The Inner Structure of Time-Dependent Signals

The Inner Structure of Time-Dep enden t Signals David N. Levin Dept. of Radiology , Univ ersity of Chicago, 1310 N. Ritchie Ct., Unit 26 AD, Chicago, IL 60610 Email: d- lev in @uchicago .edu http://rad iology.uchicago. ed u/directory/david- n- levin Abstract. This pap er sh o ws how a time series of measuremen ts of an evol vin g system can b e pro cessed to create an ”inner” time series that is unaffected by any instantaneous invertible, p ossibly nonlinear t ransfor- mation of th e measurements. A n inner time series contains information that does not dep end on t h e nature of the sensors, whic h the o b server chos e to monitor the system. Instead, it enco des information that is intrinsi c to t h e evolution of the observed system. Because of its sensor- indep endence, a n inner time series ma y pro duce fewer false negativ es when it is used to detect even ts in the p resence of sensor drift. F urther- more, if the observed physica l system is comprised of non-interacting subsystems, its in ner time series is separable; i.e., it consists of a collec- tion of time series, each one b eing the inner time series of an isolated subsystem. Because of this p rop erty , an inner t ime series can b e used to detect a sp ecific b ehavior of one of t he indep endent subsystems with- out using blind source separa tion t o disentangle that su bsystem from the others. The method is illustrated b y applying it to: 1) an analytic example; 2) th e audio wa veform of one speaker; 3) video images from a mo vin g camera; 4) mixtures of au d io w av eforms of t wo speakers. Keywords: time series, nonlinear signal processing, inv arian ts, sensor, calibration, channel equalization, blind source separation 1 In tro du ct ion Consider a physical system that is b eing observed with a set of sens ors. The time s eries of raw sensor measurements con tains information ab out the evolu- tion of the s ystem of in terest, mixed with information about the nature of the sensors. F or example, video pictures contain infor mation ab o ut the evolution of the scene of interest, but they are als o influenced b y sens or-dep endent factors such as the p osition, ang ular orientation, field of view, a nd sp ectral r esp onse of the ca mera. Likewise, audio measurements may de s crib e the evolution of an acoustic source, but they a re also influenced by extrinsic factor s such as the po sitions and frequency respo ns es o f the microphones. Calibration pr o cedures can b e us e d to tra nsform measurements created with one set o f senso rs so that they can b e compar ed to measurements made with a different s et of sensor s ([1],[2],[3]). Ho wev er , there are situations in which it is inco nv enient, awkward, 2 David N . Levin or imposs ible to calibrate a mea s urement apparatus. F or example: 1) the calibr a- tion pro cedure ma y take too muc h time; 2) the ca libration pro cess may in ter fere with the evolution of the system b eing observed; 3) the observer may no t hav e access to the measuring device (e.g., b ecause it is at a remote lo catio n). This pap er descr ib e s how a time s eries of mea surements can be pro cesse d to derive a purely sensor-indep endent description of the evolution of the underly- ing physical system. Specifically , consider a n evolving ph ys ical system with N degrees o f freedom ( N ≥ 1), and s upp os e that it is b eing observed b y N sen- sors, who se output is denoted by x ( t ) ( x k ( t ) for k = 1 , . . . , N ). F or simplicit y , assume that the sensor s’ output is in vertibly related to the sys tem states. In other words, assume that the sensor measurements repres e nt the s ystem’s state in a co o rdinate sys tem defined by the nature of the sensors . Section 4 describ es how measur ements can be c hosen to ha ve this in vertibilit y pr op erty . Now, sup- po se that the same system is also be ing observed by another s et of sensor s, whose output, x ′ ( t ), is in vertibly rela ted to the s ystem sta tes and, therefore, is inv ertibly related to x ( t ). F or example, x ( t ) a nd x ′ ( t ) could be the outputs of calibrated and uncalibrated sensors , res p ectively , as they s imultaneously observe the same system. Or, they could be th e outputs of sensor s that detect different t yp es of energ y (e.g., infrar ed light vs ultraviolet light). Under these conditions , we show how to pro cess x ( t ) in o rder to der ive an ”inner” time series, w ( t ) ( w k ( t ) for k = 1 , . . . , N ). W e then demonstrate that the same inner time series will r esult if the other set of sensor outputs, x ′ ( t ), is sub jected to the same pr o- cedure. Because of its sensor-indep endence, an inner time series may pro duce few er false negatives when it is used to detect even ts in the presence of sen- sor drift. In mathematical terms, x ( t ) a nd x ′ ( t ) repre s ent the evolving system’s state in different co ordina te systems on state space, and the inner time series is a co ordinate- system-indep endent description of the system’s velo city in state space. T o derive this sensor- independent time series, the time ser ies of sens or mea - surements, x ( t ), is statistically pro cess ed in order to constr uct N lo cal vectors at eac h point in state space. The system’s path through state space c an then be describ ed by a successio n of s mall displacement v ecto r s, each of which is a weigh ted super p osition of the lo cal vectors. The inner time series is compris ed of these time-dep endent weigh ts, w ( t ), which are co ordinate-sy s tem-indep e nden t and, therefore , sensor - indepe nden t. Thus, any t wo observers will describ e the system’s e volution with the sa me inner time series, even though they utilize different sensors to monitor the sy s tem. Es sentially , an inner time ser ies is a ”canonical” fo r m of a measurement time series , created by normalizing the mea- surements with resp ect to their own statistica l prop erties. No matter what linear or nonlinea r transformatio n has b een a pplied to a sequence of measurements, its canonical form (i.e., its inner time series) is the same. An inner time series is roughly analogo us to the principa l compo nent s of a data se t, which represent the data in the same ”canonical” w ay , no matter w ha t rotation a nd/or tr anslation has bee n applied to them. The Inner Structure of Time-Dep en dent Signals 3 There ar e man y wa ys of using a time series of meas ur ements to define lo cal vectors on the system’s state space , and each o f these methods can b e used to create a sensor -indep endent description of the system’s evolution. Ho wev er, the lo cal vectors describ ed in this pap er ha ve an unusually attractive pro p er ty: namely , they pro duce sep ar able sensor -indep endent descriptions of sy s tems that are comp osed o f non-interacting subsystems. Sp ecifica lly , cons ider a s y stem that is composed of tw o statistically independent subsy stems, and supp ose that the raw measurements of it are linear or nonlinear mixtures of the state v ar iables of its non-interacting s ubsystems. It can be shown that eac h component of the inner time series of the comp osite system is als o a comp onent of the inner time series of an isola ted subsystem. In other words, each compone nt of the inner time serie s of the comp os ite sys tem is a stream of informa tion ab out just one of the subsystems, even though it may hav e been derived from measurements sensitive to several subsy stems. Because of this pro per ty , an inner time se r ies can be used to detect a sp ecific behavior o f one subsystem, which is evolving in the presence o f other subs y stems. I n contrast to blind sour c e separ ation pro cedures ([4], [5 ], [6]), this is do ne without finding the mixing function, which rela tes the raw measurements of the comp osite sys tem to the states of its subsystems. Reference [7] describ es a different wa y o f cr eating sensor- indep endent repre- sentations of ev olving systems. Fir st, the second- order co rrelatio ns of the sys - tem’s loc al v elo city distr ibutions are used to define a Riemannian metric and affine connectio n o n the manifold of mea s urements. Then, each incr emental dis- placement along the system’s path through state spa c e is describ ed a s a su- per p osition of reference vectors, parallel tra nsferred from the beginning of the path. Such a description will b e co o rdinate-sys tem-indep e ndent (and, therefore, sensor-indep endent), if it includes a co or dina te-system-indep endent wa y of iden- tifying the reference vectors at the initial p oint of each path of in ter est. In contrast, the metho d pr op osed in the current pap er do es not re quire reference vectors; instead it utilizes loca l vectors that are pr op erties of the loca l velo city distributions of the system’s past tra jectory . F urthermore, the methodo logy in [7] do es not provide a simple description of comp osite systems. In co nt r a st, the metho d prop osed here always creates a sensor-indep endent description of a co m- po site system, consisting of a colle c tion of the sensor-indep endent des criptions of the indep endent subsystems. The next section describ es the pro ce dur e for computing an inner time series from a time s e ries of r aw measurements. It also demonstrates that the inner time series of a co mpo site system consists of a collection of the inner time series of its constituent parts. Section 3 illustr a tes the metho d by applying it to: 1) a n analytic example; 2) the audio wav eform of one sp eaker; 3) video images fro m a moving camera; 4) mixtures of audio w aveforms of t wo sp eakers. The last section discusses the implications of this appr oach. 4 David N . Levin 2 Metho d The following subsection outlines how a time series of sensor measurements ca n be pro cessed in order to der ive lo cal vectors at each point in the state space o f the observed system. This pro cedure is only presented in outline form here, b ecause detailed des criptions ca n b e found in [8] a nd [9]. It is then shown ho w these vectors can be used to create an inner description of the system’s path through state space. In the se cond subsection, the sys tem is a ssumed to b e comp osed of t wo statistically indep endent subsystems. It is shown that the inner tim e series of the comp o site system is a simple collection of the inner time series of its subsystems. 2.1 Deriv ation of inner time series The fir st step is to construct seco nd-order and four th-order lo ca l corr elations of the data’s velocity ( ˙ x ) C kl ( x ) = h ( ˙ x k − ¯ ˙ x k )( ˙ x l − ¯ ˙ x l ) i x (1) C klmn ( x ) = h ( ˙ x k − ¯ ˙ x k )( ˙ x l − ¯ ˙ x l )( ˙ x m − ¯ ˙ x m )( ˙ x n − ¯ ˙ x n ) i x (2) where ¯ ˙ x = h ˙ x i x , where the br a ck et denotes the time average ov er the tra jectory’s segments in a small neighborho od of x , a nd wher e all subscr ipts are integers betw een 1 and N with N ≥ 1. Next, let M ( x ) be any lo cal N × N matrix, and use it to define M -transformed velocity corr elations, I kl and I klmn I kl ( x ) = X 1 ≤ k ′ , l ′ ≤ N M kk ′ ( x ) M ll ′ ( x ) C k ′ l ′ ( x ) , (3) I klmn ( x ) = X 1 ≤ k ′ , l ′ , m ′ , n ′ ≤ N M kk ′ ( x ) M ll ′ ( x ) M mm ′ ( x ) M nn ′ ( x ) C k ′ l ′ m ′ n ′ ( x ) . (4) Because C kl ( x ) is ge ne r ically p os itive definite at any p o int x , it is almost alwa ys po ssible to find a particular form of M ( x ) that satisfies I kl ( x ) = δ kl (5) X 1 ≤ m ≤ N I klmm ( x ) = D kl ( x ) , (6) where D ( x ) is a diago nal N × N matrix ([8], [9]). As long as D is not degen- erate, M ( x ) is unique, up to arbitra ry lo c al p ermutations a nd/or reflections. In almost all applica tio ns of interest, the velocity corr elations will b e co nt inuous functions of x . Ther efore, in any neig hborho o d of state s pace, there will a lwa ys be a con tinuous solution for M ( x ), and this solution is unique, up to arbitra ry glob al p ermutations and/ o r r e fle c tions. The Inner Structure of Time-Dep en dent Signals 5 In any other co o rdinate sy stem x ′ , the most g eneral solution for M ′ is g iven by M ′ kl ( x ′ ) = X 1 ≤ m, n ≤ N P km M mn ( x ) ∂ x n ∂ x ′ l , (7) where M is a matrix that satisfies (5) and (6) in the x co or dinate system and where P is a pr o duct of pe rmutation, reflection, and identit y matrices ([8], [9]). By construction, M is no t singular. Notice that (7) s hows t ha t the rows o f M tr ansform as lo ca l cov aria nt vectors, up to a global per mut a tion and/or reflection. Likewise, the same equatio n implies that the columns of M − 1 transform as lo cal contrav ar iant vectors (denoted as V ( i ) ( x ) for i = 1 , . . . N ), up to a global p er m utatio n and/o r reflection. Because these vectors are linea rly indep endent, the measuremen t velo city at each tim e ( ˙ x ( t )) ca n b e r epresented by a weigh ted super p osition of them ˙ x ( t ) = X 1 ≤ i ≤ N w i ( t ) V ( i ) ( x ) , (8) where w i are time-dependent weigh ts. Becaus e ˙ x and V ( i ) transform as con- trav ar iant vectors (except for a possible global per mut a tio n and/or re flec tion), the w eights w i m ust transform a s s c alars o r inv a riants; i.e., they are indepen- dent of the co o r dinate system in which they are computed (except for a po s sible per mutation and/or reflection). Therefor e, the time-dep endent weights, w i ( t ), provide an inner (co ordinate-system-indep endent) description of the system’s velocity in state s pace. Two observers, who use differen t senso rs (and, therefor e, different state space co ordinate s ystems), will derive the same inner time ser ies, except for a p ossible global p ermutation and/ or r eflection. This equation can b e integrated o ver the time in ter v al [ t 0 , t ] to give an ex- pression for the system’s state during that time interv al x ( t ) = x ( t 0 ) + Z t t 0 X 1 ≤ i ≤ N w i ( t ) V ( i ) [ x ( t )] dt, (9) This is an integral equation for c onstructing x ( t ) on the interv a l [ t 0 , t ] from the weigh t time ser ies, w i ( t ), on the same time interv a l. Note that, g iven a set of lo ca l vectors, ther e is a many-to-one corresp ondence b etw een the set o f measurement time series and cor resp onding inner time series. Sp ecifically , (8) shows that each measurement time ser ies maps o nt o just o ne weigh t time series. How ever, as shown b y (9), one w eight time ser ies maps on to multiple time series of sensor measurements, differing by the choice of the initial po int, x ( t 0 ). It should als o b e men tioned that it may b e difficult to use this equation to numerically compute the measurement time series, c orresp o nding to a given weight time series, because error s will tend to accumulate as one integrates the r ight side. 2.2 Inner time series of comp os ite systems Now, consider the special cas e in which the observed sys tem is co mpo site (or separable) in th e sense that it consists o f t wo statistically indep e nden t subs ys- 6 David N . Levin tems. Specifica lly , assume that there is a state spa ce coo r dinate system, s , in which the state co mpo nents, s k ( t ) for k = 1 , . . . , N , c a n b e partitioned into tw o groups, s (1) = ( s k for k = 1 , . . . , N 1 ) and s (2) = ( s k for k = N 1 + 1 , . . . , N ), that are statistically indep endent in the following s ense ([8], [9]). Let ρ S ( s, ˙ s ) b e the PDF in ( s, ˙ s )-space. Namely , ρ S ( s, ˙ s ) dsd ˙ s is the f r a ction of total time that the lo ca tion and velocity of s ( t ) a re within the volume element dsd ˙ s at lo cation ( s, ˙ s ). The subs ystem state v ariables, s (1) and s (2) , are assumed to b e sta tis ti- cally independent in the sense that the density function of the system v a riable is the pro duct of the density functions of the tw o subsystem v ariables ; i.e., ρ S ( s, ˙ s ) = Y a =1 , 2 ρ a ( s ( a ) , ˙ s ( a ) ) . (10) This separability cr iterion in ( s, ˙ s )-space is stronger than the con ven tiona l for- m ulatio n in s -space, and references [8] and [9] argue that this makes it preferable to the co nv entional criterion. In the fo llowing paragraphs, it is shown that, if the data ar e separa ble in the ab ov e sense, the co mp one nts of the inner time series o f the comp osite s ystem ca n b e partitioned in to tw o gro ups, each of which provides a n inner description of one of the subsys tems. Although these results are demonstrated here for s ystems with t wo indep endent subsystems, they can be easily gener alized to systems with any nu mber of subsystems. T o show this, the first step is to transform (8) into the s co ordinate system, by multiplying each side b y ds/dx . Bec ause the V ( i ) transform as co ntra v ariant vectors (up to a p ossible p ermutation a nd/or reflectio n), it follows that ˙ s ( t ) = X 1 ≤ i, j ≤ N w i ( t ) P ij V S ( j ) , (11) where V S ( j ) is V ( j ) in th e s coor dinate system and P is a p oss ible p ermutation and/or reflection. By definition, the V S ( i ) are the lo cal vectors, which ar e derived from the lo ca l distribution of ˙ s in the sa me wa y tha t the V ( i ) were derived from the lo cal distribution of ˙ x . Specifically , V S ( i ) is the i th column of M − 1 S , where M S is the M matrix that is derived from the se c ond- and four th-order velocity correla tions in the s co or dinate system. The next step is to show tha t the matrix M S has a simple block-diago nal form. In particular, [8] and [9] show that M S is given by M S ( s ) =  M S 1 ( s (1) ) 0 0 M S 2 ( s (2) )  . (12) where each submatrix , M S a for a = 1 , 2 , satisfies (5) and (6) for correlations betw een comp onents of s ( a ) . Observe that e ach vector V S ( i ) v anishes except where it pass e s through one of the blocks of M − 1 S . Therefo re, equation (11) is equiv alent to a pair of equatio ns, which are formed by pro jecting it onto eac h blo ck co rresp onding to a subsystem state v ariable. F o r exa mple, pro jecting bo th sides of (11) onto blo ck a gives the res ult ˙ s ( a ) ( t ) = X 1 ≤ i ≤ N j ∈ block a w i ( t ) P ij V S ( j a ) . (13) The Inner Structure of Time-Dep en dent Signals 7 Here, V S ( j a ) is the pro jection of V S ( j ) onto blo ck a ; i.e., it is the co lumn of M − 1 S a that coincides with column j of M − 1 S , as it passes throug h blo ck a . This means that the vectors, V S ( j a ) for j ∈ bl ock a , ar e the lo cal vectors on the s ( a ) manifold, which are der ived from the lo c al distr ibution of ˙ s ( a ) in the sa me wa y tha t the V ( i ) were derived from the local distribution of ˙ x . Notice that ea ch time- de p endent weigh t, w i ( t ), describ es the evolution of just one subsys tem. In o ther words, the weigh ts do not co ntain a mixture of infor mation ab out the evolution of the t wo subsystems. This is tr ue despite the fact that they ca n b e derived from raw measurements that may be complicated unknown mixture s of the state v ariables of b oth subsystems. Next, define group 1 (group 2) to b e the set of weigh ts app ear ing in the expression X 1 ≤ i ≤ N w i P ij (14) for j ∈ b l ock 1 (for j ∈ bl ock 2 ). Equation (13) shows that the weigh ts in g roup 1 (group 2) compris e a sensor- indepe ndent descriptio n of the velo city of subsys tem 1 (subsy stem 2). E quation (13) also sugg e sts that the weigh ts in gr oup 1 must b e statistically indep endent of the weigh ts in g r oup 2 . Sp ecifically , (13) implies that the w eights in each group can be computed from: 1) the time co urse of the state v ariable of the corresp onding subsystem; 2) the local vectors of the corresp onding subsystem, which themselv es are constructed from the time cours e of the state v ariable of the corresp onding subsystem. Because the w eig hts in group 1 and group 2 are derived fro m s 1 ( t ) and s 2 ( t ), respe c tively , and b eca use the latter are sta tistically independent, it is likely that the former are a lso statistically independent. 3 Analytic and E xp erimen tal E xamples In this sectio n, the metho dology of Section 2 is illustra ted by applying it to: 1 ) an analytic example (namely , a time series equal to a sine wav e); 2) th e audio wa veform of a single sp eaker; 3) video data from a camer a moving with tw o degrees of freedom; 4) nonlinear mixtures of the wa veforms of tw o sp eakers. 3.1 Analytic example: a sine wa ve In this subsection, the pro p osed metho dology is applied to a mea surement time series, s imu la ted by a sine wav e. Its inner time s eries is derived analytically , b e- fore and a fter it is transfo rmed by an ar bitrary monotonic function. The trans - formed data, which simulate the o utput of a second sensor, are sho wn to ha ve the same inner time series as the untransformed data fr o m the fir st s ensor. Suppo se the measured senso r signa l is x ( t ) = a sin( t ) (15) 8 David N . Levin where a is any real num b er and −∞ ≤ t ≤ ∞ . Beca use of the per io dicity of the signal, the lo cal second-o rder velocity co rrelatio n can b e shown to b e C 11 ( x ) = a 2 − x 2 . (16) The 1 × 1 “ ma trix”, M , is M 11 ( x ) = ± 1 / p a 2 − x 2 , (17) and the one-component lo ca l vector, V (1) ( x ), is V (1)1 ( x ) = ± p a 2 − x 2 . (18) Either sig n can be chosen in (17 ) and (18) beca us e M is o nly determined up to a global reflectio n. Substituting (15) and (18) into (8) shows that the weigh t time series is w 1 ( t ) = ± sg n [ a cos( t )] . (19) Thu s, for this simple p er io dic signal, the inner time series is the sign of the signal’s time der iv ative. As shown in the follo wing subsections, a m uch larg er amount of information is contained in the inner time s eries of more co mplex one-comp onent signa ls. The senso r-indep endence (or co or dinate-system-indep endence) o f the inner time serie s ca n b e demonstrated explicitly by computing it fr om measurements that hav e been tra nsformed by a monotonic function, f ( x ), whic h simulates the re lative resp o nse of a different senso r. Sp ecifically , consider the trans formed measurements given by x ′ ( t ) = f [ a sin( t )] , (20) where f is monotonic. The lo ca l seco nd-order co r relation of the v elo c ity of these measurements is C ′ 11 ( x ′ ) =  d f dx a cos( t x ′ )  2 , (21) where d f /dx is ev aluated at x = a sin( t x ′ ) and where t x ′ is any solution of f [ a sin( t x ′ )] = x ′ . Beca use the measurements have just one co mpo nent, the 1 × 1 ”matrix” M ′ is e q ual to M ′ 11 ( x ′ ) = ± 1 / q C ′ 11 ( x ′ ) , (22) and the local vector is V ′ (1)1 ( x ′ ) = ± q C ′ 11 ( x ′ ) . (23) Substituting (20) and (23) into (8) shows that the weight function is w ′ 1 ( t ) = ± sg n [ a cos( t ) ] = w 1 ( t ) . (24) Thu s, the transfor med and untransformed measurements ((20) and (15 )) hav e the s ame inner time se r ies (up to a reflection), This shows tha t the weigh ts are sensor-indep endent (and co or dinate- system-indep endent), a fact that w as prov ed in Sec tio n 2. The Inner Structure of Time-Dep en dent Signals 9 3.2 The audio signal of a singl e sp eak er In this subsection, the prop os ed method is applied to the audio w aveform of a single sp eaker, b efor e and after it has bee n transformed by a nonlinear monotonic function, which simulates the r elative resp onse of another sensor. The inner time series of the untransformed and transformed signa ls are shown to b e almost the same. The male s pe aker’s audio wa veform, x ( t ), was a 3 1.25 s excer pt from a n audio bo ok reco rding. This wav eform was sampled 16 ,000 times p er second with t wo bytes of depth. The thin black line in Fig ure 1 shows the sp eaker’s wa veform during a short (31.25 ms ) in terv al. The thic k gray line in Figure 1, x ′ ( t ), simu la tes the output of another sensor , which is related to x ( t ) by the monotonic nonlinear transformatio n in Figure 2. Fig. 1: The thin black line shows a 31.25 ms excerpt of x ( t ), the audio wa veform of a spea ker. The thick gray line shows x ′ ( t ), the same wa veform, after it has bee n trans formed b y the monotonic nonlinear tra nsformation shown in Figure 2. The technique in Subsection 2 .1 w a s applied to 500,00 0 samples of x ( t ) and x ′ ( t ), in o rder to derive the one-co mpo nent vectors, V (1) ( x ) and V ′ (1) ( x ′ ), in an array o f 128 bins on the x a nd x ′ manifolds, r esp ectively . Thes e vectors are display ed in Figur e 3 . Then, these vectors and equation (8) were used to compute the inner time series, w 1 ( t ) and w ′ 1 ( t ), corresp o nding to the tw o measurement time ser ie s, x ( t ) and x ′ ( t ), resp ectively . The r esulting time series o f weigh ts ar e s hown in Figure 4. Notice that the tw o inner time series ar e almost the same, despite the fact that they w ere derived from sensor measurements, which differed b y a nonlinear transformatio n. This demonstrates the sensor -indep endence of the weigh ts, a prop erty that was proved in g e neral in Subse c tion 2.1. 10 David N . Levin (a) (b) Fig. 2: The left panel shows the mo notonic nonlinear transformation, x ′ ( x ), whic h was applied to the sensor measurements, x ( t ), in order to cr eate x ′ ( t ) (Figure 1). The la tter time se ries simulates the output of a differen t senso r. The rig ht panel is a magnified view of the central p ortion of the left panel. (a) (b) Fig. 3: The left a nd rig ht panels s how the lo cal vectors, V (1) ( x ) and V ′ (1) ( x ′ ), which were derived from 500 ,0 00 samples of x ( t ) and x ′ ( t ), resp ectively . The Inner Structure of Time-Dep en dent Signals 11 Fig. 4: The thin black line and thick gray line sho w the inner time ser ies, w 1 ( t ) and w ′ 1 ( t ), resp ectively , during the 31 .25 ms time interv a l depicted in Figure 1. When e ither inner time series was played a s an audio file, it sounded lik e a completely intelligible v er sion o f the or iginal audio wa veform, x ( t ). No se ma nt ic information was lost, although the proso dy of the signal may ha ve been mo dified. Therefore, in this exp eriment, almos t all of the sig nal’s information co nten t was preserved by the pro cess of deriving its inner time ser ies. 3.3 Video data from a mo ving camera In this subsection, the pr o cedure in Subsection 2.1 is used to derive the inner time serie s of a sequence of video images, re corded by a camera moving in an office. W e also computed the inner time series of the same image sequence, after each image was s ub jected to a nonlinear transfor mation, ther eby s im ula ting the output of a different sens o r (i.e., a different video camera). The t wo inner time series were a lmost the same, despite the fa c t tha t they w er e derived from the outputs of dramatically different sensor s. The original (i.e., untransformed) images were recor ded by a cell phone v ideo camera as it was moved in an irregula r fa shion over a p ortion of a spher ical sur- face, having a r adius of a pproximately 2 5 cm. The plane of the camera was ori- ent ed so that it was alwa ys tange ntial to the surface, a nd the camer a’s low er edge was k ept par allel to the flo o r a t all times. In this wa y , the camera was mo ved with t wo degrees of freedo m; i.e., it w as mov ed through a ser ies of configuratio ns (p o- sitions and orientations) that formed a t wo-dimensional manifold. The camera recorded thirty frames p er seco nd over the co urse o f approximately 70 minutes, pro ducing a total of 126,0 36 fra mes. Each frame consis ted of a 320 × 240 ar ray of pixels, in whic h the RGB res po nses were measured with one byte of depth. The top r ow of Fig ure 5 displays a typical s eries of ima ges, subsampled at 1.6 7 s interv als over the course of 17 s. The second time s eries of images was created by sub j ecting ea ch recorded image to a nonlinear transforma tion. Sp ecifically , each pixel with image co o rdi- nates ( h, v ) in a giv en recorde d frame was mapped to the lo ca tion with image 12 David N . Levin Fig. 5: The top r ow shows sample images , which w ere r e c orded by a moving video camera a t 30 fra mes p er second and then subsampled at 1 .67 s in terv als. The bo ttom row show the images in the to p row, after they were sub j ected to the nonlinear transfor mation depicted in Fig ure 6 co ordinates ( h ′ ( h ) , v ′ ( v )) in the corresp onding transformed frame, wher e h ( h ′ ) and v ( v ′ ) are shown in Figure 6. It is eviden t that this tra nsformation turns each image upside down and backwards, in addition to stretching or compressing each image near its bo rders. The b ottom r ow of Figure 5 shows the images tha t were pro duced by nonlinearly tra nsforming the corres po nding r ecorded frames in the top ro w. These ima ges sim ulate the output of a different senso r (e.g., a video camera, which was ”wearing” gogg les having inv erting/ distorting le nses). (a) (b) Fig. 6: The nonlinear transforma tio n betw een ( h, v ), the co or dinates of a pixel in eac h reco rded image, and ( h ′ , v ′ ), the co ordinates of the c o rresp onding pixel in the transformed image. Because the video was reco r ded as the camera moved through a tw o-dimens ional manifold of configura tions, the resulting imag es were exp ected to form a tw o - dimensional manifold in which each frame w as represented by a p oint. A co or- dinate sys tem, x , was imp osed o n this manifold in the follo wing manner . First, we co mputed six num be rs consisting of the cen tro ids of the R, G, and B com- The Inner Structure of Time-Dep en dent Signals 13 po nents for each recorded image. Then, we did a pr incipal comp onents analy- sis of the collection of six-dimensional multiplets for all recorded video frames. This showed that these m ultiplets were in or close to a tw o-dimensional pla- nar subspa ce, which contained 99% of their v ariance. B e cause this subspace did not self-in ter s ect, its p oints w ere in vertibly rela ted to the configurations of the camera. The x co o rdinates of each image w er e taken to b e the first tw o v ariance- normalized principal comp onents of the corre s po nding multiplet. The sa me pro - cedure was applied to the collection of transformed images in order to assign a t wo-comp onent coo rdinate, x ′ , to ea ch one. The thin black lines in Figure 7 show the measuremen t time series, x ( t ), derived from the images recorded during a t ypica l 17 s time in terv al. The thick g ray lines in the same figure show the se ns or measurements, x ′ ( t ), derived fr om the sequence of transformed images during the same time int er v al. The x ( t ) and x ′ ( t ) time series can be co nsidered to be the measurements that were produce d by tw o observers who were watching the same ph ys ical s y stem with differen t sensors (i.e., with an ordinary video camera and with a ca mera having distorting/inv er ting lenses, resp ectively). Alternatively , x ′ ( t ) can b e considered to b e the mea surements x ( t ), a fter they have b een trans- formed to a no ther coo rdinate system ( x ′ ) o n the tw o- dimensional manifold of images. (a) (b) Fig. 7: The thin blac k lines and the thic k gray lines show the sens or measure- men ts, x ( t ) and x ′ ( t ), derived from the sequences of un tra nsformed and trans- formed images, resp ectively , dur ing a 1 7 s time interv al. The 126,0 36 measurements, x ( t ), derived from the seque nce of un tra nsformed images, were a ssigned to bins in a 4 × 4 ar r ay . Then, the pro cedure in Subsection 2.1 was used to compute the lo cal vectors in each bin ( V ( i ) ( x ) for i = 1 , 2 ). The same pro cedure w a s applied to measurements x ′ ( t ), derived from the transformed images, in order to compute the lo cal vectors, V ′ ( i ) ( x ′ ). These lo cal vectors a re shown in the left and right panels of Figure 8. 14 David N . Levin (a) (b) Fig. 8: The thin black lines and the thic k gray lines in the left panel show the lo cal vectors, V (1) and V (2) , r esp ectively , derived from the sequence of unt r a nsformed images. The rig ht panel sho ws the lo ca l vectors derived fro m the transformed images. The black p oints in each panel show the coo rdinates of a ra ndom sa mple of the measurements, x ( t ) and x ′ ( t ), derived from the recor ded and transfor med images, resp ectively . The meas urement time series, x ( t ), and the cor r esp onding lo cal vectors, V ( i ) , were substituted in (8) in o rder to derive the inner time series, w i ( t ), co rresp ond- ing to the seque nce of untransformed images. Likewise, the measurement time series, x ′ ( t ), a nd the cor resp onding lo cal v ectors, V ′ ( i ) , were used to der ive the inner time series, w ′ i ( t ), co rresp onding to the sequence o f transformed images . The thin black lines and the thic k gray lines in Figure 9 show the weights, w i ( t ) and w ′ i ( t ), resp ectively , during the time interv al depicted in Figure 7, a fter w ′ i ( t ) was mult iplied by a global p ermutation and reflection. Notice that the inner time series are near ly the same, despite the fact that they were derived from the outputs of drama tica lly differen t sensor s. In other w or ds, the inner time series are sensor - indepe nden t, as prov ed in Subsectio n 2.1. These r esults lo o sely mimic the findings of the well-known psychoph ysical ex- per iments ([12]) in which sub jects, who wore in verting/distorting gog gles, even- tually learned to per ceive the world as it was per ceived befor e wearing the gog- gles. Similarly , Figure 9 shows that the observer, whose camera was ”w earing ” goggle s , p erceived the inner properties of the image time series to be the same (thic k g ray lines) as they w ere p erceived b efor e w earing the go ggles (thin black lines). 3.4 Nonlinear mixtures of t wo audio w a v eforms In this subse c tion, the system co nsists of tw o spe akers, whose uttera nces are statistically indep endent and are observed in tw o wa ys : 1) as a pa ir of unmixed signals, each one b eing o ne sp eaker’s w aveform; 2) as a pair o f nonlinear mixtures The Inner Structure of Time-Dep en dent Signals 15 (a) (b) Fig. 9: The thin blac k lines and t he thic k gray lines show the inner time series, w i ( t ) and w ′ i ( t ), deriv ed from the sequences of untransformed and transformed images, resp ectively , dur ing the 17 s tim e interv al depicted in Figure 7. of the unmixed signals. The unmixed and mixed pair s of signals simulate mea- surements made b y tw o observers who were using differen t sensors. The pro ce- dure in Subsection 2.1 was applied to der ive the inner time series , corresp onding to the unmixed and mixed s ignals. These inner time series a re s hown to be al- most the same, there b y demonstrating their sensor indep endence. F urthermor e, the time series of each weigh t comp onent, derived from the signal mixtures, is almost the sa me as the time series of a weigh t comp onent, derived from one of the unmixed signals. This demonstrates that the inner tim e series of a com- po site system is simply a collection o f the inner time series of its statistically independent subsystems, as proved in Subsection 2.2. The unmixed signals were e x cerpts from a udio b o o k recor dings of tw o ma le sp eakers, who w ere reading different texts. The t wo audio w avef o r ms, denoted x k ( t ) for k = 1 , 2, were 31.25 s long and w ere sampled 16,000 times per second with tw o bytes of depth. Figure 10 shows the tw o speakers’ w aveforms during a short (31.25 ms) interv al. These wa veforms were then mixed by the nonlinear functions µ 1 ( x ) = 0 . 7 63 x 1 + (958 − 0 . 0225 x 2 ) 1 . 5 µ 2 ( x ) = 0 . 1 53 x 2 + (3 . 75 ∗ 10 7 − 763 x 1 − 229 x 2 ) 0 . 5 , (25) where − 2 15 ≤ s 1 , s 2 ≤ 2 15 . This is one of a v ariety o f nonlinea r transformatio ns that were tried with s imilar res ults. The mixed mea s urements, x ′ k ( t ), were taken to b e the v aria nce-normaliz ed, principal comp onents of the wa veform mixtures, µ k [ x ( t )]. Figur e 11 shows how this nonlinear mixing function mapp ed an evenly- spaced Cartesian grid in the x coor dinate sys tem on to a w arp ed grid in the x ′ co ordinate s ystem. Notice that the mapp ed grid do es not ”fold over” o nto itself, showing that it is an invertible mapping. The lines in Fig ure 1 2 s how the time course of x ′ ( t ). When either wa veform mixture ( x ′ 1 ( t ) o r x ′ 2 ( t )) was play ed as a n 16 David N . Levin audio file, it sounded like a confusing superp os itio n of tw o v oices, which w ere quite difficult to understand. (a) (b) Fig. 10: The unmixed audio w aveforms of the t wo sp ea kers during a 31.2 5 ms time int er v al. The metho d in Section 2 was then applied to these data as follows: 1. The 500,000 measurement s of the first unmixed wav eform, consisting of x 1 and ˙ x 1 at each sampled time, were sorted into an arr ay of 1 6 bins in x 1 -space. Then, the ˙ x distr ibution in each bin was used to compute lo ca l velo c it y correla tions, and these were used to de r ive the one - comp onent lo cal vector, V (1) ( x 1 ), in each bin in x 1 -space. The left panel of figure 13 shows these lo cal vectors a t each p oint. These vectors and the ˙ x 1 time ser ies were substituted in (8) in o rder to compute the inner time series , w 1 ( t ), for the first unmixed wa veform, The result is shown by the thin black line in the left panel o f Figure 14. 2. The same pro cedure w as applied to the second unmixed wav eform in o rder to compute its inner time ser ie s, w 2 ( t ). The re sult is shown by the thin black line in the right panel of Figure 14. 3. The 50000 0 s a mples o f the mixed wa veform, x ′ ( t ), were sorted int o a 1 6 × 16 array of bins in x ′ -space, and the distribution of velocities, ˙ x ′ , in each bin was used to compute the lo cal v ectors , V ′ ( i ) ( x ′ ), a t each p oint. These are shown in the right panel of Figure 13. These vectors a nd the v elo city tim e s eries, ˙ x ′ ( t ), were substituted in (8 ) to compute the inner time series, w ′ i ( t ), of the mixed wa veforms. These are depicted b y the thick gray lines in Figure 14, after they ha d b een multiplied by an ov erall p ermutation/reflection matrix . It is evident that the unmixed a nd mixed wa veforms hav e inner time series that are almost the same. This demonstra tes that a n inner time series is not The Inner Structure of Time-Dep en dent Signals 17 Fig. 11: A w ar p ed gr id in the x ′ co ordinate system, obtained b y applying the nonlinear mixing function in (25) to a regular Car tesian gr id in the x co ordinate system. (a) (b) Fig. 12: The mixed audio w av efor ms of the t w o spea kers, obtained b y applying the nonlinear mixing function in (25) to the unmixed wav eforms in Figure 10. 18 David N . Levin (a) (b) (c) Fig. 13: The left and middle panels show the one-comp onent lo cal vectors derived from the unmixed w av efor ms, x 1 ( t ) and x 2 ( t ), excerpts of whic h are illustr ated in Figure 1 0. The line segments in the r ight panel show the lo cal vectors derived from the mixed wa veforms, x ′ ( t ), excerpts o f which ar e illustrated in Figure 12. These line segments hav e b een unifor mly res caled fo r the pur po se of display . The small blac k points in the righ t panel sho w the distribution of randomly chosen samples of the mixed wa veforms, x ′ ( t ). (a) (b) Fig. 14: The thin black line s a nd the thick gr ay lines show the inner time ser ies, w i ( t ) and w ′ i ( t ), derived from the unmixed and mixed w av eforms, respec tively , during the 31.25 ms time interv a l depicted in Figure 10 and 12. The Inner Structure of Time-Dep en dent Signals 19 affected by trans fo rmations of the measurement time serie s. In o ther words, the inner time series enco des sensor -indep endent information. When each inner time s eries w as pla yed as an audio file, it sounded lik e a completely in telligible recording of one of the sp ea kers. In each case, the other sp eaker was not heard, except for a faint buzzing sound in the background. Thus, each inner time series contained all of the semantic informatio n in the unmixe d w av eform. Notice that this comp osite system has an inner time series , w ′ i ( t ), which is equal to the collection of the inner time s eries of its statistically independent subsystems, w 1 ( t ) a nd w 2 ( t ). This demonstra tes the s eparability prop erty of the inner time ser ies of a compo site system, which w as proved in Subsection 2 .2. Also, no tice that the correla tion betw een the time s eries, w ′ 1 ( t ) and w ′ 2 ( t ), is quite low (-0.0016 ). As discussed in Subsection 2.2, this is exp ected beca use these are inner time series of tw o statistically indep endent subsystems . 4 Conclusion This pap er descr ib es how a time series o f senso r measur ement s can b e pro cessed in order to cr eate an inner time series, which is unaffected by the nature of the sensors. Spe cifically , if a system is observed b y tw o sets of sensor s, each s et of measurements will lead to the same inner time serie s if the tw o sets of measur e- men ts ar e rela ted by a ny instan taneo us, invertible, different iable tra nsformation. In effect, an inner time series enco des infor ma tion ab out the intrinsic nature of the observed system’s evolution, without dep ending o n extrins ic facto rs, such a s the observer’s choice o f sensors. An inner time series is created b y statistically pro cessing the lo ca l distributions o f measuremen t velocities in o r der to derive vectors at eac h p oint in measur ement space. The system’s velocity can then be describ ed as a weigh ted sup erp o sition o f the lo cal v ector s at each p oint. Thes e time-dep e ndent weigh ts comprise the inner time series. Because they are inde- pendent of the co ordina te system in measure ment spa c e , they repr esent sensor- independent information ab out the system’s velo cit y in state space . The inner time s eries may b e useful in ce r tain pr actical applications. F or instance, it ma y be used to reduce false negatives in the detection o f ev ents o f int er est. T o see this, ima gine that the ob jective is to detect certa in ”targeted” mov ements of a system as it mov es thr ough state space, and supp o s e that this is being done by using a pa ttern recognition technique to monitor the output of sensors that a re observing the system. If the patter n recognition soft ware is trained o n the output o f calibr ated se ns ors, subsequent sensor drift will cause false negatives to o ccur. This can be avoided if the patter n rec ognition algorithm is trained on the inner time series, instead of the time s eries of ra w measurements. As long as the lo cal v ector s are co mputed from recent data fr om the drifted sensors, the inner time serie s will b e unaffected b y sensor drift, and this pro cedure will sens itively detect the targeted mov ements. How ever, it should b e no ted that this pro cedure may be accompanied b y so me false p o sitives. This is beca us e a given inner tim e series corresp o nds to multiple measure men t time series, which 20 David N . Levin describ e tra jectories in different reg ions o f the measurement spa ce, as ment io ned in Subsection 2.1. As an example, cons ide r the output of the moving camera in Subsection 3.3, and s upp os e that our o b jectiv e is to detect camera movemen ts tha t pro duce the sensor output shown b y the thin black lines in Figure 7. Imagine that a pattern rec o gnition algor ithm is trained to detect this par ticular tra jectory seg- men t. How ever, supp os e that the camera’s lens subsequently ” drifts” so that the targeted ca mera mov ements pro duce the signal shown by the thic k g r ay line in Figure 7. In that case, the drifted data will not b e recognized, and false negatives will o ccur . Now, supp ose tha t the patter n recognition so ft ware was trained to recognize the inner time ser ies (Figure 9) corres p o nding to the ta rgeted ca mera mov ements. Then, sensor drift will not cause false negatives, as long as the time series to b e recog niz e d is pro ces s ed with lo cal v ectors, computed fro m recen tly acquired data from the drifted sensor s. As describ ed in Subsection 2.2, a n inner time series has another attractive prop erty , in addition t o its s e nsor independence. Namely , it automatically pro- vides a separa ble description of the evolution o f a system that is compo site in the sense of (10). T o see this, consider the sensor s, which observe suc h a com- po site system. They ma y be sensitive to the mov ements of many subsystems, causing the raw senso r outputs to be unknown, po ssibly nonlinear, mixtures of many subs ystem state v aria bles. Now, supp os e that we compute the time s e - ries of m ulti-co mpo nent weigh ts derived from suc h mixture measurements. As prov ed in Subsection 2.2, ea ch comp onent of the inner time ser ies of the co m- po site system is the same as a component o f the inner time series o f one o f its subsystems. In other w ords, the inner time series of a comp o site system ca n b e partitioned in to groups of comp onents, with each g roup b eing equal to the inner time ser ie s that would hav e b een derived fro m a subsystem, if it were pos s ible to observe it alone. B ecause of this separability prop erty , the inner time series may be useful for detecting a tar geted mov ement o f one pa rticular subsys tem, in the presence of other indepe ndent subsystems. In particular , a pattern r ecognition pro cedure can b e trained to determine if the comp onents of the inner time ser ies of the tar geted mov ement can b e found among the compo nents o f the inner time series derived fro m the mixed mea surements of the entire sy stem. An adv ant a g e of this pr o cedure is that it is no t necessary to use blind source separation ([4 ], [5], [6], [8], [9]) to disentangle the measurement time serie s into its indep endent comp onents. On the other hand, false p ositive detections can complicate any such attempt to recognize a targeted sig nal b y its inner time ser ies (instead of its time series of s ensor measurements). These erro rs may o ccur b eca use multi- ple different measurement time series may hav e the sa me inner time series, as describ ed in Subsection 2.1. As an illustrative example, consider the system comprised of tw o indep endent audio signals, describ ed in Subsection 3.4, and imagine that our ob jective is to detect an uttera nce of the firs t sp eaker (left panel of Figure 10), in the pre s ence of the second sp eaker (right pane l of Figure 10). It is difficult to determine if this targeted signal is present in the mixtures that ar e actually measured (Figure 1 2). The Inner Structure of Time-Dep en dent Signals 21 How ever, notice that the inner time s e r ies o f the mo vemen t o f interest, derived from the unmixed wa veforms o f a s ubs y stem (the thin black lines in Figur e 14), is almost the same as one of the inner time ser ies comp onents, derived from the mixed sig nals of the comp osite sy s tem (thick gray lines in Fig ure 14). Therefore, a pattern recognition pro cedure , which is tra ined on the inner time series of the unmixed signal, is lik ely to recognize the tar geted s ignal, even in the pres ence o f signals from other subsystems. Some c omments on thes e results: 1. As stated in Section 1, we hav e a ssumed that the sens ors pro duce measur e- men ts tha t a re inv ertibly related to the state v ar ia bles of the underlying system. This in vertibilit y pro p er ty can almost be guaranteed by obser ving the system with a sufficiently large n umber o f independent sensors: sp ecif- ically , by utilizing at least 2 N + 1 indep endent sensor s, wher e N is the dimension of the system’s state spac e . In this case, the sensors ’ output lies in an N -dimensiona l subspa c e em b edded within a space of a t le ast 2 N + 1 dimensions. B ecause an embedding theor em as serts that this subspace is very unlik ely to self-int er sect ([11]), the points in this subspace are almost certainly in vertibly related to the system’s state space. Then, dimensional reduction techn iques (e.g., [10]) can b e us ed to find the subspace co o rdi- nates ( x ) that are inv ertibly related to the state space po ints, as des ired. An example was presented in Subsection 3.3. There, the camer a configu- rations formed a t wo-dimensional subspace, em b edded in a six-dimensional space of raw sensor measuremen ts. This subspace was very unlikely to self- int er sect, giv en that 6 > 2 N + 1 = 5. Then, principal components a nalysis was used to dimensionally r e duce the descr iption of each subspace p oint from six-dimensional co or dinates to tw o-dimensio nal co or dinates ( x ). 2. An inner time ser ies contains information that is in trinsic to the evolution of the obs erved system, in the sense that it is indep endent of extrinsic fa c tors, such as the t yp e of sensors used to observe the s ystem. In o ther w o rds, a n inner time series cont a ins information ab out what is happening ” out there in the r eal w or ld”, indep endent of how the observer ch o o ses to descr ibe it or exp erience it. Ma thematically spea king, an inner time series is a co or dinate- system-indep endent pr op erty of the measurement time series; i.e., its v alues are the same no matter what meas urement co ordinate s y stem is used on the system’s state space. The lo c a l vectors ( V ( i ) ) a lso repr esent a kind o f intrinsic structure on state spa ce. These vectors ” mark” state space in a way that is analogo us to directional arrows, whic h mark a physical surface and whic h can b e used as navigational aids, no matter what co ordinate system is b eing used. 3. It is interesting to specula te abo ut the role of inner time series in sp eech per ception. B y definition, tw o people, who understand the sa me languag e, tend to p er c eive the sa me semantic conten t o f an utterance in that lan- guage. Remark ably , this listener-indep endenc e o ccurs despite the fact that the listeners ma y b e using significantly differen t senso rs to ma ke measure- men ts o f that utterance (e.g., differen t outer , middle, and inner ears ; differen t 22 David N . Levin co chleas; different neural arc hitectures of the acoustic cortex). This senso r- independenc e o f s p eech p erception sugges ts that the semantic conten t of sp eech may b e an inner prop erty; i.e., it may b e encoded in the inner time series of sp eech ( w i ( t )). Sp ecifically , a ssume that the tw o listener s hav e pa st exp osure to statistically similar colle c tions of spee ch-lik e sounds. Then, they will p erceive the speech-sound manifold to be “ marked” b y the same lo cal vectors ( V ( i ) ( x )), even though they ma y r epresent those v ectors in differen t co ordinate systems on the speech-sound manifold. Therefor e , when the t wo listeners use (8) to deco de an utterance, they will derive the same inner time series, and they will p erceive the sa me s emantic conten t. 4. It is equally r emark able th a t sp eech p erc eption is larg ely s p e aker-inde p endent . Namely , a single listener will insta nt ly r ecognize that t wo speakers are ut- tering the same text. This is true despite the fac t that the tw o sounds were pro duced by significantly different vocal tracts a nd may have trav erse d dif- ferent r e gions of the sp eech-sound ma nifold. This sp eaker-indep endence will o ccur as long as long as eac h sp eaker and the listener have past exp os ure to statistically similar collectio ns o f sp e ech sounds. In that case, b ecause of the ab ov e-mentioned listener- indepe ndenc e , each sp eaker a nd the listener will derive the same inner time ser ie s when they listen to the spea ker’s utter- ance. Therefor e, if the tw o sp e a kers hav e enco ded the s ame semantic conten t (i.e., the same inner time series ) in their utterances , the listener will immedi- ately pe r ceive that they are saying the same thing. Notice that t wo spea kers’ utterances, which ha ve the sa me semantic conten t, may corresp ond to t wo different speech-sound tr a jectories, which ha ve the same inner time series. Thu s, in this sp ecula tive sce nario, the fact that the same inner time series may b e encoded in man y measurement time series (see the discussion fol- lowing (9)) cor resp onds to the fact tha t the same semantic con tent can b e expressed by many different voices. References 1. C. Elb ert, C alibr ation T e chnolo gy . (Suddeut sc her V erlag, Mun ich, 2012). 2. S. A. Morain and A. M. Bud ge, Post-L aunch Calibr ation of Satel lite Sensors . (CRC Press, 2004). 3. G. E. Bottomley , Channel Equalization for Wir eless Communic ations . (Wiley , New Y ork, 2012). 4. P . Comon and C. Jutten (eds), Handb o ok of Blind Sour c e Sep ar ation, Indep endent Comp onent Ana l ysis and Applic ations . (Academic Press, O xford, 2010). 5. C. Jutt en and J. Karhunen, “Adv ances in blind source separation (BSS) and inde- p endent comp onent analysis (ICA) for nonlinear mixtures,” International J. Neur al Systems , vol. 14, pp. 267-292, 2004. 6. L. Almeida, ”Nonlinear source separation”, in Synthesis L e ctur es on Signal Pr o c ess- ing , vol. 2, Morgan and Claypo ol Publishers, 2006. 7. D. N. Levin, “Channel-indep end ent and sensor-indep enden t stim ulus representa- tions,” J. Applie d Physics , vol. 98, art. no. 104701, 2005. 8. D. N. Levin, ”Model-indep en dent analytic nonlinear blind source separation,” http://a rx iv.org/abs/17 03.01518 (Marc h, 2017) The Inner Structure of Time-Dep en dent Signals 23 9. D. N. Levin, ”Mod el-indep endent method of nonlinear blind source separation,” In: Tichavsky, P., Barb aie-Zadeh, M., Mi chel, O., and Thirion-Mor e au, N. (e ds.), L atent V ariable Analysis and Signal Sep ar ation, L e ctur e Notes in Computer Scienc e, Springer , vol. 10169, pp. 310-319, 2017. 10. S. T. Row eis and L. K.Saul, “Nonlinear dimensionalit y redu ct ion b y lo cally linear em b ed d ing,” Scienc e , vol. 290, p p. 2323-2326, 2000. 11. T. Sauer, J. A . Y ork e, M. Casdagli, “Embedology ,” J. St atistic al Phy sics , vol. 65, pp. 579-616, 1991. 12. R . Held and R. Whitman, Per c ept ion: Me chanisms and Mo dels. San F rancisco, Califo rnia: F reeman, 1972.