# What is Extrapolation: Definition and 31 Discussions

In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a (usually conjectural) knowledge of the unknown (e.g. a driver extrapolates road conditions beyond his sight while driving). The extrapolation method can be applied in the interior reconstruction problem.

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1. ### Python Why backpropagation dominates neural networks instead of interpolation

Hi guys, I was learning machine learning and I found something a bit confusing. When I studied physics I saw the method of least squares to find the best parameters for the given data, in this case we assume we know the equation and we just minimize the error. So if it is a straight line model...
2. ### I Hawking Radiation Extrapolation: A Conjecture

The intense gravity near the event horizon causes complementary particles to pop into existence spontaneously. As local space-time is continuous through the EV, the same would be happening just inside the EV, only more so as the gravity field and gradient is greater. So near the singularity...
3. ### Numerical Analysis - Richardson Extrapolation on Riemann Sum

I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?). For example sin(x) for [0,1], n with trapezoidal rule = 0.420735... With my own...
4. ### B Extrapolation of relationship of temperature and volume for gas

Hi, Originally, the absolute temperature was thought to be around -273 Celsius around 1750 and it was the result extrapolation of of ideal gas law as shown below. I find it hard to phrase my question. But the question is how come they were so confident that the relationship between the volume...
5. ### I Is the backward extrapolation of the expansion of the Universe valid?

Summary: The universe is expanding, so at one point in the past it must have been all concentrated into a single point. But is this really an accurate observation? Hi, This is my second probably naive question that's been on my mind as a lay-scientist for a long time (the other will hopefully...
6. ### MHB Extrapolation and interpolation in line search optimization

hi can you tell me these equations: A = 6*(f2-f3)/z3+3*(d2+d3); % cubic fit B = 3*(f3-f2)-z3*(d3+2*d2); z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A; % numerical error in MATLAB fmincg.m...
7. ### B Extrapolation back to T zero, before 1980

Hello. I have a question about how close (in time) the Hot Big Bang theories formulated before 1980 could be reliably extrapolated to T zero. In his book... https://en.wikipedia.org/wiki/The_Inflationary_Universe ...Alan Guth recounts a lecture by Robert Dicke in 1978 which set the benchmark...
8. ### Is there a philosophical analysis of scientific extrapolation?

Hi, Could you recommend a book/essay/anything else about the philosophical approach of scientific extrapolation?
9. ### Matlab Extrapolation: Find Approximate Size of Matrix n∗

Let A be a random n×n matrix, x = (1,1,...,1)⊤ be an n-vector of ones and b = Ax be the right-hand side vector. As in class, let z = (zj) ∈ Rn be the result of solving the system Ax = b in finite precision using the backslash command. To measure the error between x and z, we let δ= max |xj−zj|...
10. ### A Help understanding this extrapolation technique

Trying to understand what is going on here, and how this technique removes the dependence of sample rate from the solution.
11. ### Extrapolation for Two Dimensional Lookup Table

Hello, I am writing a program for a two dimensional look up table. The idea is that the user will enter values for x and y and the program will look up the corresponding output value from the table. For example, consider the table shown below. For this table, if the user enters x = 2 and y...
12. ### Non-linear extrapolation graph

Hi, Please note: this is not a homework question! It is a real world problem I am trying to solve. I have some values in mA and tonnes which I need to extrapolate but they are not linear. I know it the mA curve drops off the higher the tonne values go. I have plotted the values in excel...
13. ### Curve extrapolation: polynomial or Bézier?

On a stationary, non-periodic signal (black) a smooth causal filter is calculated (green/red). It is sampled discretely (every distance unit of 1 on the X-axis). My goal is to find which "path" it is "travelling" on so I can extrapolate the current shape until it is completed (reaches a...
14. ### Richardson Extrapolation with 3 steps?

Homework Statement [/B] use richardson extrapolation to estimate the first derivative y=ln(x), x=5 using steps of 2, 1, 0.5. Four decimal points. obtain true relative error for the last estimate and comment on its value. Homework Equations [/B] deriv ln(x)=1/xThe Attempt at a Solution I know...
15. ### Issue Finding the Mean Squared Error / Polyfit MATLAB

Homework Statement Homework Equations Given above The Attempt at a Solution I used polyfit, but my mean swuare errors are way bigger than they should be- don't see what is wrong with my code! My code is ugly btw, my apologies. %Hw 7 clear all close all y3=[1960; 1965; 1970; 1975; 1980...
16. ### Richardson extrapolation matlab error

Hi everyone, I am trying to differentiate a function by means of Richardson Extrapolation in MATLAB. However, I could not run the program. Here are the outputs and its codes. Could anyone explain to me what is my mistake? The codes are based on "Numerical Methods Using MATLAB by John H. Matthews...
17. ### What is the next step for part iii in Taylor Series Extrapolation?

Hi could anyone give me pointer as to where to go with part iii please?
18. ### Construct extrapolation table with optimal convergence

Construct extrapolation table with optimal rates of convergence Homework Statement Let S be a cubic spline interpolant that approximates a function f on the given nodes x_{0},x_{1},...,x_{n} with the boundary conditions: S''(x_{0})=0 and S'(x_{n})=f'(x_{n}). Use S to estimate f(0.1234567)...
19. ### Retarded positions, extrapolation vs. hitting a wall

I'm a compsci guy reading Feynmann's lectures leisurely. I find the retarded-field equations somewhat shocking due to the consequences of nature "extrapolating" trajectories to find the fields. Is my understanding of the following scenario correct? Let's say I have an "electrostatic compass"...
20. ### MHB R - Richardson Extrapolation for Accurate Estimates

Hey, I was hoping someone could help me with this question I can't get at all. If $$\phi{h}=L-c_1h^{\frac{1}{2}}-c_2h^{\frac{2}{2}}-c_3h^{\frac{3}{2}}-...$$ , then what combination of $$\phi{h}$$ and $$\phi(\frac{h}{2})$$ should give a more accurate estimate of L. Thanks for any help.
21. ### MHB Richardson's Extrapolation for Derivative Estimation

From Randy Blythe's Yahoo Answers Question: Use Richardson’s extrapolation, using a centred difference scheme for the initial estimates, to estimate the first derivative of y=(x^3)sinx at x=2 with step sizes h1=0.5 and h2=0.25. Calculate the actual derivative and the percentage error
22. ### Richardson Extrapolation to check convergence

Hi, I need to know how one can check space and time convergence using Richardson Extrapolation. Does anyone know any good references. I have a slight idea... the thing I am wondering about is how using this method can eliminate the need for further simulations using smaller time steps or a...
23. ### Evolutionary reason for 3-D extrapolation

Can anyone explain why we humans have the innate ability to extrapolate 3-dimensional images from 2-dimensional images. I am not interested in how it works (i.e., how the brain accomplishes this feat). Instead, I would like to know why it was so important in the evolutionary process. In other...
24. ### Error on using a curve of best fit for extrapolation

Hello everyone, I did a quick search but could not find this in the forums. I have quite a basic situation. I have been gathering data points from an experiment and was able to fit an exponential curve of best fit to it. What I want to do is approximate some values between my data points...
25. ### Ridiculus position-time graph extrapolation

I'm working on some physics lab homework and one particular graph is obnoxious. It's hard to tell whether it's intended as a compound curve, or if it's just a piss-poor sketch. I've contacted my professor but he hasn't responded for days, and it's due tomorrow. If I'm not mistaken, the...
26. ### Is the Big Bang Theory a Foolhardy Extrapolation?

Today I have been traveling across the night sky with Google Earth a bit. As always I'm surprised about how inspiring the cosmos is for thinking about physics. But I've always been quite skeptical about the big bang theory. Nowadays we're unable to simulate even a single hadron. And the...

Hi, I need to extrapolate vector v_{-1} from v_{0}, v_{1} and v_{2} (see attached pic), so that if v_{2} is on the right/left (2D case for simplicity) of v_{1}-v_{0}, v_{-1} would also be on the right/left. My initial solution was like this: v_{2} - v_{1} = v_{1} - v_{0} + dv, v_{1} - v_{0}...
28. ### Interpolation and Extrapolation

Hi, I've a table of data in the following format: x--> 0.1 3.767 4.395 5.0223 y 0.1 | 1 1.5 2.0 2.0---->z 0.6764 | 1 2.0 2.2 3.599-->z 1.10146 | 2 2.2 2.5 3.686-->z 1.3855 | 2.5...
29. ### Interpolation vs Extrapolation

Could someone help me understand why interpolation rather than extrapolation should be used for calibration curves (with the exception of Standard additions). I know that extrapolation is less precise, but the book I've got doesn't over any more than that, and I think I need a little more detail...
30. ### Forward difference formula extrapolation

the forward difference formula can be expressed as f'(x_{0}) = \frac{1}{h} [f(x_{0} + h) - f(x_{0})] - \frac{h}{2} f''(x_{0}) - \frac{h^2}{6} f'''(x_{0}) + O(h^3) use extrapolation to derivae an O(h^3) formula for f'(x0) would i be using the taylor expansion to get the answer here? I knwo...
31. ### Is There a Simple Quantitative Technique for Projecting Values in a Series?

Hi, Is there a general extrapolation formula (or other *simple* quantitative technique) for projecting values in a given series? I have these numbers in a certain series sequence: x=1, y=-3.80, x=2, y=-4.15, x=3, y=-4.47, x=4, y=-4.77, x=5, y=-5.05, x=6, y=-5.27, -5.40 Question is...