From 9f0fda47a7d2f79fa428958740a878c565ee7ead Mon Sep 17 00:00:00 2001 From: "Daniel J. McDonald" Date: Mon, 10 Apr 2023 11:34:16 -0700 Subject: [PATCH] add time series lecture --- .../time-series/execute-results/html.json | 18 + .../time-series/figure-revealjs/plot-it-1.png | Bin 0 -> 205494 bytes .../figure-revealjs/plot-smooth-1.png | Bin 0 -> 135877 bytes .../figure-revealjs/unnamed-chunk-2-1.png | Bin 0 -> 326948 bytes .../figure-revealjs/unnamed-chunk-6-1.png | Bin 0 -> 147734 bytes index.qmd | 5 +- slides/time-series.qmd | 461 ++++++++++++++++++ .../figure-revealjs/plot-it-1.png | Bin 0 -> 205494 bytes .../figure-revealjs/plot-smooth-1.png | Bin 0 -> 135877 bytes .../figure-revealjs/unnamed-chunk-2-1.png | Bin 0 -> 326948 bytes .../figure-revealjs/unnamed-chunk-6-1.png | Bin 0 -> 147734 bytes 11 files changed, 481 insertions(+), 3 deletions(-) create mode 100644 _freeze/slides/time-series/execute-results/html.json create mode 100644 _freeze/slides/time-series/figure-revealjs/plot-it-1.png create mode 100644 _freeze/slides/time-series/figure-revealjs/plot-smooth-1.png create mode 100644 _freeze/slides/time-series/figure-revealjs/unnamed-chunk-2-1.png create mode 100644 _freeze/slides/time-series/figure-revealjs/unnamed-chunk-6-1.png create mode 100644 slides/time-series.qmd create mode 100644 slides/time-series_files/figure-revealjs/plot-it-1.png create mode 100644 slides/time-series_files/figure-revealjs/plot-smooth-1.png create mode 100644 slides/time-series_files/figure-revealjs/unnamed-chunk-2-1.png create mode 100644 slides/time-series_files/figure-revealjs/unnamed-chunk-6-1.png diff --git a/_freeze/slides/time-series/execute-results/html.json b/_freeze/slides/time-series/execute-results/html.json new file mode 100644 index 0000000..782c9db --- /dev/null +++ b/_freeze/slides/time-series/execute-results/html.json @@ -0,0 +1,18 @@ +{ + "hash": "2e8fe82b2b0a0e932ec57a2b93084dc7", + "result": { + "markdown": "---\ntitle: \"Time series, a whirlwind\"\nformat: revealjs\nknitr: \n opts_chunk: \n cache: true\n---\n\n\n\n\n## The general linear process\n\n* Imagine that there is a noise process\n\n$$\\epsilon_j \\sim \\textrm{N}(0, 1),\\ \\textrm{i.i.d.}$$\n\n* At time $i$, we observe the sum of all past noise\n\n$$y_i = \\sum_{j=-\\infty}^0 a_{i+j} \\epsilon_j$$\n\n* Without some conditions on $\\{a_k\\}_{k=-\\infty}^0$ this process will \"run away\"\n* The result is \"non-stationary\" and difficult to analyze.\n* Stationary means (roughly) that the marginal distribution of $y_i$ does not change with $i$.\n\n## Chasing stationarity\n\n\n::: {.cell hash='time-series_cache/revealjs/unnamed-chunk-1_6ffa35a74d509b91c6ce299ee24dfed3'}\n\n```{.r .cell-code}\nn <- 1000\nset.seed(12345)\nnseq <- 5\ngenerate_ar <- function(id, n, b) {\n y <- double(n)\n y[1] <- rnorm(1)\n for (i in 2:n) y[i] <- b * y[i - 1] + rnorm(1)\n tibble(time = 1:n, y = y, id = id)\n}\nstationary <- map_dfr(1:nseq, ~ generate_ar(.x, n = n, b = .99))\nnon_stationary <- map_dfr(1:nseq, ~ generate_ar(.x, n = n, b = 1.01))\n```\n:::\n\n::: {.cell layout-align=\"center\" hash='time-series_cache/revealjs/unnamed-chunk-2_63defcfeb1a8516a5be3d1485626fb58'}\n::: {.cell-output-display}\n![](time-series_files/figure-revealjs/unnamed-chunk-2-1.png){fig-align='center' width=1152}\n:::\n:::\n\n\n## Uses of stationarity\n\n* Lots of types (weak, strong, in-mean, wide-sense,...)\n* not required for modelling / forecasting\n* But assuming stationarity gives some important guarantees\n* Usually work with stationary processes\n\n## Standard models {.smaller}\n\n### AR(p)\n\nSuppose $\\epsilon_i$ are i.i.d. N(0, 1) (distn is convenient, but not required)\n\n$$y_i = \\mu + a_1 y_{i-1} + \\cdots + a_p y_{i-p} + \\epsilon_i$$\n\n* This is a special case of the general linear process\n* You can recursively substitute this defn into itself to get that equation\n\nEasy to estimate the `a`'s given a realization. \n\n\n::: {.cell hash='time-series_cache/revealjs/unnamed-chunk-3_5c4d7d14513b8eeb69692b46ce5025a9'}\n\n```{.r .cell-code}\ny <- arima.sim(list(ar = c(.7, -.1)), n = 1000)\nY <- y[3:1000]\nX <- cbind(lag1 = y[2:999], lag2 = y[1:998])\nsummary(lm(Y ~ X + 0))\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n\nCall:\nlm(formula = Y ~ X + 0)\n\nResiduals:\n Min 1Q Median 3Q Max \n-3.6164 -0.6638 0.0271 0.6456 3.8367 \n\nCoefficients:\n Estimate Std. Error t value Pr(>|t|) \nXlag1 0.66931 0.03167 21.134 <2e-16 ***\nXlag2 -0.04856 0.03167 -1.533 0.126 \n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\nResidual standard error: 0.9899 on 996 degrees of freedom\nMultiple R-squared: 0.4085,\tAdjusted R-squared: 0.4073 \nF-statistic: 344 on 2 and 996 DF, p-value: < 2.2e-16\n```\n:::\n:::\n\n\n## AR(p)\n\n* The estimate isn't [that]{.secondary} accurate because the residuals (not the $\\epsilon$'s) are correlated. \n* (Usually, you get `1/n` convergence, here you don't.)\n* Also, this isn't the MLE. The likelihood includes $p(y_1)$, $p(y_2 | y_1)$ which `lm()` ignored.\n* The Std. Errors are unjustified.\n* But that was easy to do.\n* The correct way is\n\n\n::: {.cell hash='time-series_cache/revealjs/unnamed-chunk-4_6f2097b1b1d750be0eac51c9a6da3fb2'}\n::: {.cell-output .cell-output-stdout}\n```\n\nCall:\narima(x = y, order = c(2, 0, 0), include.mean = FALSE)\n\nCoefficients:\n ar1 ar2\n 0.6686 -0.0485\ns.e. 0.0316 0.0316\n\nsigma^2 estimated as 0.9765: log likelihood = -1407.34, aic = 2820.67\n```\n:::\n:::\n\n\n* The resulting estimates and SEs are identical, AFAICS.\n\n## MA(q)\n\nStart with the general linear process, but truncate the infinite sum.\n\n$$y_i = \\sum_{j=-q}^0 a_{i+j} \\epsilon_j$$\n\n* This is termed a \"moving average\" process.\n* though $a_0 + \\cdots a_{-q}$ don't sum to 1.\n* Can't write this easily as a `lm()`\n\n\n::: {.cell hash='time-series_cache/revealjs/unnamed-chunk-5_40a116039422e68a0129e2caf6236ddd'}\n\n```{.r .cell-code}\ny <- arima.sim(list(ma = c(.9, .6, .1)), n = 1000)\narima(y, c(0, 0, 3), include.mean = FALSE)\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n\nCall:\narima(x = y, order = c(0, 0, 3), include.mean = FALSE)\n\nCoefficients:\n ma1 ma2 ma3\n 0.9092 0.6069 0.1198\ns.e. 0.0313 0.0380 0.0311\n\nsigma^2 estimated as 0.8763: log likelihood = -1353.41, aic = 2714.82\n```\n:::\n:::\n\n\n## MA(q) as an AR(1) hidden process\n\nLet $X_j = [\\epsilon_{j-1},\\ \\ldots,\\ \\epsilon_{j-q}]$ and write\n\n$$\n\\begin{aligned}\nX_i &= \\begin{bmatrix} a_{i-1} & a_{i-2} & \\cdots & a_{i-q}\\\\ 1 & 0 & \\cdots & 0\\\\ & & \\ddots \\\\ 0 & 0 & \\cdots & 1\\end{bmatrix} X_{i-1} + \n\\begin{bmatrix} a_{i}\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix} \\epsilon_i\\\\\ny_i &= \\begin{bmatrix} 1 & 0 & \\cdots 0 \\end{bmatrix} X_i\n\\end{aligned}\n$$\n\n* Now $X$ is a $q$-dimensional AR(1) (but we don't see it)\n* $y$ is deterministic conditional on $X$\n* This is the usual way these are estimated using a State-Space Model\n* Many time series models have multiple equivalent representations\n\n## AR[I]{.secondary}MA\n\n* We've been using `arima()` and `arima.sim()`, so what is left?\n* The \"I\" means \"integrated\"\n* If, for example, we can write $z_i = y_i - y_{i-1}$ and $z$ follows an ARMA(p, q), we say $y$ follows an ARIMA(p, 1, q).\n* The middle term is the degree of differencing\n\n## Other standard models\n\nSuppose we can write \n\n$$\ny_i = T_i + S_i + W_i\n$$\n\nThis is the \"classical\" decomposition of $y$ into a Trend + Seasonal + Noise.\n\nYou can estimate this with a \"Basic Structural Time Series Model\" using `StrucTS()`.\n\nA related, though slightly different model is called the STL decomposition, estimated with `stl()`.\n\nThis is \"Seasonal Decomposition of Time Series by Loess\"\n\n(LOESS is \"locally estimated scatterplot smoothing\" named/proposed independently by Bill Cleveland though originally proposed about 15 years earlier and called the Savitsky-Golay Filter)\n\n## Quick example\n\n\n::: {.cell layout-align=\"center\" hash='time-series_cache/revealjs/unnamed-chunk-6_d97b9f35fac8193dadc47553fcbe08f3'}\n\n```{.r .cell-code}\nsts <- StructTS(AirPassengers)\nbc <- stl(AirPassengers, \"periodic\") # use sin/cos to represent the seasonal\ntibble(time = seq(as.Date(\"1949-01-01\"), as.Date(\"1960-12-31\"), by = \"month\"),\n AP = AirPassengers, StrucTS = fitted(sts)[,1], STL = rowSums(bc$time.series[,1:2])) %>%\n pivot_longer(-time) %>%\n ggplot(aes(time, value, color = name)) +\n geom_line() +\n theme_bw(base_size = 24) +\n scale_color_viridis_d(name = \"\") +\n theme(legend.position = \"bottom\")\n```\n\n::: {.cell-output-display}\n![](time-series_files/figure-revealjs/unnamed-chunk-6-1.png){fig-align='center' width=960}\n:::\n:::\n\n\n## Generic state space model\n\n:::: {.columns}\n::: {.column width=\"50%\"}\n\n


\n\n$$\\begin{aligned} x_k &\\sim p(x_k | x_{k-1}) \\\\ y_k &\\sim p(y_k | x_k)\\end{aligned}$$\n\n:::\n::: {.column width=\"50%\"}\n\n* $x_k$ is unobserved, dimension $n$\n\n* $y_k$ is observed, dimension $m$\n\n* $x$ process is the [transition]{.tertiary} or [process]{.tertiary} equation\n\n* $y$ is the [observation]{.tertiary} or [measurement]{.tertiary} equation\n\n* Both are probability distributions that can depend on parameters $\\theta$\n\n* For now, assume $\\theta$ is [KNOWN]{.secondary}\n\n* We can allow the densities to vary with time.\n\n:::\n::::\n\n\n\n## GOAL(s)\n\n1. Filtering: given observations, find $$p(x_k | y_1,\\ldots y_k)$$\n\n1. Smoothing: given observations, find $$p(x_k | y_1,\\ldots y_T), \\;\\;\\ k < T$$\n\n1. Forecasting: given observations, find $$p(y_{k+1} | y_1,\\ldots,y_k)$$\n\n\n## Using Bayes Rule\n\nAssume $p(x_0)$ is known\n\n$$\n\\begin{aligned}\np(y_1,\\ldots,y_T\\ |\\ x_1, \\ldots, x_T) &= \\prod_{k=1}^T p(y_k | x_k)\\\\\np(x_0,\\ldots,x_T) &= p(x_0) \\prod_{k=1}^T p(x_k | x_{k-1})\\\\\np(x_0,\\ldots,x_T\\ |\\ y_1,\\ldots,y_T) &= \\frac{p(y_1,\\ldots,y_T\\ |\\ x_1, \\ldots, x_T)p(x_0,\\ldots,x_T)}{p(y_1,\\ldots,y_T)}\\\\ &\\propto p(y_1,\\ldots,y_T\\ |\\ x_1, \\ldots, x_T)p(x_0,\\ldots,x_T)\\end{aligned}\n$$\n\nIn principle, if things are nice, you can compute this posterior (thinking of $x$ as unknown parameters)\n\nBut in practice, computing a big multivariate posterior like this is computationally ill-advised.\n\n\n\n## Generic filtering\n\n* Recursively build up $p(x_k | y_1,\\ldots y_k)$.\n\n* Why? Because if we're collecting data in real time, this is all we need to make forecasts for future data.\n\n$$\\begin{aligned} &p(y_{T+1} | y_1,\\ldots,y_T)\\\\ &= p(y_{T+1} | x_{T+1}, y_1,\\ldots,y_T)\\\\ &= p(y_{T+1} | x_{T+1} )p(x_{T+1} | y_1,\\ldots,y_T)\\\\ &= p(y_{T+1} | x_{T+1} )p(x_{T+1} | x_T) p(x_T | y_1,\\ldots,y_T)\\end{aligned}$$\n\n* Can continue to iterate if I want to predict $h$ steps ahead\n\n$$\\begin{aligned} &p(y_{T+h} | y_1,\\ldots,y_T)= p(y_{T+h} | x_{T+h} )\\prod_{j=0}^{h-1} p(x_{T+j+1} | x_{T+j}) p(x_T | y_1,\\ldots,y_T)\\end{aligned}$$\n\n\n\n## The filtering recursion\n\n1. Initialization. Fix $p(x_0)$.\n\nIterate the following for $k=1,\\ldots,T$:\n\n2. Predict. $$p(x_k | y_{k-1}) = \\int p(x_k | x_{k-1}) p(x_{k-1} | y_1,\\ldots, y_{k-1})dx_{k-1}.$$\n\n3. Update. $$p(x_k | y_1,\\ldots,y_k) = \\frac{p(y_k | x_k)p(x_k | y_1,\\ldots,y_{k-1})}{p(y_1,\\ldots,y_k)}$$\n\n\nIn general, this is somewhat annoying because these integrals may be challenging to solve.\n\nBut with some creativity, we can use Monte Carlo for everything.\n\n\n\n## What if we make lots of assumptions?\n\nAssume that $$\\begin{aligned}p(x_0) &= N(m_0, P_0) \\\\ p_k(x_k\\ |\\ x_{k-1}) &= N(A_{k-1}x_{k-1},\\ Q_{k-1})\\\\ p_k(y_k\\ |\\ x_k) &= N(H_k x_k,\\ R_k)\\end{aligned}.$$\n\nThen [all the ugly integrals have closed-form representations]{.secondary} by properties of conditional Gaussian distributions.\n\n## Closed-form representations\n\n:::: {.columns}\n::: {.column width=\"50%\"}\n\nDistributions:\n\n$$\n\\begin{aligned}\np(x_k | y_1,\\ldots,y_{k-1}) &= N(m^{-}_k, P^{-}_k)\\\\\np(x_k | y_1,\\ldots,y_{k}) &= N(m_k, P_k)\\\\\np(y_{k} | y_1,\\ldots,y_{k-1}) &= N(H_k m^-_k, S_k)\\\\\n\\end{aligned}\n$$\nPrediction:\n$$\n\\begin{aligned}\nm^-_k &= A_{k-1}m_{k-1}\\\\\nP^-_k &= A_{k-1}P_{k-1}A^\\mathsf{T}_{k-1} + Q_{k-1}\n\\end{aligned}\n$$\n\n:::\n::: {.column width=\"50%\"}\n\nUpdate:\n$$\n\\begin{aligned}\nv_k &= y_k - H_k m_k^-\\\\\nS_k &= H_k P_k^- H_k^\\mathsf{T} + R_k\\\\\nK_k &= P^-_k H_k^\\mathsf{T} S_k^{-1}\\\\\nm_k &= m^-_k + K_{k}v_{k}\\\\\nP_k &= P^-_k - K_k S_k K_k^\\mathsf{T}\n\\end{aligned}\n$$\n\n:::\n::::\n\n\n\n## Code or it isn't real (Kalman Filter)\n\n:::: {.columns}\n::: {.column width=\"50%\"}\n\n\n\n::: {.cell hash='time-series_cache/revealjs/kalman-filter_92f068993b3a8e47da9e73680f6cd594'}\n\n```{.r .cell-code}\nkalman <- function(y, m0, P0, A, Q, H, R) {\n n <- length(y)\n m <- double(n+1)\n P <- double(n+1)\n m[1] <- m0\n P[1] <- P0\n for (k in seq(n)) {\n mm <- A * m[k]\n Pm <- A * P[k] * A + Q\n v <- y[k] - H * mm\n S <- H * Pm * H + R\n K <- Pm * H / S\n m[k+1] <- mm + K * v\n P[k+1] <- Pm - K * S * K\n }\n tibble(t = 1:n, m = m[-1], P = P[-1])\n}\n\nset.seed(2022-06-01)\nx <- double(100)\nfor (k in 2:100) x[k] = x[k - 1] + rnorm(1)\ny <- x + rnorm(100, sd = 1)\nkf <- kalman(y, 0, 5, 1, 1, 1, 1)\n```\n:::\n\n\n:::\n::: {.column width=\"50%\"}\n\n\n\n::: {.cell layout-align=\"center\" hash='time-series_cache/revealjs/plot-it_148dfd48b73002b1b4905d52eb0f31d2'}\n::: {.cell-output-display}\n![](time-series_files/figure-revealjs/plot-it-1.png){fig-align='center' width=768}\n:::\n:::\n\n\n:::\n::::\n\n\n## Important notes\n\n* So far, we assumed all parameters were known.\n* In reality, we had 6: `m0, P0, A, Q, H, R`\n* I sort of also think of `x` as \"parameters\" in the Bayesian sense\n* By that I mean, \"latent variables for which we have prior distributions\"\n* What if we want to estimate them?\n\n[Bayesian way]{.tertiary}: `m0` and `P0` are already the parameters of for the prior on `x1`. Put priors on the other 4.\n\n[Frequentist way]{.tertiary}: Just maximize the likelihood. Can technically take \n`P0` $\\rightarrow\\infty$ to remove it and `m0`\n\n. . .\n\nThe Likelihood is produced as a by-product of the Kalman Filter. \n\n$$-\\ell(\\theta) = \\sum_{k=1}^T \\left(v_k^\\mathsf{T}S_k^{-1}v_k + \\log |S_k| + m \\log 2\\pi\\right)$$\n\n\n\n## Smoothing \n\n* We also want $p(x_k | y_1,\\ldots,y_{T})$\n* Filtering went \"forward\" in time. At the end we got, \n$p(x_T | y_1,\\ldots,y_{T})$. Smoothing starts there and goes \"backward\"\n* For \"everything linear Gaussian\", this is again \"easy\"\n* Set $m_T^s = m_T$, $P_T^s = P_T$. \n* For $k = T-1,\\ldots,1$, \n\n\n$$\\begin{aligned}\nG_k &= P_k A_k^\\mathsf{T} [P_{k+1}^-]^{-1}\\\\\nm_k^s &= m_k + G_k(m_{k+1}^s - m_{k+1}^-)\\\\\nP_k^s &= P_k + G_k(P_{k+1}^s - P_{k+1}^-)G_k^\\mathsf{T}\\\\\nx_k | y_1,\\ldots,y_T &= N(m^s_k, P_k^s)\n\\end{aligned}$$\n\n\n## Comparing the filter and the smoother\n\n* Same data, different code (using a package)\n\n\n::: {.cell hash='time-series_cache/revealjs/kalman-smoothing_ccceb6e0c50936da76aff1a4b2891dab'}\n\n```{.r .cell-code}\nlibrary(FKF)\nfilt <- fkf(\n a0 = 0, P0 = matrix(5), dt = matrix(0), ct = matrix(0), \n Tt = matrix(1), Zt = matrix(1), HHt = matrix(1), GGt = matrix(1), \n yt = matrix(y, ncol = length(y)))\nsmo <- fks(filt)\n```\n:::\n\n::: {.cell layout-align=\"center\" hash='time-series_cache/revealjs/plot-smooth_6ca57c1d680d726f2dc16c5fae7b99c5'}\n::: {.cell-output-display}\n![](time-series_files/figure-revealjs/plot-smooth-1.png){fig-align='center' width=960}\n:::\n:::\n\n\n\n\n## What about non-linear and/or non-Gaussian\n\n$$\\begin{aligned} x_k &\\sim p(x_k | x_{k-1}) \\\\ y_k &\\sim p(y_k | x_k)\\end{aligned}$$\n\nThen we need to solve integrals. This is a pain. We approximate them. \n\nThese all give [approximations to the filtering distribution]{.secondary}\n\n* Extended Kalman filter - basically do a Taylor approximation, then do Kalman like\n* Uncented Kalman filter - Approximate integrals with Sigma points\n* Particle filter - Sequential Monte Carlo\n* Bootstrap filter (simple version of SMC)\n* Laplace Gaussian filter - Do a Laplace approximation to the distributions\n\n\n## The bootstrap filter\n\n* Need to **simulate** from the transition distribution (`rtrans`)\n* Need to **evaluate** the observation distribution (`dobs`)\n\n\n::: {.cell hash='time-series_cache/revealjs/bootstrap-filter_bec4a7eae231c7447bd05087a76a9dec'}\n\n```{.r .cell-code}\nboot_filter <- \n function(y, B = 1000, rtrans, dobs, a0 = 0, P0 = 1, perturb = function(x) x) {\n n <- length(y)\n filter_est <- matrix(0, n, B)\n predict_est <- matrix(0, n, B)\n init <- rnorm(B, a0, P0)\n filter_est[1, ] = init\n for (i in seq(n)) {\n raw_w <- dobs(y[i], filter_est[i, ])\n w <- raw_w / sum(raw_w)\n selection <- sample.int(B, replace = TRUE, prob = w)\n filter_est[i, ] <- perturb(filter_est[i, selection])\n predict_est[i, ] <- rtrans(filter_est[i, ])\n if (i < n) filter_est[i + 1, ] <- predict_est[i, ]\n }\n list(filt = filter_est, pred = predict_est)\n }\n```\n:::\n", + "supporting": [], + "filters": [ + "rmarkdown/pagebreak.lua" + ], + "includes": { + "include-after-body": [ + "\n\n\n" + ] + }, + "engineDependencies": {}, + "preserve": {}, + "postProcess": true + } +} \ No newline at end of file diff --git a/_freeze/slides/time-series/figure-revealjs/plot-it-1.png b/_freeze/slides/time-series/figure-revealjs/plot-it-1.png new file mode 100644 index 0000000000000000000000000000000000000000..8d6199198f3600b92b55bbdd7a4435f569634f27 GIT binary patch literal 205494 zcmeFZby!qw_cpwT4hazi0R`zs>1HfKQbIZuknYYIZd8yGq@+<01nKUO?(USvp*v^Z z?W5k`58uc4`}aGJ@BITDhnc<kBFZgmI;T#S6SjO-W*j$X@WaYEbfRY% z(Ksk`IhO|Ki&~ePnnau$y!!~e`#9M{qu8RDAXE!q>n1j~Xj^7R56g~Friff+X-L5I 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[Unit tests](slides/unit-tests.qmd) 1. [Bootstrap](slides/bootstrap.qmd) 1. [Cluster computing](slides/cluster-computing.qmd) -1. Model selection -1. Testing non-nested models, contrasts, dummy variables, interpreting coefficients in non-Gaussian regression. -1. Regularization and modern methods +1. [Time series](slides/time-series.qmd) +1. Causal inference 1. Reading statistical papers diff --git a/slides/time-series.qmd b/slides/time-series.qmd new file mode 100644 index 0000000..30b01b5 --- /dev/null +++ b/slides/time-series.qmd @@ -0,0 +1,461 @@ +--- +title: "Time series, a whirlwind" +format: revealjs +knitr: + opts_chunk: + cache: true +--- + +```{r setup, include=FALSE} +source("qmd-config.R") +``` + +## The general linear process + +* Imagine that there is a noise process + +$$\epsilon_j \sim \textrm{N}(0, 1),\ \textrm{i.i.d.}$$ + +* At time $i$, we observe the sum of all past noise + +$$y_i = \sum_{j=-\infty}^0 a_{i+j} \epsilon_j$$ + +* Without some conditions on $\{a_k\}_{k=-\infty}^0$ this process will "run away" +* The result is "non-stationary" and difficult to analyze. +* Stationary means (roughly) that the marginal distribution of $y_i$ does not change with $i$. + +## Chasing stationarity + +```{r, echo=TRUE} +n <- 1000 +set.seed(12345) +nseq <- 5 +generate_ar <- function(id, n, b) { + y <- double(n) + y[1] <- rnorm(1) + for (i in 2:n) y[i] <- b * y[i - 1] + rnorm(1) + tibble(time = 1:n, y = y, id = id) +} +stationary <- map_dfr(1:nseq, ~ generate_ar(.x, n = n, b = .99)) +non_stationary <- map_dfr(1:nseq, ~ generate_ar(.x, n = n, b = 1.01)) +``` + +```{r, echo = FALSE, fig.align='center', fig.width=12} +bind_rows(stationary = stationary, + non_stationary = non_stationary, .id = "type") %>% + mutate(replicate = as.factor(id)) %>% + ggplot(aes(time, y, color = replicate)) + + geom_line() + + theme_bw(base_size = 24) + + facet_wrap(~ type, scales = "free_y") + + scale_color_viridis_d() + + theme(legend.position = "bottom") +``` + +## Uses of stationarity + +* Lots of types (weak, strong, in-mean, wide-sense,...) +* not required for modelling / forecasting +* But assuming stationarity gives some important guarantees +* Usually work with stationary processes + +## Standard models {.smaller} + +### AR(p) + +Suppose $\epsilon_i$ are i.i.d. N(0, 1) (distn is convenient, but not required) + +$$y_i = \mu + a_1 y_{i-1} + \cdots + a_p y_{i-p} + \epsilon_i$$ + +* This is a special case of the general linear process +* You can recursively substitute this defn into itself to get that equation + +Easy to estimate the `a`'s given a realization. + +```{r, echo=TRUE} +y <- arima.sim(list(ar = c(.7, -.1)), n = 1000) +Y <- y[3:1000] +X <- cbind(lag1 = y[2:999], lag2 = y[1:998]) +summary(lm(Y ~ X + 0)) +``` + +## AR(p) + +* The estimate isn't [that]{.secondary} accurate because the residuals (not the $\epsilon$'s) are correlated. +* (Usually, you get `1/n` convergence, here you don't.) +* Also, this isn't the MLE. The likelihood includes $p(y_1)$, $p(y_2 | y_1)$ which `lm()` ignored. +* The Std. Errors are unjustified. +* But that was easy to do. +* The correct way is + +```{r} +arima(y, c(2, 0, 0), include.mean = FALSE) +``` + +* The resulting estimates and SEs are identical, AFAICS. + +## MA(q) + +Start with the general linear process, but truncate the infinite sum. + +$$y_i = \sum_{j=-q}^0 a_{i+j} \epsilon_j$$ + +* This is termed a "moving average" process. +* though $a_0 + \cdots a_{-q}$ don't sum to 1. +* Can't write this easily as a `lm()` + +```{r, echo = TRUE} +y <- arima.sim(list(ma = c(.9, .6, .1)), n = 1000) +arima(y, c(0, 0, 3), include.mean = FALSE) +``` + +## MA(q) as an AR(1) hidden process + +Let $X_j = [\epsilon_{j-1},\ \ldots,\ \epsilon_{j-q}]$ and write + +$$ +\begin{aligned} +X_i &= \begin{bmatrix} a_{i-1} & a_{i-2} & \cdots & a_{i-q}\\ 1 & 0 & \cdots & 0\\ & & \ddots \\ 0 & 0 & \cdots & 1\end{bmatrix} X_{i-1} + +\begin{bmatrix} a_{i}\\ 0 \\ \vdots \\ 0 \end{bmatrix} \epsilon_i\\ +y_i &= \begin{bmatrix} 1 & 0 & \cdots 0 \end{bmatrix} X_i +\end{aligned} +$$ + +* Now $X$ is a $q$-dimensional AR(1) (but we don't see it) +* $y$ is deterministic conditional on $X$ +* This is the usual way these are estimated using a State-Space Model +* Many time series models have multiple equivalent representations + +## AR[I]{.secondary}MA + +* We've been using `arima()` and `arima.sim()`, so what is left? +* The "I" means "integrated" +* If, for example, we can write $z_i = y_i - y_{i-1}$ and $z$ follows an ARMA(p, q), we say $y$ follows an ARIMA(p, 1, q). +* The middle term is the degree of differencing + +## Other standard models + +Suppose we can write + +$$ +y_i = T_i + S_i + W_i +$$ + +This is the "classical" decomposition of $y$ into a Trend + Seasonal + Noise. + +You can estimate this with a "Basic Structural Time Series Model" using `StrucTS()`. + +A related, though slightly different model is called the STL decomposition, estimated with `stl()`. + +This is "Seasonal Decomposition of Time Series by Loess" + +(LOESS is "locally estimated scatterplot smoothing" named/proposed independently by Bill Cleveland though originally proposed about 15 years earlier and called the Savitsky-Golay Filter) + +## Quick example + +```{r, echo = TRUE, fig.align='center', fig.width=10, fig.height=4} +sts <- StructTS(AirPassengers) +bc <- stl(AirPassengers, "periodic") # use sin/cos to represent the seasonal +tibble(time = seq(as.Date("1949-01-01"), as.Date("1960-12-31"), by = "month"), + AP = AirPassengers, StrucTS = fitted(sts)[,1], STL = rowSums(bc$time.series[,1:2])) %>% + pivot_longer(-time) %>% + ggplot(aes(time, value, color = name)) + + geom_line() + + theme_bw(base_size = 24) + + scale_color_viridis_d(name = "") + + theme(legend.position = "bottom") +``` + +## Generic state space model + +:::: {.columns} +::: {.column width="50%"} + +


+ +$$\begin{aligned} x_k &\sim p(x_k | x_{k-1}) \\ y_k &\sim p(y_k | x_k)\end{aligned}$$ + +::: +::: {.column width="50%"} + +* $x_k$ is unobserved, dimension $n$ + +* $y_k$ is observed, dimension $m$ + +* $x$ process is the [transition]{.tertiary} or [process]{.tertiary} equation + +* $y$ is the [observation]{.tertiary} or [measurement]{.tertiary} equation + +* Both are probability distributions that can depend on parameters $\theta$ + +* For now, assume $\theta$ is [KNOWN]{.secondary} + +* We can allow the densities to vary with time. + +::: +:::: + + + +## GOAL(s) + +1. Filtering: given observations, find $$p(x_k | y_1,\ldots y_k)$$ + +1. Smoothing: given observations, find $$p(x_k | y_1,\ldots y_T), \;\;\ k < T$$ + +1. Forecasting: given observations, find $$p(y_{k+1} | y_1,\ldots,y_k)$$ + + +## Using Bayes Rule + +Assume $p(x_0)$ is known + +$$ +\begin{aligned} +p(y_1,\ldots,y_T\ |\ x_1, \ldots, x_T) &= \prod_{k=1}^T p(y_k | x_k)\\ +p(x_0,\ldots,x_T) &= p(x_0) \prod_{k=1}^T p(x_k | x_{k-1})\\ +p(x_0,\ldots,x_T\ |\ y_1,\ldots,y_T) &= \frac{p(y_1,\ldots,y_T\ |\ x_1, \ldots, x_T)p(x_0,\ldots,x_T)}{p(y_1,\ldots,y_T)}\\ &\propto p(y_1,\ldots,y_T\ |\ x_1, \ldots, x_T)p(x_0,\ldots,x_T)\end{aligned} +$$ + +In principle, if things are nice, you can compute this posterior (thinking of $x$ as unknown parameters) + +But in practice, computing a big multivariate posterior like this is computationally ill-advised. + + + +## Generic filtering + +* Recursively build up $p(x_k | y_1,\ldots y_k)$. + +* Why? Because if we're collecting data in real time, this is all we need to make forecasts for future data. + +$$\begin{aligned} &p(y_{T+1} | y_1,\ldots,y_T)\\ &= p(y_{T+1} | x_{T+1}, y_1,\ldots,y_T)\\ &= p(y_{T+1} | x_{T+1} )p(x_{T+1} | y_1,\ldots,y_T)\\ &= p(y_{T+1} | x_{T+1} )p(x_{T+1} | x_T) p(x_T | y_1,\ldots,y_T)\end{aligned}$$ + +* Can continue to iterate if I want to predict $h$ steps ahead + +$$\begin{aligned} &p(y_{T+h} | y_1,\ldots,y_T)= p(y_{T+h} | x_{T+h} )\prod_{j=0}^{h-1} p(x_{T+j+1} | x_{T+j}) p(x_T | y_1,\ldots,y_T)\end{aligned}$$ + + + +## The filtering recursion + +1. Initialization. Fix $p(x_0)$. + +Iterate the following for $k=1,\ldots,T$: + +2. Predict. $$p(x_k | y_{k-1}) = \int p(x_k | x_{k-1}) p(x_{k-1} | y_1,\ldots, y_{k-1})dx_{k-1}.$$ + +3. Update. $$p(x_k | y_1,\ldots,y_k) = \frac{p(y_k | x_k)p(x_k | y_1,\ldots,y_{k-1})}{p(y_1,\ldots,y_k)}$$ + + +In general, this is somewhat annoying because these integrals may be challenging to solve. + +But with some creativity, we can use Monte Carlo for everything. + + + +## What if we make lots of assumptions? + +Assume that $$\begin{aligned}p(x_0) &= N(m_0, P_0) \\ p_k(x_k\ |\ x_{k-1}) &= N(A_{k-1}x_{k-1},\ Q_{k-1})\\ p_k(y_k\ |\ x_k) &= N(H_k x_k,\ R_k)\end{aligned}.$$ + +Then [all the ugly integrals have closed-form representations]{.secondary} by properties of conditional Gaussian distributions. + +## Closed-form representations + +:::: {.columns} +::: {.column width="50%"} + +Distributions: + +$$ +\begin{aligned} +p(x_k | y_1,\ldots,y_{k-1}) &= N(m^{-}_k, P^{-}_k)\\ +p(x_k | y_1,\ldots,y_{k}) &= N(m_k, P_k)\\ +p(y_{k} | y_1,\ldots,y_{k-1}) &= N(H_k m^-_k, S_k)\\ +\end{aligned} +$$ +Prediction: +$$ +\begin{aligned} +m^-_k &= A_{k-1}m_{k-1}\\ +P^-_k &= A_{k-1}P_{k-1}A^\mathsf{T}_{k-1} + Q_{k-1} +\end{aligned} +$$ + +::: +::: {.column width="50%"} + +Update: +$$ +\begin{aligned} +v_k &= y_k - H_k m_k^-\\ +S_k &= H_k P_k^- H_k^\mathsf{T} + R_k\\ +K_k &= P^-_k H_k^\mathsf{T} S_k^{-1}\\ +m_k &= m^-_k + K_{k}v_{k}\\ +P_k &= P^-_k - K_k S_k K_k^\mathsf{T} +\end{aligned} +$$ + +::: +:::: + + + +## Code or it isn't real (Kalman Filter) + +:::: {.columns} +::: {.column width="50%"} + + +```{r kalman-filter, echo=TRUE} +kalman <- function(y, m0, P0, A, Q, H, R) { + n <- length(y) + m <- double(n+1) + P <- double(n+1) + m[1] <- m0 + P[1] <- P0 + for (k in seq(n)) { + mm <- A * m[k] + Pm <- A * P[k] * A + Q + v <- y[k] - H * mm + S <- H * Pm * H + R + K <- Pm * H / S + m[k+1] <- mm + K * v + P[k+1] <- Pm - K * S * K + } + tibble(t = 1:n, m = m[-1], P = P[-1]) +} + +set.seed(2022-06-01) +x <- double(100) +for (k in 2:100) x[k] = x[k - 1] + rnorm(1) +y <- x + rnorm(100, sd = 1) +kf <- kalman(y, 0, 5, 1, 1, 1, 1) +``` + +::: +::: {.column width="50%"} + + +```{r plot-it, fig.height=6, fig.width=8, fig.align='center'} +ggplot(bind_cols(kf, x = x, y = y), aes(t)) + + geom_ribbon(aes(ymax = m + 2 * P, ymin = m - 2 * P), alpha = .2) + + geom_line(aes(y = m, color = "kf_mean"), linewidth = 2) + + geom_line(aes(y = x, color = "x"), linewidth = 2) + + geom_point(aes(y = y, color = "y"), size = 2) + + theme_bw(base_size = 24) + + scale_color_manual( + name = "", + breaks = c("kf_mean", "x", "y"), + values = c(kf_mean = primary, x = tertiary, y = secondary)) + + theme(legend.position = "bottom", axis.title=element_blank()) +``` + +::: +:::: + + +## Important notes + +* So far, we assumed all parameters were known. +* In reality, we had 6: `m0, P0, A, Q, H, R` +* I sort of also think of `x` as "parameters" in the Bayesian sense +* By that I mean, "latent variables for which we have prior distributions" +* What if we want to estimate them? + +[Bayesian way]{.tertiary}: `m0` and `P0` are already the parameters of for the prior on `x1`. Put priors on the other 4. + +[Frequentist way]{.tertiary}: Just maximize the likelihood. Can technically take +`P0` $\rightarrow\infty$ to remove it and `m0` + +. . . + +The Likelihood is produced as a by-product of the Kalman Filter. + +$$-\ell(\theta) = \sum_{k=1}^T \left(v_k^\mathsf{T}S_k^{-1}v_k + \log |S_k| + m \log 2\pi\right)$$ + + + +## Smoothing + +* We also want $p(x_k | y_1,\ldots,y_{T})$ +* Filtering went "forward" in time. At the end we got, +$p(x_T | y_1,\ldots,y_{T})$. Smoothing starts there and goes "backward" +* For "everything linear Gaussian", this is again "easy" +* Set $m_T^s = m_T$, $P_T^s = P_T$. +* For $k = T-1,\ldots,1$, + + +$$\begin{aligned} +G_k &= P_k A_k^\mathsf{T} [P_{k+1}^-]^{-1}\\ +m_k^s &= m_k + G_k(m_{k+1}^s - m_{k+1}^-)\\ +P_k^s &= P_k + G_k(P_{k+1}^s - P_{k+1}^-)G_k^\mathsf{T}\\ +x_k | y_1,\ldots,y_T &= N(m^s_k, P_k^s) +\end{aligned}$$ + + +## Comparing the filter and the smoother + +* Same data, different code (using a package) + +```{r kalman-smoothing, echo = TRUE} +library(FKF) +filt <- fkf( + a0 = 0, P0 = matrix(5), dt = matrix(0), ct = matrix(0), + Tt = matrix(1), Zt = matrix(1), HHt = matrix(1), GGt = matrix(1), + yt = matrix(y, ncol = length(y))) +smo <- fks(filt) +``` + +```{r plot-smooth, fig.height=4, fig.align='center'} +tibble(time = seq_along(x), x = x, filter = c(filt$att), smooth = c(smo$ahatt)) %>% + pivot_longer(-time) %>% + ggplot(aes(time, value, color = name)) + + geom_line(linewidth = 2) + + theme_bw(base_size = 24) + + scale_color_viridis_d() + + theme(legend.position = "bottom", axis.title = element_blank(), + legend.title = element_blank()) +``` + + + +## What about non-linear and/or non-Gaussian + +$$\begin{aligned} x_k &\sim p(x_k | x_{k-1}) \\ y_k &\sim p(y_k | x_k)\end{aligned}$$ + +Then we need to solve integrals. This is a pain. We approximate them. + +These all give [approximations to the filtering distribution]{.secondary} + +* Extended Kalman filter - basically do a Taylor approximation, then do Kalman like +* Uncented Kalman filter - Approximate integrals with Sigma points +* Particle filter - Sequential Monte Carlo +* Bootstrap filter (simple version of SMC) +* Laplace Gaussian filter - Do a Laplace approximation to the distributions + + +## The bootstrap filter + +* Need to **simulate** from the transition distribution (`rtrans`) +* Need to **evaluate** the observation distribution (`dobs`) + +```{r bootstrap-filter, echo = TRUE} +boot_filter <- + function(y, B = 1000, rtrans, dobs, a0 = 0, P0 = 1, perturb = function(x) x) { + n <- length(y) + filter_est <- matrix(0, n, B) + predict_est <- matrix(0, n, B) + init <- rnorm(B, a0, P0) + filter_est[1, ] = init + for (i in seq(n)) { + raw_w <- dobs(y[i], filter_est[i, ]) + w <- raw_w / sum(raw_w) + selection <- sample.int(B, replace = TRUE, prob = w) + filter_est[i, ] <- perturb(filter_est[i, selection]) + predict_est[i, ] <- rtrans(filter_est[i, ]) + if (i < n) filter_est[i + 1, ] <- predict_est[i, ] + } + list(filt = filter_est, pred = predict_est) + } +``` diff --git a/slides/time-series_files/figure-revealjs/plot-it-1.png b/slides/time-series_files/figure-revealjs/plot-it-1.png new file mode 100644 index 0000000000000000000000000000000000000000..8d6199198f3600b92b55bbdd7a4435f569634f27 GIT binary patch literal 205494 zcmeFZby!qw_cpwT4hazi0R`zs>1HfKQbIZuknYYIZd8yGq@+<01nKUO?(USvp*v^Z z?W5k`58uc4`}aGJ@BITDhnc<kBFZgmI;T#S6SjO-W*j$X@WaYEbfRY% z(Ksk`IhO|Ki&~ePnnau$y!!~e`#9M{qu8RDAXE!q>n1j~Xj^7R56g~Friff+X-L5I ztFdgM4!U_007=$ds0YWmch-hG`urHTNz^KVcAliM1kj*JBHAwOjSc4sFnGklk`;_> znhd~Rp+D5dz;@i83>D-5oWn}j`KVpsOPnJ8V2$4ytn|^eWz(de(%Ygt-lVhW<%cnB z4$1iBTpM=D0ibfL>MgD+jzCi6$d;&uZ#br_3zD{(AO|p;%Mm zSwA=D2C`xwC)BRmoHy` zf#$u*X=*Mq+9&+!Mc_S-&7g(dt4}i|nlv(#V(woUaF&s_Fp9s{Pq^aLcE|br;M7GU 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