Questions tagged [nonlinear]

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How to test a product of a OLS estimators?

I estimated a vector $\beta $ using OLS $\hat \beta =(\hat e_1 \hat e_2 \hat e_3)$ and I have the covariance matrice $\hat v(\hat \beta )$. How can I test this null hypothesis $H_0 : \hat e_1 \hat ...
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Estimating the coefficients of a non-linear regression

I am trying to estimate the coefficients $\lambda, \alpha, \beta_1, \beta_2, \gamma, \eta$ in the below equation using Python and some financial data $$ \lambda \times \text{(participation %)} \times \...
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Is Ordinal logistic regression linear or nonlinear?

Quick question, is ordinal logistic regression a linear or nonlinear model? Finding different sources supporting the other, and the more I read the more I get confused myself. Perse, it should fall ...
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Advice on fitting curve to three-pooled plant decomposition model in r

I have just finished running a plant decomposition experiment measuring the decomposition of pine needles across climate and lithological types. We have mass loss, plant chemistry data (c, n, labile ...
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How do I fit a nonlinear mixed effects model with multiple observations from the same sample units using nlmer()?

I am trying to fit a nonlinear mixed-effects model with lme4. However, I am not sure how to adequately account for my sampling design. The model is supposed to ...
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Need a predictive (binary outcome) model for a set of binary variables and one continuous variable

I have a data set of about 1600 binary results, that I want to predict from 9 binary variables and a continuous variable. The relationship between the continuous variable and the result variable by ...
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1 answer
58 views

Fitting a cubic-like curve to data in R

I tried using nls() in R to fit the following expression to a set of data: where g=9.8, alpha & B_0 are unknown, a = 0.01, z_0 = 0.3 such that: theory <- as.formula(V~-(9.8)ab*(pi)(0.02^2)(T)(...
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Differentiate One-step ahead prediction, Simulation (or free run simulation), Open Loop architecture and close loop architecture in NARX

I'm trying to figure out the difference of the following terms from one another: One-step ahead prediction; Simulation (or free run simulation); Open Loop architecture; and close loop architecture I'm ...
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20 views

Distribution of product of samples from two Normal distributions [duplicate]

Hello! I have two normally distributed random variables $M \sim N(\mu_1,\sigma_1^2)$ and $V \sim N(\mu_2,\sigma_2^2)$ having physical meaning as $M$ - mass and $V$ - volume. I need to figure out the ...
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Additive error model for non-linear case

I have checked other questions regarding additive noise model, but I could not convince myself for non-linear case. Assume the data vector $\mathbf{d}$ is described by a possibly non-linear ...
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Highly correlated linear and quadratic slope in Growth model; cov matrix not definite positive

I'm running a a Growth model in R using the package lavaan, function growth.mi(), on an imputed dataset with four measurement points. The model estimates an intercept, a linear slope, and a quadratic ...
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Kaplan's Non-Linearity Test

I've been searching for an R package or function that applies the Kaplan non-linearity test to univariate time series but they are nowhere to be found. Such test has been widely applied in the ...
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1 vote
1 answer
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Possible non-linearity pspline Cox model

Our working group ran a Cox regression with a p-spline to model the possible non-linearity of continuous variables. However, I'm a bit confused with the interpretation of the linearity or non-...
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Efficient metrics for measuring interaction and Building features

Long time user of tree models, rediscovered the benefits of Linear models recently. Now I want to transfer as much non linearity as possible into my features so to mimic as much as possible the ...
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Auto-correlation for irregular non-linear sample

I have irregular sampled residuals resulting from solving a non-linear optimization problem. Currently I am gridding them into a regular grid and calculating the auto-correlation to check the ...
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2 votes
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186 views

How to fit logistic regression to circular data?

I've made a script that can do normal logistic regression with sigmoid(linear model). However, I have data that has a circular decision boundary and looks like this. My question is how I can modify ...
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Degrees of freedom for a $\chi^2$ with non-linear polynomial model

I have a $\chi^2$ below for some model function $F$: $$ \chi^2 = \sum_{i=1}^{i=M} \frac{\left(y_{i}-F\left(x_i;\vec{a}\right)\right)^2}{\left(\Delta y_{i}\right)^2} $$ I know that non-linear model ...
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If X^2 is not significant but X is significant, do I have to remove X^2 and run again the regression analysis?

Model DV ~ W+PDSR+Corr+(FAGDP1+FAGDP2)+log(PCGDP)+Exp+Pop+Health FAGDP2=FAGDP1^2 Result: After removing FAGDP2(FAGDP1^2) from the model, FAGDP1 turns to be insignificant Am I right in removing FAGDP^...
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9 votes
2 answers
2k views

RMSE vs MSE loss function - the optimization solutions are equivalent?

If we optimize a function $f$ with respect to loss $L$, which is defined as RMSE; Are we going to get the same solution as optimizing MSE ? Even, if the function $f$ is non-linear (e.g. a neural ...
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31 views

Parameter estimation of a model with exponential almon lag structure

Suppose I have the following model: $$y_t = \beta_0\sum_{i=0}^p w(\delta;i)x_{t-i}$$ Where $\displaystyle w(\delta;i)=\frac{\exp(\delta_1 i+ \delta_2 i^2)}{\sum_{i=0}^p \exp(\delta_1 i+ \delta_2 i^2)}$...
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Quantile regression with an exponential function

The following equation: y = a*x**b where y is a nonlinear function of x. By taking logs, the equation can be expressed as: ln(y) = ln(a) + bln(x). I would like to run a quantile regression instead of ...
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standar error of non-linear regression

I'm trying to compute the standar error of a non-linear regression fit, I find the answer in this post Non-linear regression confidence interval But i did't find that formula in any other place,and I ...
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"Nonlinear" random spatial field: an example

I want to generate a "nonlinear" random spatial field in the sense that the autocorrelation function in function of the lag/distance $h$, $\rho(h)$, should be not equal to the $R(h)$ ...
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1 vote
2 answers
216 views

Isn't ReLU just a linear function? [duplicate]

I am doing Andew Ng's Deep Learning course and he says that ReLU is better than Sigmoid, but it makes no sense to me at all. The biggest advantage of activation functions are too get a non-linear ...
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2 votes
1 answer
51 views

Proof of how to calculate the coefficient of variation of a nonlinear function

I'm struggling to find the proof of Eq. D.14b shown in the figure below: The variable $b$ represents the “Least Squares” best-fit to the slope between $r_e$ and $r_t$ which both depend on $X_1$ to $...
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Why do we use Relu if it's mostly linear

We use activation functions in neural nets to introduce some non-linearity. Now I understand that Relu is a non-linear function and I had no problems with it. But today I learned that when the output ...
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Simultaneous interaction model with nonlinear equations

I observe many groups of 3 individuals, and I want to estimate an interaction model of the form: $$\begin{array}{rl} y_1&=f_1(x, y-1)+e_1\\ y_2&=f_2(x, y-2)+e_2\\ y_3&=f_3(x, y-3)+e_3 \end{...
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3 votes
1 answer
72 views

Non-normal data transformation - what does it imply exactly and what does my results mean?

I am missing some understanding here. I am inspecting the relationship between the heart rate variability (HRV) and errors in the Sustained Attention to Response Task. When I conduct a basic linear ...
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1 answer
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Bayesian fitting of a nonlinear model [closed]

Some years ago I developed a nonlinear model and java fitting engine that performs well enough to be useful, but is definitely a hack. I would like to modernize and publish an open-source tool (...
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6 votes
2 answers
222 views

Regression with flexible functional form

I am assuming a model of the form $$Y_i=\alpha+\beta X_i+g(\mathbf{Z}_i)+\epsilon_i,$$ here $\mathbf{Z}_i$ is an $m$ dimensional vector and $\epsilon_i$ is i.i.d. white noise. I would like to ...
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1 answer
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Correlation is a symmetric measure, but scatter plot matrix shows asymmetric dependence

The correlation matrix demonstrates that correlation is a symmetric measure: $\rho(X,Y) = \rho(Y,X)$ since the lower off-diagonals are mirror images of the upper off-diagonals. The scatterplot matrix ...
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How to use a scatter plot to detect non-linear co-dependence between two variables?

A scatter plot matrix of 3 time series (financial returns data) are shown below. It is a multivariate representation of a scatter plot, with the individual pairs shown in the off-diagonals. The ...
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4 votes
2 answers
847 views

Can non-linear dependence be detected between two variables by regressing them?

Linear regression is meant for linear relationships right, so, if I don't trust linear correlation and want to find out if random variables $y$ and $x$ have a non-linear relationship, can't I just ...
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17 votes
4 answers
2k views

What is the best programmatic way for determining whether two variables are linearly or non-linearly or not even related

What is the best programmatic way for determining whether two predictor variables are linearly or non-linearly or not even related, maybe using any of the packages scipy/statsmodels or anything else ...
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0 answers
25 views

Does a low correlation level warrant the need to try a different dependence measure?

Can it be determined somehow whether the correlation coefficient between two random variables would be insufficient for assessing the level and type of co-dependency between them, and that another ...
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1 vote
1 answer
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Solving a system of nonlinear equations involving only products of unknowns

I would like to find the solution of a system of equations of the form: $A = w_1 F(x_k) + w_2 F(x_l) + w_3 F(x_m) +...$, where the unknowns are the $w_i$ and the function $F$, while my data are the $A$...
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Stopped by zero step from line search - R stops optimization early

I am trying to minimize an objective function, $J(\theta)$, with respect to $\theta$, a 19-dimensional parameter vector. $J(\theta)$ is a smooth nonlinear function so I have tried various gradient-...
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1 answer
407 views

How come covariance can pick up non-linear relationships but correlation can't? [closed]

correlation is computed from covariance so how come covariance can pick up non-linear relationships between variables $X$ and $Y$ but (Pearson's) correlation can't?
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3 votes
1 answer
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Neural Network: Matlab uses different activation functions for different layers - why?

I have trained on matlab an Artificial Neural Network with one input layer, one hidden layer and one output layer (my output is values between zero and one, which I turn into 0 or 1 according to a ...
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How to decompose entropy into linear and non-linear components?

Mutual information is the entropy between two random variables, $X$ and $Y$, based on their probabilities. It captures both the linear and non-linear interactions between $X$ and $Y$, whereas ...
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3 votes
2 answers
74 views

Non-Linear Time Series Book

I am looking for a book to study non linear time series for both univariate and multivariate topics. Your suggestions will be greatly appreciated. Thanks.
2 votes
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Augmented Dickey Fuller and Nonlinear Time Series

Since the Augmented Dickey Fuller test assumes the time series to be an autoregressive-moving average model, I was wondering if it would be possible to explicitly construct a nonlinear time series ...
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2 votes
0 answers
135 views

How to determine the influence of a variable on the relationship between another two variables?

I have three continuous variables X, Y, and Z in the form of timeseries at several geographic locations. All variables have a skewed distribution because the timeseries mostly have zeros in them ...
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2 votes
0 answers
166 views

What are options if Rainbow test fails after Linear Regression

I tested with commonly available iris dataset and ran OLS with following formula: PW ~ SW + PL + SL + Species Output is as follows: ...
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8 votes
2 answers
1k views

Why not always use generalized estimating equations (GEE) instead of linear mixed models?

I read about generalized estimating equations (GEE) here, here and at other sites. It is mentioned in first of above links that "the parameter estimates are nearly identical" for linear ...
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1 vote
0 answers
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Difference between sequential/simultaneous nonlinear partial least squares and NIPALS algorithm

I've been reading about nonlinear partial least squares, and according to the below study, there are two types of NLPLS: sequential NLPLS and simultaenous NLPLS. https://www.sciencedirect.com/science/...
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2 votes
0 answers
169 views

Nonlinear minimum Mean Square Error estimation

Let say I have the following parameter to estimate: $$ \theta = \frac{1}{\mu^2 - 1} \ .$$ The observed measurement is $x \sim \mathcal{N}(\mu,\sigma)$. The mean $\mu$ is unknown. There are two cases I ...
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0 votes
0 answers
58 views

When does a Linear regression stop being a good fit?

I'm testing a couple of hypotheses and I have a non normally distributed continuous response variable (residuals are not normally distributed as well). I have been given mixed suggestion on what model ...
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4 votes
2 answers
714 views

Effect of nonlinear transformations on the mean

Suppose I have a continuous random variable $X$ and a random variable $Z = f(X)$, where $f$ is a nonlinear monotonic transformation. How can I prove the following relation between the mean and the ...
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34 views

Statistical Testing of Non-linear Model Parameters

I have produced three non-linear 4-parameter logistic curves based on some experimental observations using three different measurement devices. Each curve shows the measurement of density decline ...
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