analysis, forecasting and mining information from time

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 RepublicVilem.Novak@osu.cz

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

Time series

X : T× Ω −→ R

T = 0,1, . . . ,N;T = [a,b];T = R

Fixed ω ∈ Ω: Realization

X (t) | t ∈ T

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?

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

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

Applied to :

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.

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.

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

Fuzzy Natural Logic

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)

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〉

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〉

Extension of evaluative expressions

Small Medium BigExtremely

0 4 10

VeSm? Small?Big ?

0 4 10

Roughly

Intension: W −→ F (R)

various contexts extensions

( ([0,1]))wFWIntension:

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

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

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.

Simple demonstration of PbLD

LFLC 2000 (University of Ostrava)

Simple demonstration of PbLD

LFLC 2000 (University of Ostrava)

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

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

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

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

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).

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).

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?

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?

Trend-cycle using inverse F-transform

Assumption

X (t) = TC(t) +r∑

Pj ei(λj t+ϕj )

︸︷︷︸S(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]

Assumption

X (t) = TC(t) +r∑

Pj ei(λj t+ϕj )

︸︷︷︸S(t)

F[X ] = F[TC] + F[S] + F[R]

Assumption

X (t) = TC(t) +r∑

Pj ei(λj t+ϕj )

︸︷︷︸S(t)

F[X ] = F[TC] + F[S] + F[R]

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

h = d T

λ =2πd

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

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

Artificial time series

- - - - - Original trend-cycleInverse F-transform of X

Estimation of trend-cycle

Estimation of trend

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

Forecasting methodology

Symmetric Mean Absolute Percent Error

SMAPE =1|T|

∑t∈T

|X (t)− X (t)||X(t)|+|X(t)|

SMAPE∈ [0,200]

The best trend-cycle and seasonal predictor is chosenaccording to SMAPE error on the validation set TV

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 ]

F[X ] = (Fn+1[X ], . . . ,Fn+`[X ])

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

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

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

Sj = (S((j − 1)p + 1),S((j − 1)p + 2), . . . ,S(jp))

Sθ =θ−1∑j=1

dj · Sθ−j

Sj = (S((j − 1)p + 1),S((j − 1)p + 2), . . . ,S(jp))

Sθ =θ−1∑j=1

dj · Sθ−j

Sj = (S((j − 1)p + 1),S((j − 1)p + 2), . . . ,S(jp))

Sθ =θ−1∑j=1

dj · Sθ−j

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%

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%

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

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

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

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

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)

X [h] = β01 , . . . , β

0n−1.

X [h] = β01 , . . . , β

0n−1.

X [h] = β01 , . . . , β

0n−1.

Original time series, 186 and its inverse F0-transform, h = 3Reduced time series, components β0

1 , . . . , β062

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

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

(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

(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.

∑n−1k=1 |β1

β2k =

∫ ck+1ck−1

((t − ck )2 − I2)2Ak (t)dt

I2 = 1h

∫ ck+1ck−1

sufficiently large.

∑n−1k=1 |β1

β2k =

∫ ck+1ck−1

((t − ck )2 − I2)2Ak (t)dt

I2 = 1h

∫ ck+1ck−1

sufficiently large.

∑n−1k=1 |β1

β2k =

∫ ck+1ck−1

((t − ck )2 − I2)2Ak (t)dt

I2 = 1h

∫ ck+1ck−1

sufficiently large.

Example:

Linguistic evaluation of the direction of trend

Automatically generated linguistic expressions

negligibly decreasing stagnating slightlydecreasing

somewhatdecreasing

Trend of the (whole) time series is stagnating

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

Trend of X in period T is 〈special hedge〉Inc/Dec .

Example

Trend of X in period T is slightly increasing

Example

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)

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

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

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

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)

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

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

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

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)

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

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

Conclusions

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