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Questions tagged [exponential-smoothing]

A basic forecasting technique for time series data, optionally including trend and/or seasonality, but (usually) excluding causal influences.

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Errors and residuals in simple exponential smoothing (state space form) in FPP textbook

I am reading Hyndman & Athanasopoulos "Forecasting: Principles and Practice" 2nd edition (FPP2). (I am aware that 3rd edition exists.) In the chapter about exponential smoothing, section ...
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Adam's $\beta_1$ fixed in practice but required to depend on $t$ for convergence proofs

In the paper ADAM: A METHOD FOR STOCHASTIC OPTIMIZATION, the exponential moving average parameter $\beta_1$ is set to $0.9$ as default in most ML/DL APIs but the convergence proof requires that $\...
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Is the ETS function from forecast appropriate for skewed environmental time series data? does the time-series data need to meet certain criterion? [closed]

I have environmental data that is usually just analysed with the nonparametric man-kendall and sens slope, because it is skewed. Can the ETS function which picks what exponential smoothing model to ...
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ETS Confidence Intervals in R are several orders of magnitude larger than the time series itself?

I am forecasting a time series with confidence intervals using the ets model in R. Here is the time series: Running the following R code: ...
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Estimating Exponential Decay Base

Suppose I have time series of noisy observations $\hat{Y_t}$, with the true underlying but not observable $Y_t$ ~ $N(\hat{Y_t},\sigma)$. I want to estimate $Y_t$ with the basic model $Y = mean(\hat{Y}...
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Heteroscedasticity

I am trying to build a regression model to explain variations in mortgage volumes using variations in different mortgage rates. To account for the drastic change in macroeconomic environment: from ...
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Is Brown's double exponential smoothing a special case of Holt's exponential smoothing?

I am currently reading the book "Forecasting with Exponential Smoothing: The State Space Approach". On the bottom of page 14, it says that “In the special case where $\alpha=\beta^*$, Holt’s ...
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Derivation of Double (Brown) Exponential Smoothing

I am learning the double (Brown) exponential smoothing. Comparing to the simple exponential smoothing, the Brown exponential smoothing smooths the output sequence $Y_t$ twice, such that \begin{align} ...
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I cannot replicate the slope for an ETS model with alpha = 0.5 and beta =0.5 using the model's formula. Misunderstanding or a bug?

When I fit an "AAN" ETS model with alpha and beta both equal to 0.5, the level is correct as per the Level equation, but the slope as independently calculated by the Trend equation is not. I ...
Filby's user avatar
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ETS (error, trend, seasonal) formulation

Does someone know if there is (clever) way to formulate mathematically all the following models below: in a unique (system) of equations?
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Local linear trend-exponential smoothing duality with non-Gaussian likelihood?

Say we have the following state space/structural time series model: $$ y_t \mid \mu, \sigma \sim \text{Normal(} \mu = \mu_t, \sigma^2_\varepsilon) \\ \mu_{t+1} = \mu_t + \eta_t $$ where $$ \eta_t \sim ...
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Interpretation of Moving Average coefficient in ARIMA and its equivalance with exponential smoothing parameter

On Running an ARIMA (0,0,1) model from statsmodels.api on a differenced time series say X(t). I get the following output const = 5.34e-06 ma.L1 = 0.8934 How should I interpret the above MA model with ...
Math lover's user avatar
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Trying to understand log-likelihood estimation for exponential smoothing models in R forecast function ets()

I am doing work on AIC comparisons. For this purpose, I am trying to understand how log-likelihood is calculated for exponential smoothing models (ETS models) in different R packages. In particular, <...
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Estimating the decay parameter in Exponentially Weighted Moving Average (EWMA) model

Given the data $y_t$, $t=1, \cdots ,N$; I would like to estimate the decay parameter $\lambda$ in Exponentially Weighted Moving Average (EWMA) model, such that $y_{t+1} = \sum_{k=0}^K \lambda^k y_{t - ...
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Pre-processing (standardization and exponential smoothing) in expanding window cross-validation for time series data

I am currently working on a time series forecasting task using an expanding window cross-validation approach. My dataset is created using the sliding window technique (Window: 4, Horizon: 1). During ...
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Switching between alpha, half-life and span in exponential moving average

I have been reviewing documentation here on exponential moving averages. I am having a hard time being able to analytically move between the definition of an exponential moving average specified in ...
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Detailed derivation of Multiplicative and Additive Holts & Winter triple exponential smoothing forecasting variance

I'm not able to found any website or textbook or scientific paper with the detailed derivation of of Multiplicative and Additive Holts & Winter triple exponential smoothing forecasting variance... ...
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Exponential Averaging using Simple averaging

Mathematically what is the expression that is close to a 10 Day Exponential moving average (span of 10 means decay factor of 0.818181) that is created using averaging over Simple moving averages. E.g. ...
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Holt-Winters forecasting method

Holt-Winters forecasting equations for quarterly observations are $\alpha_t = \alpha.\frac{y_t}{s_{t-4}} + (1-\alpha) . (\alpha_{t-1} + g_{t-1} )$ $g_t=\gamma.(\alpha_{t}-\alpha_{t-1})+(1-\gamma).g_{t-...
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exponential moving average taking into account different time intervals

i want to calculate the exponential moving average with the following formula EMAt = valt * α + EMAt - 1 * (1 - α) but i don't have all the data, i only have some data with a big gap in time. while ...
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Confusion between the meaning of seasonality and seasonal patterns in time-series forecasting

According to Forecasting: Principles and Practice Seasonality is always of a fixed and known frequency. If it is a fixed and known frequency, does that mean every series with monthly or quarterly ...
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Why does Exponential Smoothing model on all data produces NAN forecasts when given parameters from Training and Testing model in Python s

I am using statsmodels.tsa.holtwinters.ExponentialSmoothing to perform Holt Winters' Additive method, first on training dataset and later on the whole dataset. After training and testing, I take the ...
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Forecasting - Review Volume & Average Rating

I just want to confirm - our review volume has an upward trend but no seasonality. I can use holt's exponential smoothing. Right? for average ratings, no trend, and no seasonality. simple exponential ...
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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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What is $h$ in the Holt-Winters model as denoted in Hyndman, R.J., & Athanasopoulos, G. (2018)?

The Holt-Winters additive method model is defined to be \begin{align*} \hat{y}_{t+h|t} &= \ell_{t} + hb_{t} + s_{t+h-m(k+1)} \\ \ell_{t} &= \alpha(y_{t} - s_{t-m}) + (1 - \alpha)(\ell_{t-1}...
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How to fix exponential smoothing straight line with R

I'm a novice in using R and in forecasting. Right now I'm using a dataset with daily precipitation(mm) data from 2001 to mid 2022. Using STL decomposition seems to suggest the data has a yearly ...
Guilherme's user avatar
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How to use the initial values computed for the Holt-Winters model to update the model to time t=n?

I have followed the technique for determining the initial level, trend and seasonal components for the Holt-Winters model as detailed by Rob Hyndman on https://robjhyndman.com/hyndsight/hw-...
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Calculating the Standard Deviation for EWMA

I'm building an EWMA chart based on the predicted mortality of patients admitted to hospital. I'm working with some previous R code to calculate the standard deviation of the EWMA which work, but I ...
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Exponential time series model usecases

When do we need to use following ETS models: additive seasonality model, multiplicative seasonality model, additive error model, multiplicative error model? Is there any study available regarding when ...
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Time series model in production - Re-train on the fly as as batch process?

Let's say I've a time series of phone calls per day over the last three years. I could train a model using exponential smoothing (e.g. HoltWinters) for predicting the future amount of phone calls per ...
Constantin Müller's user avatar
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Tuning ARIMA/ETS for univariate time series

I am running auto.arima/ETS models from the forecast package in R on monthly seasonal time series. I see the following fitted ...
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517 views

Variance for exponential smoothing

I want to obtain the analytical expression of variance for simple exponential smoothing . Please help verify and see if the expression could be further simplified , thanks . Assume the discrete time ...
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Does seasonality frequency matter in exponential smoothing?

I'm pretty new to time-series forecasting and I hope that someone could help me out. Referring to Holt-Winters' multiplicative method in https://otexts.com/fpp2/holt-winters.html, how does frequency ...
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Exponential fitting in R with fixed minimal value

I need approximate datapoints by exponential function with some type of lower limit (variable "y" is price in time and I need fixed minimal value, so asymptote of exponential function cant ...
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Good libraries for exponential time series smoothing

I've a pandas series which contains the daily load consumption of a city for a year. I wish to forecast the load consumption for next year.As a result , I'm making use of exponential time series. The ...
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Why does the level update equation change in triple exponential smoothing (but the trend equation does not)?

I'm confused on the level and seasonal update equations in Holt-Winters (aka Triple) Exponential Smoothing. Namely, the equations are as follows (in additive form): Level: $l_t =\alpha(y_t - s_{t-m}) +...
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How is the decay rate in exponential smoothing optimized?

For the sake of simplicity, I just want to focus on single/level exponential smoothing. When alpha, the decay rate, is near 1, the most recent observation has the highest weight and influence of ...
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How is the recursive function in exponential smoothing evaluated?

In exponential smoothing models, the most recent observation is weighted most heavily, while observations further back receive a smaller and smaller portion of weight. An alpha parameter will inform ...
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Correct form of multiplicative Holt-Winters

In forecasting principles and practice, the update equations are given as: $$l_t = \alpha\frac{y_t}{s_{t-m}} + (1-\alpha)(l_{t-1} + b_{t-1})$$ $$b_t = \beta^*(l_t-l_{t-1}) + (1-\beta^*)b_{t-1}$$ $$s_t ...
ignoring_gravity's user avatar
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1 answer
512 views

Problem with ets function diagnostic for model with trend and seasonality

I have been meaning to fit an exponential smoothing model to a monthly series that looks like the one below: When I decompose the series it is almost evident that we have seasonality and also there ...
Anoushiravan R's user avatar
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What to do when model fails Ljung-Box test?

I have been learning time-series forecasting recently and I am trying to understand the procedure. I would like to find the best model for a daily time series, so far I tried exponential smoothing ...
Anoushiravan R's user avatar
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Estimating average of a random variable with lower variance is faster to converge than a variable with higher variance?

I have a fallowing problem: I have to estimate average of 2 random variables $X$ and $Y$ based on average of two other. Where the $X$ and $Y$ are some $n$ by $n$ matrices. $$ X = 0.5\cdot(A+B) $$ $$ Y ...
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Hyperparameters of Exponential Smoothing

I have a large number of time series to forecast on which I would like to evaluate the potential of Exponential Smoothing. However, I am faced with a problem of parameter selection: I dont know how to ...
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Appropriate forecasting methods for only 20 observations [duplicate]

I am trying to forecast the regional GDP growth of our region in the next five years, I only have 20 observations in my data which is yearly, what forecasting model is appropriate? I tried ARIMA in r ...
Anisah's user avatar
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Differentiating data before exponential smoothing?

I know that to perform exponential smoothing you don't have to make your time series stationary, but I seem to get better results when I do it. Do you know anything about it?
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(How) Can you combine moving average and exponential smoothing filters to get smoother trends?

Goal: Have a machine perform smoothing on time series data set to have a smooth looking trend. The correctness of the trend is balanced by being optimal i.e., minimize the MSE (Mean Standard Error). ...
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Point Estimates using forecast in R for Multi-Step TS Forecast -- Sometimes Same/Sometimes Not -- Why?

I am using the simple forecast(data, h = 6) function in R - as I work through Hyndman's 'Forecasting: Principles and Practice" textbook - which returns forecasts from the ETS algorithm. I'm not ...
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Why does the exponential moving average equation divide with 1+(1-⍺)+...?

I am trying to understand an exponential moving average, reading its Wikipedia page: https://en.wikipedia.org/wiki/Moving_average#Weighted_moving_average In the middle of the explanation, the page ...
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What does span mean in exponential moving average?

To borrow from the documentation of pandas' ewm function: the center of mass, span, halflife and alpha of an exponential moving ...
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ets() function does not minimize AICc?

I have a question that is similar to this question: ETS function in forecast package is not choosing minimized AICc I see what the author of that question misunderstood but I basically have a reverse ...
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