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You can write each prediction in vector form: $$\hat{x}_{t+1}=\underbrace{\begin{bmatrix}x_t \cdots x_{t-M+1} \end{bmatrix}}_{\mathbf{a}_{t,M}}\begin{bmatrix}\beta_1\\\vdots\\\beta_M\end{bmatrix}=\mathbf{a_{t,M}}\mathbf{\beta}$$ If you concatenate them vertically for each $t$, you'll obtain $A$, i.e. $$\begin{bmatrix}\hat{x}_{M+1} \\\vdots\\ \hat{x}_N\end{... 3 You asked "Is it a general thing that an ARIMA(0,0,0) becomes a MA(1) when the series is differenced?" Yes if the differencing is unwarranted ... as in this example where Y(t) is a white noise series. If Y(t)=A(t) and you difference Y you get [1-B]Y(t)= A(t)[1-1.0B] where B is the backshift operator. thus you have injected structure by differencing and ... 2 @IrishStat's suggestion to use SMAPE is probably the most correct. A log(x+1) transform could also work. It would be useful to know your application. What is the proportion of 0's? I am curious if you have an intermittent time series. Intermittent time series are often modeled with Crostons method or the Poisson distribution, the benchmark forecast ... 1 Another interesting measure that can deal with zero actuals: Relative Total Absolute Error: mean of all absolute errors devided by the mean of the absolute actuals. This measure is a MAPE where the sum is moved into the fraction. An advantage is that the RTAE can deal with zero actuals. That said, SMAPE and the RTAE are useful when there are a few zeroes ... 4 Symmetric mape (SMAPE) will be useful for you . Pursue https://stats.stackexchange.com/search?q=SMAPE . 0 I just looked at the difficulties you seem to have with your ARIMA(0,0,1) model (or is it ARIMA(0,1,1), since you mention differencing?). I'd strongly suggest you compare your model to some very simple alternatives, which surprisingly often are quite competitive with ARIMA and other more complex methods. For instance, an overall mean is an ARIMA(0,0,0) with ... 2 I am not aware of any forecasting methods that work off uncertain inputs, although it's an interesting question. A simple (though possibly prohibitively expensive, performance-wise) method would be to draw a random time series, one observation from each corresponding past density (or a full series from the joint density, whatever you have). Fit a model to ... 1 A VAR(p) model has multiple dependent variables:$$ y_t=A_1 y_{t-1}+\dots+A_p y_{t-p}+\varepsilon_t $$where A_1,\dots,A_p are coefficient matrices. The current values of the dependent variables are y_t; this is a vector of length k: (y_{1,t},\dots,y_{k,t}). The current values depend on past values (vectors) y_{t-1}, \dots, y_{t-p}. In addition, ... 1 For this type of problem, it should be possible to make a prediction of the total donations by predicting the infinite tail of donations, and adding this to the observed donations. To facilitate our analysis, suppose we let M_t denote the donation received on day t, and let U denote the total remaining donations, and V denote the total donations (... 0 I would not classify all of those things as "methods", at least not in the same sense as AR and MA. A Naive forecast could be done in many ways; it is a principle, that forecast accuracy shouldn't be evaluated in a vacuum but rather should be evaluated within a forecast value added framework, comparing each forecasting step back to a "naive approach" and ... 0 Simple Models If you are looking for simple models I would test these: Linear Regression Exponential Smoothing with seasonality Seasonal ARIMA models Rule of thumb methods: seasonal naive method, seasonal mean, simple mean. These models should help you in assessing a baseline for the model performance. I believe models 1-2-3 are already implemented in ... 0 I would start by looking through the excellent free online book Forecasting: Principles and Practice (2nd ed.) by Athanasopoulos & Hyndman. It leverages the equally excellent forecast package for R, which allows you to automatically build ARIMA or Exponential Smoothing models and forecasts very easily. With only 36 data points, there is really not much ... 1 1. I am not sure why using weekly data (around 2500 data points) is giving very high error but using monthly data points (only 31 data points) was giving decent result (although not very good). What am I doing wrong here ? @IrishStat is likely right. The 53rd week issue is rough. It is usually easier to get clear seasonality with monthly data or daily ... 0 with 23 values , I would simply form a useful model using all 23 ... then I would use that model from period 18 ..estimating parameters based upon the 18 historical values and predict the 19th and the 20th .. similarly i would use the useful model and use 19 values to estimate parameters and predict the 20th and the 21st ... everything being reasonable I ... 0 36 months of data can often be useful in forming a model. Initially, you might want to develop arima type model that incorporates trends, level shifts, and memory while dealing with unusual values. Secondly, you might investigate adding user-suggested causal variables to your model. These models are an extension and are often referred to as Dynamic ... 1 I used the 44 values . Visually there appears to be more variability at higher values . This lead the automatic analysis to suggest a log xform using AUTOBOX , a piece of software that I have helped to develop. Following is the Actual/Fit and Forecast from a useful model in log space (3,0,0)(0,0,0) The equation is here and here The model residuals are ... 0 As a general rule you don't want to do piece-fitting using previous results to feed the next stage without a simultaneous optimization. In this case after identifying the regime shift (visually obvious) , we treat the most recent set of values reflecting homogeneous structure. I took your very interesting ( to me ! ) time series which visually appeared to ... 0 I took your 981 daily values and used AUTOBOX ( a piece of forecasting software that I have helped to develop) . The original data visually suggests level shifts ( up at period 560 down at period 801 ) which was confirmed here in a useful model also containing German holiday effects AND monthly effects and here The Actual/Fit and Forecast graph is here ... 0 A good model can detect and effectively replace anomalies using methods called Intervention Detection . See http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html 0 Yes, it is possible but lags can make it less accurate. If your data is periodic over a time interval then I would suggest you to use ARIMA modeling. If your data is non periodic than choose RNN for regression. 1 Typically, loss functions for forecasts are computed for each horizon separately. This is partly because performance usually degrades the further out you go, and any improvements in the short term loss might be difficult to pick up when the long term loss is numerically much larger. Each horizon is effectively a different target and it makes sense to ... 1 ROUND TWO: You asked " how do I do this with the log-link function and quasi(Poisson) errors? )" . I say put aside your priors suggesting a particular fixed model and use a data-driven empirical process to identify the (possible) memory model , refining parameters and testing both necessity and sufficiency . When you only have 29 days ( 4 seasons of daily ... 0 Let x_t denote the value of the time series of interest at time t. Let \Delta x_t:=x_t-x_{t-1} denote the increment in x_t from t-1 to t. Let hats (\widehat{}) denote predictions. If you have x_t and \widehat{\Delta x}_{t+1},\dots,\widehat{\Delta x}_{t+5} available, then it is straightforward to obtain \hat x_{t+5}:$$ \hat x_{t+5}=x_t+\...

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You asked 2 questions ... 1)Could you please help me to see if my prediction method is accurate and how to handle errors in R. I wouldn't think so because you didn't fully extract a sufficient equation as per Help me about using ARIMA forecasting rainfall 2) Could you please help me run the code in R to predict rainfall according to the above data can'...

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I believe that our problem is that we are jumping directly to ARIMA model without trying the traditional models. for this reason, you can find the model is not giving the needed results. In your case, I tested your data, I found that there is a seasonality every 12 months which is clear for you, but also I found that a simple moving average of 3 terms ...

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The problem with using a simple arima approach is that model identification is done when it is based upon the tacit assumption that there is no deterministic structure in the data and model parameters are invariant over time. Your rainfall data (216 monthly values) has two significant seasonal dummies ( October and November) and a few pulses ( 6 of them all ...

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I took your 29 days (oldest to newest) and found that there were 3 unusual days thus the following equation with Actual/Fit and Forecast here All models are wrong ... but some are useful .... . It is fundamentally an autoregressive process of order 1 after one has adjusted for the three "unusual data points " see for a clear support for the anomaly ...

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If you regress your probabilities against your predictor as you suggest, you may find that your fitted model predicts outside of the $[0,1]$ interval for some values of the predictor $X_i$, which makes interpreting your model awkward. You could alternatively logit-transform the probabilities, regress these transformed probabilities against your predictor, ...

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After giving it some thought, I think the straight forward answer is that VAR does not resolve the circular function situation as framed in my question. And, frankly I don't think any other methodology can. You can't use A to forecast B at the same time as you use B to forecast A once you go beyond any related lagged variables. The key is "at the same ...

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To properly answer your question regarding automatic model differences requires a little bit of history https://autobox.com/pdfs/econometrics.pdf to explain some different approaches to ARIMA model identification. Model identification has and always will be an iterative process much like peeling an onion where clues are found and followed and possibly ...

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The 5 predictors future values may be the cause of your "increasing forecast" OR a trend coefficient tat is unwarranted . How are you specifying the values for these (X) series into the future ? Additionally your arima model might be questionable as it has redundant arima structure . Also note that AUTOBOX used differences of all series as triggered by the ...

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Aside from what has been suggested, you might use an state-space model (see packages such as dlm, KFAS and others in R). State-space models are quite tolerant to NA values in the data.

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I'm an idiot; I could have just used a nested foreach loop via %:% to store each pair-wise combination of K-value and each of my series/models in a nested list. By not using the .combine argument, things are left simpler (IMO) to have a nested list where each element of our models can be examined (i.e. checking coefficients, AICc/BIC values). For those who ...

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An answer to a very similar (but not the actual $\color{red}{^*}$) question is, White's Reality Check and Hansen's Superior Predictive Ability (SPA) Test. See Section 17.5.2 in Elliott & Timmermann (2016) for a summary of both tests. Below I give a summary of the summary (consisting in no small part of direct quotes): White (2000) asks how confident we ...

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you have to generate the psi weights which can be obtained by expressing the model as a pure long-lagged ma model https://rdrr.io/cran/tswge/man/psi.weights.wge.html . You also need the estimate of the variance. If you have identified anomalies in the data set you need to employ monte-carlo procedures and re-sample the error process.

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Your daily data is here . This is a cumulative series which will have SIGNIFICANT autocorrelation due to the INDUCED structure as a result of your accumulation. Building an arima model for this is essentially attempting to reverse the user-driven autocorrelation to identify the fixed effects. I took your cumulative and first differenced it to obtain what ...

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