Questions tagged [extrapolation]

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

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Prove boundary of error when extrapolating origin of function

Given are $n \in \mathbb N, h > 0$ and $a \in C^{n+1}(0, h)$. Furthermore, let $(x_k)_{k = 0, \dots, n}$ be a monotonic decreasing sequence of positive numbers with: $$x_0 \leq h \;\;\; \text{ and }...
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How do I calculate the expected number of duplicates in a set based on the number of duplicates found only in a sample?

I have a set of 10,000 points of data. I know that some of those points of data are identical to other points of data (i.e. duplicates), but (for now) there are no triplicates, quadruplets, etc. This ...
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$[1]$ Does good $R^2$ necessarily tells us that the experiment went right? $[2]$ Is not extrapolation dangerous when used for a far point?

In an experiment, the relation between $x$ and $y$ is linear. $x_\text{actual} = \alpha_x x_\text{observed} + \beta_x + \gamma_x$. [Here, $\alpha$ is the slope, $\beta$ is the intercept, and $\gamma$ ...
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Extrapolating the equation of a Gaussian function from 3 points

I have a system with a moving sensor, and the measured value is in the form of a gaussian function. Thus, I'd like to model this sytem with a gaussian. Naming Ms the measured value, I'd like to ...
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Error of Richardson Extrapolation with composite midpoint rule

My goal is to find an (exact) error expression for Richardson extrapolation applied to the composite midpoint rule. I know that the error for this rule is $$\displaystyle\frac{(b-a)h^2}{6}f^{''}(\xi)$$...
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Is there a general form of a logical formula with N variables?

Let N = 2. Then there are 16 possible non-equivalent N variable logical formulas, listed below. False, A ∧ B, ¬(A → B), A, ¬(B → A), B, A ⊕ B, A v B, ¬(A v B), ¬(A ⊕ B), ¬B, B → A, ¬A, A → B, ¬(A ∧ B),...
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Testing linear extrapolation for validity

How can we assess the validity of a linear extrapolation? Is there a standard way of measuring the soundness of an extrapolation, like "statistical significance"? The only way I can think of ...
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What "tool" to extrapolate traffic data on graph used for routing (Open Street Maps).

Background (non-math): I'm planning to use Open Street Maps data to find the fastest route by car between points. Data is a directed graph where vertices represent locations, and edges are routes. ...
Łukasz Patecki's user avatar
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How to create synthetic data for a decaying curve in order to extrapolate it beyond some point? [closed]

In the following curve , I would like to extend the measurements beyond $x$=1 in order to have a better estimate of the green curve compared to red line. Note: I do not have the analytical form of the ...
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How do I extrapolate probability distributions from a single distribution?

If I have 3 fair 6 sided dice, and 1 throw consists of rolling all 3 of them, how do I extrapolate the distribution of roll values of each individual die for a specific sum from the distribution of ...
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linear extrapolation to $\infty$

In my "Numerical methods in Linear Algebra" course I have to calculate the eigenvalues and eigenfunctions/eigenvectors for the 1-D stationary Schrodinger equation. The interval where the $x$ ...
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Maximum of $\sum_{k=0}^{P} (-1)^{k+P}\binom{P}{k} k^N $ with $P\leq N$

For a combinatorics problem, I would need to identify the value of $P$ that maximizes the function $$\Omega(P) = \sum_{k=0}^{P} (-1)^{k+P}\binom{P}{k} k^N$$ with $P\leq N$. Numerically, I see that the ...
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Extrapolating a smooth curve

I am looking for a method for extending a curve, like the ones shown in the figure below. The known part of the curve is defined only for integer arguments, which however span a range from one to ...
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Working out probabilities of a singular instance and extrapolating to larger numbers

I'm trying to work out out that if, for example, you were doing a test, and the chances of the test returning a false positive or false negative were worked out to be a specific percentage, say 20%, ...
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Asymmetrical step size to apply Richardson Extrapolation to improve Runge-Kutta order 2 solution

I'm trying to solve a series of problems related to approximations of ODEs with Runge-Kutta that have their approximation to values improved by using the Richardson Extrapolation. Some of these ...
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What does "judicious extrapolation" mean?

In the article There’s more to mathematics than rigour and proofs, there is a term "judicious extrapolation": It is of course vitally important that you know how to think rigorously, as ...
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Extrapolating a limit from finite numerical data

Simply for fun, I'm wondering if there are techniques (rigourous or not) for estimating a sequential limit from finitely many terms (under certain niceness assumptions, e.g., there is only negligible ...
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Romberg Integration Midpoint

I've seen multiple sources, claiming that the Richardson Extrapolation for the Romberg Integration (Trapezoidal method) is... $$R_{i,j}=\frac{4^{j-1}R_{i+1,j-1}-R_{i,j-1}}{4^{j-1}-1}$$ if the ...
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How to estimate best fit for factors

I have a table of data that looks like this: Target R1 R2 R3 3.70 0.7863000000 0.8313000000 1.1019000000 3.40 0.8750000000 0.9000000000 0.9994000000 3.40 0.7719000000 0.9500000000 0.9994000000 3....
Be Kind To New Users's user avatar
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Extrapolation using Taylor series - giving negative results for increasing positive inputs

I would like to use a 2nd order Taylor series expansion to perform an extrapolation to predict points outside of a known range. I am using the following formulation: \begin{equation*} d(N+ \Delta N) =...
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Why is polynomial fit not a good choice or a good extrapolation technique?

I would like to know why an polynomial fit is divergent on the boundaries of the fit interval (as shown on the doc site of the matlab polyfit function). And why a polynomial fit on a data set is a ...
Paul Shon's user avatar
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Generating completely new vector based on other vectors

Assume I have four-vectors (v1,v2,v3,v4), and I want to create a new vector (vec_new) that is not close to any of those four-vectors. I was thinking about interpolation and extrapolation. Do you think ...
sezar sampaio's user avatar
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Young’s inequality implies $a^ {n/(n+1)}\leq 2a+(1/n^2)$ for any $a>0$ and $n\geq 1$

I’ve seen this statement in a Yano’s article and I can not prove it. I take the Young’s inequality $a^{1/p}b^{1/q} \leq a/p + b/q$ where $1/p +1/q =1$. I’ve prove it in the case $a\geq 1$. In the ...
Laura SPD's user avatar
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Lagrange polynomial extrapolation errors

Lagrange polynomial interpolation error term is: $$E(x)=\frac{f^{(n+1)}(ζ)}{(n+1)!}π_{n+1}(x)$$ where $π(x)=(x−x_0)…(x−x_n)$ and $ζ∈(a,b)$. However, the theorem doesnot include the situation when $x \...
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At what point is extrapolation reasonable?

Are there generally accepted thresholds (number of data points, length of extrapolation, etc.) of what is reasonable to extrapolate? ex·tra·po·la·tion noun the action of estimating or concluding ...
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Richardson extrapolation formula for Runge-Kutta method

We use the 4-stage Runge Kutta method and we have computed $y_{n+1}^{(h)}$ and $y_{n+2}^{(h / 2)}$, two approximations of $y\left(t_{n}+h\right).$ Develop a formula for the Richardson extrapolation by ...
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Inequality for decreasing functions

While trying to prove some extrapolation theorem for $B_{p}$ weights, I tried to prove the following: Suppose that $0<p\leq 1 $ and $w(x)$ is a non-negative and belongs to $B_{p}$, that is \begin{...
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Error term for backward differentiation formula

I am trying to derive the error term for the backward differentiation formula $f'(x) \approx \frac{3f(x)-4f(x-h)+f(x-2h)}{2h}$. By Taylor expansion $f(x-h)=f(x)-hf'(x)+\frac{h^2}{2}f''(x)-\frac{h^3}...
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Computing derivative as accurately as possible by finite difference formula

The central difference formula is $D_h(x)=\frac{f(x+h)-f(x-h)}{2h}$. In the form $(x,f(x))$ am given the data points $(0.40, 2.3987), (0.44, 2.4182), (0.48, 2.4377), (0.52, 2.4571), (0.56, 2.4764)$. ...
Azamat Bagatov's user avatar
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Richardson Extrapolation on forward differencing formula

After deriving this with forward differencing formula & Taylor series: $$ D_1 f = f'(x_{i}) = \frac{f(x_{i+1}) - f(x_{i})}{h} - \frac{f''(x_{i})h}{2!} $$ which could perhaps also be written as: ...
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Interpolating values using a fitted function

I have data for $x,y$ where $y=[0,0,0.016,0.65,0.97,0.99,1]$ and $x=[1,2,3,4,5,6,7]$. For this data I fitted a sigmoid type function as $f(x)=a/(1+exp(-b*x)+c)$. I fitted this using matlab cftool, and ...
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How to extrapolate the data?

The question demands me to find "rupture lifetime" for 300MPa load at a temperature of 649$^{\circ}$ C.How do I solve it?All that I know is to apply interpolation and extrapolation which I ...
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Finding higher order method using Richardson Extrapolation with integration

Suppose that the interval of integration $[a,b]$ is divided into equal subintervals of length $h$ each such that $r = \frac{b-a}{h}$ is even. Denote by $R_1$ the result of applying the composite ...
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Extrapolate $y^2=4x^2-4x^4$ from plot

I am trying to extrapolate a function based on the distribution of eigenvalues of certain matrices I am working with. In a simple case I successfully described the data with the function $y^2=4x^2-4x^...
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Wrong coefficients when deriving Simpson's Rule using Richardson Extrapolation

Ok, so I am going with Wikipedia's definition of the Richardson Extrapolation, rather than the recursive definition that I know is floating out there. One step at a time. Given $A(h)$ is an estimate ...
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function approximation - identifying suitable functions to evaluate

I want to approximate best fitting function using first 3 inputs of X,Y values and then extrapolates for X=[5,20]. X=[2,20] and Y=[0,100], in the attached plot. which kind of function is likely to fit ...
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Which is better: Least Squares Regression or some Unknown Method I found my coworker using?

It's been a while since I did some good old regression, but I've been given a dataset with some simple x and y data and need to ...
Thomas Doyle's user avatar
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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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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 ...
Twesigye Boaz Michael's user avatar
4 votes
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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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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: ...
Laurence Lord's user avatar
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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 ...
Rafael 's user avatar
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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 ...
glowstonetrees's user avatar
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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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How to find more accurate numerical integration result using Richardson's Extrapolation given midpoint and trapezoidal conditions?

This question has been confusing me. Any help would be very appreciated.
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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) + ...
Robert's user avatar
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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 ...
Albert Renshaw's user avatar
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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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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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