sub-daily data over several years? How to iden fy unusual pa erns/incidents? How to forecast sub-daily data taking account of public holidays and special events? 4
sub-daily data over several years? How to iden fy unusual pa erns/incidents? How to forecast sub-daily data taking account of public holidays and special events? 4 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec M T W T F S S M T W T F S S M T W T F S S M T W T F S S
sub-daily data over several years? How to iden fy unusual pa erns/incidents? How to forecast sub-daily data taking account of public holidays and special events? Di Cook Earo Wang Mitchell O’Hara-Wild 5
for fast iden fica on of forecas ng models? How to generate new me series with specified feature vectors? What can we say about the feature space of me series? 8
for fast iden fica on of forecas ng models? How to generate new me series with specified feature vectors? What can we say about the feature space of me series? Kate Smith-Miles George Athanasopoulos Thiyanga Talagala 8
behaviour within streaming data? How to define an anomaly in a large mul variate data set? Kate Smith-Miles Mario Andr´ es Mu˜ noz Acosta Sevvandi Kandanaarachchi Dilini Talagala 12
to reconcile household demand forecasts with state and na onal demand forecasts? How to iden fy unusual demand pa erns? How to measure forecast accuracy when forecasts are probability distribu ons within a hierarchy? 16
to reconcile household demand forecasts with state and na onal demand forecasts? How to iden fy unusual demand pa erns? How to measure forecast accuracy when forecasts are probability distribu ons within a hierarchy? Souhaib Ben Taieb Cameron Roach 16
Australian tourism demand by state, region, zone. 18 Total A AA AAA AAB AAC AB ABA ABB ABC AC ACA ACB ACC B BA BAA BAB BAC BB BBA BBB BBC BC BCA BCB BCC C CA CAA CAB CAC CB CBA CBB CBC CC CCA CCB CCC
Bo om level typically has thousands or millions of me series How to define coherence probabilis cally? How to visualize so many me series? 19 George Athanasopoulos Anastasios Panagiotelis Shanika Wickramasuriya Puwasala Gamakumara Earo Wang
1998:Q1 – 2013:Q4 From Na onal Visitor Survey, based on annual interviews of 120,000 Australians aged 15+, collected by Tourism Research Australia. Split by 7 states, 27 zones and 76 regions (a geographical hierarchy) Also split by purpose of travel Holiday Visi ng friends and rela ves (VFR) Business Other 304 bo om-level series
, yA,t , yB,t , yC,t ] = 1 1 1 1 0 0 0 1 0 0 0 1 yA,t yB,t yC,t 21 yt : observed aggregate of all series at me t. yX,t : observa on on series X at me t. bt : vector of all series at bo om level in me t.
, yA,t , yB,t , yC,t ] = 1 1 1 1 0 0 0 1 0 0 0 1 S yA,t yB,t yC,t 21 yt : observed aggregate of all series at me t. yX,t : observa on on series X at me t. bt : vector of all series at bo om level in me t.
, yA,t , yB,t , yC,t ] = 1 1 1 1 0 0 0 1 0 0 0 1 S yA,t yB,t yC,t bt 21 yt : observed aggregate of all series at me t. yX,t : observa on on series X at me t. bt : vector of all series at bo om level in me t.
, yA,t , yB,t , yC,t ] = 1 1 1 1 0 0 0 1 0 0 0 1 S yA,t yB,t yC,t bt yt = Sbt 21 yt : observed aggregate of all series at me t. yX,t : observa on on series X at me t. bt : vector of all series at bo om level in me t.
aggrega on constraints can be wri en as yt = Sbt where yt is a vector of all series at me t bt is a vector of the most disaggregated series at me t S is a “summing matrix” containing the aggrega on constraints. 22
of ini al h-step forecasts, made at me n, stacked in same order as yt. (In general, they will not “add up”.) Reconciled forecasts must be of the form: ˜ yn (h) = SPˆ yn (h) for some matrix P. 23
of ini al h-step forecasts, made at me n, stacked in same order as yt. (In general, they will not “add up”.) Reconciled forecasts must be of the form: ˜ yn (h) = SPˆ yn (h) for some matrix P. P projects base forecasts ˆ yn (h) to bo om level. 23
of ini al h-step forecasts, made at me n, stacked in same order as yt. (In general, they will not “add up”.) Reconciled forecasts must be of the form: ˜ yn (h) = SPˆ yn (h) for some matrix P. P projects base forecasts ˆ yn (h) to bo om level. S adds them up 23
SPˆ yn (h) Bias Reconciled forecasts are unbiased iff SPS = S. Variance Let error variance of h-step base forecasts ˆ yn (h) be Wh = Var[yn+h − ˆ yn (h) | y1 , . . . , yn ] Then error variance of the reconciled forecasts is Var[yn+h − ˜ yn (h) | y1 , . . . , yn ] = SPWh P S 24
yn (h) Theorem: MinT Reconcilia on If P sa sfies SPS = S, then minP = trace[SPWh P S ] has solu on P = (S W−1 h S)−1S W−1 h . Reconciled forecasts Base forecasts 25 ˜ yn (h) = S(S W−1 h S)−1S W−1 h ˆ yn (h)
that Wh = kh W1 to simplify computa ons. WLS solu on Approximate W1 by its diagonal. GLS solu on Es mate W1 using shrinkage to the diagonal. 26 ˜ yn (h) = S(S W−1 h S)−1S W−1 h ˆ yn (h)
. , yT, observed at frequency m, we generate aggregate series y[k] j = jk t=1+(j−1)k yt , for j = 1, . . . , T/k k ∈ F(m) = {factors of m}. A single unique hierarchy is only possible when there are no coprime pairs in F(m). 35
. , yT, observed at frequency m, we generate aggregate series y[k] j = jk t=1+(j−1)k yt , for j = 1, . . . , T/k k ∈ F(m) = {factors of m}. A single unique hierarchy is only possible when there are no coprime pairs in F(m). Mk = m/k is seasonal period of aggregated series. 35
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