Business intelligence III:Feature generation and model selectionfor multiscale time series forecasting

Vadim Strijov

Moscow Institute of Physics and Technology

AINL FRUCT2016, 10 - 12 of November

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Model selection for time series forecasting

The Internet of things is the world of networking devices (portables, vehicles, buildings)embedded with sensors and software.

I Environment and energy monitoringI Medical and health monitoringI Consumer support, sales monitoringI Urban management and manufacturing

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Case 1. Energy consumption and price forecasting, 1-day ahead hourly

The components of multivariate time series with periodicity

Time series:I energy price,I consumption,I daytime,I temperature,I humidity,I wind force,I holiday schedule.

Periodicity:I one year seasons

(temperature,daytime),

I one week,I one day (working

day, week-end),I a holiday,I aperiodic events.

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Energy consumption one-week forecast for each hour

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The autoregressive matrix, five weeks

2 4 6 8 10 12 14 16 18 20 22 24

5

10

15

20

25

30

35

40

Hours

Days

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The autoregressive matrix and the linear model

X∗(m+1)×(n+1)

=

sT sT−1 . . . sT−κ+1

s(m−1)κ s(m−1)κ−1 . . . s(m−2)κ+1

. . . . . . . . . . . .snκ snκ−1 . . . sn(κ−1)+1

. . . . . . . . . . . .sκ sκ−1 . . . s1

=

sT1×1

xm+11×n

ym×1

Xm×n

.

In terms of linear regression:y = f(X,w) = Xw,

ym+1 = sT = 〈xm+1, w〉.

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The one-day forecast: expected error is 3.1% working day, 3.7% week-end

5 10 15 20

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

x 104

Hours

Valu

e

History

Forecast

The model y = f(X,w) could be a linear model, neural network, deep NN, SVN, . . .7 / 68

Structure of energy consumption

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Similarity of daily consumption

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Sunrise bias: one-year daytime and consumption

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Biased and original daytime to fit consumption over years

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One-hour line, day-by-day during a year: autoregressive analysis

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Daily similarity

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The model performance criteria and forecast errors

Stability:

I the error does not change significantly following small changes in time series,I the distribution of the model parameters does not change.

Complexity:

I the number of parameters (elements in superposition) is minimal,I the minimum description length principle holds the William Ockham’s rule.

Error: the residue εj = yj − yj for

I mean absolute error and (symmetric) mean absolute percent error

RSS =r∑

j=1

ε2j , MAPE =1

r

r∑

j=1

|εj ||yj |

, sMAPE =1

r

r∑

j=1

2|εj ||yj + yj |

.

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Design matrix

Forecast is a mapping from p-dimensional objects space to r -dimensional answers

space.

X∗ =

[x

1×ny

1×r

Xm×n

Ym×r

]

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Rolling validation

Z =

. . . . . .

xval,k1×n

yval,k1×r

Xtrain,kmmin×n

Ytrain,kmmin×r

. . . . . .

xk

The rolling validation procedure1) construct the validation vector x∗val,k for time series of the length ∆tr as the first

row of the design matrix Z,2) construct the rest rows of the design matrix Z for the time after tk and present it as3) optimize model parameters w using Xtrain,k ,Ytrain,k and compute

residues εk = yval,k − f(xvalk ,w) and MAPE,4) increase k and repeat.

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Case 2. Sales planning: to forecast the goods consumption

Retailers’ daily routines:I custom inventory,I calculation of optimal insurance stocks,I consumer demand forecasting.

I There given historical time series of the volume off-takes: foodstuff.I Let the time series be homoscedastic: its variance is time-constant.I Minimizing the loss function one must forecast the next sample.

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Custom inventory

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Excessive forecast and insufficient forecast lead to loss

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The performance criterion is minimum loss of money

Error functions: quadratic, linear, asymmetric.

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The time series of residues and its histogram

There given historgam

H = {Xi , gi}mi=1

and loss function

L = L(Z ,X ).

The optimal forecast is

X = arg minZ∈{X1...Xm}

m∑

i=1

giL(Z ,Xi )

of this convolution.

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Candies: the seasonality and trend weekly over three years

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Beverage: the week periodicity daily over four weeks

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Sparkling wine: holidays weekly over three years

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Promotional actions

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Promotional profile extraction

I Hypothesis: the shape of the profile (excluding the profile parameters) does notdepend of duration of the action.

I Problem: to forecast the customer demand during the promotional action.

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Forecast the residues to boost performance

1. Model f forecasts n(g) history ends x ft , ..., xft−n(g)+1 for one sample.

2. Compute n(g) residues εt , ..., εt−n(g)+1 as εt−k = xt−k − x ft−k .3. Function g forecasts residues εt+i ahead max(i) time-ticks.4. Combine forecasts x f ,gt+i = x ft+i + εt+i computing f for each sample x ft+i .

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Case 3. Forecasting volumes of Russian railways freight transportation

Keep a hierarchical structure of time series without loosing performance

Forecast with hierarchical aggregation of

I types of freight inI stations, regions, and roads,I for a day, week, month, and quarter,I counting all combinations above.

Satisfy the conditions:

I minimize error,I incorporate important external factors,I respect hierarchical structure,I do not exceed physical bounds of forecast values. 28 / 68

The railroad map counts ∼ 78 regions, ∼ 4000 stations, and ∼ 100 rail-yards

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Independent forecasts might be inconsistent with aggregated ones

Time

50 100 150 200

Tim

eseries

-0.4

-0.2

0

0.2

0.4

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Hierarchical data χ, independent forecasts χ and reconciliated forecasts ϕ

χt =

xt(:, :). . .

xt(n, 1). . .

xt(n,m)

, χ =

x(:, :). . .

x(n, 1). . .

x(n,m)

, ϕ =

y(:, :). . .

y(n, 1). . .

y(n,m)

.

Consistency condition Sχt = 0, t = 1, . . . ,T , where the

link matrix S of size (2 + n + m)× (1 + n + m + nm) has the form

S =

−1 1 . . . 1 0 . . . 0 0 0 . . . 0 . . . 0 0 . . . 0−1 0 . . . 0 1 . . . 1 0 0 . . . 0 . . . 0 0 . . . 00 −1 . . . 0 0 . . . 0 1 1 . . . 1 . . . 0 0 . . . 0. . . . . . . . . . . . . . .0 0 . . . −1 0 . . . 0 0 0 . . . 0 . . . 1 1 . . . 10 0 . . . 0 −1 . . . 0 1 0 . . . 0 . . . 1 0 . . . 0. . . . . . . . . . . . . . .0 0 . . . 0 0 . . . −1 0 0 . . . 1 . . . 0 0 . . . 1

.

The regions

xt(:, :) =n∑

i=1

xt(i , :);

The freights

xt(:, :) =m∑j=1

xt(:, j);

Freights given a region

xt(i , :) =m∑j=1

xt(i , j),

i = 1, . . . n;

Regions given a freight

xt(:, j) =n∑

i=1

xt(i , j),

j = 1, . . .m;

t = 1, . . . ,T .

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Performance of independent and reconciliated forecastsThere given link matrix S, admissible sets A, Band independent forecasts χ 6∈ A, χ ∈ Bto make reconciliated forecast ϕ subject to

I consistency ϕ ∈ A, A = {χ ∈ Rd | Sχ = 0},I physical limitations ϕ ∈ B,

I precision lh(χT+1, ϕ) ≤ lh(χT+1, χ) for anyhierarchical data χT+1 ∈ A ∩ B.

Theorem [Maria Stenina, 2014]Given listed conditions, the projection vector

ϕ = χproj = arg minχ∈A∩B

lh(χ, χ)

is guaranteed to satisfy the requirements ofconsistency, physical limitations and precision.

The solution of the optimizationproblem ϕ = arg min

χ∈A∩B‖χ− χ‖22

demonstrates decrease in loss for allcontrol samples.

0 20 40 60 80 100−6

−5

−4

−3

−2

−1

0x 10

6

Control point

Lt

Lt = ‖χt − ϕ‖22 − ‖χt − χ‖2232 / 68

Case 4. Internet of things, multiscale dataset for vector forecasting

Days

Energy

Max T.

Min T.

Precipitation

Wind

Humidity

Solar

Energy

Solar

τ′

τ

ti ti+1 t′

i= iτ

′ t′

i+1

Each real-valued time series s = [s1, . . . , si , . . . , sT ], si = s(ti ), 0 ≤ ti ≤ tmax

is a sequence of observations of some real-valued signal s(t) with its own sampling rate τ .

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To boost the forecast quality include external variables

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Generate features with nonparametric transformation functions

Univariate

Formula Output dimension√x 1

x√x 1

arctan x 1ln x 1x ln x 1

Bivariate

Plus x1 + x2Minus x1 − x2Product x1 · x2Division x1

x2x1√x2

x1 ln x2

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Nonparametric aggregation: sample statistics

Nonparametric transformations include basic data statistics:I Sum or average value of each row xi , i = 1, . . . ,m:

φi =n∑

j=1

xij , or φ′i =1

n

n∑

j=1

xij .

I Min and max values: φi = minj xij , φ′i = maxj xij .I Standard deviation:

φi =1

n − 1

√√√√n∑

j=1

(xij −mean(xi ))2.

I Data quantiles: φi = [X1, . . . ,XK ], wheren∑

j=1

[Xk−1 < xij ≤ Xk ] =1

K, for k = 1, . . . ,K .

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Nonparametric transformations: Haar’s transform

Applying Haar’s transform produces multiscale representations of the same data.

Assume that n = 2K and init φ(0)i,j = φ

′(0)i,j = xij for j = 1, . . . , n.

To obtain coarse-graining and fine-graining of the input feature vector xi , fork = 1, . . . ,K repeat:

I data averaging step

φ(k)i ,j =

φ(k−1)i ,2j−1 + φ

(k−1)i ,2j

2, j = 1, . . . ,

n

2k,

I and data differencing step

φ′(k)i ,j =

φ′(k−1)i ,2j − φ′(k−1)i ,2j−1

2, j = 1, . . . ,

n

2k.

The resulting multiscale feature vectors are φi = [φ(1)i , . . . ,φ

(K)i ] and φ′

i = [φ′(1)i , . . . ,φ

′(K)i ].

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Monotone functionsI By grow rate

Function name Formula ConstraintsLinear w1x + w0

Exponential rate exp(w1x + w0) w1 > 0

Polynomial rate exp(w1 ln x + w0) w1 > 1

Sublinearpolynomial rate

exp(w1 ln x + w0) 0 < w1 < 1

Logarithmic rate w1 ln x + w0 w1 > 0

Slow convergence w0 + w1/x w1 6= 0

Fast convergence w0 + w1 · exp(−x) w1 6= 0

I OtherSoft ReLu ln(1 + ex)

Sigmoid 1/(w0 + exp(−w1x)) w1 > 0

Softmax 1/(1 + exp(−x))

Hiberbolic tangent tanh(x)

softsign |x |1+|x | 38 / 68

Collection of parametric transformation functions

Functionname

Formula Outputdim.

Num.of args

Num.of pars

Add constant x + w 1 1 1Quadratic w2x

2 + w1x + w0 1 1 3Cubic w3x

3 + w2x2 + w1x + w0 1 1 4

Logarithmicsigmoid

1/(w0 + exp(−w1x)) 1 1 2

Exponent exp x 1 1 0

Normal 1w1

√2π

exp((x−w2)2

2w21

)1 1 2

Multiply byconstant

x · w 1 1 1

Monomial w1xw2 1 1 2

Weibull-2 w1w2xw2−1 exp−w1x

w2 1 1 2Weibull-3 w1w2x

w2−1 exp−w1(x − w3)w2 1 1 3. . . . . . . . . . . . . . .

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Parametric transformations

Optimization of the transformation function parameters b is iterative:

1. Fix the vector b, collected over all the primitive functions {g}, which generatefeatures φ:

w = arg minS(w|f(w, x), y

), where φ(b, s) ⊆ x.

2. Optimize transformation parameters b given model parameters w

b = arg min S(b|f(w, x), y

).

Repeat these steps until vectors w, b converge.

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Markup the time series, two types of marks: Up and Down

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Parameters of the local models

More feature generation options:I Parameters of SSA approximation of the time series x(q).I Parameters of the FFT of each x(q).I Parameters of polynomial/spline approximation of each x(q).

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Parameters of the local models: SSA

For the time series s construct the Hankel matrix with a period k and shift p, so thatfor s = [s1, . . . , sT ] the matrix

H∗ =

sT . . . sT−k+1...

. . ....

sk+p . . . s1+p

sk . . . s1

, where 1 > p > k .

Reconstruct the regression to the first column of the matrix H∗ = [h,H] and denote itsleast square parameters as the feature vector

φ(s) = arg min ‖h−Hφ‖22.

For the orignal feature vector x = [x(1), . . . , x(Q)] use the parameters φ(x(q)),q = 1, . . . ,Q as the features.

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SSA: principal components

Compute the SVD of covariance matrix of H

1

NH

TH = VΛV

T, Λ = diag(λ1, . . . , λN)

and find the principal components yj = Hvj .

−5 0 5

−6

−4

−2

0

2

4

6

8

Principal component, y1

Principalcomponent,y2

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Models and features

Models:I Baseline method: si = si−1.I Multivariate linear regression (MLR) with l2-regularization. Regularization coefficient: 2I SVR with multiple output. Kernel type: RBF, p1: 2, p2: 0, γ: 0.5, λ: 4.I Feed-forward ANN with single hidden layer, size: 25I Random forest (RF). Number of trees: 25 , number of variables for each decision split: 48.

Feature combinations:I History: the standard regression-based forecast with no additional features.I SSA, Cubic, Conv, Centroids, NW: history + a particular feature.I All: all of the above, with no feature selection.I PCA and NPCA: all generation strategies with feature selection.

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Feature analysis

Histo

rySSA

Cub

ic

Con

v

Cen

troid

sN

W All

PCA

NPC

A

MLR

MSVR

RF

ANN

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Test subset MAPE

Ratio of times each combination of model and feature performed best for at least oneof the time series (7) or error functions (6), all (6) data sets (6× 7× 6 = 252 cases).

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Best models

Energ

y

Max

T.

Min

T.

Preci

pita

tion

Win

d

Hum

idity

Solar

ANNConv

MLRHistory

MSVRSSA

RFCubic

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Test subset residues

Energ

y

Max

T.

Min

T.

Preci

pita

tion

Win

d

Hum

idity

Solar

MLRHistory

MSVRSSA

RFCubic

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Standard deviations of test subset residues

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Case 5. Classification of human physical activity with mobile devices

3D-projection of acceleration time series to spatial axis

x = {accx(t); accy (t); accz(t)}nt=1 7→ y ∈ RS .

Slow walking

0 1 2 3 4 5

−0.5

0

0.5

1

1.5

2

Time t, s

Acceleration,x,y,z

Jogging

0 1 2 3 4

−1

0

1

2

3

4

Time t, sAcceleration,x,y,z

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Local transformations for deep learning neural network

Model f = a(hN(. . .h1(x)))(w) contains autoencoders hk and softmax classifier a:

f(w, x) =exp(a(x))∑j exp(aj(x))

, a(x) = WT

2 tanh(WT

1 x), hk(x) = σ(Wkx + bk),

where w minimizes the error function.

Feature generation by local transformations:

I parameters of SSA approximation of the time series x,I FFT of x,I parameters of polynomial/spline approximation,

could reduce complexity this model down to complexity of logistic regression.

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Parameters of the local models: SSA

For time series s construct the Hankel matrix witha period k and shift p, so that for s = [s1, . . . , sT ]the matrix

H∗ =

sT . . . sT−k+1...

. . ....

sk+2 . . . s2sk . . . s1

.

−5 0 5

−6

−4

−2

0

2

4

6

8

Principal component, y1

Principalcomponent,y2

Reconstruct the regression to the first column of the matrix H∗ = [h,H] and denote itsleast square parameters as the feature vector

φ(s) = arg min ‖h−Hφ‖22.

For the original feature vector x = [x(1), . . . , x(Q)] use the parameters φ(x) as an item.

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Human gate detection with time series segmentation

Find dissection of the trajectory of principal components yj = Hvj , where H is theHankel matrix and vj are its eigenvectors:

1

NH

TH = VΛV

T, Λ = diag(λ1, . . . , λN).

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Metric features: distances to the centroids of local clusters

Apply kernel trick to the time series.1. For objects xi from X compute k-mean centroids c.2. Use distance function ρ to combine feature vector

φi = [ρ(c1, xi ), . . . , ρ(cp, xi )] ∈ Rp+.

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Computing distances to centroids of the time series: sources

0 20 40 60 80 100 1200

1

2

Walking

t

x(t)

0 20 40 60 80 100 1200

1

2

Walking Upstairs

t

x(t)

0 20 40 60 80 100 1200

1

2

Walking Downstairs

t

x(t)

0 20 40 60 80 100 1200.98

1

1.02

Sitting

t

x(t)

0 20 40 60 80 100 1201.03

1.04

1.05

Standing

t

x(t)

20 40 60 80 100 120

0.99

1

Laying

t

x(t)

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Computing distances to centroids of the time series: aligned series

0 20 40 60 80 100 1200

1

2

Walking

t

x(t)

0 20 40 60 80 100 1200

1

2

Walking Upstairs

t

x(t)

0 20 40 60 80 100 1200

1

2

Walking Downstairs

t

x(t)

0 20 40 60 80 100 1201

1.05

Sitting

t

x(t)

0 20 40 60 80 100 1201.02

1.04

Standing

t

x(t)

20 40 60 80 100 120

0.98

1

Laying

t

x(t)

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Computing distances to centroids of the time series: centroids

0 20 40 60 80 100 1200

1

2

Walking

t

x(t)

0 20 40 60 80 100 1200

1

2

Walking Upstairs

t

x(t)

0 20 40 60 80 100 1200

1

2

Walking Downstairs

t

x(t)

0 20 40 60 80 100 120

1

1.05

Sitting

t

x(t)

0 20 40 60 80 100 1201

1.05

Standing

t

x(t)

20 40 60 80 100 120

1

1.02

Laying

t

x(t)

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Performance of the human physical activities classification model

1 2 3 4 5 6 7 8 9 10 11 12Class labels

0

5

10

15

20

Ob

ject

snu

mb

er

99.5% 97.8%98.7% 98.8% 97.5%100.0%99.4% 93.4% 97.0% 99.9% 99.4% 97.2%

Mean Accuracy: 0.98231) walk forward2) walk left3) walk right4) go upstairs5) go downstairs6) run forward7) jump up and down8) sit and fidget9) stand

10) sleep11) elevator up12) elevator down

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Case 6. How many parameters must be used? Relations of 24 hourly models

5 10 15 20

2

4

6

8

10

12

14

16

18

20

22

Hours

Pa

ram

ete

rs

57 / 68

Selection of a stable set of features of restricted size

The sample contains multicollinear χ1,χ2 and noisy χ5,χ6 features, columns of the designmatrix X. We want to select two features from six.

Stability and accuracy for a fixed complexityThe solution: χ3,χ4is an orthogonal set of features minimizing the error function.

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Multicollinear features and the forecast: possible configurations

Inadequate and correlated Adequate and random

Adequate and redundant Adequate and correlated59 / 68

Model parameter values with regularization

Vector-function f = f(w,X) = [f (w, x1), . . . , f (w, xm)]T ∈ Ym.

0 2 4 6 8 10

−15

−10

−5

0

5

10

15

20

25

30

Regularization, τ

Parameters,w

S(w) = ‖f(w,X)− y‖2 + γ2‖w‖20 2 4 6 8

−2

−1

0

1

2

3

Parameters sum,∑

i

|wi|

Parameters,w

S(w) = ‖f(w,X)− y‖2, T (w) 6 τ60 / 68

Empirical distribution of model parameters

There given a sample {w1, . . . ,wK} of realizations of the m.r.v. w and an errorfunction S(w|D, f). Consider the set of points {sk = exp

(−S(wk |D, f)

)|k = 1, . . . ,K}.

0

0.2

0.4 0

0.2

0.4

0.02

0.04

0.06

0.08

w2w1

exp( −S

(w))

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100

w1

w2

x- and y -axis: parameters w, z-axis: exp(−S(w)

). 61 / 68

Minimize number of similar and maximize number of relevant features

Introduce a feature selection method QP(Sim, Rel) to solve the optimization problem

a∗ = arg mina∈Bn

aT

Qa− bT

a,

where matrix Q ∈ Rn×n of pairwise similarities of features χi and χj is

Q = [qij ] = Sim(χi ,χj) =

∣∣∣∣∣∣Cov(χi ,χj)√Var(χi )Var(χj)

∣∣∣∣∣∣

and vector b ∈ Rn of feature relevances to the target is

b = [bi ] = Rel(χi ),

where elements bi equal absolute values of the sample correlation coefficient betweenfeature χi and the target vector y.Number of correlated features Sim→ min, number of correlated to the target Rel→ max.

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Evaluation criteria for the diesel NIR spectra data set

Method Cp RSS ln λ1λn

SVD VIF BIC

QP (ρ, ρ) (τ = 10−9) −110 1.37 · 10−18 −25.7 6.43 · 106 548.38

Genetic −110.88 7.68 · 10−30 −24 8.13 · 105 534.19

LARS 3.22 · 1021 2.07 · 10−7 −28.3 7.94 · 107 529.47

Lasso 2.5 · 1028 1.61 −27.72 1.03 · 1021 1712.92

ElasticNet 2.51 · 1028 1.61 −27.72 1.03 · 1021 1712.92

Stepwise 3.66 · 1029 23.56 −36.78 1.94 · 1022 1919.23

Ridge 1.59 · 1028 1.02 −36.22 1.07 · 1022 1.79 · 103

Dependence of residual norm on the number ofselected features QP(Sim, Rel).

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ECoG brain signals to forecast upper limbs’ movements

I Neurotycho.org food-tracking dataset: 32 epidural electrodescaptures brain signals of the monkey (ECoG),

I 11 sensors track movements of the hand, contralateral to theimplant side,

I experiment duration is 15 minutes,I the experimenter feeds the monkey ≈ 4.5 per minute,I ECoG and motion data were sampled at 1KHz and 120Hz,

respectively.64 / 68

Weather forecasting and geo (spatio) temporal dataset

Monthly Ecosystem Respiration by M. Reichstein, GHG-Europe 65 / 68

Spatio temporal dataset, frequency versus time, given spatial position

Electric field measurement, the Van Allen probes by I. Zhelavskaya, Skoltech 66 / 68

Подробнее о вышеизложенных задачахАвторегрессионное прогнозирование и выбор признаков

I Катруца А., Стрижов В. Проблема мультиколлинеарности при выборе признаков в регрессионных задачах //Информационные технологии, 2015, 1 : 8-18.

I Нейчев Р.Г., Катруца А.М., Стрижов В. Выбор оптимального набора признаков из мультикоррелирующего множества взадаче прогнозирования // Заводская лаборатория. Диагностика материалов, 2016, 3.

I Мотренко А.П., Рудаков К.В., Стрижов В.В. Учет влияния экзогенных факторов при непараметрическомпрогнозировании временных рядов // Вестник Московского университета. Серия 15. Вычислительная математика икибернетика, 2016. A

Классификация временных рядов акселерометраI Ignatov A., Strijov V. Human activity recognition using quasiperiodic time series collected from a single triaxial accelerometer

// Multimedia Tools and Applications, 2015, 17.05.2015 : 1-14.

I Motrenko A.P., Strijov V.V. Extracting fundamental periods to segment human motion time series // Journal of Biomedicaland Health Informatics, 2016.

I Попова М.С., Стрижов В.В. Выбор оптимальной модели классификации физической активности по измерениямакселерометра // Информатика и ее применения, 2015, 9(1) : 79-89.

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Model selection for time series forecasting

Thanks to the Chair of Intelligent Systems, MIPT

I Anastasia Motrenko

I Mikhail Kuznetsov

I Alexandr Aduenko

I Arsentiy Kuzmin

I Maria Stenina

I Alexandr Katrutsa

I Oleg Bakhteev

I Maria Popova

I Andrey Kulunchakov

I Mikhail Karasikov

I Radoslav Neychev

I Alexey Goncharov

I Roman Isachenko

I Maria Vladimirova

http://machinelearning.rustrijov@phystech.edu

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