I have calculated data from a computer simulation for a wireless network as shown below, here P is an algorithm tuning parameter, 'C' and 'D' are % decrease in collisions and % decrease in delay respectively. Frame lenght is 20%. P   C   D    Frame Lenght 0.3 184 249   20% 0.4 137 241   20%  0.5 168 230   20% 0.6 221 210   20% 0.7 148 219   20% 0.8 80  219   20%  I need to derive a cost function to provide an output (value of 'P') based on 'C', 'D' and Frame Length. The value of 'P' should be such that max % decrease in 'C' and 'D' is obtained, but 'C' has priority.  In this case 'P'=0.6 should be the output of the cost function. I also need to perform simulations by varying Frame length like 40% and 80% to obtain optimum value of 'P' from cost function that provides high %decrease in 'C' . Therefore is it also possible to use Frame length in cost function. Can any one tell how can I derive cost function for 'P'.
Goal: I have a function $f(x,y)=z$ (two variables for illustration only) which I know almost nothing about--it has a compact domain which I can determine, it is non-negative, and bounded above.  My goal is to find the maximum value $f(x,y)$ takes on its domain.  I can query the function and I know the values at a few points: $f(x_1, y_1)=z_1, f(x_2, y_2)=z_2,..., f(x_n, y_n)=z_n$.  Optimization via Linear Regression:  I have used linear regression (and related methods like the lasso) to model functions like this in the past; this is the method that I am very comfortable with.  Typically, I might try to find a polynomial (of degree $k$) which relatively accurately models the data. The end result would be to say that:   $$f(x,y)\approx \beta_1 + \beta_2 x + \beta_3 y + \beta_4 x^2 + \beta_5 xy + \beta_6 y^2 + ... + \beta_{m}y^{k}.$$  Determining the coefficients is done by solving the .  The "best degree" for the polynomial might be determined using $k$-fold cross validation.  Once the "best" polynomial has been determined, one can choose any of a number of methods to optimize it (Newton's method, gradient descent, hill climbing, etc).  Then, if all goes well, the optimal value on the polynomial corresponds to an optimal value of $f(x,y)$.  Bayesian Optimization:  Recently, I was describing this technique (linear regression) to a colleague who asked: "What is the difference between that and Bayesian Optimization?"  I've been attempting to answer this question for the last couple days, and so far I don't have anything really conclusive.  After reading , trying to read  and getting stuck reading  I think I understand the following about BO:   Start with a prior (a guess as to what family of functions the black box belongs to... or at least some family of functions which you believe might be able to describe $f$ well).  It looks using a  as a prior is popular. Specify whatever is needed to build your prior (for GP this is a covariance function). As you go along, you maintain two evolving functions, a "mean" function $m(x,y)$ and a "confidence interval" function $c(x,y)$.  Basically, you can predict the value at any point by evaluating $m(x,y)$ and you are confident that $f(x,y)\in [m(x,y)-c(x,y), m(x,y)+c(x,y)]$. Bayesian optimization provides a systematic way of updating/exploiting the model by alternatively selecting points where the confidence interval is wide (you are very uncertain about the value) or at optimal points on the model $m(x,y)$.   My Question: Is it fair/correct to say that the major difference between Bayesian Optimization and simply optimizing the linear regression model is that Bayesian Optimization provides the systematic way of sampling with the goal of optimizing the model and reducing uncertainty whereas the linear regression model ignores the idea of uncertainty and doesn't explicitly provide a means of updating the model?  Said another way, is the linear regression model an example of what might be a "mean" function $m(x,y)$ in Bayesian Optimization?  If I stick with linear regression, specify how to compute the uncertainty, and provide a means of updating/improving the model (exploit/explore) am I just doing Bayesian Optimization (without calling it that)?  Background: My background is in pure mathematics (graph theory) but my new job has me working with a lot of data (so I'm self learning statistics).  With that in mind, my statistics vocabulary is somewhat limited (so I appreciate answers that either avoid technical terminology or briefly explain any technical terminology used)--for example, Monday was the first time I heard/read the word "prior" used in a stats sense and I'm still not 100% sure I understand that correctly.
Can I turn off macOS Catalina's internal display without close the lid while using an external monitor? Show that $R/I\otimes_R R/J\cong R/(I+J)$ Let $R$ be a commutative ring with identity and $I$ and $J$ be ideals in $R$.     Show that $R/I\otimes_R R/J\cong R/(I+J)$ as $R$-modules.   This is what I tried.  Define a map $f:R/I\times R/J\to R/(I+J)$ as $f(\bar r, \bar s)=rs+(I+J)$. This is a well-defined bilinear map.  By the universal property of tensor products, we get an $R$-linear map $F:R/I\otimes_R R/J\to R/(I+J)$ which sends $\bar r\otimes \bar s$ to $rs+(I+J)$.  This map is surjective.  So all we need to show is that the kernel of $F$ is $0$.  But I am unable to do this. How do I install Citrix receiver? Has anyone managed to get the Citrix receiver client (icaclient) working in Ubuntu? If you are hurrying to reply, System → Administration → StartUp Disk Creator -- no, that's not what I'm talking about.   I want to try Ubuntu 11.04's Unity without touching my existing Ubuntu install.  To do this, I need to install the nVidia drivers first (sigh).  To do this, I need changes to persist a reboot.  To do this, I need to really install Ubuntu on a USB key.  How do you do that?    What I tried   I tried to make a USB key from , then boot from it, then choose "Install Ubuntu." The installer refused to install to the installation media itself. I tried, from my installed copy of Ubuntu:  sudo kvm /dev/sdb --cdrom .cache/testdrive/iso/ubuntu_natty-desktop-i386.iso   ...but the installer didn't detect the disk properly. Can I bind a (large) block of addresses to an interface? I know that the ip tool lets you bind multiple addresses to an interface (eg, ). Right now, though, I'm trying to build something on top of IPv6, and it would be really useful to have an entire block of addresses (say, a /64) available, so that programs could pick any address from the range and bind to that. Needless to say, attaching every IP from this range to an interface would take a while.  Does Linux support binding a whole block of addresses to an interface? Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $ File has different name in terminal and Finder I've just downloaded the driver for my preferred trackball from Logitech. In the Finder, the Installer's filename is Logitech Control Center Installer.app. In the Terminal, ls shows LCC Installer.app. How did this happen? Calling Table with custom iterator Vertically align text next to an image? The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$ (NBHM_2006_PhD Screening Test_Analysis)     The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the   point on the curve farthest from the line $x = y$   how should i solve this problem Can we get rid of the threshold of 15 chars for comments? I understand that there is an upper limit. But why do you have to type at least 15 characters to post a comment? It is easy to work around it anyway (like adding dots or other useless things). Boot Error - ACPI PCC Probe Failed ACPI PCC Probe Failed is the error that pops up instead of the default GRUB loader after I updated my kernel to 3.19.  I searched around and couldn't find much information except that it is a known error and it is registered in kernel.org's launchpad . My question is, is anyone else facing it?, what are the impacts of this? And if only an update in the future can fix it, any suggestions or safety tips till then? I'm getting this error:  ! Undefined control sequence. l.64 ...p(- \frac{E_j - E_i}{k_B T}) &amp; \quad \text                                                   {:  E_j &gt; E_i}\\ ?  ! Undefined control sequence. l.65     1 &amp; \quad \text                         {:  E_j&lt;E_i}\\ ?    which is from this:  \begin{equation}\label{eq:hop} w_{i,j}=w_0 exp(-2 \gamma R_{i,j})\left\{    \begin{array}{l l}     exp(- \frac{E_j - E_i}{k_B T}) &amp; \quad \text{:  E_j &gt; E_i}\\     1 &amp; \quad \text{:  E_j&lt;E_i}\\   \end{array} \right. \end{equation}   But I don't understand why I get the error. This is my header:  \documentclass[11pt]{iopart} %\newcommand{\gguide}{{\it Preparing graphics for IOP journals}} %Uncomment next line if AMS fonts required \usepackage{iopams} \usepackage{graphicx, wrapfig} \usepackage{fixltx2e} \graphicspath{{images/}} \DeclareGraphicsExtensions{.pdf,.png,.jpg,.gif} \makeatletter \g@addto@macro\@floatboxreset\centering %\def\@cite#1{\textsuperscript{[#1]}} \makeatother   And my class file is the standard IoP one from   Can anyone help me get read of this error? Pre & post increment operator behavior in C, C++, Java, & C# Incremental (delta) backup of a encrypted data I have several files that I need to keep encrypted on my home hard disk.  I currently use a TrueCrypt file container, currently around 100MB in size. I need to backup this file daily, and I'm looking for an incremental backup solution.  I read that on encrypted volumes even a small change in a file would change the whole volume and prevent the use of an efficient incremental backup solution. Any ideas how to backup in this case in an efficient way which would allow incremental solution?  I currently use TrueCrypt as my backup solution and CrashPlan as my backup utility, but I'm willing to experiment with alternatives. Order of operations using Object Initializer Syntax Does the order in which I set properties using the object initializer syntax get executed in the exact same order?  For instance if I do this:  var s = new Person { FirstName = "Micah",                      LastName = "Martin",                      IsLoaded = true                    }   will each property get set in the same order? Can I set max_retries for requests.request? Why does Gandhi want to nuke me? Different civilizations obviously have different priorities. For example, Gandhi always seems to rush nuke technology and the Mongols will most likely start invading and spread like mad. Are these variables public knowledge? Can you tell by who you are playing what strategy they will adopt during the game? Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$ I want to prove this equation, $$ \binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a} $$ I thought of proving this equation by prove that you are using different ways to count the same set of balls and get the same result. But I'm stuck. Help me please.   (Presumptive) Source: Theoretical Exercise 1.14(a), P19, A First Course in Pr, 8th Ed, by S Ross. Prove that a group is infinite and nonabelian. I'm trying to solve a question of Hungerford's Algebra:     Show that the group defined by generators $a,b$ and relations   $a^2=b^3=e$ is infinite and nonabelian.   I guess a good approach would be to find an infinite and nonabelian group with two generators satisfying the same conditions and let Van Dyck do all the work. I've been thinking about the group of permutations $S_{\mathbb N}$, some groups of matrices, but I couldn't get it. Could anyone give me a hint? Thanks in advance!