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Questions tagged [smoothing]

Smoothing methods in data analysis, like splines or kernel smoothers, also regression smoothers like lowess.

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How to generate mean curve of non-function?

I am currently working on curves generated in tensile tests of polymer specimens. Here, I try to generate a mean curve of five data sets generated at the same composition of the samples. Unfortunately,...
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294 views

Is there a name for a moving average when it is done not across time but some other variable?

The moving average is defined as A method of smoothing a time series to reduce the effects of random variation and reveal any underlying trend or seasonality. (Oxford Dictionary of Statistics, ed. ...
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Smoothing detected vehicle positions from camera given my own vehicle's location, velocity and acceleration

I have a dashcam which detects the position and estimated depth of other vehicles on the road relative to my own. Given my own vehicle's global co-ordinates, I can convert the detected vehicle's ...
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51 views

Detecting trend in panel data, smoothing techniques and outlier detection

I'm conducting an analysis on a Landsat scene to detect trends for change detection phenomena (forest disturbances) over a time series of 20 years. I identified on the image the pixels that are ...
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30 views

Why 2nd derivative is “squared” to represent wigglyness in GAM?

In David Miller's presentation (here, slide 21), he drew 1st derivative and 2nd derivative of a function. Then he said (slide 22) that grey part can is : $ \int (\frac{\partial ^{2}f(x)}{\partial x^{...
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15 views

How we handle unknown bigrams in bigram probability model with Good-Turing discounting?

Assume Good-Turing discounting. Assume number of unknown words is equal to the number of known words in our event space. Let $s$ be a sentece such that $s=w_1,w_2,\ldots,w_n$. We know the probability ...
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15 views

Smoothing with non-regular observations

If we have data that changes continuously in time, and we sample this data at regular intervals - i.e. we get samples $x_0$, $x_1$, $\dots$, where the time $\Delta t$ between taking samples $x_i$ and $...
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49 views

Retrieving time series from a smoothed periodogram

If I were to smooth a periodogram and then filter out low level frequencies, how can I derive the filtered time series? For example, in the case of a non-smoothed periodogram: https://folk.uib.no/...
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66 views

Natural spline term in GAM

Is it advisable to use natural regression spline basis? I learned that in R the supported smoothers in gam are the lo, ...
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0answers
13 views

Time series regression with temporal smoothing of regressors

I'm trying to forecast a discrete time series $Y_t$ with many observations of a very noisy covariate $X_t$. I could simply use a large number of lags: $$ \mathrm{E}[Y_t] = \beta_0 + \sum_{i=1}^k \...
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22 views

Wavelet smoothing or regression with scattered data

Suppose we have a data set $\{x_i, y_i\}_i$ where $x_i$ is a multi-dimensional tuple and scattered (not on a equally spaced regular lattice). How does one regress or smooth such a scattered data set ...
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27 views

Reproducing Holt-Winters analysis in the Cowpertwait-Metcalfe book

I am having trouble reproducing some output for some R code in the time series book by Cowpertwait and Metcalfe. There are quite a few typos in their code throughout the book, but in other cases I ...
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21 views

Good turing smoothing for unigram LM

I was wondering if it is at all possible to use good turing smoothing for unigram language model? I know that this smoothing technique helps distribute the weights from most occurring words to less ...
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22 views

Transforming a categorical distribution up side down

I have the following categorical distribution: $$ original = (2, 7, 3, 5, 0, 1, 4, 6, 3, 8, 8)\\ \sum original = 47 $$ I want to transform it to its upside down distribution step by step, while ...
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15 views

Computing a moving average when data points arrive one at a time

I am sure there's a cool python/numpy/pandas way to do this. I am receiving one data point at a time. I would like to compute something like a moving average over the last n observations, even better ...
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1answer
75 views

Random effect in GAM - what are the smooth functions used?

In the GAM package in R created by Simon Wood there is a selection of the smooth function basis. I sort of understand the options such as bs='tp', bs='cr', etc. But bs='re' seems odd... that does ...
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30 views

What do the outputs of the function dlmSmooth from R's dlm package mean?

In the vignette for the R package dlm: link on page 12, the author runs the function dlmSmooth to smooth the data and the function returns an object which is ...
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Fixed-Delay Kalman smoother with/without augmented measurements

There are several algorithms regarding fixed-lag Kalman smoothing. In most cases, an augmented state vector is defined in which the elements are the current and delays of the original state vector. ...
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Smoothing of experimental data for PCA

I am applying PCA to a set of spectrophotometric measurements with the aim of differentiating two groups of substances. The many small wiggles on the right-hand side of the curves (the region above ~...
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1answer
46 views

Is a Kalman filter ever the optimal way to estimate a dynamic value given a full history of measurements?

I'm trying to get some intuition for Kalman filtering, and I conceived this toy example: Say that I have a sensor that tracks a moving 1-dimensional target. Say that the measurements from the sensor ...
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4answers
296 views

Measure of smoothness

I have an image that has artefacts which I am using a specific process to remove. I want to show that the new image is improved by that process. To compare the two images I am using data from a ...
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3answers
206 views

How to fit a robust step function to a time series?

I have a somewhat noisy time series that hovers around different levels. For example, the following data: I have the solid line data available, and I would like to obtain an estimate for the dashed ...
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44 views

Is there a filter function that performs similar to moving average but does not loose data?

I need to smooth a time series using a low pass filter. A simple moving average is working fine for me, however, using a moving average causes an inevitable loss of data a the beginning of the ...
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28 views

Expected Value of Cross Validation approximates Predictive Square Error

In the context of Smoothing Splines, Im trying to show that the expected value of the cross-validation can approximate the predictive square error. More specifically, I want to show that $$E[(y_i - \...
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51 views

Natural Splines and Smoother Matrix

In the context of smoothing splines, one can show that the Reinsch form is given by: $ \hat{y} = N (N^{T}N +\lambda \Omega)^{-1}N^{T} y = (I+ \lambda K)^{-1}y $ where (1) $K = (N^{T})^{-1}\Omega N^{-...
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1answer
52 views

How to smooth a curve by learning location and shape of 4 Gaussian kernels?

I have a data set of 365 daily curves $f_i(t)$ and want to smooth them by positioning four radial basis functions $r_i(t)$. Each daily curve should then be approximated by $$ \hat{f}_i(t) = w_{i1} ...
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52 views

RKHS norm and Fourier transform link

In the notes here, it is stated that norms of some reproducing kernel Hilbert spaces can be written in terms of Fourier transforms, and this is often used to argue that a higher RKHS norm implies a ...
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1answer
27 views

Partial spline. Reference

I have a well done and perfectly working protocol to smooth my experimental data. I do the following: I have a variable of size 1000. Iteratively I choose random 100 points and spline them using the ...
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1answer
61 views

What does “middle of the data” really mean?

I have some noisy time series data for different climate variables and I want to know overall if they are increasing or decreasing with time. From this Water Resources Statistics Guide, the LOWESS ...
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1answer
58 views

Show that solution to cubic smoothing spline reduces to regular least squares minimization as $\lambda$ approaches infinity

I am asked to show that the solution to a smoothing splines problem of the form $$ \text{PRSS}(f,\lambda) = \sum_{i=1}^N\left[y_i-f(x_i)\right]^2 + \lambda \int f''(t)^2 dt, $$ with $$ f(x) = \sum_{j=...
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1answer
82 views

Generalized additive models - formula for basis functions

I'm trying to understand the basics of GAMs. Wood's book "Generalized Additive Models: an introduction with R" introduces GAMs via a cubic spline basis $b_j(x)$ (see p. 122), where $b_1(x)=1, b_2(x)=...
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How can I do monotonic smoothing in R? [duplicate]

I would like to create a monotonic smoothing function to transform vectors such that I reduce their spikes (i.e., noise) while maintaining their trend. The problem I am encountering with standard ...
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80 views

Generalized Additive Models: How to fit models with the LMS method by Cole?

I have some problems to understand the univariate generalized models that were proposed by Cole (1992). My question is how the fitting procedure works described in the appendix. More specific, how to ...
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1answer
217 views

Functional Data Analysis and Splines Regression

I'm new to Functional Data Analysis so my question will'be very simple for an expert in this topic. Operatively, when I fit a model like this, in R : model<- lm( y ~ bs( x, df = k, degree = l ) )...
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1answer
105 views

Improve estimate of zeros and how to deal with concurvity in GAM NB

I'm trying to model the number of deaths for each age, for 4 regions, across 5 years. I fitted several models, but end up with the following model: ...
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36 views

Back-Off with Good-Turing Discounting

How do I handle backing off when using Good-Turing discounting? Let's say I want to score $p(A|B)$. If $A$ has been seen with the word $B$ as its history, I can compute the probability as $p(A|B) = \...
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1answer
373 views

Exponential Smoothing & Seasonality

I am relatively new to stats and forecasting and was hoping to tap the wisdom of this community in regards to a question. One of my colleagues insists that exponential smoothing presumes seasonality. ...
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62 views

mgcv optimization: why optimize **log** smooth parameter?

In mgcv package, when optimizing smoothing parameters (in the outer iteration), one take the partial derivative of objective function with respect to $\rho = log(\lambda)$. My question is why not take ...
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1answer
528 views

Find good smoothing spline factor

I'd like an automatic way to find the "best" smoothing factor s for a spline fit to a given set of data points. Here's a sample visualization of some data and the ...
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1answer
55 views

Exchangeability and data smoothing

If a non-i.i.d sequence of a continuous random variable that is exchangeable, is smoothed by taking rolling average, is the resulting sequence exchangeable? My intuition suggests that it is not. I'll ...
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108 views

Estimating signal-to-noise ratio after smoothing data and covariance estimates

I have data for two variables, let's say $A$ and $B$, which have distributions that are roughly centered at 0. I am computing a value, let's say $C = \sqrt{A^2 + B^2}$, from this data. I also have a ...
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3answers
236 views

Positive smoothing with the fda-package (Functional data analysis)

In the book Functional Data Analysis with R (Ramsay&Silverman) there is described the possibility to do the "positive smoothing" if it’s needed instead of the "normal smoothing". In the books ...
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1answer
56 views

STL function in R: is it possible to make it backward looking?

I would like to smooth modeling input time series data using the stl function in R. Based on my understanding, the function performs a local regression using data ...
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60 views

transformation for binary and categorical independent variables

Friends, I have a large dataset in which only Y and one of the independent variables are continuous. there are 12 binary independent variables and 2 other categorical independent variables (each with ...
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1answer
266 views

Probabilistic interpretation of Thin Plate Smoothing Splines

TLDR: Do thin plate regression splines have a probabilistic/Bayesian interpretation? Given input-output pairs $(x_i,y_i)$, $i=1,...,n$; I want to estimate a function $f(\cdot)$ as follows \begin{...
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1answer
239 views

GAM : smoothing splines

Here are 2 questions. Your expertise on those issues would be highly appreciated. In the GAM approach, it makes sense to start with a highly flexible approach and then apply penalties to achieve the ...
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1answer
256 views

Invertibility in Reinsch form Derivation (Smoothing Splines)

In Element of Statistical Learning II, in the context of smoothong splines, $\pmb{S_{\lambda}}$ is defined as $$ \pmb{S_{\lambda}} = \pmb{N}(\pmb{N}^T\pmb{N} + \lambda \pmb{\Omega_N})^{-1}\pmb{N}^T $$ ...
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92 views

some examples of filtering the noise out of a data set

I have a data set which measures 60 data points in a second (60Hz). Clearly, I do not really need all 60 points in one second since this only generates some noise. ABOUT MY DATA: So my data sets ...
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1answer
186 views

GAM automatic smoothness selection [closed]

I'm trying to use gam in MGCV to automatically select my level of smoothness using cross validation or some equivalent method for my generalized additive model. reading around online it looks like ...
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What is the density of $X$ under fixed design?

We observe an i.i.d. sample $(X_1, Y_1), \ldots (X_n, Y_n).$ Let $m(x) = E(Y|X=x),$ $\sigma^2(x) = \operatorname{Var}(Y|X=x)$ and let $f(\cdot)$ be the density of $X.$ Under some regularity ...