonal to its corresponding eigenvector. | medium | 2,165 |
Control Theory, Linear System.
However, is some other control input is able to control the system dynamics in that direction, the system could still be controllable. I don’t claim that if the system is uncontrollable, B has to be along an eigenvector. However, if B is along an eigenvector, the controllability matrix has only one | medium | 2,166 |
Control Theory, Linear System.
linearly independent column, and the control has influence only along one direction. Let us revisit the solution of the state equation. Note that by the Cayley-Hamilton theorem, The coefficients are obtained from the characteristic equation of the matrix. Since A^n and onwards can be expressed in | medium | 2,167 |
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