# What is Time series: Definition and 63 Discussions

In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
Time series are very frequently plotted via run charts (a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.
Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. Interrupted time series analysis is used to detect changes in the evolution of a time series from before to after some intervention which may affect the underlying variable.
Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility).
Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).

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1. ### I Stationary time series with seasonality...

Hello, I was under the impression that a time series with trend, seasonality, and cyclic component would automatically be nonstationary. Stationarity means constant mean, variance, and autocorrelation (weakly stationary). However, it seems that we could have a stationary time-series that has a...
2. ### I Time series and why would we remove seasonality and trend?

Hello, I understand a few things about time series but I am unclear on other main concepts. Hope you can help me get on the right track. A time series is simply a 1D signal with the variable time ##t## on the horizontal axis and another variable of choice ##X## on the vertical axis. The time...
3. ### I Looking for advice in clusterization

Hello everyone. I have a machine with a series of sensors. All sensors send a signal each minute. I want to know if any of those sensors are redundant. The data is available as an Excel file, where the columns are the variables and the rows are the measurements. I have 1000 rows. To do this, I...
4. ### I Fit a non-linear function to this time series

I have an experimantally obtained time series: n_test(t) with about 5500 data points. Now I assume that this n_test(t) should follow the following equation: n(t) = n_max - (n_max - n_start)*exp(-t/tau). How can I find the values for n_start, n_max and tau so as to find the best fit to the...
5. ### I Regression Prediction with Time Series Data

Hi, I am not sure what the correct forum is for this question. Question: When do we need to remove seasonality from time series data to do a regression analysis? Context: I am planning to conduct a prediction analysis where I want to find out how a device performs. I hope to estimate a...
6. ### Stationary time series Help

Hello, I'm trying to solve this, any idea please? Basically: Demonstrate for the next three processes if the Time Series would be stationary, if not, it should establish the conditions for it to be stationary. Thanks
7. ### A Reduction of Heteroscedasticity in Time Series

Hi, I have some crucial questions belong to statistics: First, How can we derive the variance function with respect to mean for a given data? Secondly, I would like to ask: what method should we employ if the variance in time series behaves like a high order (such as ##𝑎𝑢_𝑡^5+𝑏𝑢_𝑡^4+𝑐𝑢_𝑡^3##...
8. ### A Differential equation involving a time series

Hey all, it's been awhile since done any calculus or DE's but was trying out some modelling (best price price per item for bulk value deals as a function of time and amount), in the last line i have f(n,t) implicitly. Any pointers or techniques for solving such things?
9. ### MHB Book Recommendation, Please: Time Series Analysis

I am looking for a book recommendation. I've been looking for something like this on Amazon without success. I want a book on Time Series analysis that includes the following topics: ARMA/ARIMA, ARCH/GARCH, LSTM and deep learning, filters, state spaces, and any other main categories of...
10. ### I Attractors of a time series

Hi, I am a beginner and I don't speak very well... So I'm really sorry for my poor scientific language... I work on 1-Dimension time series of a same system measured at different periods. In these periods, time series have different chaotic characteristics as their lyapunov exponent are...
11. ### A Interpreting a paper on spectral analysis

I do a fair bit of spectral analysis of time series in my research, but to date my experience in the topic is almost exclusively from an engineering perspective rather than the more statistical approach. Of course I am aware that ultimately they are equivalent, but it means that my familiarity...
12. ### A Robustness of time series analysis

I have a time series model constructed by using ordinary least square (linear). I am supposed to provide some general comments on how one would improve the robustness of the analysis of a time series model (in general). Are there any general advice apart from expanding data, making it more...
13. ### A Reasonable length of forecast horizon in a time series

Suppose we have monthly totals of observed data for last 35 years. That data is of inflow of a river in a reservoir and monthly demands from the reservoir. We are interested to check the effect of construction of a dam in the upstream. The effect is, whether the downstream reservoir will have...
14. ### A Regression analysis and Time Series decomposition

If we can use Regression analysis to forecast, why do we use “Time Series Decomposition”? What's the difference between the 2? Thanks
15. ### Extinction Coefficient from Time series data

I have some time series data of the absorbance of Br2 formation using UV Vis spectroscopy and I need to figure out the extinction coefficient/ absorptivity. The overall reaction is BrO3-+5Br- +6H+-->3Br2+3H2O which is expcted to go to completion I know that the equation relating absorbance to...
16. ### I Extracting characteristics from time series data

hi I have a random set of time series data that is calculated after applying an algorithm to a main random time serie data, and really need to extract all the possible characteristics from the set. The goal is to measure those characteristics and perform some statistical graphs based on those...
17. ### I Mutual information between two time series.

So I am studying chaotic dynamical systems and I need to find mutual information between two chaotic time series say x(t) and y(t). Any help would be much appreciated.
18. ### A Generate Time Series with specific ACF and multiple LAG

Hello Everyone, I will try to explain what am I doing here and I hope someone will understand. ACF - autocorrelation function I'm doing a research about non-parametric methods utility. How they fit and are useful in a different environment. I'm generating time series with different sizes of...
19. ### Time Series - Autoregressive process and Probability Limit

Homework Statement Calculate: PLIM (probability limit) \frac{1}{T} \sum^T_{t=2} u^2_t Y^2_{t-1} Homework Equations Y_t = \rho Y_{t-1} + u_t, t=1,...T, |\rho| <1 which the autoregressive process of order 1 E(u_t) = 0, Var(u_t) = \sigma^2 for t cov(u_j, u_s) = 0 for j \neq s The Attempt...
20. ### Generating 1/f^b colored time series

Please excuse (and ignore) this if this is not the right place to ask this. I am an ecologists and need to generate a time series with a specific color or frequency spectra. I never learned how Fourier transforms work in class and while I get the gist from reading there are so many subtleties...
21. ### Time Series: What's Going On Here?

http://xkcd.com/1473/
22. ### MHB Mean and variance of difference operators on a time series process

\text{Consider the following decomposition of the time series }{Y}_{t}\text{ where }{Y}_{t}={m}_{t}+{\varepsilon}_{t},\text{ where }{\varepsilon}_{t}\text{ is a sequence of i.i.d }\left(0,{\sigma}^{2}\right)\text{ process. Compute the mean and variance of the process }{\nabla}_{2}{Y}_{t}\text{...
23. ### MHB Prove Weakly Stationary Process w/$E(X^2_t ) < ∞$

Not 100% sure if this is the right board. My question is to Show that a strictly stationary process with $E(X^2_t ) < ∞$ is weakly stationary. So weakly stationary implies two things: - the mean value function $u_t$ does not depend on time $t$ and - the autocovariance $\gamma_x(t+h,t)$ is...
24. ### Conditional probability and time series

I really need some help here, will appreciate any effort. I calculated time series of tidal stresses. It turned out that the probability of having positive tidal stress is 0.4 and negative - 0.6 (I counted up number of hours when the stress was positive/negative and divided by the total number...
25. ### MHB Calculating a smooth 90% limit for differences in a time series

I have 50 data sets. Each set has three related time series: fast, medium, slow. My end purpose is simple, I want to generate a number that indicates a relative degree of change of the time series at each point. That relative degree of change should range between 0-1 for all the time series and...
26. ### Probability and time series

hi I'm a physician and really need help from somebody who is good at probability. I calculated time series of tidal stresses. It turned out that the probability of having positive tidal stress is 0.45 and negative - 0.55 (I counted up number of hours when the stress was positive/negative and...
27. ### Wiener Hopf filter in time series.

Homework Statement I have a signal where w(n) is white noise u(n) = .4s(n) +.7s(n-1)-.1s(n-1) +w(n) and where variance of w(n) = .003 I want to find the cross correlation matrix against the optimal delay which I found to be 6. Homework Equations The Attempt at a Solution...
28. ### MHB Time Series: weakly stationary or non-constant ACF?

I am taking an introductory course in Time Series and our initial study of ARMA processes has proven to be challenging for me. The math we are asked to do is quite simple but recognizing various attributes is tricky. We are given 5 time series and are asked to label them from a given list of...
29. ### How to calculate general trendline for a time series

Hi there, Given a time series with data points x_1, x_2, x_3,...,x_n I want to be able to extrapolate its future behaviour. I can see three options: \sum_i (x_(i+1)-x_i)/x_i count of how many terms of (x_(i+1)-x_i) are positive and negative Assume linearity and calculate m for the best...
30. ### Prob/Stats Time Series Analysis by Hamilton

Author: James Douglas Hamilton Title: Time Series Analysis Amazon Link: https://www.amazon.com/dp/0691042896/?tag=pfamazon01-20
31. ### First Post: How to Smooth End point of Finite Data Series time series

I wish it wasn't out of desperation that I'm making this first post! I have a neural network that is making predictions, the next 5 time points per training. Back testing consists of appending these 5 point sets together to produce a data set that spans time over a much longer period...
32. ### Introductory Time Series Analysis Textbook

I'll be taking an introductory course on time series analysis in the spring, and we will be using the instructor's online notes as the "textbook". My previous experiences with such instructor's notes have been that they contain only the essentials of the course and aren't really useful as...
33. ### Time Series Analysis Prep for Undergrad | Self-Study Guide

Hi I am an undergrad interested in learning time series analysis by himself. Other than the obvious prerequisite courses - 2nd year Calculus and Statistics - what else should I teach myself in prepreation for learning time series analysis? Thanks
34. ### Which Dickey-Fuller test should I apply to this time series?

I have a time series of climate data that I'm testing for stationarity. Based on previous research, I expect the model underlying the data to have an intercept term, a positive linear time trend, and some normally distributed error term. In other words, I expect the underlying model to look...
35. ### Time series understanding a proposition

Homework Statement Proposition: If C(0)>0 and C(h)->0 as h-> infinity, then the covariance matrix gamma_n =[c(i-j)] for i,j- 1,2,...n of (x_!,...x_x)' is non singular for every n. I want to convince myself that the converse is not true. (ie I want a counter example of a stationary process...
36. ### High-frequency time series database

I'm choosing a database to write high-frequency time series data onto and have narrowed it down to MongoDB, Kyoto Cabinet or HDF5. I will be inserting 1200 rows of 8 entries per second, cumulating about 5 GB of data per day I'm estimating. Does anyone have experience between the three and...
37. ### Solving stochastic differentials for time series forecasting

I am trying to reproduce results of a paper. The model is: dS = (v-y-\lambda_1)Sdt + \sigma_1Sdz_1 \\ dy = (-\kappa y - \lambda_2)dt + \sigma_2 dz_2 \\ dv = a((\bar{v}-v)-\lambda_3)dt + \sigma_3 dz_3 \\ dz_1dz_2 = \rho_{12}dt \\ dz_1dz_3 = \rho_{13}dt \\ dz_2dz_3 = \rho_{23}dt \\...
38. ### Correlation of time series - is there a better indicator?

Is correlation between 2 time-series a useful indicator? In currency pairs for example, sometimes the correlation between 2 pairs (e.g. EUR/USD and GBP/USD) for the past x days is strong, but it can weaken very fast. Should I just take the average over time and go with it, or is there a better...
39. ### Uncertainty in time series

Hello all, I am new to this forum and was wondering if any of you could help me out. I have this interview scheduled for next week for which I have to prepare a 10 min presentation on Uncertainty and how to determine it in a time series. Now, there is a wealth of information of the net...
40. ### Stochastically find dissimilarity between time series data and model prediction

I have many time series, with each featuring fifteen data points (x,t). I wish to fit a parameterised model to each data series. At present I'm using particle swarm optimisation (PSO) for this purpose. Within my objective function I quantify fitness using the Bray Curtis distance between a)...
41. ### Time Series with Normal curve

I am trying to analyse a past series of numbers that flucuates between 107&210 with a normal frequency distribution of mean 162. What is the way to model and project short term future range for trendless but cyclical type of time series?
42. ### Time series, normal distribution?

Homework Statement If I have a time series model x_1 = my + epsilon_1 x_i = my + a(x_{i-1} - my) + epsilon_i epsilon_i are iid standard normal. Can I then say that y = a_1 x_1+a_2 x_2 + ... a_n x_n is multi normal? The Attempt at a Solution All the x-es are normal...
43. ### Software to extract statistics (per month, per year) from a time series

Hi. I work with rain (precipitation) time series, and would like to extract time statistics: - precipitations per month - precipitations per year - precipitations per month and year - precipitations per hour - precipitations per month and hour - mean precipitation, deviation, ... I...
44. ### MHB Is Weak Stationarity of Y_k Achievable in AR(1) Process?

Consider AR(1) process $$X_t=bX_{t-1}+e_t$$ where $$e_t$$ with mean of 0 and variance of $$\sigma^2$$ and |b| <1 Let $$a_k$$ be a recursive sequence with $$a_1$$ =1 and $$a_{k+1} = a_k + P_k +1$$ for $$k = 1, 2 ,...,$$ where $$P_k$$ is Poisson iid r.v with mean = 1 also, assume $$P_t$$...
45. ### Removal of Polynomials From Raw Aperture Flux Time Series?

Hello! This is my first post so forgive any errors of decorum. :o) I am a student working toward a degree in astrophysics but I'd like to jump a few years ahead when it comes to the study of exoplanets. While examining some data about the new discovery of Kepler-22b, I noticed a plotted data...
46. ### Time Series: ARCH model properties

Consider an ARCH(1) model: Xt = σtZt, where Zt~ i.i.d. N(0,1) σt2 = w0 + w1 Xt-12 Find (i) E(Xt) and (ii) the autocovariance function γX(h) for h=0,1,2,3,..., assuming the process is second-order stationary. Solution: (i) E(Xt) = E[E(Xt|σt2)] =E[E(σtZt|σt2)] =E[σtE(Zt|σt2)] =...
47. ### Time Series: Residuals of ARMA model

Time Series: "Residuals" of ARMA model To check whether the white noise {at} are uncorrelated, we usually look at the residuals (which are sample estimates of the white noise {at}) and residual plots. But I just don't understand the meaning of "residuals" in the context of ARMA model...
48. ### Time Series: Partial Autocorrelation Function (PACF)

Consider a stationary AR(2) process: Xt - Xt-1 + 0.3Xt-2 = 6 + at where {at} is white noise with mean 0 and variance 1. Find the partial autocorrelation function (PACF). I searched a number of time series textbooks, but all of them only described how to find the PACF for an ARMA process...
49. ### Time Series: stationary AR(1) -> MA(infinity)

Theorem: A stationary AR(1) model can be expressed in terms of MA(infinity). Proof: Now I don't understand how they get from the second last line to the last line. Where did the term Yt-m go? I understand you can keep doing the substitution iteratively, but you always have to end...
50. ### Time Series Analysis: 3x Exp. Smoothing

I have 6 years of data which has both a 3-month and a 12-month seasonality, it exhibits a trend and is very noisy. I implemented the triple exponential smoothing procedure and changed my seasonality, trend, and smoothing parameters until the difference between the forecasted data and the actual...