I want to track a single non-stationary real valued variable that increases over time using a Kalman filter. However, I don't know the expected measurement/process/ error covariance apriori.   Is it a bit of an overkill to use a Kalman filter? I had hoped it would do a better job than guessing appropriate parameters for a moving average. However, after playing around with a couple of , it seems to be quite sensitive to incorrect measurement/process covariance assumptions.
122926
I recently implemented a Kalman filter on the simple example of measuring a particles position with a random velocity and acceleration.  I found that Kalman filter worked well, but I then asked myself what's the difference between this and just doing a moving average?  I found that if I used a window of about 10 samples that the moving average outperformed the Kalman filter and I'm trying to find an example of when using a Kalman filter has an advantage to just using the moving average.    I feel like a moving average is far more intuitive than the Kalman filter and you can apply it blindly to the signal without worrying about the state-space mechanism.  I feel like I am missing something fundamental here, and would appreciate any help someone could offer.
72308
I've been reading over this , trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate a dozen or so hours to refreshing my linear algebra knowledge, it's out of my reach for the time being. that being said, I'm looking for a conceptual explanation for that these equations represent: $$\mu_{1|2} = \mu_1 + \Sigma_{1,2} * \Sigma^{-1}_{2,2}(x_2 - \mu_2)$$ I read the first as &quot;Take $\mu1$ and augment it by some factor, which is the covariance scaled by the precision (measure of how closely $X_2$ is clustered about $\mu_2$, maybe?) and projected onto the distance of the specific $x_2$ from $mu_2$.&quot; $$\Sigma_{1|2} = \Sigma_{1,1} - \Sigma_{1,2} * \Sigma^{-1}_{2,2} * \Sigma_{1,2}$$ I read the second as, &quot;take the variance about $\mu_1$ and subtract some factor, which is covariance squared scaled by the precision about $x_2$.&quot; In either case, the precision $\Sigma^{-1}_{2,2}$ seems to be playing a really important role. A few questions:  Am I right to treat precision as a measure of how closely observations are clustered about the expectation? Why is the covariance squared in the latter equation? (Is there a geometric interpretation?) So far, I've been treating $\Sigma_{1,2} * \Sigma^{-1}_{2,2}$ as a ratio, (a/b), and so this ratio acts to scale the (second) $\Sigma_{1,2}$, essentially accounting for/damping the effect of the covariance; I don't know if this is valid. Anything else you'd like to add/clarify?
36588
I was reading an article about the polar motion of the Earth: . A regression is performed between observed and modelled complex time series of data - the type of regression is not specified. The regression coefficient is then calculated for different values of the estimated parameter (quality factor); its amplitude and phase are shown in Figure 2 (I don't know if I am allowed to post it directly here). My question is why can the amplitude of the regression coefficient be as high as 2.3? I thought that the absolute value of the regression coefficient, for real series, had to be in the range 0-1. Therefore I was expecting a maximum amplitude of $\sqrt{2}$ for a complex series. The definition of regression coefficient is not mentioned in the article. At first I thought it may be a linear regression since if the model is correct, the modelled and observed series should be very close; however, this seems incompatible with a value of 2.3. Is there a more general definition that I'm missing?
105373
What are some typical approaches to handling incomplete data in the Kalman Filter? I'm talking about the situation where some elements of the observed vector $y_t$ are missing, distinct from the case where an entire observed vector $y_t$ is missed. Another way of thinking about this would be that the dimension $p$ of the observed vector is different for each time point.  To explain my context a bit further, the observations are estimated parameters from a logistic regression performed at each time point. While each logistic regression includes the same covariates, sometimes the estimates are undefined due to collinearities in the data for that time point.
10191
I am currently investigating the impact of certain indicators such as GDP and inflation on the stock market. However some of my variables are non-stationary and some stationary in levels. All variables are stationary in first differences.  My question:   Since I have a mix of I(0) and I(1), I need to take the first difference of the I(1) variables and then use VAR. Say that I have a dependent variable $y$ which is I(0) and independent variables $x_1$ which is I(1) and $x_2$ which is I(0). Do I take the first differences of only $x_1$ and then apply the VAR model? Is that correct?  If I take the first differences, then might I lose the long-run relationship between the variables? Is ARDL the better method in this case?   If you have any references regarding the above I would be grateful.
83784
I have two series of measurements values, first series is X and second is Y. I need to model Y as a function of X, where I know the method that was used to measure X is two times better then the method used for measure Y in terms of error variance.  I red about regression with error term in the independent variable, but I'm not sure what to do when there is also error term in the dependent variable and they are different.  Which technique/tools/constraints should I use for this sort of case?
113692
In case you suspect your time series has a linear trend, what is the best way to prove it? If you just regress it against time, you ignore the auto correlation of the time series so I assume that is a bit foolish.  We learned to do unit root tests, and in those test you remove the auto correlation by including lags of the time series. So that problem is solved.  But I am struggling a bit with the unit root test which includes a trend (  ur.df(type="trend") in R ) , because the intercept is also included. But this intercept is like a drift, doesn't it remove the significance of the trend? Why don't we test for unit root with only the presence of a trend and not the intercept?
121167
Let’s say you have a 3000 lb vehicle moving very slow at 1 kilometer per hour. It seems obvious that you could simultaneously determine its position and momentum at any point along the way but does the uncertainty principle still apply even here?  You should be able to calculate this at a future point. If not then why not? If so then at what point do you revert back to uncertainty?
83642
I'm trying to use ARIMA process to predict the behaviour of a time series, the probleme I face is that I can't get the order of each component of ARIMA, the lag is between 0 and 1, same goes for the acf. The time series has no tendency, and I guess no seasonality.     My question is : Does this mean that ARIMA orders are all 0s, and if so, is the HoltWinters a good (or the best) solution for modelling this kind of TS ?
109616
Suppose we have $N$ measurable variables, $(a_1, a_2, \ldots, a_N)$, we do a number $M &gt; N$ of measurements, and then wish to perform  on the results to find the axes of highest variance for the $M$ points in $N$-dimensional space.  (Note: assume that the means of $a_i$ have already been subtracted, so $\langle a_i \rangle = 0$ for all $i$.)  Now suppose that one (or more) of the variables has significantly different characteristic magnitude than the rest.  E.g. $a_1$ could have values in the range $10-100$ while the rest could be around $0.1-1$.  This will skew the axis of highest variance towards $a_1$'s axis very much.  The difference in magnitudes might simply be because of an unfortunate choice of unit of measurement (if we're talking about physical data, e.g. kilometres vs metres), but actually the different variables might have totally different dimensions (e.g. weight vs volume), so there might not be any obvious way to choose "comparable" units for them.  Question:  I would like to know if there exist any standard / common ways to normalize the data to avoid this problem.  I am more interested in standard techniques that produce comparable magnitudes for $a_1 - a_N$ for this purpose rather than coming up with something new.  EDIT:  One possibility is to normalize each variable by its standard deviation or something similar.  However, the following issue appears then: let's interpret the data as a point cloud in $N$-dimensional space.  This point cloud can be rotated, and this type of normalization will give different final results (after the SVD) depending on the rotation.  (E.g. in the most extreme case imagine rotating the data precisely to align the principal axes with the main axes.)  I expect there won't be any rotation-invariant way to do this, but I'd appreciate if someone could point me to some discussion of this issue in the literature, especially regarding caveats in the interpretation of the results.
3556
I noticed that sometimes the moving average process is written in two different ways. I would like to know which one is correct.   $y_t = \epsilon_t + \theta \epsilon_{t-1}$  $y_t = \epsilon_t - \theta \epsilon_{t-1}$
57544
I am trying to fit a model using wind data (0, 359) and time of day (0, 23), but I am concerned that they will poorly fit into a linear regression because they are not themselves linear parameters. I would like to transform them using Python. I have seen some mention of calculating a vector mean by way of taking the sin and cos of the degrees, at least in the wind case, but not a whole lot.  Is there a Python library or relevant method that might be helpful?
43549
We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into  $$\boldsymbol\mu = \begin{bmatrix}  \boldsymbol\mu_1 \\  \boldsymbol\mu_2 \end{bmatrix} $$ $${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\  {\boldsymbol y}_2 \end{bmatrix}$$  with a similar partition of $\Sigma$ into $$  \begin{bmatrix} \Sigma_{11} &amp; \Sigma_{12}\\ \Sigma_{21} &amp; \Sigma_{22} \end{bmatrix} $$ Then, $({\boldsymbol y}_1|{\boldsymbol y}_2={\boldsymbol a})$, the conditional distribution of the first partition given the second, is  $\mathcal{N}(\overline{\boldsymbol\mu},\overline{\Sigma})$, with mean $$ \overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\Sigma_{22}}^{-1}({\boldsymbol a}-\boldsymbol\mu_2) $$ and covariance matrix $$ \overline{\Sigma}=\Sigma_{11}-\Sigma_{12}{\Sigma_{22}}^{-1}\Sigma_{21}$$  Actually these results are provided in Wikipedia too, but I have no idea how the $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ is derived. These results are crucial, since they are important statistical formula for deriving Kalman filters. Would anyone provide me a derivation steps of deriving $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ ? Thank you very much!
1439
The denominator of the (unbiased) variance estimator is $n-1$ as there are $n$ observations and only one parameter is being estimated.  $$ \mathbb{V}\left(X\right)=\frac{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}}{n-1} $$  By the same token I wonder why shouldn't the denominator of covariance be $n-2$ when two parameters are being estimated?  $$ \mathbb{Cov}\left(X, Y\right)=\frac{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)\left(Y_{i}-\overline{Y}\right)}{n-1} $$
40177
Let $X,A,B,C,D$ be time-series variables and the covariance between any two pairs of these are known.  Suppose we want to find $\textrm{cov}(X,aA + bB + cC + dD)$, where $a,b,c,d$ are constants.  Is there any way of doing this without expanding out $E[(X-E[X])(aA+......)]$?
123429
Is it possible to extract data points from moving average data?  In other words, if a set of data only has simple moving averages of the previous 30 points, is it possible to extract the original data points?  If so, how?
45122
I have a time series that shows a nonstationary seasonal autoregressive component as well as known heteroshedasticity. In order to model the series, I have fit a seasonal ARIMA model for the mean with the auto.arima model in the forecast package in R and a GARCH model on the residuals of the ARIMA model.   Is the procedure of sequentially estimating ARIMA and GARCH model correct or would it have been better to jointly model the mean and the variance of the series? If this were correct, is there a (possibly R) function to do it?