Questions tagged [calculus]

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How to differentiate the hinge loss?

I'm asked to differentiate the following hinge loss term. $$ \dfrac{1}{n}\sum _{\left( x_{i},y_{1}\right) \in S}\sum _{j'=1}L\left( w^{j'};\left( x_{i},y_{i}\right) \right) $$ where $$ L\left( w^{j'};\...
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14 views

Approximate / Standardize value in certain range

I have table with numeric values like ...
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33 views

Backpropogation Implementation [closed]

I have been working to understand the backpropogation algorithm, and I finally think I have gained some grasp of the concept. So I have now begun trying to implement the algorithm, and have so far had ...
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1answer
16 views

Backpropogation Derivatives

I've been working on trying to understand the backpropogation algorithm and the calculus behind it, and in my work I have stumbled across a sort of odd situation. I am just practicing on a 1 input, 1 ...
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1answer
16 views

What is the mean average of $y=kg^t$ from $t=a$ to $t=b$ [closed]

Mean average of $y$ in $y=kg^t$ from $t=a$ to $t=b$. $g$ is a constant, $t$ varies. I have looked this up in textbooks and online and all I can find is the mean average of a function where $t$ is a ...
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Is there a smart algorithm of finding the maximum of $X^{\top}a$ with $X$ and $a$ both belong to some compact convex set? [closed]

Suppose $X\in\mathcal{X}\subset R^k$ and $a\in\mathcal{A}\subset R^k$, where $\mathcal{X}$ and $\mathcal{A}$ are both compact convex set. Is there a systematic way of finding the maximum of $X^{\top}a$...
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Book recommendations needed - building foundational knowledge for ISL - Introduction to Statistical Learning (by Gareth James)

I'm trying to build a data science base from scratch. I started a book called Introduction to Statistical Learning by Gareth James and found that there are many mathematical & statistical concepts ...
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1answer
77 views

Differentiating $ (y-X\beta)^T(y - X \beta) $ with respect to $\beta$

How do I differentiate $$ (y-X\beta)^T(y - X \beta) $$ with respect to $\beta$. The result I saw was $$X^T(y - X\beta)$$
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12 views

How does truncated power basis function imposed the continuity constraint for each knot in splines

This was sort of answered in this post Truncated power basis function and continuity in b-splines Using high-enough powers in [truncated power][1] functions allows you to "match up" not ...
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16 views

Evaluating the integral using conjugate distributions

Hello I just want to verify that I am evaluating this integral correctly. When I implement it in code values seem to be incorrect. could be my implementation though. Thank you for the second pair of ...
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19 views

Backpropagation through time for stacked RNNs

I was able to find the partial derivative of the cost function with respects to a single variable without much difficulty. However, this requires propagating backwards through the network for each ...
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31 views

Evaluation of Limit involved in the proof of Asymptotic Unbiasedness

We know that $S^{2}$ is an unbiased estimator of $\sigma^{2}$ and $S$ is a biased estimator of $\sigma$. But if $n\rightarrow\infty$, then $S$ is an asymptotically unbiased estimator of $\sigma$. I ...
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1answer
183 views

Derivation of M step for Gaussian mixture model

Summary So to summarize my question, how can I take \begin{align} = \sum_{i=1}^{n}W_{i1} \left(log (1-\sum_{j=2}^{K}\pi_j) -\frac{1}{2} log(|\Sigma_1|) -\frac{d}{2} log(2\pi) -\frac{1}{2}(x_i-\mu_1)^{...
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Question about Functions of Several Random Variables

In the Mathematical Statistics and Data Analysis by John Rice, it states that for random variables $U,V$ which are functions of random variables $X,Y$, we have: We know that $$f_{UV}(u,v) = f_{XY}(h_1(...
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175 views

Correlation between normal and log-normal variables

(This is not a homework question.) Let $(X_1 \sim N(\mu_1,\sigma_1), X_2 \sim N(\mu_2, \sigma_2))$ be a bivariate normal random variable with the correlation between $X_1$ and $X_2$ given by $\rho$. ...
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54 views

Maths for deeply understanding backpropgation

I have been trying to develop a deeper understanding of Neural Networks so I can understand the libraries such as tensorflow and others. I have had good success with pereceptron models, and have a ...
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2answers
51 views

How does one compute a variational derivative?

The expected regression loss is given as:$$E[L]=\int\int \{y(\mathbf x)-t\}^2 p(\mathbf x,t)d\mathbf xdt$$ To minimise the expected loss,Euler Lagrange equation is used which goes like this in the ...
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1answer
72 views

Expected Predicted Error (EPE) with L1 loss

In Element of Statistical learning it is saying on page 20, equation 2.18. That using the L1 norm instead of the usual L2 norm leads to an $f(X)$ optimising the EPE being the median instead of the ...
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32 views

Why does the dimension of gradient and Hessian matrix not conform for this function?

The function is $f(\mathbf{x}) = e^{-\frac{1}{2}\mathbf{x^TAx}}$, where $\mathbf{A}$ is a square symmetric matrix, and $\mathbf{x}$ is an n-vector. What I found were: $$ \begin{align*} \nabla f ...
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32 views

How Hessian matrix is helping in taking big step towards minimization and how is it better than usual Gradient Descent? [duplicate]

I know what Hessian is and $θ:=θ−H^{-1}f′(θ)$ this relation too from Newton Raphson but what i dont understand is how Hessian is really helping with big step and also how is this efficient in ...
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40 views

What is the first order derivative of linear regression's cost function using matrix calculus?

For linear regression's cost function J(b), where X is a n*m matrix, b is a m*1 vector and y is n*1 vector: First order derivative with respect to vector b (coefficients) is shown to be Using the ...
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144 views

On the convergence of infinite sum of a hypergeometric function resulted from a nested sum

I am interested in finding the CDF of the sum $U=\sum_{i=0}^N U_i$ where: $$F_{U_i}(x)=\sum_{n=0}^\infty \frac {2(-1)^nR^{k\alpha+n\alpha}x^{k+n}}{\Gamma(k)n!\theta^{k+n}(k+n)(k\alpha+n\alpha+2)}$$ ...
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28 views

How to approximate expectation and variance of an integral from a discrete Time series financial dataset?

I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this. ...
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36 views

Calculus with quadratic B-spline

After fitting a quadratic B-spline in R with the cobs package : Rbs <- cobs(x,y, constraint= "decrease", pointwise = con) I would like to do some calculus on ...
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50 views

what is the relation between change of variables in probability and calculus?

I want to know if the change of variables formula in probability is a special case of the change of variables in calculus, or something different. Trying to think for myself, first write the CoV for ...
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1answer
36 views

Symmetry between integrals including absolute value

So I came across below symmetry in my probability course that I can't understand. I understand how the lower bound changes when removing the absolute value operator, but how does the 2 disappear?
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17 views

A maximization problem involving random variables: a special case

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \color{blue}0 \\ \mu_Y \end{pmatrix} , \...
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28 views

A maximization problem involving random variables

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix} , \begin{...
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13 views

How to calculate logarithmized average monthly returns

I am currently reading the following paper https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2750064 . On page 57 the researchers state that they calculate the logarithmized average monthly returns ...
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1answer
103 views

Find marginal distribution

The random vector $(X,Y)$ is uniformly distributed over $$D=\{(x,y): 0 \leq x \leq 2 , 0 \leq y \leq 2-x\}.$$ Find the marginal distribution of the random variables $X$ and $Y$. For the radom vector $...
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1answer
65 views

Derivative of expectation where the variable appears in the integration limit and in the integrand?

I want to calculate the derivative of $$\varphi(\mu) = \int_{-\infty}^{\mu} r(x-\mu) f(x)dx,$$ wrt to $\mu$, where $r$ is a function and $f$ is a density function. How can I account for the presence ...
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2answers
157 views

Why does Judea Pearl call his causal graphs Markovian?

In his texts on causality, Judea Pearl always refers to the simplest graphs he uses, i.e. the acyclic graphs with independent confounders, as Markovian. I don't see why these graphs contain anything ...
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230 views

Solving integrals in R

I would like to write an R function for solving the following equation: Essentially I would like to be able to set or vary the parameters values of "m" and "s" and those parameters in "p(t)" ...
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916 views

How to manually calculate odds ratio for continuous variables?

In school, long before learning about logistic models, I've been taught how to calculate odds ratios by hand. Formula was based on a contingency table, just like this: This is very easy to ...
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1answer
191 views

Properties of Kernel Density Estimators

Given Let $X \in \mathbb{R}$ be a real-valued random variable with theoretical probability density function (pdf) $f(x)$ and corresponding cumulative distribution function (cdf) $F(x)$. Let $X_1, X_2,...
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9 views

Threshold optimization with two parameters where first parameter need to be minimum and second value to be maximum

We have threshold values ranging from 1 to 10 where attributes p1, p2 increase with increase in threshold. Our intention is to find threshold with minimum p1 and maximum p2. It would be of great help ...
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1answer
312 views

Law of Iterated Expectations Example

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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1answer
25 views

Expectation / Summation inequality with multiple indices

Part of a solution to an exercise in the book Stochastic Processes From Application to Theory (exercise 87) is the following summations: $$\sum_{i\ge1}P\left(I_i \ge i \right) = \sum_{j \ge i\ge1}P\...
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1answer
696 views

Relationship between first and second order condition of convexity

Suppose we have a function $f(\boldsymbol{x})$ and its hessian, i.e $\nabla_{\boldsymbol{x}}^2f(\boldsymbol{x})$, equals $\mathbf{0}$. We know that for convexity $\nabla_{\boldsymbol{x}}^2f(\...
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268 views

What does gradient with respect to a function mean [duplicate]

I am trying to understand this paper better Greedy function approximation: A gradient boosting machine, but I start having difficulty at around Equation (6) and (7). What does a gradient w.s.t. to an ...
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1answer
42 views

Simple Appplication of Law of Iterated Expectation

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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0answers
32 views

Verifying sub-exponential property when a random variable is not sub-gaussian pro

I am referring to the Example 2.4 (page 16) in this book chapter https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf Suppose $Z \sim N(0,1)$ and random variable $X=Z^2$. ...
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1answer
250 views

Expected Loss Calculation

How do you solve the integral $$E(L(\theta_A,\theta_B)) = \int_0^1\int_{\theta_B}^1(\theta_A - \theta_B)f(\theta_A)f(\theta_B)d\theta_Ad\theta_B$$ where $\theta_A \sim Beta(\alpha_1, \beta_1)$ and $\...
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2answers
59 views

Simple Log-likelihood question

I've got a simple question about deriving log-likelihoods. I am stumped by the following--> If the log-likelihood is: 𝑙(𝜆1,𝜆2) = 𝑦1 log(𝜆1𝐹1)−𝜆1𝐹1 −log((𝑦1)!)+𝑦2 log(𝜆2𝐹2) −𝜆2𝐹2 −log⁡ (...
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0answers
18 views

Maximize profits by setting price

I am working on a model to set a price that maximizes profits. The equation for profits is: Profits=price x (# sold) - (fixed cost) x (# sold) I have models ...
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1answer
3k views

Finding the slope at different points in a sigmoid curve

This is my data. x <- c(0.5,3.0,22.2,46.0,77.3,97.0,98.9,100.0) plot(x, pch = 19) I want to fit a curve through these points and then calculate the slope at ...
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0answers
51 views

Can Regularization by achieved using Relative Sensitivity?

In a Mathematical Model we measure the sensitivity of the output with respect to the parameters and it is desirable that a small change in a parameter doesn't lead to wild fluctuations in the output ...
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1answer
5k views

Under the 0-1 loss function, the Bayesian estimator is the mode of the posterior distribution

My notes are rather light when it comes to this topic. I understand that the bayesian estimator, defined as (for sample space $\hat{x}$): $E[\Theta | \hat{x}] = \int_{ \forall \Theta}yf_{\theta|\hat{...
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1answer
774 views

gradient descent momentum vs step size

In the gradient descent method, the learning rate (which is multiplied by the results of the gradient on each weight) identifies the size of the step (steep down) that the algorithm takes in each ...
3
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1answer
427 views

Help with derivation of Mean Field Variational Inference

I am studying Variational Inference using Bishop's book: Pattern Recognition and Machine Learning. At the moment, I am struggling to understand the Lower Bound derivation for the Mean-Field ...