Topic is Machine Learning Supervise Learning Regression Clas

Topic is Machine Learning Supervise Learning: Regression (Classification or Logistic Regress: Analysing the Bias and Variences)

Analysis of Error: Hypothesis Complexity and Sample Size. Assume that the input variable x is uniformly distributed in the interval [-1, 1] and the target function is known f(x) = x^2. Of course, the target function is normally unknown to us and it is assumed to be known here so that we can carry out the analytical as well as numerical analysis on error, i.e., bias and variance. The data set consists of 2 points {x_1, x_2}. Thus, the full data set is D = {(x_1, x^2_1), (x_2, x^2_2)}. The learning algorithm of least squares fit will return the best function, g^D(x), out of the function space described by our hypothesis below that best fits the data set D. Hypothesis 1 - Constant Functions: H_0: h(x) = a Hypothesis 2 - Linear Functions: H_1: h(x) = ax + b For each of the hypotheses H_0 and H_1, Derive the analytical expression for the average function bar g(x) = (g^D(x))D where (...)D stands for average over data sets. In general, g^D(x) is a function of x, x_1, and x_2 for a two-point data sets. Note that we don\'t need to state explicitly g^D(x) also uses y_1 and y_2 since y_1 and y_2 themselves are functions of x_1 and x_2 respectively. Thus bar g(x) = (g^D(x))_D = integral^x_1=1_x_1 = -1 integral^x_2=1_x_2 = -1 g(x, x_l, x_2) p(x_1) p(x_2) dx_l dx_2 where p(x_1) = 1/2 and p(x_2) = 1/2 (i.e., uniform distribution over [-1, 1]). Calculate Bias. Variance, and MSE. Recall each term is defined as Bias^2 = ([f(x) - bar g(x)]^2) = integral^x=1_x=-1 [f(x) - bar g(x)]^2 p(x) dx Var = (([g^D(x) - bar g(x)]^2)D)_x MSE = (([f(x) - g^D(x)]^2)D)_x Of course, the bias-variance decomposition shows MSE = Bias^2 + Var. Based on your results, discuss which hypothesis works out better for the two-point data set. Run numerical simulation (using whatever package that you are comfortable with, e.g., R, Python) on three-point, four-point, and five-point data sets. Show how MSE from each hypothesis behaves for these different data sets.

Solution

A scripting or script language could be a programing language that supports scripts, programs written for a special run-time setting that alter the execution of tasks that would instead be dead one-by-one by a person\'s operator.[1] Scripting languages area unit usually taken (rather than compiled). Primitives area unit typically the elementary tasks or API calls, and also the language permits them to be combined into additional complicated programs. Environments that may be machine-controlled through scripting embody package applications, websites inside an internet browser, the shells of operational systems (OS), embedded systems, similarly as various games. A scripting language will be viewed as a domain-specific language for a selected environment; within the case of scripting associate application, this is often additionally called associate extension language. Scripting languages are generally mentioned as terribly high-level programming languages, as they operate at a high level of abstraction, or as management languages, significantly for computer programme languages on mainframes.

The term \"scripting language\" is additionally used loosely to ask dynamic high-level all-purpose languages, like Perl,[2] Tcl, and Python,[3] with the term \"script\" usually used for tiny programs (up to a number of thousand lines of code) in such languages, or in domain-specific languages like the text-processing languages sed and AWK. a number of these languages were originally developed to be used inside a selected setting, and later developed into moveable domain-specific or all-purpose languages. Conversely, several all-purpose languages have dialects that area unit used as scripting languages. this text discusses scripting languages within the slender sense of languages for a selected setting.

The spectrum of scripting languages ranges from terribly little and extremely domain-specific languages to all-purpose programming languages used for scripting. normal samples of scripting languages for specific environments include: Bash, for the {unix|UNIX|UNIX system|UNIX operational system|operating system|OS} or Unix-like operating systems; ECMAScript (JavaScript), for net browsers; and Visual Basic for Applications, for Microsoft workplace applications. Lua could be a language designed associated wide used as an extension language. Python could be a all-purpose language that\'s additionally usually used as associate extension language, whereas ECMAScript continues to be primarily a scripting language for net browsers, however is additionally used as a all-purpose language. The Emacs Lisp non-standard speech of Lisp (for the Emacs editor) and also the Visual Basic for Applications non-standard speech of Visual Basic area unit samples of scripting language dialects of all-purpose languages. Some game systems, notably the Second Life virtual world and also the Trainz franchise of Railroad simulators are extensively extended in practicality by scripting extensions. In alternative games like Wesnoth, the range of actual games vie by players area unit scripts written by alternative users.