Questions tagged [function]
A mapping between a set of inputs and a set of outputs.
232
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Understanding notation for input space
I'm taking the free Caltech machine learning course. I'm having trouble understanding the notation on one of the problems:
In this problem, you will create your own target function f and data set D ...
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Sufficient/complete statistic $\leftrightarrow$ injective/surjective map?
I can't understand the paragraph in Completeness (statistics) - Wikipedia:
We have an identifiable model space parameterised by $\theta$, and a statistic $T$. Then consider the map $f:p_{\theta }\...
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Solving integrals using incomplete gamma function (upper gamma rule)
I am attempting to integrate this function, ∫15x^(0.28) * e^(-0.21x) dx and am struggling with what techniques to apply. The lower boundary is 0 and the upper boundary is infinity. From research, I ...
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Map from Normally distributed Variable to Gamma distributed Varaible [duplicate]
I need to find some function $f:\mathbb{R} \rightarrow \mathbb{R}^+$ such that
If $\; \; x \sim \mathcal{N}(x; \mu, \sigma^2) \; \;$ then $\; \; f(x) \sim \mathcal{G}(y; \alpha, rate=\beta)$
Where $\...
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Is it possible to learn a mapping between unpaired time-series data?
I have two time series. One is an actual recording of physiological signals, the other is a binary representation of activations (0 or 1) based on a simulation of the process underlying the ...
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Transform data domain and maintain mean
I have a dataset, $y$, that is on some arbitrary range. I would like to transform this data to be on the range [-1,1]. This is accomplished using a linear transformation, such as the one described ...
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1
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Standard deviation of a function of random variables
I was a bit confused recently with why two jointly distributed variables will exhibit a smaller scatter as the correlation between the random variables increases (assuming only positive values for the ...
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Is there a way to stretch sigmoid function output
I have an array of values and a value that lies outside of array's max value:
arr = [10, 15, 20, 30]
value = 150
and I want to make that value less of an outlier, ...
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Describe a geometrical shape as a piecewise function
Consider a cube filled with random particles. Let's say the particles in cube are rotated around the z-axis through the center of the cube. Here, the rotation is proportional to the height of the cube,...
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Estimating the sample variance and expected value of a maximization function of multiple random variables
Suppose I have three i.i.d. random variables $X$, $Y$, and $Z$. A model is used to generate $i$ outcomes for each variable that are used to estimate the sample mean and sample variance for each of the ...
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Rewriting the expectation of f(x) by means of its derivative
I have a question regarding this proposition.
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an a.e. differentiable function so that $\int \frac{\left|f^{\prime}(x)\right|}{(1+|x|)^s} d x<\infty$ ($...
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Why complete data log likelihood in M-step of EM algorithm
In Bishop's Pattern Recognition and Machine Learning book(page440), it talks about the M-step in EM algorithm of Gaussian Mixture Model. I am confused about the likelihood function of M-step. By ...
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Intersection of matern covariance functions with different parameters
I am trying to prove some ideas about the identifiability of covariance function parameters for small samples. From various numerical experiments, it seems that the graphs of two matern covariance ...
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Convergence of moment of functional of random variable
Define $X_n$ a continuous random variable that converges in distribution to $X$. Morever, we know that $E[|X_n|^p] \rightarrow E[|X|^p]$ for some $p > 0$.
Then, could we prove that for any ...
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How to train a network with multiple inputs and multiple outputs [closed]
How to train a network with multiple inputs and multiple outputs
I have a dataset of (400,256) 400 examples with 256 features.
and each example has its corresponding output array
(400,256) 400 ...
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Is there a substitute for an "average" which best inherently represents data, rather than reducing it to one or another aspect?
I was thinking about the arithmetic average as a type of metric or a representation of data. Maybe there is a mathematical argument for the non-arbitrariness of that function. I think something like ...
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Extract the functional mapping between input and output from a machine learning model
A lot of ML models, such as neural networks, are Universal Function Approximators. But when evaluating ML models, we use usually metrics, such as MSE or accuracy, to assess the performance of a ML ...
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Do error terms of a value that is part of a normal distribution also follow a normal distribution?
I asked this yesterday, but was not very concise. Hoping to get better results with different wording. I have a question related to the error term in a normal distribution. Let's assume that we are ...
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Function that maps low values to [0,1] [closed]
I'm currently working on a predator prey simulation and have to parameters that are codependent, the attack rate and the survival rate.
The attack rate is mutated within the simulation and determines ...
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Best way to analyse two experiments
I'm looking for help/indication on approaching a data integration problem.
I have a dose-response curve that is described by a log curve. Because of measurement difficulties, the dose and response can'...
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Method to model exponential fit line coefficients based on main function parameters
I have a quite complicated trigonometric equation I was able massage into a parametric function that takes four parameters as inputs/arguments, say: $a$, $b$, $c$, and $d$. Regardless of the ...
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Problem with probabilities of functions of continuous random vector
I have problems with a true or false exercise: if the joint pdf of the continuous random vector $(X,Y)$ is
$$f(x,y)=
\begin{cases}2x,& 0 \le x \le 1, 0 \le y \le 1,\\
0,& \text{otherwise}
\end{...
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Function that receive two variables and decides which statistical analysis should be done
I am trying to create a function that receives two variables (dependent and independent) and figures out according to their class (numeric / factor) and number of groups, which statistical test should ...
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I need to derivate the maximum likelihood of GLR function [closed]
I am trying to do the partial derivative of this function but I don't know how to include the $\theta$ inside my function to be able to apply the partial derivative in $\theta$.
The likelihood ...
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How to decompose the component that is influenced by X from time-series Y?
There is a time series Y, which is the remaining part of the initial time series after removing the trend and seasonal periodic components. And there is also a time series X, which represents the ...
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How do I find a mapping f : X -> Y to characterize the relationship between 2 time-series X and Y? [closed]
I have a time series Y, and another time series X related to Y.
The value of Y is affected by X and some noise.
How can I separate the effect of X on Y ?
That is, how do I find a mapping f : X -> Y ...
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Is sequence of probability mass functions always uniformly bounded
Say that we have a sequence of discrete random variables, $\left\{X_n\right\}_{n \in \mathbb{N}}$, which converges to a random variable, $X$, with a continuous distribution, e.g., the Normal (Gaussian)...
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Using output of categorical distribution in Pymc3 as index for selection of down stream parameters
The code that I would like to generate is something like:
I want to have a categorical variable that maps to another value different from the index given by the categorical distribution, and fits as a ...
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How do you get this bifurcation diagram of this 2D discrete time model?
I want to replicate the results of "Chaos in Duopoly Pricing" by T. Puu, which can be found here: https://www.sciencedirect.com/science/article/pii/096007799190045B#section-cited-by. The ...
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How to find expected value from cumulative distribution function?
Hello everyone, I'm currently doing research based on the model in Competitive fit-revelation sampling and mixed pricing strategy (Wu &Deng, 2021). And I don't understand how they can conclude the ...
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Determine expected goals from two normal distributions
Assume Team A scores an average of 3 goals per game with a standard deviation of 1.0, and assuming Team B allows an average of 2 goals per game with a standard deviation of 0.5.
How would you ...
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Problem formulation classification task
I would like to know if it is correct for a classification task in a supervised learning to say the model we are looking for is a function from RxR to a discrete space
$$
f:\mathbb{R}\times\mathbb{R} \...
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SMOTE vs Stratified Sampling in highly imbalanced dataset - classification
I am working on a project with the goal of predicting Cerebral strokes from brain arteries data (speed of blood, resistance etc. of one artery and of the neighboring ones).
I have a dataset with ...
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Difference between Predictive Inference and Causal Inference
I am looking for functional mathematical notation to explain the difference between Predictive Inference and Causal Inference? I list an example model. I also list links further down that give ...
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Equal intervals on log scale
I want to make a plot in which the horizontal axis is log scaled, ranging from 1 to 1000. I want to divide the horizontal axis into $n$ equal intervals, where "equal" means that it appears ...
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distance between regression models
Consider two multivariate linear regression models (vector inputs and outputs) with the same domain observations. Namely, let:
$X \in \mathcal{R}^{a \times N}$ be a matrix of domain observations (...
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How to calculate exponential growth when the value is a percentage or probability?
I have a variable x that takes values between 1 and 0 (0<x<1). I want to calculate value of ...
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1
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Property of unbiased estimators
If $f(x)$ and $f(y)$ are both unbiased estimators of $\mu$, aka $E[f(x)]$ = $E[f(y)]$ = $\mu$, is it possible that $f((x+y)/2)$ is also an unbiased estimator of $\mu$?
We know $f((x+y)/2)$ would be ...
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Function fit to skewed data and non-zero beginning of the function
I would like to find a function that would represent the best fit to represent this type of biological data. More precisely, I would like to estimate expected daily egg production by an insect, based ...
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Prove that every linear regression predictor is a linear function
A function $f : \mathbb{R}^D \rightarrow \mathbb{R}$ is linear if both of the following
conditions hold.
(1) For all $\textbf{x}, \textbf{y} \in\mathbb{R}^D, f(\textbf{x} + \textbf{y}) = f(\textbf{x}) ...
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How to use statistics to speed up row-wise computations on a data.frame?
I have a data frame with 10,000 rows and 40 columns. I am trying to apply a function to each of these rows. For each row, I am expecting to return a scalar which is the value of the statistic I am ...
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How to weight ranks?
I have a set of responses ranked by my participants. For example, they gave responses A, B, C and ranked them as 3, 2, 1 (or C, B, A). I computed relative frequencies of each responses (A, B, C) and ...
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What is the equation of a line fitting a log-log model computed in R?
I am currently stuck, wanting to extract a line function from my fitted line on my log-log model.
...
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Why does non-parametric approach break down when the joint distribution is estimated by a finite data sample?
I am currently reading the paper on Gradient Boosting Machines - J. H. Friedman, “Greedy function approximation: A gradient boosting machine,” Ann. Stat., vol. 29, no. 5, pp. 1189–1232, 2001, doi: 10....
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replacement error when feeding data from function into matrix in R [closed]
I am trying to write a function that creates simulated datasets.
The function takes the argument size, being the size of a group, and it is meant to produce a matrix with 100 columns with values 0 or ...
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Other functions with sigmoid shape [duplicate]
Apart from the classic logistic or sigmoid function, are there other interesting functions that map real x into a positive space? Maybe they also have an S shape.
Maybe inspired by some nature ...
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$f$ is a decreasing function whose integral converges. Does $\lim_{x \to \infty}xf(x) = 0$?
My finals are over and I cannot help but ruminate over this particular problem. Could anyone help prove this?
Suppose $f$ is a continuous decreasing function on $[0,\infty)$ and $\int_0^\infty f(t)\, ...
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Which type of sums of squares does lm-function in R use?
I ran a two-way ANCOVA in R:
ancova = lm(DV ~ IV1*IV2 + CV1 + CV2 + CV3, data = Data)
summary.aov(ancova)
Anybody know if this uses type III sums of squares?
I ...
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What is the 'feature function'? [duplicate]
The term 'feature function' is very frequently used in the context of machine learning, but I'm still not sure what it really is. Could anyone give the precise definition? Can it be understood as a ...
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Get Continuous Distribution from Discrete Variable: Problem 6.77 of Wackerly, Mendenhall, Schaeffer, 5th Ed
Problem Statement:
$\newcommand{\szdp}[1]{\!\left(#1\right)}$
Let $v$ denote the volume of a three-dimensional figure. Let $Y$
denote the number of particles observed in volume $v,$ and assume that $Y$...
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