3e:null 3f:["$","title",null,{"children":"Seasonal Trend Loess Filtering | Christian Klettner | Observable"}] 45:T6f1,# Seasonal Trend Loess Filtering ## Using Seasonal loess Loess filter in d3r.regressionLoess is of degree 1 Algorithm -- Inner loop Iteratively calculate trend and seasonal terms. Run ${tex` n_{(i)}`} times: 1. **Detrend**: ${tex` Y_\nu − T_\nu^{(k)}`} where ${tex`k`} is the loop number. In the first pass the trend is calculated using a moving average. 2. **Cycle-subseries smoothing**: The detrended time series is broken into cycle-subseries. For example, monthly data with a periodicity of twelve months would yield twelve cycle-subseries, one of which would be all of the months of January. Each cycle-subseries is then loess smoothed with ${tex`q = n_{(s)}`} and ${tex`d=1`}. The smoothed values yield a temporary seasonal time series ${tex` C^{k+1}`}. 3. **Low-pass filter**: The low pass filter on ${tex` C^{k+1} `} yields ${tex`L^{k+1}`}. This filter is the application of two moving averages of lag equal to three followed by loess filtering with ${tex` q = n_{(l)}`} and ${tex`d=1`}. ${tex`n_{(l)}`} is defaulted the smallest odd integer greater than the period (e.g. 13 for monthly data). The output of the low-pass filter is ${tex`L^{k+1}`}. 4. **Detrending of smoothed cycle-subseries**: ${tex`S^{k+1} = C^{k+1} − L^{k+1}`}. This is the ${tex`k+1`}th estimate of seasonal component. Importantly, the low-pass filter causes this seasonal time series to average to be nearly zero. 5. **Deseasonalizing:** Remove the seasonal trend from the original series ${tex`Y − S^{k+1}`} 6. **Trend smoothing:** Loess smooth the deseasonalized time series with ${tex`q=n_{(t)}`}. Results in ${tex`T^{k+1}`}, the ${tex`k+1`}-th estimate of the trend component. ## Example The time series, Australian Unemployment Rate, appears to have significant seaonality 46:T47d,Plot.plot({ title: "Time Series Trends per Month", height: 400, x: {axis:null}, y: { ticks: 5 }, fy: { label: null, tickFormat: null }, fx: { label: null, tickFormat: null }, marks: [ Plot.gridY([-0.5,0.5,1]), Plot.ruleY([0]), Plot.text(["Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", "Dec"], { x: 0, dx: 10, y: 0, dy: -70, text: d => d, fy: (_,i) => Math.floor(i / 4), fx: (_,i) => i % 4, fontWeight: "bold" }), Plot.ruleY(d3.rollups(firstpass_movav(obs).map((d,i) => ({ dt: obs[i] - d, i: i, c: i % 12 })), v => d3.mean(v, d => d.dt), v=> v.c), { y: d => d[1], fy: (_,i) => Math.floor(i / 4), fx: (_,i) => i % 4, stroke: "blue", strokeWidth: 2 }), Plot.ruleX( d3.rollups(firstpass_movav(obs).map((d,i) => ({ dt: obs[i] - d, i: i, c: i % 12 })), v => detrend_lowpass(v.map(d => d.dt), v.map(d => d.i), bw), v => v.c).flatMap((d,i) => d[1].map(d => ({ ...d, c: i, fy: Math.floor(i / 4), fx: i % 4 }))) , { x: "i", y: "s", fx: "fx", fy: "fy" }) ]})47:[] 43:{"id":"291c23537a608412","slug":null,"title":"Seasonal Trend Loess Filtering","thumbnail":"871f800babaf4880b988414faa23e0de3271b820354f7bee75aedd331aa3b9eb","access_level":"public","create_time":"$D2025-10-24T02:34:52.401Z","update_time":"$D2025-10-25T21:22:31.772Z","view_time":null,"first_public_time":"$D2025-10-25T04:03:22.296Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"1e6bbbc1fea19b9b","login":"klettsch","name":"Christian Klettner","avatar_url":"https://avatars-next.observableusercontent.com/a2034439cf44a563da383de835ee021f6b6b0fe384e6da3aac36176d6f7b8a0a","tier":"starter_2024","bio":"Risk, finance, economic modelling, forecasting and stress testing","following":false},"version":925,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"9c99afc3bb345c32","login":"klettsch","name":"Christian Klettner","avatar_url":"https://avatars-next.observableusercontent.com/0d0f4aae093b36282a186ea737feb66dc7f5bb69d65e30cc5a159c34f0a77f47","tier":null,"bio":"Risk, finance, economic modelling, forecasting and stress testing","following":false}],"members":[{"type":"team","id":"1e6bbbc1fea19b9b","login":"klettsch","name":"Christian Klettner","avatar_url":"https://avatars-next.observableusercontent.com/a2034439cf44a563da383de835ee021f6b6b0fe384e6da3aac36176d6f7b8a0a","tier":"starter_2024","role":"editor"},{"type":"individual","id":"9c99afc3bb345c32","login":"klettsch","name":"Christian Klettner","avatar_url":"https://avatars-next.observableusercontent.com/0d0f4aae093b36282a186ea737feb66dc7f5bb69d65e30cc5a159c34f0a77f47","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Seasonal Trend Loess Filtering","cells":[{"id":0,"value":"$45","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":109,"value":"Plot.lineY(data, { y: \"Original\", x: \"Date\" }).plot() ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":378,"value":"But that seasonality has changed significantly over time. Months which were previous strongly positive are now weaker (Jan). Months that were positive are now negative (Nov). Overall seasonal unemployment in Australia has declined significantly. \n\nThe bandwidth determines the amount of smoothing applied in the seasonal smoother. Constant seasonal adjustment (the blue line) would generally overstate the seasonal effects if applied to current data. Some form of adaptive seasonal calculation is required. The same series decomposed with simple period averages is illustrated [here](https://observablehq.com/d/9f25a55dd68db256) ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":749,"value":"viewof bw = Inputs.range([0,1])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":311,"value":"$46","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":804,"value":"A function to implement the multi-pass loess smoother. This follows the implementation in R of the [stl smoother](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/stl). It is simplified to highlight the impact of trend and seasonal smoothing. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":73,"value":"function stl_npass(data, yvar, timevar, bw_trend, bw_season, npass=3) { \n var obs = data.map(d => d[yvar]); \n //step 1 detrend series \n var detrend = firstpass_movav(obs).map((d,i) => \n ({ i: i, y: obs[i], c: i % 12, t: d, dt: obs[i] - d })); \n var deseasonal = []; \n for (let j = 0; j < npass; j++ ) { \n //step 2 + 3 + 4 cycle subseries smoothing \n deseasonal = d3.rollups(detrend, \n v => detrend_lowpass(v.map(d => d.dt), v.map(d => d.i), bw_season).\n map(d => ({ ...detrend[d.i], ...d, ds: obs[d.i] - d.s })), \n v => v.c).\n flatMap(d => d[1]).\n sort((a,b) => a.i - b.i); // step 5 deseasonal y\n // step 6 detrend the deseasonal series \n detrend = loess_smooth(deseasonal.map(d => d.ds), bw_trend).\n map((d,i) => ({ ...deseasonal[i], t: d, dt: obs[i] - d, n: j })); \n } \n let x = data.map(d => d[timevar]); \n return detrend.map((d,i) => ({ ...d, x: x[i], r: d.dt - d.s })); \n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":858,"value":"To following illustrates the decomposed components of original time series after applying trend and seasonal loess smoothing with the bandwidths below. In general you want a relative low trend bandwidth, so that the series fits the overall long term trend of the data well, and you want a relative high seasonal bandwidth, so that there is not too much change in seasonal patterns over time. You want to pick up structural changes in seasonal patterns, rather than any short term noise. \n\nBandwidth ...","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":816,"value":"viewof bwt = Inputs.form([\n Inputs.range([0,1], { label: \"Trend\", value: 0.1 }), \n Inputs.range([0,1], { label: \"Seasonal\", value: 0.9 }) \n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":807,"value":"stldata = stl_npass(data, \"Original\", \"Date\", bwt[0], bwt[1], 2)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":384,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":877,"value":"Plot.plot({ \n color: { legend: true, \n domain: [\"Original\", \"Deasonal: mean\", \"Deseasonal: loess\"] }, \n marks: [\nPlot.gridY(),\nPlot.ruleY([d3.min(stldata, d => d.y)*0.95]),\nPlot.lineY(stl_1pass(data, \"Date\", \"Original\", bwt[0]), \n { x: \"x\", y: d => d.y - d.s, stroke: () => \"Deasonal: mean\" }),\nPlot.lineY(stldata, { x: \"x\", y: \"y\", stroke: () => \"Original\" }), \nPlot.lineY(stldata, { x: \"x\", y: d => d.y - d.s, stroke: () => \"Deseasonal: loess\" })\n]})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":792,"value":"---- \n\n## Other functions and data \n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":788,"value":"function loess_smooth(x, bw) { \n return d3r.regressionLoess().x((d,i) => i).y(d => d).bandwidth(bw)(x).map(d => d[1]); \n }","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":790,"value":" function firstpass_movav(obs, freq=12) { \n let n = obs.length; \n if (n <= freq) { return \"Time series too short\"; }\n let ma1 = d3.range(freq-1,n-1).map(d => d3.mean(obs.slice(d-(freq-1),d)));\n let ma2 = d3.range(freq,n).map(d => d3.mean(obs.slice(d-(freq-1),d)));\n let mama = d3.range(0,n-freq).map(d => (ma1[d]+ma2[d])/2 ); \n return d3.range(0,n).map(d => d < (freq/2) ? mama[0] : d >= (n-(freq/2)) ? \n mama.slice(-1)[0] : mama[d-(freq/2)]); \n } ","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":802,"value":" function detrend_lowpass(x, i, bw) { \n if (bw <= 1) { \n let c = loess_smooth(x, bw); //step 2\n //let l = lowpass_loess(c); //step 3\n return c.map((d,j) => ({ i: i[j], s: d })); //step 4\n } else { \n return x.map((d,j) => ({ i: i[j], s: d3.mean(x) })); \n }}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":924,"value":"function lowpass_loess(x) {\n let n = x.length; \n let nl = 13/n; \n let ma1= d3.range(n).map(d => d3.mean(x.slice(Math.max(0, d-1), Math.min(n, d+2)))); //3\n let ma2= d3.range(n).map(d => d3.mean(ma1.slice(Math.max(0, d-1), Math.min(n, d+2)))); //3\n return loess_smooth(ma2, 0.9);\n }","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"import {data} from \"49370ac3ca475c45\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":799,"value":"obs = data.map(d => d.Original)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":141,"value":"d3r = require(\"d3-regression\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":890,"value":"import {stl_1pass} from \"9f25a55dd68db256\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W47","stdlib":"1"}} 44:"$43:body"