time series

high-level

basics

• usually assume points are equally spaced
• modeling - for understanding underlying process or predicting
• nice blog, nice tutorial, Time Series for scikit-learn People
• noise, seasonality (regular / predictable fluctuations), trend, cycle
• multiplicative models: time series = trend * seasonality * noise
• additive model: time series = trend + seasonality + noise
• stationarity - mean, variance, and autocorrelation structure do not change over time
• endogenous variable = x = independent variable
• exogenous variable = y = dependent variable
• changepointe detection / Change detection - tries to identify times when the probability distribution of a stochastic process or time series changes

high-level modelling

• common methods
• decomposition - identify each of these components given a time-series
• ex. loess, exponential smoothing
• frequency-based methods - e.g. look at spectral plot
• (AR) autoregressive models - linear regression of current value of one or more prior values of the series
• (MA) moving-average models - require fitting the noise terms
• (ARMA) box-jenkins approach
• moving averages
• simple moving average - just average over a window
• cumulative moving average - mean is calculated using previous mean
• exponential moving average - exponentially weights up more recent points
• prediction (forecasting) models
• autoregressive integrated moving average (arima)
• assumptions: stationary model

similarity measures

• An Empirical Evaluation of Similarity Measures for Time Series Classification (serra et al. 2014)
• lock-step measures (Euclidean distance, or any norm)
• can resample to make them same length
• feature-based measures (Fourier coefficients)
• euclidean distance over all coefs is same as over time-series, but we usually filter out high-freq coefs
• can also use wavelets
• model-based measures (auto-regressive)
• compare coefs of an AR (or ARMA) model
• elastic measures
• dynamic time warping = DTW - optimallt aligns in temporal domaub ti nubunuze accumulated cost
• can also enforce some local window around points
• Every index from the first sequence must be matched with one or more indices from the other sequence and vice versa
• The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match)
• The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match)
• The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i are indices from the first sequence, then there must not be two indices l > k in the other sequence, such that index i is matched with index l and index j is matched with index k , and vice versa
• edit distance EDR
• time-warped edit distance - TWED
• minimum jump cost - MJC

book1 (A course in Time Series Analysis) + book2 (Intro to Time Series and Forecasting)

ch 1

• when errors are dependent, very hard to distinguish noise from signal
• usually in time-series analysis, we begin by de-trending the data and analyzing the residuals
• ex. assume linear trend or quadratic trend and subtract that fit (or could include sin / cos for seasonal behavior)
• ex. look at the differences instead of the points (nth order difference removes nth order polynomial trend). However, taking differences can introduce dependencies in the data
• ex. remove trend using sliding window (maybe with exponential weighting)
• periodogram - in FFT, this looks at the magnitude of the coefficients (but loses the phase information)

ch 2 - stationary time series

• in time series, we never get iid data
• ex. the process has a constant mean (a type of stationarity)
• ex. the dependencies in the time-series are short-term
• autocorrelation plots: plot correlation of series vs series offset by different lags
• formal definitions of stationarity for time series ${X_t}$
• strict stationarity - the distribution is the same across time
• second-order / weak stationarity - mean is constant for all t and, for any t and k, the covariance between $X_t$ and $X_{t+k}$ only depends on the lag difference k
• In other words, there exists a function $c: \mathbb Z \to \mathbb R$ such that for all t and k we have $c(k) = \text{cov} (X_t, X_{t+k})$
•  strict stationary and $E X_T^2 < \infty \implies$ second-order stationary
• ergodic - stronger condition, says samples approach the expectation of functions on the time series: for any function $g$ and shift $\tau_1, … \tau_k$:
• $\frac 1 n \sum_t g(X_t, … X_{t+\tau_k}) \to \mathbb E [g(X_0, …, X_{t+\tau_k} )]$
• causal - can predict given only past values (for Gaussian processes no difference)

ch 3 - linear time series

note: can just assume all have 0 mean (otherwise add a constant)

• AR model $AR(p)$: $X_t = \sum_{i=1}^p \phi_i X_{t-i}+ \varepsilon_t$
• $\phi_1, \ldots, \phi_p$ are parameters
• $\varepsilon_t$ is white noise
•  stationary assumption places constraints on param values (e.g. processes in the $AR(1)$ model with $\phi_1 \ge 1$ are not stationary)
• looks just like linear regression, but is more complex
• if we don’t account for issues, things can go wrong
• model will not be stationary
• model may be misspecified
•  $E(\epsilon_t X_{t-p}) \neq 0$
• this represents a set of difference equations, and as such, must have a solution
•  ex. $AR(1)$ model - if $\phi < 0$, then soln is in terms of past values of {$\epsilon_t$}, otherwise it is in terms of future values
• ex. simulating - if we know $\phi$ and ${\epsilon_t}$, we still need to use the backshift operator to solve for ${ X_t }$
•  ex. $AR(p)$ model - if $\sum_j \phi_j$< 1, and $\mathbb E \epsilon_t < \infty$, then will have a causal stationary solution
• backshift operator $B^kX_t=X_{t-k}$
• solving requires using the backshift operator, because we need to solve for what all the residuals are
• characteristic polynomial $\phi(a) = 1 - \sum_{j=1}^p \phi_j a^j$
• $\phi(B) X_t = \epsilon_t$
• $X_t=\phi(B)^{-1} \epsilon_t$
• can represent $AR(p)$ as a vector $AR(1)$ using the vector $\bar X_t = (X_t, …, X_{t-p+1})$
• note: can reparametrize in terms of frequencies
• MA model $MA(q)$: $X_t = \sum_{i=1}^q \theta_i \varepsilon_{t-i} + \varepsilon_t$
• $\theta_1 … \theta_q$ are params
• $\varepsilon_t$, $\varepsilon_{t-1}$ are white noise error terms
• harder to fit, because the lagged error terms are not visible (also means can’t make preds on new time-series)
• $E[\epsilon_t] = 0$, $Var[\epsilon_t] = 1$
• much harder to estimate these parameters
• $X_t = \theta (B) \epsilon_t$ (assuming $\theta_0=1$)
• ARMA model: $ARMA(p, q)$: $X_t = \sum_{i=1}^p \phi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i} + \varepsilon_t$
• ${X_t}$ is stationary
• $\phi (B) X_t = \theta(B) \varepsilon_t$
• $\phi(B) = 1 - \sum_{j=1}^p \phi_j B^j$
• $\theta(B) = 1 + \sum_{j=1}^{q}\theta_jz^j$
• causal if $\exists { \psi_j }$ such that $X_t = \sum_{j=0}^\infty \psi_j Z_{t-j}$ for all t
• ARIMA model: $ARIMA(p, d, q)$: - generalizes ARMA model to non-stationarity (using differencing)

ch 4 + 8 - the autocovariance function + parameter estimation

• estimation
• pure autoregressive
• Yule-walker
• Burg estimation - minimizing sums of squares of forward and backward one-step prediction errors with respect to the coefficients
• when $q > 0$
• innovations algorithm
• hannan-rissanen algorithm
• autocovariance function: {$\gamma(k): k \in \mathbb Z$} where $\gamma(k) = \text{Cov}(X_{t+h}. X_t) = \mathbb E (X_0 X_k)$ (assuming mean 0)
• Yule-Walker equations (assuming AR(p) process): $\mathbb E (X_t X_{t-k}) = \sum_{j=1}^p \phi_j \mathbb E (X_{t-j} X_{t-k}) + \underbrace{\mathbb E (\epsilon_tX_{t-k})}{=0} = \sum{j=1}^p \phi_j \mathbb E (X_{t-j} X_{t-k})$
• ex. MA covariance becomes 0 with lag > num params
• can rewrite the Yule-Walker equations

• $\gamma(i) = \sum_{j=1}^p \phi_j \gamma(i -j)$
• $\underline\gamma_p = \Gamma_p \underline \phi_p$
• $(\Gamma_p)_{i, j} = \gamma(i - j)$
• $\hat{\Gamma}_p$ is nonegative definite (and nonsingular if there is at least one nonzero $Y_i$)
• $\underline \gamma_p = [\gamma(1), …, \gamma(p)]$
• $\underline \phi_p = (\phi_1, …, \phi_p)$
• this minimizes the mse $\mathbb E [X_{t+1} - \sum_{j=1}^p \phi_j X_{t+1-j}]^2$
• use estimates to solve: $\hat{\underline \phi}_p = \hat \Sigma_p^{-1} \hat{\underline r}_p$
• the innovations algorithm
• set $\hat X_1 = 0$
• innovations = one-step prediction errors $U_n = X_n - \hat X _n$
• mle (ch 5.2)
• eq. 5.2.9: Gaussian likelihood for an ARMA process
• $r_n = \mathbb E[(W_{n+1} - \hat W_{n+1})^2]$

multivariate time-series ch 7

• vector-valued time-series has dependencies between variables across time
• just modeling as univariate fails to take into account possible dependencies between the series