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Questions tagged [extrapolation]

For question on extrapolation, the process of estimating, beyond the original observation interval.

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Natural Cubic Spline beyond boundary guarantees constant slope?

my understanding is that for if I have five points, I can interpolate a natural cubic spline out of any five points with increasing x, because degree of freedom is five. But when I try the python ...
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Propagating Uncertainties on Interpolated Data

I have a data set of 2000 $[x, F(x), \delta F(x)]$ triples, where $x$ is exact and $F$ is a measured value with an uncertainty $\delta F$. I can interpolate/fit the function however needed, and this ...
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Is filling in grid corners extrapolation or interpolation?

Let's say I measure temperature at locations on a regular 3x5 grid but I am missing data at the 4 grid corners. If I use the available measurements to estimate the grid corner values, am I performing ...
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Error in Pareto extrapolation of power function

How can we derive error in extrapolation if one tries to extrapolate a power function with Pareto distribution(which in itself is a power function) below a certain threshold?
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Analytically get amount of points sampled from a polygon extrapolated into another polygon

I am sampling points (x,y) uniformly from a polygon $P_1$ and extrapolating them according to $$ \begin{cases} x_{ex} = x + s\cdot \cos (h) \\ y_{ex} = y + s\cdot \sin (h) \end{cases} $$ where $s$ is ...
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Numerical Analysis - $n$-sided polygon tangential

i need help with this question..I'm not so sure how to go about the arguments. Any help would be appreciated. Consider a regular $n$-sided polygon tangential to and enclosing the unit circle to ...
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polynomial approximation around the point

I'm trying to figure out how to approximate a function by polynomial around the point. I'm aware of Newton's series and it is almost what I need, but it uses either forward or backward differences, ...
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How can I derive the dense output of ode45?

I'm currently looking at the implementation of the Dormand-Prince 5(4) Runge-Kutta algorithm (also known as Dopr5, or ode45) in ...
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Romberg's algorithm for integration

what does the row and the column in Romberg's algorithm for integration indicate? I assume that the column represents the integration value with smaller intervals considered for calculating the ...
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87 views

How to find a formula relating three values?

I haven't studied maths since high-school (20 years ago) and would like to find a formula to relate these values: ...
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How to extrapolate daily values from a set of 30-day stats

I use a system (Slack) which provides analytics on user activity - namely, messages sent within a time window of the last 30 days. These 30-day totals are recalculated every 24h. I have these daily-...
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34 views

Extrapolation of points by a function

I have got a positive series convergent series $w_j$, which is convergent. I can calculate a finite amount of terms $\sum_{j=0}^{j^*}w_j$ and would be interested in the estimation of the sum of the ...
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Please explain how Richardson Extrapolation is used in this example

I am having a hard time understanding what on earth Richardson Extrapolation is trying to do. Consider the example of approximating $\pi$ by inscribing regular polygons in a unit circle. The ...
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167 views

Norm of laplacian in dual space $H^{-2}$

how we can prove the estimate $\|\Delta u\|_{H^{-2}} \leq C \|u\|_{L^2(\Omega)}$, where $\Omega$ is a bounded open set of $\mathbb{R}^n$, and $H^{-2}$ is the dual space of $H^{2}\cap H_0^1$. Can we ...
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Trying to use Taylor series to find a formula for a Richardson extrapolation of order 6

I am trying to use Taylor series to find the formula for the Richardson extrapolation based on $f'(x)$ of order $O(h^6)$. I know the formula should have the form of: $$\Delta_1(h) - 20\Delta_1(h/2) + ...
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When extrapolating for projections, how do you know which function-form to use?

When you have a set of raw data, there are many models to extrapolate a projection. Linear, Quadratic, Logarithmic, Etc. How does one know which to use? I'm assuming its by the nature of the data ...
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120 views

Improving accuracy of f''(x) by Richardson extrapolation

I have been given the following Question, I literally have no Idea where to start with it, I understand that the original expression comes from central difference approximation, that makes sense. I ...
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270 views

Numerical Differentiation- Central Difference approximation & Richardson extrapolation

I have been told to derive "the central difference approximation for f''(x) accurate to O(h^4) by applying Richardson extrapolation to the central difference approximation of O(h^2). I have done this ...
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When to extrapolate?

I know extrapolating is probably dependent on the question being asked. But are there some guidelines on when extrapolation is "okay"? If you can't be as certain that a set of data will maintain a ...
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Extrapolate a sum using partial sums at powers of two

In an online textbook for MIT OCW 18.013a, Calculus with Applications, the author uses residue calculus to derive the well-known formula $$\sum_{n>0} n^{-2} = \frac{\pi^2}{6}$$ (See Some Special ...
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236 views

Application of Richardson Extrapolation

Apply Richardson extrapolation to $ S_{2h} = \frac{2h}{3}[f(x_0) + 4f(x_2) + f(x_4)]$ $S_h =\frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)] $ to derive a Newton–Cotes formula based on the ...
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extrapolation to $\infty $ without knowing the exponent

I obtained a curve numerically (see below). Apparently, the curve will converge to some value at $\infty $. How can I extract this limit value? The problem is that I do not know how this curve ...
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How to extrapolate smoothly, based on discrete values

I am trying to figure out how to value a property (piece of land) based on the original purchase price, and shifted proportional to irregular government valuations. This is for the purpose of a legal ...
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770 views

Finding correct step-size for the Euler method

I don't understand how to find the correct step-size $h$ for the Euler method. My script says the following: One method consists in computing the numerical solution for an arbitrary $h$ and then $...
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371 views

Linear Interpolation/Extrapolation

I have the following grid of data. The first row represents a value called $m$ that ranges between 0.8 and 1.2, and the first column represents days that can be anywhere between 0 and $\infty$ (but ...
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419 views

Extrapolation methods for ODE

While preparing for an exam I revisited extrapolation methods for ODEs and I noticed that I don't understand it properly. Although I have an exercise-related question, there is a soft question: why do ...
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9k views

1e+0.8= What? What does E mean? [duplicate]

Hello I came across a math equation and I was wondering what did the "e" stand for? the equation is : y=47931x-1E+0.8 Can someone please help me by showing what the E stands for and the answer for my ...
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How to test hypothesis where population size is unknown?

Assume that somebody claims that driving a car or being an airplane pilot is just like riding a bike and anybody can do it with just a little practice and therefore it is unnecessary to have a license....
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extrapolate terms of sequence, knowing sums

I'm not familiar with filters etc, so I'm asking here if anyone knows an existing method to do the following. I have a sequence of observations $S_n$ and I know that they satisfy $S_n=\sum_{i=1}^m a(n,...
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173 views

Composite rule of Newton cotes? Simpson + Trapzoid

What is the best combination of Trapezoidal rule and Simpson's rule to solve this problem? I used: Trapz from 1 to 2 1/3 Simpson's from 2 to 4.5 Trapz from 4.5 to 6 3/8 Simpson's from 6 to 9 1/3 ...
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279 views

Proof that using Richardson extrapolation once yields Simpson's rule

Given that $\int_{a}^{b}f(x)\,dx \approx\frac{h}{2}\left[f(b) + f(a) \right]$, it must be shown that $\int_{a}^{b}f(x) \, dx \approx \frac{h}{6}\left[f(b) + 4f\left[\frac{a+b}{2} \right ] + f(a) \...
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126 views

Generalized application of Young's inequality

Let $a,b\geq 0$ and let $(p,q)$ be conjugate exponents, which means $p,q>0$ and $p^{-1}+q^{-1}=1$. Then, we have the so-called Young's inequality: $$ab\le \frac{a^{p}}{p}+\frac{b^{q}}{q}$$ Now, ...
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496 views

Non-linear Extrapolation formula

Given a set of at least 180 data-points (dates as X and associated non-negative values as Y) - how would one go about predicting/forecasting future data-points? I'm not even sure whether what I'm ...
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759 views

How to extrapolate data using the second order rate equation?

I would like to model the second order rate equation using the following criteria: \begin{align} A_\text{initial}&=8\,500\,000,\\ A_\text{final} &=1\,200\,000,\\ t &=38. \...
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Name of the field of study that details extrapolation of a series based on subset sample data

Apologies if my notation and/or terminology is way off - I'm not well versed in mathematics. I'm looking for the name of the field of mathematics that might help me solve my problem. Here's my problem:...
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45 views

Extrapolation of $c^{-3/2}$. Optimal data points?

I have a computer code that does some complicated calculations and results in a real number. At the end I had to perform a infinite sum, $$ S=\sum_i^\infty x_i \;, $$ which I approximate by a finite ...
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58 views

Extrapolating the sequence of derivatives of an analytic function

Suppose that $f(x)$ is an analytic complex or real function on the real domain and we only know $f(x_0)$ and the first $n$ derivatives of $f(x)$ at $x=x_0$. Is there an algorithm whose accuracy ...
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Box-Jenkins and related extrapolation methods

I have been reading ARMA MODELS AND THE BOX JENKINS METHODOLOGY by S. MAKRIDAKIS AND M. HISON 95/45/TM . I seem to be just a little stuck on the idea of using "trends". I think of a "trend" as a $y$ ...
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Find formula with Richardson Extrapolation based on centered difference formula

I'm preparing for my exams next week, and I'm making exercises as a preparation. Now, I'm asked to derive the following formula using Richardson Extrapolation based on the centered difference formula: ...
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Richardson extrapolation for second derivative

So we know what the Richardson Extrapolation for a first derivative looks like using a recursive formula like this: $$ D_{m\Delta x}^{1}=\frac{f(x+m\Delta x)-f(x-m\Delta x)}{2m\Delta x}$$ $$ D_{m\...
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Most accurate finite difference result for first derivative [closed]

I got a problem in my assignment: obtain the most accurate finite difference results possible for the first derivative of f (x) = exp(cos(x)) at x=1, h = 0.5, 0.25, 0.125,...2^{16}. I have to do this ...
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426 views

Taylor series and extrapolation 2

Hey guys, I've finished the first three parts, but I have no idea how to approach part d and part e. Any hints would help! Thanks! Update: I've finished the first three parts. For part d, I found the ...
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Finding a curve given only its basic form and its tangent line

The basic form is $f(x)=k\sqrt{x}$ and the tangent line is $4x+36$. I've spent over half and hour with wolfram alpha and my notes on this problem: I've tried $(k\sqrt{x})'=4x+36$ and got $k=8\sqrt{x}(...
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Estimating landing position for a slowly falling object using latitude, longitude, and altitude.

I have a weather balloon project, in which I intend to use GPS to locate the payload when it finally comes down again. I will make the computer send coordinates to a server every minute or so, as ...
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Smoothest function which passes through given points?

I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function ...
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Estimating the value of a variable based on the existing variable?

Need your help with a math problem I having trouble to solve. Let's say when we buy an item we need to pay 100 and the tax would 10 percent. So my buying cost becomes 110. And when I go to sell it ...
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101 views

Extrapolate data points from a series of averages

Given a series of data points: $$\begin{array}{c|c|c|c|c|c|c|c|} & \text{Monday} & \text{Tuesday} & \text{Wednesday} & \text{Thursday} & \text{Friday} & \text{Saturday} &...
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706 views

Richardson extrapolation: how to derive an approximation for $f'(x)$ with Euler backward/forward methods

We are told that the starting point is the difference $ f(x+h) - f(x-h) $ (each of which represents it's own series; the last an alternating series) for getting the Richardson extrapolation. When I ...
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548 views

Extrapolation and Splines

If you have a smooth curve, and at a certain point in time you want to predict the next turning point, and you assume it is a non-periodic, stationary, smooth process, then what would be the best way ...