# Questions tagged [taylor-series]

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### Quadratic Approximation of the binary logistic regression

I am using https://web.stanford.edu/~hastie/Papers/glmnet.pdf package to solve my optimization problem for the Binary Logistic Regression. On page 10 it is stated that the quadratic approximation of ...
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### Multivariate Taylor series for moments of a random variable

In the expectation propagation for the generative aspect model, Minka uses Taylor series for the parameter estimation of the topics $p(w\mid a)$ eq 31. I am a little confused in the last equation. He ...
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### Taylor Series Power Function in R

I'm trying to recreate a compartmental model from Yu et al (2017). In this model there is a expanded taylor series power function that is used that I am not understanding how to code. where W(t) = 0, ...
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### The question of Taylor expansion of loss function in XGBoost [duplicate]

I am learning XGBoost from documentation, but there are a few questions in the derivation of it. In the part of ...
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### xgboost tayler expansion detail [duplicate]

This is the objective function for Xgboost. I have no idea where $g_{i}$ and $h_{i}$ came from is some one explain how this two terms came form? or direct me to the related tutorial page then I ...
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### Derive taylor series expansion of df

I was trying to understand ito's lemma. When I came across the taylor series expansion of df(x). df(x) = f'(x) dx + (1/2!) f''(x) (dx)^2 + ... I searched everywhere for the derivation of this but ...
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### How to use derivatives of a function to better estimate its variance over the domain?

How to use derivatives of a function to better estimate its variance over the domain? I have a scalar smooth function $f(x)$ and a multivariate random variable $x$ with known distribution (e.g. ...
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### Iterated estimation of Taylor series

Say your data generating process is given by the function $y=f(x|\theta)$, where $y$ and $x$ represent variables (data) and $\theta$ represent parameter(s). For convergence reasons (e.g. $f(\cdot)$ is ...
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### Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

X-Posted on math.stackexchange, apologies, though I thought this was equally relevant to both communities. I'm wondering if there exists any higher-order SVD for dimensionality reduction. Note that ...
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### Exponential Kth Moment Derivation

I have essentially a mathematical question, relating to deriving the formula for the kth moment of an exponential. I can't seem to work out how we get from the 2nd line to the 3rd line; i.e. the ...
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### Taylor expansion in xgboost [duplicate]

I'm reading through the math of xgboost: https://xgboost.readthedocs.io/en/latest/model.html Under the ADDITIVE TRAINING section of the objective function, I saw that in the derivation of the ...
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### Higher order delta / taylor series approximation relationship to normal distribution?

For a normally distributed variable X, one can call on the delta method to provide an asymptotically normally distributed variable for a non-linear function of it, g(X). This is based on a linear ...
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### Estimating polymomial coefficient in R

I used a Taylor series to expand log(1 - ax) so I could estimate the value of parameter 'a'. The expansion becomes -ax - a^2*x^2/2 - a^3*x^3/3 . . . Now I need to estimate the parameter 'a' using ...
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### Expansion of Cumulant Generating Function of Negbin

Let $X \thicksim Negbin(r,p)$ where $(0\lt p \lt 1)$ I want to derive skewness and kurtosis of $X$ by getting the Cgf of X. First, since Followance of Negative Binomial equals to the distribution ...
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### What is the intuition behind neural networks?

Everywhere in the theory of neural networks, authors saying that idea came about by observing the work of the human brain. But I can not believe in that. I guess, everything is much simpler and neural ...
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### Taylor Expansion of Power of Cumulative Log Normal Distribution Function - Show Lagrange Remainder tends to Zero

QUESTION I am looking to find a simplification of the expression below. I have attempted this using the Taylor series. The question then remains if we can show the Lagrange remainder goes to zero. I ...
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### Approximating $\log( E(X))$

I was casually reading an article (in economics) which had the following approximation for $\log(E(X))$: $\log(E(X)) \approx E(\log(X))+0.5 \mathrm{var}(\log(X))$, which the author says is exact if ...
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### Best basis set for polynomial expansion

I want to do a regression of x onto y: $$f(y)=c_{1}x+c_{2}x^{2}+c_{3}x^{3}\cdots$$ Obviously a plain Taylor expansion as above is suboptimal since the coefficients will not be orthogonal/...
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### Why is Fisher Scoring easier to compute?

In practice, the observed information matrix (Newton-Raphson) is usually replaced by its expectation, known as Fisher scoring. Link: https://en.wikipedia.org/wiki/Scoring_algorithm#Fisher_scoring ...
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### Help with Taylor expansion of log likelihood function

I've reed the following part of a sketch of the proof that the maximum likelihood estimator is asymptotically normal: "Sketch of the second part of the proof. Recall that we may write the likelihood ...
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### Express $E(x^{\alpha})$ in terms of $E(e^{-\zeta x})$? to a 1st or second order?

I have a random variable, $X$, and am able to find $\mathbf{E}(e^{-\zeta X})$ for many $\zeta$ (through the Laplace transform solving an ODE as this actually evolves over time) Is there any way I can ...
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### Asking help with Taylor approximation of expectation of ratio

I am trying to understand how I should approach the problem of a Taylor approximation to the expectation of the ratio of two random variables. In my particular problem I am concerned with the ...
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### Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used

I am trying to understand a proof of quite a long theorem that I report completely for the sake of completeness. This is From Jensen and Rahbek Asymptotic Inference for Nonstationary GARCH (2004). My ...
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### linearization of an estiamtor

Suppose we have two variables $x$ and $y$ defined in some population, with all values of $x$ known. A Poisson sample is drawn, with corresponding inclusion probabilities $\pi_k$ that are proportional ...
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### How can you cluster a set of functions with unknown functional forms?

Say you've $N$ functions $f_N(x)$ defined on a regular grid $x$. You don't know the form of $f(x)$, you've only got several realizations of it. The different functions are related to each other ...
Letting $\Phi$ be the CDF of the standard normal, and $\Phi^{-1}$ be the quantile function of the normal, I am looking for the Taylor series expansion of  \Phi^{-1}\left(\Phi\left(x\right) + \...