Back to the past: the historical roots of labor-saving automation - Eurasian Business Review

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Back to the past: the historical roots of labor-saving automation
Eurasian Business Review volume 11, pages 27–57 (2021) Cite this article
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Abstract
This paper, relying on a still relatively unexplored long-term dataset on U.S. patenting activity, provides empirical evidence on the history of labor-saving innovations back to early nineteenth century. The identification of mechanization/automation heuristics, retrieved via textual content analysis on current robotic technologies by Montobbio et al. (Robots and the origin of their labour-saving impact, LEM Working Paper Series 2020/03), allows to focus on a limited set of CPC codes where mechanization and automation technologies are more prevalent. We track their time evolution, clustering, eventual emergence of wavy behavior, and their comovements with long-term GDP growth. Our results challenge both the general-purpose technology approach and the strict 50-year Kondratiev cycle, while they provide evidence of the emergence of erratic constellations of heterogeneous technological artefacts, in line with the development-block approach enabled by autocatalytic systems.
Introduction
The existence of labor-saving (hereafter, LS) heuristics driving the rate and direction of technological change is a documented pattern, since the inception of the First Industrial Revolution. Reducing time of operations during Taylorism, increasing saturation of takt-times during Toyotism, and speeding up processes and executions of functions remotely tracking operators’ intervention nowadays, are core drivers of mechanization and automation.
In the tradition of the economics of innovation, the First Industrial Revolution had been a combination of time-saving heuristics, enabled by the mechanization process, and the division of labor inside factories, together with the emergence of innovative artefacts. The role played by time-saving heuristics in shaping the direction of mechanization has been emphasized by von Tunzelmann ( 1995 ) with reference to the cotton industry in the British Industrial Revolution: the massive increase in labor productivity resulted from the use of innovation and discovery through which a spinner was able to produce in a day as much yarn as previously required by a full year of work, without mechanization. Footnote 1
On top of that, Freeman ( 2019 ) conceptualizes the First Industrial Revolution as a paradigmatic shift emerging from the combination of time-saving heuristics on the one hand, and the new clear demarcation between working- and life-time for wage laborers on the other hand, an attitude absent in pre-industrial societies (Thompson 1963 ), allowing workers discipline and ensuring their participation to productive activities, e.g. by turning Monday into a working, rather than a drinking, day. As corroborating evidence, using a detailed and quite granular report, the Hand and Machine Labor Study commissioned by the Department of Labor in 1899 to detect the impact of mechanization on labor productivity, Atack et al. ( 2020 ) estimate that only one-third of the increase in labor productivity (measured as time spent in a given operation) in the late nineteen century was due to ‘inanimate power’, while the rest unexplained component remains attributed to other factors, among which division of labor plays a prominent role.
Speeding up the production process clearly maps into the need of reducing human active participation to the process itself. Therefore, time-saving and LS heuristics have been considered by economic historians as potential focusing devices (Rosenberg 1976 ) guiding the search process, however of a very particular type. In general, in the development of a new artefact, inventors face technical trade-offs and bottlenecks which have to be overridden. Search heuristics might have various nature and directions (Cohen et al. 1996 ), going from the ‘make it smaller’ for microprocessors, to the ‘make it faster’ for aircraft, even to the ‘make it more exclusive’ for smartphones (e.g. Apple’s iPhone). Indeed, focusing devices are rather heterogeneous among inventors, and as such they locally guide the search and discovery process, defining the technological trajectory, inside a given established paradigm (Dosi 1982 ). This is not the case for LS heuristics which, rather than local, appear as a generalized feature in the history of innovation and in general of capitalism.
“In England, strikes have regularly given rise to the invention and application of new machines. Machines were, it may be said, the weapon employed by the capitalists to equal the result of specialized labor. The self-acting mule, the greatest invention of modern industry put out of action the spinners who were in revolt. If combinations and strikes had no other effect than of making the efforts of mechanical genius react against them, they would still exercise an immense influence on the development of the industry.”
[Marx ( 1955 ), p. 161]
Are these LS heuristics empirically detectable? Attempts to infer heuristics and knowledge bases appear e.g. in Castaldi et al. ( 2009 ) at the artefact level, focusing on the tank technology and the evolution of its attributes over time, but also in Martinelli ( 2012 ), who uses patent-citation networks to infer the emergence of new paradigms by changes in bottlenecks and search heuristics, therefore at the so-called knowledge level. Recently, Taalbi ( 2017 ), relying on specialistic trade journals, collected information about drivers of innovative activities supposed to be relevant by innovators, and investigates eventual distinct patterns across industry and over time.
Currently, heuristics are usually inferred from the technical engineering literature and related case-studies. However, patents and their textual content also provide a good source of information to detect codified knowledge and ensuing search heuristics. Relatedly, the use of textual analysis techniques enables a comprehensive study of large scale textual dataset. By looking at the textual contents of robotic patents over the last decade, Montobbio et al. ( 2020 ) are able to isolate those ones which clearly embed a LS trait. The identification of LS patents, done by natural language processing which includes probabilistic topic modeling, lead to a clear definition of the set of technological artefacts behind LS robotic patents published by the USPTO between 2009 and 2018. Two insightful excerpts from LS patents follow:
“Automated systems, such as robotic systems, are used in a variety of industries to reduce labo[u]r costs and/or increase productivity. Additionally, the use of human operators can involve increased cost relative to automated systems.”
[US20170178485A1]
“The use of the technology [robots] results in improved management of information, services, and data, increased efficiency, significant reduction of time, decreased manpower requirements, and substantial cost savings.”
[US20100223134A1]
After identifying patents explicitly containing LS heuristics, Montobbio et al. ( 2020 ) infer the type of human activities that the technology contained in LS patents is intended to replace, by capturing both the formal technological content of the invention, using patent classification codes, and the substantial purpose of broader robotic innovations, using the vector of words which characterizes each topic. Thanks to this twofold analysis, they describe those fields and activities that are more exposed to LS innovations. LS patents appear to be concentrated in particular in the following fields: (i) Transport, Storage and Packaging, (ii) Diagnosis and Therapy, (iii) Transmission of Digital Information, (iv) Optical elements, (v) Chemical or Physical Laboratory Apparatus (measuring and testing in chemistry), and (vi) Moving Parts.
The authors propose a taxonomy wherein it emerges that the typical tasks on which LS research effort is focused include (i) dexterity and manipulations, as in packing, storing, conveying, and handling packages in the logistics industry; (ii) activities entailing social intelligence, such as caretaking patients and the elders; (iii) activities requiring cognitive intelligence and complex reasoning, e.g. the ability of predicting, learning, classifying and evaluating, typical of high-level professional segments. Notably, the analysis shows that the overall bundle of technologies behind LS heuristics is not simply related to robots stricto sensu, but it encompasses a wider set of technologies, functions and operations. In this respect, rather than interpreting the new wave of LS technologies as the next GPT (Trajtenberg 2019 ), to genuinely account for the unfolding of the latest wave of LS technologies, a ‘technological constellation’ perspective à la Freeman and Louçã ( 2001 ) would be more informative.
In this paper, we intend to move ahead by delving into the past, i.e. by adopting a ‘historical technological constellation perspective’ and looking at the emergence and evolution of the bundle of technologies behind current LS heuristics detected in robotic innovations. Indeed, as we shall show, mechanization and automation are not the result of a single dominant product design, but rather of a bundle of technological artefacts, which experience patterns of comovements, anti-comovements, explosion, and dissipation. Our empirical investigation, which looks at historical patent data over the period 1836–2019 vindicates, first, the underlying technological complexity, in terms of bundles of output, behind LS technologies; second, the increasing historical relevance of those technological artefacts entailing mechanization and automation; last, the absence of a neat recurrence of periodic waves of innovations. In fact, although we identify the emergence of long waves, they are hardly periodically recurrent.
Our findings clearly parallel the Schumpeterian reading of capitalist systems (Schumpeter 1939 ) which epitomizes the perspective of long waves of technological innovations, with phases of upswings and downswings, clustering of heterogeneous innovations and patterns of interdependence among them, giving rise to upsurge and transformation, alternated with phases of slackening and declines.
Building upon the Schumpeterian perspective, according to Freeman and Louçã ( 2001 ) the history of modernization is punctuated by distinct phases, characterized in terms of dominant technological systems, or better techno-economic paradigms. They go beyond the deterministic Kondratiev wave approach put forward by Schumpeter and propose the notion of ‘constellations of major technical innovations’, by far more complex than the popularized GPT version entailing the diffusion of unique technologies, say steam, electricity, ICTs, and now AI, which for their pervasiveness encompass all sectors of the economy, and therefore turn out to entirely characterize the process of economic growth.
Differently, constellations pertain to the notion of autocatalytic mechanisms, entailing development blocks of technological artefacts (Dahmén 1988 ). Therefore, according to this perspective (Nuvolari 2019 ), the development block underlying the British Industrial Revolution consisted of machinery, machine tools, steam engines, coal, and iron production techniques, while the one underlying the Third Industrial Revolution consists of semiconductors, computers, software, and networking equipment.
The periodic cycle approach proposed by Kondratiev, rephrased by Schumpeter, and then endorsed by Perez ( 1983 ), has been challenged by the empirical literature and questioned particularly by Silverberg ( 2007 ) who highlights a series of drawbacks characterizing the empirical detection of long cycles. The latter pertains, first, to the non-stationary nature of long-term time series and, related, to the distortion imposed by making the series stationary using whatever detrending technique; second, to the short time horizon characterizing the majority of the analyses, with many series lasting exactly fifty years, and therefore over-imposing the Kondratiev wave; third, to the absence of a dataset for true innovations; fourth, to arbitrary trimming of the dataset.
These drawbacks have been taken into account by Silverberg and Verspagen ( 2003 ) who dismiss the long cycle perspective and opt for a more neutral detection of clustering of innovations by fitting a Poisson model, under the hypothesis of absence of clustering, versus a negative binomial model, allowing for clustered events, and therefore for a variance component. Although innovation clustering is verified, any periodic deterministic cluster hardly emerges:
“Innovations may indeed cluster, but not in any deterministic sense, and their pattern may shed light on a unified mechanism explaining a range of their properties. Aggregate economic activity, simultaneously with certain patterns of structural change, may obey certain laws that dialectically intertwine chance and necessity and produce robust patterns, but ones that do not lend themselves to any very simple forecasting. It is on this note that I hope long waves will long be with us as a field of scientific research.” [Silverberg ( 2007 )]
In the following, we shall proceed by explicitly addressing the major drawbacks pointed out by Silverberg ( 2007 ) by going beyond limited spectral analysis and filtering techniques, and resorting to a non-stationary resilient methodology, namely wavelet analysis, which will be performed upon a well defined set of technological innovations, i.e. patents published since the 1790, and therefore also overcoming the shortness of the data structure, and avoiding to super-impose the long cycle identification. As said above, we focus on a particular subset of the overall technological artefacts, namely those who have been recognized to currently involve explicit LS heuristics.
Linking (i) the evolutionary literature studying the employment impact of technical change, theoretically discussing different compensation mechanisms balancing labor-saving effects of innovation (Freeman and Soete 1987 ; Vivarelli 1995 ; Simonetti et al. 2000 ; Piva and Vivarelli 2018 ; Calvino and Virgillito 2018 ; Dosi et al. 2021 ), Footnote 2 (ii) the study of knowledge bases embedded in technology (Dosi 1988 ) and (iii) the emergence of long waves or alternatively of clusters of innovations (Silverberg and Verspagen 2003 ), our contribution departs from the literature in terms of both the novelty of the empirical analysis, by fully exploiting the long-run historical dimension of the USPTO dataset, still relatively unexplored, the use of wavelet analysis to study patent data, and ultimately enriches our understanding of the long run history of the constellations of artefacts behind current LS robotic technologies. Indeed, in the wake of a purported Fourth Industrial Revolution Footnote 3 and of the over-abused statement ‘this time is different’, the understanding of the evolution in the bundle of technologies behind current explicit LS heuristics might allow a thorough and deeper policy action to counteract labor shedding effects.
The paper is organized as follows: in Sect.  2 , we identify the long-term evolution of the constellation of technologies behind current LS innovations, we present their time trend and clustering patterns. Section  3 detects the presence of temporal cycles in the data by means of wavelet analysis and explores the extent to which the intensity of innovative activity is correlated with business cycles and recessions. Finally, Sect.  4 concludes by outlining potential avenues of further research and useful policy implications.
Back to the past: labor-saving innovations since 1830s
The first step of our empirical investigation entails the determination of technological classes which are recognized to currently present LS traits, with the aim at delving into the past and tracing a historical account of their evolution. Our main source is Montobbio et al. ( 2020 ), which investigates the presence of LS heuristics within a set of 29,789 robotic patent applications published by the USPTO between 2009 and 2018 and quantitatively identify, through a probabilistic topic model of their full-texts, the Cooperative Patent Classification (CPC) codes which bear the most relevance to underlying LS innovations. In a nutshell, the prevalence of a LS trait is identified by means of a metric pointing at those topics which are more prevalent in LS robotic patents vis-à-vis the population of generic robotic patents. Topics are then matched to CPC codes. The metric used by Montobbio et al. ( 2020 ), namely a topic relevance distribution, is also reported here in Fig. 1 for convenience (a more detailed technical summary of Montobbio et al. ( 2020 ) is also provided in Appendix  1 ).
Fig. 1
Topic relevance metrics \(\Theta _k^{\texttt {rob}}\) and \(\Theta _k^{\texttt {LS}}\) for robotic patents (blue, in descending order) and their LS subset (orange) Source: Montobbio et al. ( 2020 )
Full size image
Discussion and conclusions
This paper, relying on a long-term, still relatively unexplored, dataset on U.S. patenting activity, provides empirical evidence on the history of automation innovation, back since 1830s. The labor-saving heuristics identified by Montobbio et al. ( 2020 ) via textual analysis on current robotic technologies allow to focus on a coherent set of technological CPC classes, the historical evolution of which is analyzed in terms of timing, clustering, periodic behavior, and comovements with GDP growth. The very fact that labor-saving CPCs differ widely in their assignment dynamics challenges the so-called GPT approach postulating a unique dominant technology, while it brings support to the idea of innovation waves seen as technological constellations.
Our findings are as follows. First, mechanization and automation, or equivalently labor-saving heuristics, seem to constitute a “natural trajectory” (Nelson and Winter 1982 ) in the evolution of the capitalist system, rather than a recurrent pattern. As opposed to socio-deterministic approaches linking the upsurge of automation to contingent phases, the hypothesis of natural trajectory implies that innovative efforts in labor-saving automation act as a background collective meta-heuristics, independent of local-focusing devices operating at the individual or firm level. Nonetheless, this notion does not exclude the formation of clustering patterns of innovation. Indeed, nothing pre-empts the coexistence of a Marxian interpretation of technical change, intended to mechanize and substitute labor to increase forms of control and appropriation over the production process by capitalists, by codifying into inanimate power previously non-codified knowledge, on the one hand, and the endogenous emergence of innovative efforts concentrated over a set of technological artefacts, in the Schumpeterian sense, on the other. There exist periods of more coordinated innovative efforts resulting in upsurges and subsequent declines, highlighting some degree of technological clustering. In particular, we detect the presence of three technological clusters exhibiting distinct temporal patterns: from hump-shaped, to plateaued, to ever-increasing dynamics. Overall, the tension between invariant patterns, as expressed by socio-economic meta-routines granted by institutions and meta-institutions (Dosi et al. 2020 ), and unfolding heterogeneities characterizing historical episodes and sectors of activity (Capone et al. 2019 ), remains largely unresolved in our interpretation of the socio-economic fabric.
Second, in detecting the presence of periodic behavior via wavelet analysis, we do not confirm the presence of 50-year long Kondratiev waves. Indeed, we are not able to identify in mechanization and automation subsequent regular periodic waves, leading to new technological systems. However, the dominant CPC codes characterizing erratic technological constellations are in line with the technological system dating proposed by Freeman and Louçã ( 2001 ). The two seemingly contrasting results are instead coherent with the system block approach and the coexistence of both within-paradigm and between-paradigm trajectories (Dosi 1982 ).
Third, with respect to the existence of a relationship between economic performance and innovative activity in mechanization and automation, we hardly find any evidence thereof. Neither recession-driven nor euphoria-driven innovations are found, given the absence of correlation between GDP growth and patenting activity. When looking at the long-term cycle component, innovation and GDP growth present delinked patterns of waves, with heterogeneous troughs and peaks. Whenever comovements occur, waves in GDP growth seem to precede, rather than follow, technological innovations, although the picture gets more nuanced when looking at both time and frequency domains together. In this regard, any purported saturation of the technological frontier or of innovative ideas are not detectable from the trends in innovation directed at the mechanization and automation of tasks. Labor-saving efforts are there and involve a large set of technological artefacts, producers, and sectors of activity (Montobbio et al. 2020 ). This occurs rather independently of economic cycles at the macro-level. Our results however are not intended to deny the existence of local and discrete focusing devices or search heuristics which guide the innovative process. For instance, recent micro-evidence highlights the role of bottlenecks and opportunities shaping the innovation trajectory in Swedish manufacturing (Taalbi 2017 ). Indeed, technological trajectories remain locally bounded by technological bottlenecks and market opportunities. Additionally, there might be other variables, such as the degree of union power, wage levels, and conflictual attitudes against mechanization/automation moves, in line with the socio-technical approach (Noble 1986 ), which might trigger the innovation dynamics. Conflictual claims about labor conditions might however result also in new technology meant to increase ergonomics and safety conditions in the workplace. The direction is therefore not univocal and the aforementioned relationships may represent avenues of future research.
The main limitation of our results comes from the level of aggregation: indeed, 3-digit CPC codes are rather heterogeneous and might also include labor-friendly innovations, even in their conception phase. Additionally, whether an innovation is labor-saving or labor-friendly is a question that pertains to the use of the artefact and its implementation in the production and organizational processes occurring at the firm and sectoral level. In the present paper, we look at the direction of innovative ideas, without reaching a conclusive appraisal on the employment effect from their use. Finally, given the widely heterogeneous and complex nature of technology, our investigation focuses on labor-saving innovations uniquely derived by current robotic artefacts, therefore potentially neglecting other labor-saving innovations sprung by different artefacts, not specifically linked to robotic automation. Future research would benefit from encompassing a wider investigation across the whole set of patents.
Notes
The author refers in particular to Baines ( 1835 ).
Many contributions are emerging in the recent years looking at the impact of automation adopting a neoclassical perspective (Acemoglu and Restrepo 2018 ; Graetz and Michaels 2018 ; Acemoglu and Restrepo 2019 , 2020 ) mostly relying on sectoral and local labor market analysis. The evolutionary tradition distinguishing heterogeneous impacts of embodied vs disembodied technical change upon employment has been explored in Barbieri et al. ( 2018 ); Pellegrino et al. ( 2019 ); Van Roy et al. ( 2018 ) mostly adopting firm-level data with a panel structure, to mention a few recent studies. More established notions of process vs product innovations are explored in Van Reenen ( 1997 ); Lachenmaier and Rottmann ( 2011 ); Harrison et al. ( 2014 ).
For empirical evidence investigating current Industry 4.0 trends in the automotive industries, see Moro et al. ( 2019 ); Cirillo et al. ( 2021 ).
Codes which belong to CPC ‘raccord’ class Y are left full digit.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1
Technical summary of Montobbio et al. ( 2020 )
The contribution consists of three methodological steps. First, patents which either directly or indirectly relate to robotics technology are singled out. Second, a procedure is implemented in order to detect the underlying LS heuristics and pinpoint the set of explicitly LS patents. Finally, a probabilistic topic model is estimated in order to devise a human–machine taxonomy.
Step 1—Identification of robotic patents The analysis starts with the entire set of 3,557,435 patent applications published by the USPTO between 1st January 2009 and 31st December 2018. Robotic patents are pinpointed therein according to two distinct criteria, one based on the patent classification codes specified within applications, the other based on textual keyword search. A patent is deemed ‘robotic’ if it obeys at least one of the criteria. In particular, a robotic patent according to the first criterion (dubbed ‘CPC’) must be assigned by patent examiners at least one of a set of 174 full-digit CPC codes which reflect former U.S. Patent Classification (USPC) class 901 (“Robots”). Likewise, a robotic patent according to the second criterion (dubbed ‘K10’) must contain the word ‘robot’ in its full-text at least 10 times, including derivational and inflectional affixes. The first criterion identifies 10,929 robotic patents, while the second criterion identifies another 18,860 (after discarding robotic patents according to the first criterion). The two criteria single out a total of 29,789 robotic patents, i.e. approximately 0.84% of the original (universe) population.
Step 2—Identification of labor-saving patents Labor-saving patents constitute a subset of robotic patents, identified by a multiple word co-occurrence query at the sentence level. In particular, a patent is deemed labor-saving (after an additional manual validation step) if its full-text contains at least one sentence in which the verbal predicate, direct object, and object attribute belong to the following lists:
$$\begin{aligned} \underbrace{\begin{bmatrix} \texttt {reduc} \\ \texttt {replac} \\ \texttt {elimin} \\ \texttt {save} \\ \texttt {lower} \\ \texttt {substitut} \\ \texttt {autom} \\ \end{bmatrix}}_{\textsf {verbal\ predicate}} \quad \times \quad \underbrace{\begin{bmatrix} \texttt {labor} \\ \texttt {worker} \\ \texttt {human} \\ \texttt {employe} \\ \texttt {manpow} \\ \texttt {job} \end{bmatrix}}_{\textsf {direct\ object}} \quad \times \quad \underbrace{\begin{bmatrix} \texttt {cost} \\ \texttt {expenditure} \\ \texttt {expens} \\ \texttt {hour} \\ \texttt {intens} \\ \texttt {task} \\ \texttt {time} \\ \texttt {skill} \end{bmatrix}}_{\textsf {object\ attribute}}. \end{aligned}$$
In total, 1276 labor-saving patents are found (approximately 4.3% of all robotic patents), of which 461 (\(\approx\) 36.1%) belong to the CPC group and 815 (\(\approx\) 63.9%) belong to the K10 group.
Step 3—Probabilistic topic model and human–machine taxonomy The set of labor-saving patents is technologically characterized vis-à-vis the superset of robotic patents by leveraging the latent semantic structure of the whole collection of patents’ full-texts. The analysis proceeds along the following methodological workflow. First, a probabilistic topic model is estimated on the whole population of robotic patents, which associates a distribution \(\theta _d\) of membership over the K-dimensional set \({\varvec{\beta }}\) of topics to each patent d. Second, a distribution of CPC codes, according to the original attribution of codes to each patent by the topic proportions \(\theta _d\) found in the previous step, is associated to each topic \(\beta _k\). Finally, the relevance of each topic to the whole population of robotic patents is compared to the same relevance to the subset of labor-saving patents, in order to draw quantitative conclusions on which technologies are relatively more and less relevant in characterizing the two sets of patents. The probabilistic topic model, asked to identify \(K = 20\) topics, returns each topic \(\beta _k\) as a list of relevant keywords and a membership value \(\theta _{d,k}\) of each patent d to topic k. An aggregate measure of relevance of each topic k to an arbitrary collection of patents D (e.g. the set of robotic patents or labor-saving patents) can be defined as the simple average membership of all patents in the collection to topic k, as follows:
$$\begin{aligned} \Theta _k^{D}&:= \frac{\sum \limits _{d \in D} \theta _{d,k}}{|D|}&\forall \ k=1,\dots ,K. \end{aligned}$$
When the underlying collection of patents D is the whole set of robotic patents, \(\Theta _k^{\texttt {rob}}\) measures the relevance of each topic to robotic patents; analogously, when the underlying collection of patents D is the subset of labor-saving patents, \(\Theta _k^{\texttt {LS}}\) measures the relevance of each topic to labor-saving patents. Distributions \(\Theta _k^{\texttt {rob}}\) and \(\Theta _k^{\texttt {LS}}\) are pictured in Fig. 1 , where topics are sorted by decreasing relevance to the robotic patents collection. Finally, the relative synthetic measure \({\tilde{\Theta }}_k^{\texttt {LS}}\) mentioned in Sect. 2 is defined as
$$\begin{aligned} {\tilde{\Theta }}_k^{\texttt {LS}}&:= \frac{\Theta _k^{\texttt {LS}}}{\Theta _k^{\texttt {rob}}}&\forall \ k=1,\dots ,K. \end{aligned}$$
Appendix 2
Correlation analysis
The following picture provides a heatmap representation of the correlation matrix of target CPC intensity series (CPC codes Y10S901 and Y10T436 are discarded).
At a first glance, it is immediate to spot a first cluster of codes, B23, B25, B62, and B65, whose assignment dynamics negatively correlates (lighter pixels) with all the other codes. At a second glance, a second cluster is present, with codes A61, C12, G02, G06, H01, and H04 exhibiting a highly positive correlation (darker pixels) with one another. The remaining codes, B01, G01, and G05, form a third cluster, in that they display milder, near-zero correlation coefficients with series of the second cluster, and negative correlation coefficients with series of the first.
Appendix 3
Wavelet analysis definitions
Consider the Hilbert space \(L^2({\mathbb {R}})\) of square-integrable functions. A function \(\psi (t) \in L^2({\mathbb {R}})\) is called a mother wavelet if it satisfies the admissibility condition
$$\begin{aligned} \int _{-\infty }^{+\infty } \frac{ | \Psi ( \omega ) | }{ | \omega | } d \omega < +\infty , \end{aligned}$$
(1)
where \(\Psi ( \omega )\) stands for the Fourier transform of \(\psi (t)\). Footnote 11 Condition 1 implies that \(\Psi ( \omega )\) vanishes when frequency \(\omega\) equals zero:
$$\begin{aligned} |\Psi (\omega )| \Big |_{\omega = 0} = 0. \end{aligned}$$
(2)
In other words, the wavelet must display a band-pass like spectrum. Moreover, condition 1 also requires that the wavelet’s average value over time is zero:
$$\begin{aligned} \int _{-\infty }^{+\infty } \psi (t) dt = 0. \end{aligned}$$
(3)
Loosely speaking, Eqs. 2 and 3 together establish that \(\psi (t)\) must wiggle up and down over time, and therefore resembles a wave-like function.
The Morlet wavelet, used in the analysis of this paper and pictured in Fig. 5 , is composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope):
$$\begin{aligned} \psi _{\omega _0}(t) = \pi ^{1/4} \left( e^{i \omega _0 t} - e^{-\omega _0^2/2} \right) e^{- t^2/2}. \end{aligned}$$
It is possible to show that if \(\omega _0 \ge 5\), then \(\psi _{\omega _0}(t)\) satisfies the admissibility condition 1 . In the present analysis, we set \(\omega _0 = 6\).
A (generic) mother wavelet \(\psi (t)\) ‘gives birth’ to a family \(\psi _{s, \tau } (t)\) of so-called child wavelets or wavelet daughters by means of scaling and translation operations
$$\begin{aligned} \psi _{s, \tau }(t) = \frac{1}{\sqrt{s}}\ \psi \left( \frac{t - \tau }{s} \right) , \end{aligned}$$
(4)
where \(s \in \mathbb {R_+}\) denotes the scaling factor, which stretches/shrinks the mother wavelet, and \(\tau \in {\mathbb {R}}\) denotes the translation parameter, which shifts the mother wavelet across time.
The continuous wavelet transform (CWT) of a function of time (or time series) f(t), with respect to a mother wavelet \(\psi (t)\), is defined as
$$\begin{aligned} W_{f, \psi } (s, \tau ) = \frac{1}{\sqrt{|s|}} \int _{-\infty }^{+\infty } f(t)\ \psi ^* \left( \frac{t - \tau }{s} \right) dt, \end{aligned}$$
(5)
where the \(^*\) superscript denotes complex conjugation. The CWT provides a representation of f(t) in terms of wavelet basis functions \(\psi _{s, \tau } (t)\), by letting the scale and translation parameters vary continuously. In other words, the CWT is a convolution of the signal f(t) with the family of stretched and translated child wavelets defined in Eq. 4 .
Given Eq. 5 , the wavelet power spectrum, represented by a heatmap (scaleogram) in Figs. 6 and 7 , is computed as
$$\begin{aligned} WPS_{f, \psi }(s, \tau ) = \left| W_{f, \psi } (s, \tau ) \right| ^2. \end{aligned}$$
In the bivariate case, the cross wavelet transform of two signals f(t) and g(t) with respect to a mother wavelet \(\psi (t)\) is defined as
$$\begin{aligned} W_{f, g, \psi } (s, \tau ) = W_{f, \psi } (s, \tau ) \cdot W_{g, \psi } (s, \tau )^*\ , \end{aligned}$$
while the cross power spectrum, pictured in panels (b), (d) and (f) of Fig. 9 , is given by
$$\begin{aligned} XPS_{f, g, \psi } (s, \tau ) = \left| W_{f, g, \psi } (s, \tau ) \right| \ . \end{aligned}$$
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Staccioli, J., Virgillito, M.E. Back to the past: the historical roots of labor-saving automation. Eurasian Bus Rev 11, 27–57 (2021). https://doi.org/10.1007/s40821-020-00179-1