analysis, forecasting and mining information from time
TRANSCRIPT
Analysis, Forecasting and MiningInformation from Time Series using
F-Transform and Fuzzy Natural Logic
Vilem Novak
University of OstravaInstitute for Research and Applications of Fuzzy Modeling
NSC IT4InnovationsOstrava, Czech [email protected]
Ho Chi Minh City, October 19, 2015
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Outline
1 Introduction
2 Special soft computing techniques
3 Analysis of time series
4 Forecasting
5 Demonstration of forecasting using LFL Forecaster
6 Linguistic characterization of trend
7 Conclusions
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Time series
X : T× Ω −→ R
T = 0,1, . . . ,N;T = [a,b];T = R
Fixed ω ∈ Ω: Realization
X (t) | t ∈ T
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Standard approaches
Box-Jenkins methodology: e.g. ARMA(p,q)
X (t) = c + ε(t) +
p∑i=1
ϕiX (t − i) +
q∑i=1
θiε(t − i)
ϕ1, . . . , ϕp — autoregressive model parametersθ1, . . . , θq — moving average model parametersε — noise
Very powerful in forecasting
How to interpret the process? How to understand it?
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Standard approaches
Decomposition
X (t) = Tr(t) + C(t)︸ ︷︷ ︸TC(t)
+S(t) + R(t), t ∈ T
Tr(t) — trendC(t) — cycleS(t) — seasonal componentR(t) — random errorTC(t) —trend-cycle
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Special soft computing techniques
(i) Fuzzy Natural Logic (FNL)(ii) Fuzzy (F-)transform
Applied to :
• time series analysis• time series forecasting• mining information from time series
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Special soft computing techniques
(i) Fuzzy Natural Logic (FNL)(ii) Fuzzy (F-)transform
Applied to :
• time series analysis• time series forecasting• mining information from time series
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Special soft computing techniques
(i) Fuzzy Natural Logic (FNL)(ii) Fuzzy (F-)transform
Applied to :
• time series analysis• time series forecasting• mining information from time series
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy (F)-transform
Irina Perfilieva
Transform a continuous function f to a finite n-dimensional(real) vector of numbers (direct phase) and then approximatethe original function (inverse phase) by f .
The inverse F-transform f can be constructed to haverequired properties.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy (F)-transform
Irina Perfilieva
Transform a continuous function f to a finite n-dimensional(real) vector of numbers (direct phase) and then approximatethe original function (inverse phase) by f .
The inverse F-transform f can be constructed to haverequired properties.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy Natural Logic
Fuzzy natural logic is a mathematical theory modelingterms and rules that come with natural language and allowus to reason and argue in it
Current constituents of FNL• Theory of evaluative linguistic expressions• Theory of fuzzy/linguistic IF-THEN rules and logical
inference (Perception-based Logical Deduction)• Theory of generalized fuzzy and intermediate quantifiers;
generalized Aristotle syllogisms
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy Natural Logic
Fuzzy natural logic is a mathematical theory modelingterms and rules that come with natural language and allowus to reason and argue in it
Current constituents of FNL• Theory of evaluative linguistic expressions• Theory of fuzzy/linguistic IF-THEN rules and logical
inference (Perception-based Logical Deduction)• Theory of generalized fuzzy and intermediate quantifiers;
generalized Aristotle syllogisms
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy Natural Logic
Fuzzy natural logic is a mathematical theory modelingterms and rules that come with natural language and allowus to reason and argue in it
Current constituents of FNL• Theory of evaluative linguistic expressions• Theory of fuzzy/linguistic IF-THEN rules and logical
inference (Perception-based Logical Deduction)• Theory of generalized fuzzy and intermediate quantifiers;
generalized Aristotle syllogisms
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy Natural Logic
Fuzzy natural logic is a mathematical theory modelingterms and rules that come with natural language and allowus to reason and argue in it
Current constituents of FNL• Theory of evaluative linguistic expressions• Theory of fuzzy/linguistic IF-THEN rules and logical
inference (Perception-based Logical Deduction)• Theory of generalized fuzzy and intermediate quantifiers;
generalized Aristotle syllogisms
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Evaluative linguistic expressions
Special expressions of natural language using which peopleevaluate encountered phenomena and processes
(i) Simple evaluative expressions: small, medium, big; low,medium, high; cheap, medium expensive, expensive, verylow, more or less high, very rough increase, extremely highprofit
(ii) Fuzzy numbers: twenty thousand, roughly one thousand,about one million
(iii) Compound and negative evaluative expressions: roughlysmall or medium, high but not very (high)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Evaluative linguistic expressions
Special expressions of natural language using which peopleevaluate encountered phenomena and processes
(i) Simple evaluative expressions: small, medium, big; low,medium, high; cheap, medium expensive, expensive, verylow, more or less high, very rough increase, extremely highprofit
(ii) Fuzzy numbers: twenty thousand, roughly one thousand,about one million
(iii) Compound and negative evaluative expressions: roughlysmall or medium, high but not very (high)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Evaluative linguistic expressions
Special expressions of natural language using which peopleevaluate encountered phenomena and processes
(i) Simple evaluative expressions: small, medium, big; low,medium, high; cheap, medium expensive, expensive, verylow, more or less high, very rough increase, extremely highprofit
(ii) Fuzzy numbers: twenty thousand, roughly one thousand,about one million
(iii) Compound and negative evaluative expressions: roughlysmall or medium, high but not very (high)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Evaluative linguistic expressions
Special expressions of natural language using which peopleevaluate encountered phenomena and processes
(i) Simple evaluative expressions: small, medium, big; low,medium, high; cheap, medium expensive, expensive, verylow, more or less high, very rough increase, extremely highprofit
(ii) Fuzzy numbers: twenty thousand, roughly one thousand,about one million
(iii) Compound and negative evaluative expressions: roughlysmall or medium, high but not very (high)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Semantics of evaluative linguistic expressions
Context (possible world)
w = 〈vL, vS, vR〉 vL, vS, vR ∈ R
vL — the smallest thinkable valuevS — typical medium valuevR — the largest thinkable value
Example (Context for population of a town)
〈 vL vS vR〉Czech Republic: 〈 3 000 50 000 1 000 000〉USA: 〈30 000 200 000 10 000 000〉
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Semantics of evaluative linguistic expressions
Context (possible world)
w = 〈vL, vS, vR〉 vL, vS, vR ∈ R
vL — the smallest thinkable valuevS — typical medium valuevR — the largest thinkable value
Example (Context for population of a town)
〈 vL vS vR〉Czech Republic: 〈 3 000 50 000 1 000 000〉USA: 〈30 000 200 000 10 000 000〉
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Extension of evaluative expressions
Small Medium BigExtremely
Very
0 4 10
ExSm
VeSm? Small?Big ?
0 4 10
0 4 10
Roughly
(a)
(b)
(c)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Semantics of evaluative linguistic expressions
Intension: W −→ F (R)
֏
֏
֏
various contexts extensions
( ([0,1]))wFWIntension:
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy/linguistic IF-THEN rules and linguisticdescription
Fuzzy/linguistic IF-THEN rule
R := IF X is A THEN Y is B
X is A ,Y is B — evaluative linguistic predications
A ,B :small, very small, roughly medium, extremely big
Linguistic description R
R1 : IF X is A1 THEN Y is B1
. . . . . . . . . . . . . . . . . . . . . . . .Rm : IF X is Am THEN Y is Bm
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Fuzzy/linguistic IF-THEN rules and linguisticdescription
Fuzzy/linguistic IF-THEN rule
R := IF X is A THEN Y is B
X is A ,Y is B — evaluative linguistic predications
A ,B :small, very small, roughly medium, extremely big
Linguistic description R
R1 : IF X is A1 THEN Y is B1
. . . . . . . . . . . . . . . . . . . . . . . .Rm : IF X is Am THEN Y is Bm
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perception-based logical deduction
Consider a simple linguistic description:
R1 := IF X is small THEN Y is very bigR2 := IF X is very big THEN Y is small
Both rules provide us with a certain knowledge. Though theyare vague, we may distinguish between them.
Linguistic context of X ,Y : 〈0,0.4,1〉
“small” ≈ 0.2 (and smaller); perception of X = 0.2“very big” ≈ 0.8 (and bigger); perception of Y = 0.8
Given a value X = 0.2: we expect Y ≈ 0.8 due to R1(X is evaluated as small)
Similarly, X = 0.8 leads to Y ≈ 0.2 due to R2.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perception-based logical deduction
Consider a simple linguistic description:
R1 := IF X is small THEN Y is very bigR2 := IF X is very big THEN Y is small
Both rules provide us with a certain knowledge. Though theyare vague, we may distinguish between them.
Linguistic context of X ,Y : 〈0,0.4,1〉
“small” ≈ 0.2 (and smaller); perception of X = 0.2“very big” ≈ 0.8 (and bigger); perception of Y = 0.8
Given a value X = 0.2: we expect Y ≈ 0.8 due to R1(X is evaluated as small)
Similarly, X = 0.8 leads to Y ≈ 0.2 due to R2.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perception-based logical deduction
Consider a simple linguistic description:
R1 := IF X is small THEN Y is very bigR2 := IF X is very big THEN Y is small
Both rules provide us with a certain knowledge. Though theyare vague, we may distinguish between them.
Linguistic context of X ,Y : 〈0,0.4,1〉
“small” ≈ 0.2 (and smaller); perception of X = 0.2“very big” ≈ 0.8 (and bigger); perception of Y = 0.8
Given a value X = 0.2: we expect Y ≈ 0.8 due to R1(X is evaluated as small)
Similarly, X = 0.8 leads to Y ≈ 0.2 due to R2.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Simple demonstration of PbLD
LFLC 2000 (University of Ostrava)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Simple demonstration of PbLD
LFLC 2000 (University of Ostrava)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Learning of linguistic description
Linguistic description is learned from the dataSet linguistic context:
w = 〈vL, vS, vR〉
Learning procedure
X1 X2 . . . Xn Y999.999 999.999 · · · 999.999 999.999. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .999.999 999.999 · · · 999.999 999.999
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Learning of linguistic description
Linguistic description is learned from the dataSet linguistic context:
w = 〈vL, vS, vR〉
Learning procedure
X1 X2 . . . Xn YX1 is A11 X2 is A21 · · · Xn is An1 Y is B1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .X1 is A1m X2 is A2m · · · Xn is Anm Y is Bm
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Learning of linguistic description
Linguistic description is learned from the dataSet linguistic context:
w = 〈vL, vS, vR〉
Learning procedure
IF X1 is A11 AND X2 is A21 AND · · · AND Xn is An1THEN Y is B1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IF X1 is A1m AND X2 is A2m AND · · · AND Xn is Anm
THEN Y is Bm
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Analysis of time series
Decomposition
X (t) = Tr(t) + C(t)︸ ︷︷ ︸TC(t)
+S(t) + R(t , ω), t ∈ T
• Trend component Tr(t)• Cycle component C(t)• Seasonal component S(t)• Random noise (error) R(t , ω); µ ≈ 0, σ > 0• Trend-cycle TC(t) = Tr(t) + C(t)
Well interpretable — appropriate for alternative methods
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend and trend-cycle
General OECD specification:
Trend (tendency)Component of a time series that represents variations of lowfrequency in a time series, the high and medium frequencyfluctuations having been filtered out.
Trend-cycleComponent that represents variations of low frequency in atime series, the high frequency fluctuations having been filteredout — period longer than a chosen threshold (longer, than 1year = bussiness cycle).
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend and trend-cycle
General OECD specification:
Trend (tendency)Component of a time series that represents variations of lowfrequency in a time series, the high and medium frequencyfluctuations having been filtered out.
Trend-cycleComponent that represents variations of low frequency in atime series, the high frequency fluctuations having been filteredout — period longer than a chosen threshold (longer, than 1year = bussiness cycle).
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Predefined trend-cycle function?
Which function should be used to model the trend-cycle?
May we afford to forecast trend values just by prolongation ofthe chosen trend-cycle function?
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Predefined trend-cycle function?
Which function should be used to model the trend-cycle?
May we afford to forecast trend values just by prolongation ofthe chosen trend-cycle function?
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Assumption
X (t) = TC(t) +r∑
j=1
Pj ei(λj t+ϕj )
︸ ︷︷ ︸S(t)
+R(t)
Frequencies: λi , i = 1, . . . , rPeriodicities: Ti = 2π
λiRandom noise is a stationary stochastic process
R(t) = ξeiλt+ϕ, ξ ∈ R
Apply F-transform to X
F[X ] = F[TC] + F[S] + F[R]
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Assumption
X (t) = TC(t) +r∑
j=1
Pj ei(λj t+ϕj )
︸ ︷︷ ︸S(t)
+R(t)
Frequencies: λi , i = 1, . . . , rPeriodicities: Ti = 2π
λiRandom noise is a stationary stochastic process
R(t) = ξeiλt+ϕ, ξ ∈ R
Apply F-transform to X
F[X ] = F[TC] + F[S] + F[R]
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Assumption
X (t) = TC(t) +r∑
j=1
Pj ei(λj t+ϕj )
︸ ︷︷ ︸S(t)
+R(t)
Frequencies: λi , i = 1, . . . , rPeriodicities: Ti = 2π
λiRandom noise is a stationary stochastic process
R(t) = ξeiλt+ϕ, ξ ∈ R
Apply F-transform to X
F[X ] = F[TC] + F[S] + F[R]
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
We consider interval [0,b]
h-equidistant nodes
c0 = 0, c1 = dT , . . . ,ck−1 = (k − 1)dT , ck = kdT , ck+1 = (k + 1)dT ,
. . . , cn = ndT = b
where
h = d T
λ =2πd
h
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend and trend-cycle using inverse F-transform
TheoremIf we set the distance h between nodes properly (h = dT ) then• X (t) = TC(t) + D (T — the longest seasonal periodicity)• X (t) = Tr(t) + D (T — the longest cyclic periodicity)
D → 0
F-transform makes it possible to extract trend-cycle as wellas trend with high fidelity
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Artificial time series
X (t) = TC(t) + 5 sin(0.63t + 1.5) + 5 sin(1.26t + 0.35)+
+ 15 sin(2.7t + 1.12) + 7 sin(0.41t + 0.79) + R(t)
Trend-cycle: given by artificial data without clear periodicityOther: T1 = 10, T2 = 5, T3 = 2.3, T4 = 15.4.
T = T4, d = 1, i.e. h = 15;Width of basic functions: 2h = 30
R(t) — random noise with average µ = 0.1
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Artificial time series
- - - - - Original trend-cycleInverse F-transform of X
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Estimation of trend-cycle
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Estimation of trend-cycle
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Estimation of trend
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Trend-cycle using inverse F-transform
Estimation of trend
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting methodology
The data are split into in-samples/out-samples
Learning set Validationset
Testingset
In-samples Out-samples
In-samples = learning set TL + validation set TV
Out-samples = testing set TT
T = TL ∪ TV ∪ TT
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting methodology
Symmetric Mean Absolute Percent Error
SMAPE =1|T|
∑t∈T
|X (t)− X (t)||X(t)|+|X(t)|
2
SMAPE∈ [0,200]
The best trend-cycle and seasonal predictor is chosenaccording to SMAPE error on the validation set TV
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting trend-cycle (autoregression)
• Compute F-transform components F1[X ], . . . ,Fn[X ] andtheir differences ∆Fi [X ],∆2Fi [X ]
• Learn a linguistic description consisting of the rules suchas the following:
IF ∆Fi−1[X ] is A∆i−1 AND Fi [X ] is Ai THEN ∆Fi+1[X ] is B
The learned linguistic description is used for forecasting of theF-transform components
F[X ] = (Fn+1[X ], . . . ,Fn+`[X ])
Forecast of X (t), t ∈ TV (or t ∈ TT ) is computed from F[X ]
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting trend-cycle (autoregression)
• Compute F-transform components F1[X ], . . . ,Fn[X ] andtheir differences ∆Fi [X ],∆2Fi [X ]
• Learn a linguistic description consisting of the rules suchas the following:
IF ∆Fi−1[X ] is A∆i−1 AND Fi [X ] is Ai THEN ∆Fi+1[X ] is B
The learned linguistic description is used for forecasting of theF-transform components
F[X ] = (Fn+1[X ], . . . ,Fn+`[X ])
Forecast of X (t), t ∈ TV (or t ∈ TT ) is computed from F[X ]
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting trend-cycle (autoregression)
• Compute F-transform components F1[X ], . . . ,Fn[X ] andtheir differences ∆Fi [X ],∆2Fi [X ]
• Learn a linguistic description consisting of the rules suchas the following:
IF ∆Fi−1[X ] is A∆i−1 AND Fi [X ] is Ai THEN ∆Fi+1[X ] is B
The learned linguistic description is used for forecasting of theF-transform components
F[X ] = (Fn+1[X ], . . . ,Fn+`[X ])
Forecast of X (t), t ∈ TV (or t ∈ TT ) is computed from F[X ]
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
1 Simple prediction: The next (difference of) componentFi+1[X ] is predicted from the previous (differences of)components Fi [X ],Fi−1[X ], . . . ,Fi−k [X ] using one linguisticdescription
2 Independent models: Each following (difference of)component Fn+1[X ], . . . ,Fn+`[X ] is predicted from(differences of) components Fn,Fn−1[X ], . . . ,Fn−k [X ] usingdifferent linguistic descriptions
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
1 Simple prediction: The next (difference of) componentFi+1[X ] is predicted from the previous (differences of)components Fi [X ],Fi−1[X ], . . . ,Fi−k [X ] using one linguisticdescription
2 Independent models: Each following (difference of)component Fn+1[X ], . . . ,Fn+`[X ] is predicted from(differences of) components Fn,Fn−1[X ], . . . ,Fn−k [X ] usingdifferent linguistic descriptions
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting seasonal component (autoregression)
• θ vectors of the seasonal component:
Sj = (S((j − 1)p + 1),S((j − 1)p + 2), . . . ,S(jp))
p — period of seasonality (e.g., p = 12), j ∈ 1, . . . , θ• Finding optimal solution of the system of equations
Sθ =θ−1∑j=1
dj · Sθ−j
• Use the coefficients d1, . . . ,dθ to determine future vectorsSθ+1, . . .
Other possibility: neural networks
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting seasonal component (autoregression)
• θ vectors of the seasonal component:
Sj = (S((j − 1)p + 1),S((j − 1)p + 2), . . . ,S(jp))
p — period of seasonality (e.g., p = 12), j ∈ 1, . . . , θ• Finding optimal solution of the system of equations
Sθ =θ−1∑j=1
dj · Sθ−j
• Use the coefficients d1, . . . ,dθ to determine future vectorsSθ+1, . . .
Other possibility: neural networks
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting seasonal component (autoregression)
• θ vectors of the seasonal component:
Sj = (S((j − 1)p + 1),S((j − 1)p + 2), . . . ,S(jp))
p — period of seasonality (e.g., p = 12), j ∈ 1, . . . , θ• Finding optimal solution of the system of equations
Sθ =θ−1∑j=1
dj · Sθ−j
• Use the coefficients d1, . . . ,dθ to determine future vectorsSθ+1, . . .
Other possibility: neural networks
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Forecasting seasonal component (autoregression)
• θ vectors of the seasonal component:
Sj = (S((j − 1)p + 1),S((j − 1)p + 2), . . . ,S(jp))
p — period of seasonality (e.g., p = 12), j ∈ 1, . . . , θ• Finding optimal solution of the system of equations
Sθ =θ−1∑j=1
dj · Sθ−j
• Use the coefficients d1, . . . ,dθ to determine future vectorsSθ+1, . . .
Other possibility: neural networks
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Monthly gasoline demand in Ontario in 1960–1975
Measured by SMAPE, forecast horizon= 18 months
Validation set error: 3.38% ; testing set error: 3.04%
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Monthly gasoline demand in Ontario in 1960–1975
Measured by SMAPE, forecast horizon= 18 months
Validation set error: 3.38% ; testing set error: 3.04%
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Monthly gasoline demand in Ontario in 1960–1975context = 〈86800,154000,256000〉; partition period=24 months
Generated linguistic description
Nr. Fi [X ] ∆2Fi [X ] ⇒ ∆Fi+1[X ]
1 ze -ml me ⇒ si sm2 ve sm sm ⇒ sm3 sm sm ⇒ ve sm4 ml sm -si sm ⇒ vr sm5 qr sm vr bi ⇒ sm6 ml me -me ⇒ me7 ml me qr bi ⇒ ro sm8 vr bi -me ⇒ sm9 qr bi -ml sm ⇒ ro bi
10 ml bi ex bi ⇒ qr bi11 bi -ex sm ⇒ si bi
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Monthly gasoline demand in Ontario in 1960–1975context = 〈86800,154000,256000〉; partition period=24 months
Generated linguistic description
Nr. Fi [X ] ∆2Fi [X ] ⇒ ∆Fi+1[X ]
1 ze -ml me ⇒ si sm2 ve sm sm ⇒ sm3 sm sm ⇒ ve sm4 ml sm -si sm ⇒ vr sm5 qr sm vr bi ⇒ sm6 ml me -me ⇒ me7 ml me qr bi ⇒ ro sm8 vr bi -me ⇒ sm9 qr bi -ml sm ⇒ ro bi
10 ml bi ex bi ⇒ qr bi11 bi -ex sm ⇒ si bi
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Monthly gasoline demand in Ontario in 1960–1975context = 〈86800,154000,256000〉; partition period=24 months
Generated linguistic description
Nr. Fi [X ] ∆2Fi [X ] ⇒ ∆Fi+1[X ]
1 ze -ml me ⇒ si sm2 ve sm sm ⇒ sm3 sm sm ⇒ ve smIf average gasoline demand is SMALL in previous two years
and average accelleration is SMALL four years agothen average change in upcoming two years will be VERY SMALL7 ml me qr bi ⇒ ro sm8 vr bi -me ⇒ sm9 qr bi -ml sm ⇒ ro bi
10 ml bi ex bi ⇒ qr bi11 bi -ex sm ⇒ si bi
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Mining information from time series
1 Reduction of dimensionality2 Perceptionally important points3 Linguistic evaluation of the direction of local and global
trend; linguistic characterization of future course of TC or Tr4 Finding intervals of monotonous behavior5 Linguistic summarization of information about time series
and syllogistic reasoning
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Reduction of dimensionality
Replace the original X by the reduced one
X [h] = β01 , . . . , β
0n−1.
Advantages:1 Dimension of X [h] is significantly lower (depends on h)2 X [h] fits well the shape of the original time series X , keeps
all its salient points3 X [h] is free of all frequencies higher than λq
4 Variance of the noise of X [h] is lower than variance of theoriginal noise R(t , ω)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Reduction of dimensionality
Replace the original X by the reduced one
X [h] = β01 , . . . , β
0n−1.
Advantages:1 Dimension of X [h] is significantly lower (depends on h)2 X [h] fits well the shape of the original time series X , keeps
all its salient points3 X [h] is free of all frequencies higher than λq
4 Variance of the noise of X [h] is lower than variance of theoriginal noise R(t , ω)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Reduction of dimensionality
Replace the original X by the reduced one
X [h] = β01 , . . . , β
0n−1.
Advantages:1 Dimension of X [h] is significantly lower (depends on h)2 X [h] fits well the shape of the original time series X , keeps
all its salient points3 X [h] is free of all frequencies higher than λq
4 Variance of the noise of X [h] is lower than variance of theoriginal noise R(t , ω)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Reduction of dimensionality
Replace the original X by the reduced one
X [h] = β01 , . . . , β
0n−1.
Advantages:1 Dimension of X [h] is significantly lower (depends on h)2 X [h] fits well the shape of the original time series X , keeps
all its salient points3 X [h] is free of all frequencies higher than λq
4 Variance of the noise of X [h] is lower than variance of theoriginal noise R(t , ω)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Reduction of dimensionality
Original time series, 186 and its inverse F0-transform, h = 3Reduced time series, components β0
1 , . . . , β062
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Reduction of dimensionality
Original time series, 186 and its inverse F0-transform, h = 3Reduced time series, components β0
1 , . . . , β062
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Reduction of dimensionality
Original time series, 186 and its inverse F0-transform, h = 3Reduced time series, components β0
1 , . . . , β062
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Estimation of local trend
β1k =
∫ ck+1ck−1
X (t)(t − ck )Ak (t)dt∫ ck+1ck−1
(t − ck )2Ak (t)dt
Estimation of the first derivative of X
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perceptionally important points
F-transform provides estimation of derivatives
Main idea of the algorithm
Find periods consisting of several components β1k ,. . . with the
same sign followed by one or more components β1j that have
opposite sign and are sufficiently large
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perceptionally important points
(i) Set fuzzy partition: h = dTk for some d ∈ N and k < q.Position of nodes: sum
∑n−1k=1 |β1
k | is minimal.(ii) Find position of potentional perceptionally important points
Nodes ck with the highest |β2k |
β2k =
∫ ck+1ck−1
X (t)((t − ck )2 − I2)Ak (t)dt∫ ck+1ck−1
((t − ck )2 − I2)2Ak (t)dt
I2 = 1h
∫ ck+1ck−1
(t − ck )2Ak (t)dt
(iii) PIP: several (at least two) β1k , β
1k+1,. . . have the same sign
followed by one or more β1j with opposite sign and/or are
sufficiently large.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perceptionally important points
(i) Set fuzzy partition: h = dTk for some d ∈ N and k < q.Position of nodes: sum
∑n−1k=1 |β1
k | is minimal.(ii) Find position of potentional perceptionally important points
Nodes ck with the highest |β2k |
β2k =
∫ ck+1ck−1
X (t)((t − ck )2 − I2)Ak (t)dt∫ ck+1ck−1
((t − ck )2 − I2)2Ak (t)dt
I2 = 1h
∫ ck+1ck−1
(t − ck )2Ak (t)dt
(iii) PIP: several (at least two) β1k , β
1k+1,. . . have the same sign
followed by one or more β1j with opposite sign and/or are
sufficiently large.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perceptionally important points
(i) Set fuzzy partition: h = dTk for some d ∈ N and k < q.Position of nodes: sum
∑n−1k=1 |β1
k | is minimal.(ii) Find position of potentional perceptionally important points
Nodes ck with the highest |β2k |
β2k =
∫ ck+1ck−1
X (t)((t − ck )2 − I2)Ak (t)dt∫ ck+1ck−1
((t − ck )2 − I2)2Ak (t)dt
I2 = 1h
∫ ck+1ck−1
(t − ck )2Ak (t)dt
(iii) PIP: several (at least two) β1k , β
1k+1,. . . have the same sign
followed by one or more β1j with opposite sign and/or are
sufficiently large.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perceptionally important points
(i) Set fuzzy partition: h = dTk for some d ∈ N and k < q.Position of nodes: sum
∑n−1k=1 |β1
k | is minimal.(ii) Find position of potentional perceptionally important points
Nodes ck with the highest |β2k |
β2k =
∫ ck+1ck−1
X (t)((t − ck )2 − I2)Ak (t)dt∫ ck+1ck−1
((t − ck )2 − I2)2Ak (t)dt
I2 = 1h
∫ ck+1ck−1
(t − ck )2Ak (t)dt
(iii) PIP: several (at least two) β1k , β
1k+1,. . . have the same sign
followed by one or more β1j with opposite sign and/or are
sufficiently large.
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Perceptionally important points
Example:
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic evaluation of the direction of trend
Automatically generated linguistic expressions
negligibly decreasing stagnating slightlydecreasing
somewhatdecreasing
Trend of the (whole) time series is stagnating
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic evaluation of the direction of trend
Combination of F1-transform and fuzzy natural logic
Local perception LPerc : R×W −→ EvExpr
Linguistic context wtg = 〈0, vS, vR〉 (or (−1)wtg = 〈−vR ,−vS,0〉)
Ev := LPerc(β1[X |T],wtg) 7→ 〈special hedge〉Inc/Dec
Linguistic evaluation of the direction of trend
Trend of X in period T is 〈special hedge〉Inc/Dec .
Example
Trend of X in period T is slightly increasing
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic evaluation of the direction of trend
Combination of F1-transform and fuzzy natural logic
Local perception LPerc : R×W −→ EvExpr
Linguistic context wtg = 〈0, vS, vR〉 (or (−1)wtg = 〈−vR ,−vS,0〉)
Ev := LPerc(β1[X |T],wtg) 7→ 〈special hedge〉Inc/Dec
Linguistic evaluation of the direction of trend
Trend of X in period T is 〈special hedge〉Inc/Dec .
Example
Trend of X in period T is slightly increasing
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic evaluation of the direction of trend
Combination of F1-transform and fuzzy natural logic
Local perception LPerc : R×W −→ EvExpr
Linguistic context wtg = 〈0, vS, vR〉 (or (−1)wtg = 〈−vR ,−vS,0〉)
Ev := LPerc(β1[X |T],wtg) 7→ 〈special hedge〉Inc/Dec
Linguistic evaluation of the direction of trend
Trend of X in period T is 〈special hedge〉Inc/Dec .
Example
Trend of X in period T is slightly increasing
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic evaluation of the direction of trend
Trend of the whole series is stagnating.
Trend of time series:in Slot 1 (time 41-63) is negligibly increasingin Slot 2 (time 66-114) is slightly decreasing
Future trend of time series is clearly decreasing(Verification of real course: clear decrease)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic evaluation of the direction of trend
Trend of the whole series is stagnating.
Trend of time series:in Slot 1 (time 41-63) is negligibly increasingin Slot 2 (time 66-114) is slightly decreasing
Future trend of time series is clearly decreasing(Verification of real course: clear decrease)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic evaluation of the direction of trend
Trend of the whole series is stagnating.
Trend of time series:in Slot 1 (time 41-63) is negligibly increasingin Slot 2 (time 66-114) is slightly decreasing
Future trend of time series is clearly decreasing(Verification of real course: clear decrease)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Finding intervals of monotonous behavior
Special algorithm combining F-transform and methods of FNL
interval Ev [X |Ti ] interval Ev [X |Ti ]
T1 clear increase T6 huge decreaseT2 somewhat increase T7 clear increaseT3 huge increase T8 clear decreaseT4 huge decrease T9 clear decreaseT5 huge increase
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic comments to time series
Example• In the first (second, third, fourth) quarter of the year, the
trend was slightly decreasing (increasing, stagnating).• The longest period of stagnating (increasing, slightly
decreasing) trend lasted for T time moments (days, weeks,months, etc.)
T — the longest of the found intervals T1, . . . ,Ts in whichtrend of X is stagnating (sharply increasing, slightlydecreasing, etc.).
• The last quarter of the year, the trend was decreasing butits forecast for the next quarter is clear increase
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic comments to time series
Example• In the first (second, third, fourth) quarter of the year, the
trend was slightly decreasing (increasing, stagnating).• The longest period of stagnating (increasing, slightly
decreasing) trend lasted for T time moments (days, weeks,months, etc.)
T — the longest of the found intervals T1, . . . ,Ts in whichtrend of X is stagnating (sharply increasing, slightlydecreasing, etc.).
• The last quarter of the year, the trend was decreasing butits forecast for the next quarter is clear increase
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic comments to time series
Example• In the first (second, third, fourth) quarter of the year, the
trend was slightly decreasing (increasing, stagnating).• The longest period of stagnating (increasing, slightly
decreasing) trend lasted for T time moments (days, weeks,months, etc.)
T — the longest of the found intervals T1, . . . ,Ts in whichtrend of X is stagnating (sharply increasing, slightlydecreasing, etc.).
• The last quarter of the year, the trend was decreasing butits forecast for the next quarter is clear increase
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Summarization
GoalSummarize knowledge about one or more time series.Application of the formal theory of intermediate quantifiers (partof FNL).
Examples of intermediate quantifiers:Most, Many, Almost all, Few, A large part of
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Summarizing information about time series
One time seriesTypical summarizing proposition:
ExampleIn most (a few, many) time intervals, trend of the time series Xwas slightly increasing (stagnating, sharply decreasing).
Formalization: type 〈1〉 quantifier
(Q∀Bi VeT)((SlInc X )wT)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Summarizing information about time series
Result: set of formulas formalizing propositions, for example
In most (many, few) cases, the time series wasstagnating (sharply decreasing, roughly increasing)
Choose a formula that has the highest truth degree and thehighest level of steepness
Alternative: Use the function of local perception LPerc
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic summarization of information abouttime series
Many time series:
Example• Most (many, few) analyzed time series stagnated recently
but their future trend is slightly increasing• There is an evidence of huge (slight, clear) decrease of
trend of almost all time series in the recent quarter of theyear
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Linguistic summarization of information abouttime series
Many time series:
Example• Most (many, few) analyzed time series stagnated recently
but their future trend is slightly increasing• There is an evidence of huge (slight, clear) decrease of
trend of almost all time series in the recent quarter of theyear
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Summarization — syllogistic reasoning
Example (Valid generalized Aristotle’s syllogism)
In few cases the increase of time series is not small
In many cases the increase of time series is clearIn few cases the clear increase of time series is not small
Validity of over 105 generalized syllogisms withintermediate quantifiers was mathematically proved(Murinova, Novak)
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Summarization — syllogistic reasoning
Example (Valid generalized Aristotle’s syllogisms)
All carefully inspected time series are financialMost available time series are carefully inspectedMost available time series are financial
In no period there was increasing TS from car industryIn most periods there was inspected TS from car industryIn some periods, inspected time series was not increasing
Most financial TS were sharply decreasing last monthAll sharply decreasing TS last month are now stagnatingSome time series that are now stagnating are financial
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Summarization — syllogistic reasoning
Example (Valid generalized Aristotle’s syllogisms)
All carefully inspected time series are financialMost available time series are carefully inspectedMost available time series are financial
In no period there was increasing TS from car industryIn most periods there was inspected TS from car industryIn some periods, inspected time series was not increasing
Most financial TS were sharply decreasing last monthAll sharply decreasing TS last month are now stagnatingSome time series that are now stagnating are financial
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Summarization — syllogistic reasoning
Example (Valid generalized Aristotle’s syllogisms)
All carefully inspected time series are financialMost available time series are carefully inspectedMost available time series are financial
In no period there was increasing TS from car industryIn most periods there was inspected TS from car industryIn some periods, inspected time series was not increasing
Most financial TS were sharply decreasing last monthAll sharply decreasing TS last month are now stagnatingSome time series that are now stagnating are financial
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Conclusions
Fuzzy Transform and Fuzzy Natural Logic applied inelaboration of time series:• Analysis of time series, estimation of trend-cycle and trend• Forecasting of time series from learned linguistic
description using PbLD method• Mining information from time series
• Reduction of dimensionality• Perceptionally important points• Finding intervals of monotonous behavior• Linguistic evaluation of the direction of local and global
trend; linguistic characterization of future course of TC or Tr• Linguistic summarization of information about time series
and syllogistic reasoning
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Conclusions
Fuzzy Transform and Fuzzy Natural Logic applied inelaboration of time series:• Analysis of time series, estimation of trend-cycle and trend• Forecasting of time series from learned linguistic
description using PbLD method• Mining information from time series
• Reduction of dimensionality• Perceptionally important points• Finding intervals of monotonous behavior• Linguistic evaluation of the direction of local and global
trend; linguistic characterization of future course of TC or Tr• Linguistic summarization of information about time series
and syllogistic reasoning
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Conclusions
Fuzzy Transform and Fuzzy Natural Logic applied inelaboration of time series:• Analysis of time series, estimation of trend-cycle and trend• Forecasting of time series from learned linguistic
description using PbLD method• Mining information from time series
• Reduction of dimensionality• Perceptionally important points• Finding intervals of monotonous behavior• Linguistic evaluation of the direction of local and global
trend; linguistic characterization of future course of TC or Tr• Linguistic summarization of information about time series
and syllogistic reasoning
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Conclusions
Fuzzy Transform and Fuzzy Natural Logic applied inelaboration of time series:• Analysis of time series, estimation of trend-cycle and trend• Forecasting of time series from learned linguistic
description using PbLD method• Mining information from time series
• Reduction of dimensionality• Perceptionally important points• Finding intervals of monotonous behavior• Linguistic evaluation of the direction of local and global
trend; linguistic characterization of future course of TC or Tr• Linguistic summarization of information about time series
and syllogistic reasoning
Introduction SC techniques Analysis of TS Forecasting LFL Forecaster Trend characterization Conclusions
Conclusions
Fuzzy Transform and Fuzzy Natural Logic applied inelaboration of time series:• Analysis of time series, estimation of trend-cycle and trend• Forecasting of time series from learned linguistic
description using PbLD method• Mining information from time series
• Reduction of dimensionality• Perceptionally important points• Finding intervals of monotonous behavior• Linguistic evaluation of the direction of local and global
trend; linguistic characterization of future course of TC or Tr• Linguistic summarization of information about time series
and syllogistic reasoning