Questions tagged [transform]
If possible, use a more specific tag, such as [data-transformation], [mgf], [wavelet], or [probability-generating-fn]
86
questions
0
votes
0answers
11 views
Confusion in the domain of transformation of random variables
I have two random variables $X$ and $Y$. Let us assume both ranges from $0<x<\infty$ and $0<y<\infty$. Let is also assume both are having exponential densities.
I am making two different ...
0
votes
1answer
21 views
What distribution is this? [duplicate]
Basically, I am told that
$\varepsilon$~$N(0,1)$, and
$\omega$~$IG(\frac{v}{2}$,$\frac{v}{2})$ where $IG$ is the inverted gamma distribution
Now, I am told that the distribution of:
$\varepsilon(\frac{...
0
votes
2answers
47 views
Finding the pdf of Y from that of X, linear transformation
The question is
Let $X$ be a continuous random variable with pdf $f_X(x) = 2(1 − x)$, $0 ≤ x ≤ 1$. If $Y = 2X − 1$, find the pdf of $Y$.
I understand these steps$$F_Y(Y ≤ y) = P(2X-1 ≤ y) = P(X ≤ (y+...
3
votes
1answer
37 views
Z transform of $ 2^{-|n|} $
Dears,
I'm trying to compute the Z-transform of $$ x(n) = 2^{-|n|} $$.
My procedure is as follows:
Using definition of Z transform:
$$ X(z) = \sum_{n=-\infty}^{\infty}2^{-|n|} z^{-n} = \sum_{n=-\infty}...
0
votes
0answers
22 views
0
votes
1answer
54 views
Copula between a distribution and its univariate transformation
I'm trying to compute the copula (or joint distribution) between x and a univariate transformation, like say sin(x).
That is compute $C_{XY}$ (or $F_{XY}$) given that $x \sim U(0,1)$ and $y = sin(x)$
...
6
votes
1answer
200 views
Inverse Gaussian chi square connection
The inverse Gaussian distribution $IG(\mu,\lambda)$ is associated with the density
$$f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\,\exp\left\{-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right\}\qquad \...
1
vote
1answer
60 views
PMF of $aX_1 + bX_2$ (Bernoulli)
Let $Y_1 = aX_1 \sim \text{Bernoulli}(p)$ and $Y_2 = bX_2 \sim \text{Bernoulli}(p)$, what is the PMF of $Z = Y_1 + Y_2$ for $a > 0$, $b > 0$ and $a \neq b$?
Can somebody check my result?
$$p_{...
-1
votes
1answer
24 views
A Normal distribution variable in the power of N
If $X$ is normal distributed random variable, what is the distribution of $|X|^n$?
I am struggling to understand the distribution.
Any guidance would be appreciated.
5
votes
1answer
72 views
Why can I interpret a log transformed dependent variable in terms of percent change in linear regression?
Looking at resources such as this one and this one, you see claims like
"Exponentiate the coefficient, subtract one from this number, and
multiply by 100. This gives the percent increase (or ...
1
vote
2answers
185 views
How do you get the double sum or integral from $E(X+Y)$ (expected value)?
I was given a proof for $E(X+Y)$ = $E(X)+E(Y)$ for cases where both variables are either discrete or continuous:
Discrete:
$$
\begin{align*}
E(X+Y)
&=\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}(x+y)...
1
vote
0answers
42 views
Laplace-Stieltjes Transforms and distribution
I was going through a paper, I came across below relation,
\begin{equation}
T=\begin{cases}
C, & \text{with probability $P(H<C)$}\\
0, & \text{with probability $P(H>C)$}
\...
1
vote
0answers
41 views
Is the copula function invariant only under deterministic monotonic transformation?
I read about the following theorem (see Proposition 3 in the picture below) on the invariance of copula under monotonic transformation, my questions is:
1. Are the $T_i$ mentioned in the following ...
3
votes
1answer
124 views
Change of variables in pdf
I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(...
1
vote
1answer
53 views
How can I shift the average probability keeping constraint (0.0:1.0)?
I have a large datasets of values that range from 0 to n. I am interpreting the values as probabilities for a later pseudo-random selection process. To make the values serve as probabilities, I ...
0
votes
0answers
8 views
Knowledge Distillation - Comparing different methods
I recently got into this field and I am a little confused. For example, in this paper by Hinton and this paper, how exactly are we supposed to interpret the results? I mean sure, this kind of training ...
1
vote
1answer
118 views
How to interpret a specific data transformation?
I came across this specific data transformation in the context of a physics application, which by itself is rather complex and hence out of the scope of this question. However since this ...
2
votes
1answer
35 views
How do you work with a function of a uniform distribution? [closed]
I am struggling with parts b and c. How do you solve them?
Could you please give the solution?
0
votes
2answers
47 views
how do you transform/standardise a function to always give values between y1 and y2?
Having lost some of my math skills, I am having problems with something that I think should be fairly easy but is eluding me:
I have a plateau shaped function that I would like to standardise such ...
1
vote
0answers
46 views
Can I use Linear Regression or do I need Nonlinear Regression
I am trying to fit these two equations to data in R via regression.
First Equation:
$$y(x) = a + \frac{b}{c + x^m}.$$
This equation is constant plus reciprocal function, resulting in a hyperbolic ...
0
votes
1answer
74 views
Normalize target value for linear regression
I'm building a regression model to predict sensor value over time.
Bellow is a figure of my sensors data over time:
Based on this video about transforming nonlinear data with a log function, What ...
0
votes
0answers
30 views
Squeeze a time series to fit in a range while maintaining shape
I have the following time series with intermediate highs and lows marked by the vertical lines:
I want to transform/squeeze the series so that the resulting series would fit in a range, let's say [0,...
4
votes
1answer
115 views
What is the general second-order Taylor approximation to $\mathbb{V}(f(X))$?
If $X \sim \text{N}(0, \sigma^2)$ it is well-known that we have the second-order Taylor approximation:
$$\mathbb{V}[f(X)]
\approx f'(\mu)^2 \cdot \sigma^2 + \frac{f''(\mu)^2}{2} \cdot \sigma^4.$$
...
0
votes
0answers
25 views
Probability of a random variable being positive
Suppose $x \sim p(x); x \in R $ is a random variable. Here I do not assume any family distribution of $x$ like x is Gaussian or exponential distribution. I would like to find a clear form of a ...
1
vote
1answer
204 views
How to present Confidence Interval for Log-Transformed Means & Mean Difference?
After trying to read on this topic, I still have some clarifications remaining.
Context:
Comparing between 2 arms (categorical), measuring microbiological plate-counted bacteria concentration (...
0
votes
1answer
698 views
How to reduce kurtosis of data
I'm trying to reduce the kurtosis of my dataset and make it approximately Gaussian, with a common-sense uni-modal shape. The raw data looks like this:
I first tried ...
1
vote
0answers
63 views
What is the laplace transform of the below given PDF?
Really am interesting to know more about statistical properties of the following PDF , of the Random variable $z$:
$$F(\sigma,\mu,z)= \frac{(z-\sigma )^2 \exp \left(-\frac{(z-\sigma )^2
\sqrt{\left(...
1
vote
0answers
30 views
How to transform/convert likelihoods to scores?
I have the probability of loan default for a labeled dataset where the distribution of probabilities is heavily skewed. Labels are defined as "good/0" for no default and "bad/1" for defaults. My goal ...
2
votes
1answer
83 views
Transformation of Confidence Interval = Confidence Interval of Transformation? [duplicate]
I am wondering about the following situation: I have a confidence interval estimator $\delta(x)=[lb, ub]$, which returns valid a%-confidence intervals for a value $\theta \in \mathbb{R}$ (not ...
3
votes
1answer
178 views
Modelling exchange rates: how to log transform percentage changes?
I'm trying to model an exchange rate to test for extreme values. However, I have percentage changes from day to day. Given some changes are negative, I can't take the logarithm. Any idea how I could ...
1
vote
1answer
35 views
Is SVM RBF applied to both classes?
Lets say i have following 1D data (position on x), color is target class and I need a classifier which classifies green from red:
I decided to use SVM. Data is clearly not linearly separable, so i ...
8
votes
2answers
856 views
How can I obtain a Cauchy distribution from two standard normal distributions?
I am interested in
Let $X\sim N(0,1), Y \sim N(0,1)$ independently. Show $\frac{X}{X+Y}$
is a Cauchy random variable.
My work:
$f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty&...
21
votes
4answers
4k views
Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?
For a uniformly distributed variable between 0 and 1 generated using
rand(1,10000)
this returns 10,000 random numbers between 0 and 1. If you take the mean, it ...
1
vote
0answers
132 views
Can ADVI (Variational Inference) Induce Weak Multi Modality in a system with Uniform Priors, if a Gaussian Variational Family is Used
Question Set Up
If I have a weakly multi modal (see below in the edit) target posterior distribution which I am aiming to approximate using ADVI (Automatic Differentiation Variational Inference) with ...
0
votes
1answer
86 views
Transformation of Uniform Distribution to Real Number Line in ADVI
In the Automatic Differentiation Variational Inference (ADVI) paper, the authors claim to solve the VI problem in a transformed parameter space, which is over $\mathbb{R}$, in order to simplify the ...
2
votes
1answer
140 views
How can I generate 2 sets of variables from different distributions with a correlation between them in r? [duplicate]
I am working in R and would like to generate 40 numbers from $\mathrm{N}(0,1)$ and another 40 from $\mathrm{Uniform}(0,2)$ with a negative correlation (for example: $r = -0.45$) between them. The ...
0
votes
0answers
22 views
How to interpret hourly rate data given in one miniute intervals
I have time series data on natural gas flow, which is in units of "tonnes/hour". But the data are given in one minute intervals (each row represents a single minute of time duration). Here is a sample:...
1
vote
1answer
792 views
What does 'km' transform in cox.zph function mean?
I'm trying to understand how cox.zph function in r programming language works and I find myself not knowing what km transform ...
2
votes
0answers
565 views
Techniques to apply Discrete Wavelet Transform (DWT) to denoise and predict time series
I just started playing with wavelets and have been using this library (https://github.com/rafat/wavelib) to further my understanding and see if 'denoising' the series at all possible levels is ...
4
votes
1answer
1k views
How does the inverse transform method work in discrete r.v.?
In this question How does the inverse transform method work? it's mentioned the general procedure to generate r.v.
U <- runif(1e6)
X <- qnorm(U)
X
How ...
1
vote
1answer
677 views
CDF Variable Transformation
Let $X$ be uniform on $(-1, 2)$ and let $Y = X^2$. Find the pdf of $Y$.
So far I have noted that $F_X(x) = P(X \leq x) = \int_{-1}^x \frac{1}{3} dt = \frac{1}{3}(x+1)$.
Then, since $Y=X^2$, $y \in [...
0
votes
1answer
170 views
Transforming non-normal to normal distribution and back-transform
I would like to transform non-normal distribution to normal distribution, and back-transform to its original state (or at least close to the original state).
From this article, I've read that you can ...
2
votes
0answers
63 views
VaR/inverse cdf of transformation of normal variables
I have the following exercise to solve as good preparation for an exam:
NOTE: $VaR_p(X)$ = Value at risk = $F^{-1}_X(p)$
Consider the bivariate normal random vector $(X_1, X_2)$. The marginals are ...
2
votes
1answer
151 views
In OLS, while using log-log and linear-log transforamtions, is valid to transform some regressors only?
In OLS I was wondering if it is valid to log-transform some regressors only. Specifically, continuous regressors, because it is advised not to transform binary or categorical variables.
For instance, ...
1
vote
1answer
57 views
How to scale between a equal distribution and an empirical distribution
I am not that good at expressing things mathematically, so I'll start with the practical problem right away:
I have a set of four objects: O1, O2, O3, O4.
Now I want to assign a variable that scales ...
2
votes
1answer
5k views
Interpreting adjusted R-squared of a log transformed regression model
I am running a linear regression model where the dependent variable (Y) is log-transformed. I am struggling on how to interpret the adjusted R-squared of this log-transformed model that is meaningful. ...
6
votes
3answers
190 views
General approaches and techniques for developing good explanatory models for nonlinear data
Various recent efforts of mine on modelling some data through logistic regression have been... not successful. While there is still more data to look at, I've been wanting to explore nonlinear ...
1
vote
1answer
120 views
A transformation from uniform random variable to Gaussian mixture
I am attempting to describe a prior_transform for a multivariate Gaussian mixture in order to estimate the evidence integral of that prior convolved with another likelihood distribution. This is ...
0
votes
1answer
78 views
Likelihood of the product of a normal cdf and pdf
Suppose you had a random sample of r.vs X_i , i= 1....n . What is the likelihood of 2 * pdf(x) * cdf(x) , with pdf and cdf of the standard normal distribution?
3
votes
1answer
150 views
Variance after resampling uniformly from a sample from a normal population
I have recently been looking into the Bootstrap, and I was wondering, if I were to have a sample $X=\{x_1,x_2,\dots,x_N\}$ that has $N$ samples, all i.i.d coming from a normal distribution, $N(\mu,\...
