Distribution and Clusters of Basic Innovations

Abstract

This paper presents our recent results on the issue of the temporal distribution of basic innovations, which, as we know, it is the main driver of economic growth. Numerous scholars have addressed this issue, and we can say that the Poison distribution is the commonly accepted one. On the other hand, there is not a convincing substantiation of the fact. This article provides a retrospective analysis of statistical evidence, and present a proof of the Poisson distribution of basic innovations based on the theory of random processes. In the empirical part of the research we have used one unified supersample time series rather than smaller different datasets as previously done by other authors.

Appendices

Appendix 1

A. Proof of Statement 1

We divide the proof into several stages:

• 1. We show that, except for some simple situations, at some λ > 0, P ₀(t) = e ^−λt is satisfied.

Indeed, let p = P ₀(1) (14). We divide a time interval [0; 1] into n equal parts. Absence of events per time unit means that each short time segment of length 1/n has 0 events. Due to independence we have p = (P ₀(1/n))ⁿ (15), hence P ₀(1/n) = p ^1/n (16). It immediately follows that P ₀(k/n) = p ^k/n (17) at any k ≥ 1.

We show now that generally P ₀(t) = p ^t (18) at all t ≥ 0. For each such number and random natural n there is such a number k ≥ 1 that:

$$ \frac{k-1}{n}\le t<\frac{k}{n} $$

(19)

Function P ₀(t) does not increase on t, therefore:

$$ {P}_0\left({\scriptscriptstyle \frac{k-1}{n}}\right)\ge {P}_0(t)\ge {P}_0\left(\frac{k}{n}\right) $$

(20)

or

$$ {p}^{\frac{k-1}{n}}\ge {P}_0(t)\ge {p}^{\frac{n}{k}} $$

(21)

Let n → ∞, so that k/n → t, we obtain P ₀(t) = p ^t.(22)

There can be three cases: (a) p = 0, (b) p = 1, and (c) 0 < p < 1.

In the first case P ₀(t) = 0 at any t > 0, i.e. at least one event occurs with a probability of 1 in any time interval, however short, which is equivalent of the case when an infinite number of events occur in time intervals of any length. This resembles chain reaction of a nuclear explosion, so we shall not consider such an extremity. In case (b) P ₀(t) = 1, i.e. events never occur. Thus, we are interested only in case (c). Assume p = e ^−λ (23), here 0 < λ < ∞. Hence, we obtain P ₀(t) = e ^−λt (24). (Let us prove later, that other cases are impossible using statement 2).

• 2. We prove that at h → 0:

P ₁(h) = λh + o(h)(25), where

o(h) is a probability that more than one change occurs during (t;t+h).

To do so we use the fact that P ₀(h) = e ^−λh = 1 − λh + o(h) (26) and ${P}_0(h)+{P}_1(h)+{\displaystyle \sum_{k=2}^{\infty }{P}_k(h)}=1$.(27)

Therefore: ${P}_1(h)=1-{P}_0(h)-{\displaystyle \sum_{k=2}^{\infty }{P}_k(h)}=\lambda h+ o(h)$. (28)

• 3. In this section we show that probabilities P _k (t) satisfy a certain system of differential equations.

For t ≥ 0 and h > 0 we have

$$ {P}_k\left( t+ h\right)={\displaystyle \sum_{j=0}^k{P}_j(t){P}_{k- j}(h)} $$

(29)

(Here we go through some variants of a number of events that occur during time t and during a subsequent time interval of length h).

If h → 0, then

$$ {\displaystyle \sum_{j=0}^{k-2}{P}_j(t){P}_{k- j}(h)}\le {\displaystyle \sum_{j=0}^{k-2}{P}_{k- j}(h)}={\displaystyle \sum_{i=2}^k{P}_i(h)}= o(h) $$

(30)

Hence, there may occur even k changes at k ≥ 1 in time interval (0; t + h) by three mutually exclusive means:

• 1) No changes in time (t; t + h) and k changes in (0; t).

• 2) One change in time (t; t + h) and k − 1 changes in (0; t).

• 3) x changes in time (t; t + h) and k − x changes in (0; t).

Let us find probabilities for each of these means:

P(1) = P _k (t)(1 − λh − o(h)) (31)

P(2) = P _k − 1(t)λh + o(h))(32)

P(3) decreases faster than h

or

$$ {P}_k\left( t+ h\right)={P}_k(t){P}_0(h)+{P}_{k-1}(t){P}_1(h)+ o(h)= $$

$$ ={P}_k(t)\left(1-\lambda h+ o(h)\right)+{P}_{k-1}(t)\left(\lambda h+ o(h)\right)+ o(h)= $$

=P _k (t)(1 − λh) + P _k − 1(t)λh + o(h).(33)

Hence, we obtain:

$\frac{P_k\left( t+ h\right)-{P}_k(t)}{h}=-\lambda {P}_k(t)+\lambda {P}_{k-1}(t)+\frac{o(h)}{h}$, k = 1 , 2 ,  .  .  . , (34)

Let h → 0. As there is a limit for the right part, there is a limit for the left part as well. As a result, we obtain:

${P}_k^{\hbox{'}}(t)=-\lambda {P}_k(t)+\lambda {P}_{k-1}(t)$, k = 1 , 2 ,  .  .  . , (35)

At k = 0 probabilities (2) and (3) do not occur, hence:

P ₀(t + h) = P ₀(t)(1 − λh) + o(h), (36) remove parentheses, divide by h:

$$ {P}_0^{\hbox{'}}(t)=-\lambda {P}_0(t) $$

(37)

Initial conditions:

P ₀(0) = 1, P _k (0) = 0 we have the answer:

$$ {P}_0(t)={e}^{-\lambda t} $$

(38)

P ₀(0) = 1, P _k (0) = 0 at k ≥ 1.

• 4. Solution of a system of equations.

Let us present this differential equation as follows:

f ^' =  − λf + λf ₋₁, (39) where

f − P _k (t), f ₋₁ − P _k − 1(t), while P ₀(t) = e ^−λt(40)

Solve it using method f = uv(41)

Let f = uv, then f ^' = u ' v + uv'.(42)

Rewrite our differential equation:

$$ {f}^{\prime }+\lambda f=\lambda {f}_{-1} $$

(43)

$$ u\hbox{'} v+ uv\hbox{'}+\lambda uv=\lambda {f}_{-1} $$

(44)

$$ uv\hbox{'}+\lambda uv=0 $$

(45)

$\frac{v^{\prime }}{v}=-\lambda$ ⇒ v = e ^−λt(46)

$$ {u}^{\prime }{e}^{-\lambda t}=\lambda {f}_{-1} $$

(47)

Let k = 1 ⇒ u′e ^−λt = λe ^−λt ⇒ u′ = λ ⇒ u = λt ⇒ f = λte ^−λt(48)

Let k = 2 ⇒ u′e ^−λt = λ ² te ^−λt ⇒ u′ = λ ² t ⇒ $u=\frac{\lambda^2{t}^2}{2}$ ⇒ (49)$f=\frac{{\left(\lambda t\right)}^2}{2}{e}^{-\lambda t}$(50)

Let k = 3 ⇒ u′e ^−λt = λ ³ t ² e ^−λt ⇒ u′ = λ ³ t ² ⇒ $u=\frac{\lambda^3{t}^3}{2\cdot 3}$ ⇒ $f=\frac{{\left(\lambda t\right)}^3}{3!}{e}^{-\lambda t}$(51)

Then let us use the method of mathematical induction to prove that:

For random k: u′e ^−λt = λ ^k t ^k − 1 e ^−λt ⇒ u′ = λ ^k t ^k − 1 ⇒ $u=\frac{\lambda^k{t}^k}{2\cdot 3}$ ⇒ $f=\frac{{\left(\lambda t\right)}^k}{k!}{e}^{-\lambda t}$.(52)

B. Proof of Statement 2

Let there be a random function f which satisfies the conditions.

Assuming that y = x ₀ − x in (8), we obtain f(x)f(x ₀ − x) = f(x ₀) ≠ 0(53)

Therefore, we see that f(x) differs from 0 at all x. Moreover, replacing x in (1) and y with $\frac{x}{2}$ we find: $f(x)={\left[ f\left(\frac{x}{2}\right)\right]}^2$ (54), so f(x) is always strictly positive. Using this, take a logarithm

$$ \ln f\left( x+ y\right)= \ln f(x)+ \ln f(y) $$

(55)

If we assume ϕ(x) =  ln f(x) (56), then we have a function which is continuous and satisfies the condition:

ϕ(x + y) = ϕ(x) + ϕ(y)(57), but this is a well-known fact (this function can be linear only). ⇒ ϕ(x) =  ln (f(x) + cx, (c = const)(58)

Therefore, finally, f(x) = e ^cx = a ^x.(59).

Appendix 2

List of basic innovations

1764	Spinning machine
1775	Steam engine
1780	Automatic band loom
1794	Sliding carriage
1796	Blast furnace
1809	Steam ship
1810	Whitney’s method
1811	Crucible steel
1814	Street lighting (gas)
1814	Mechanical printing press
1819	Lead chamber process
1820	Quinine
1820	Isolated conduction
1820	Rolled wire
1820	Cartwright’s loom
1824	Steam locomotive
1824	Cement
1824	Puddling furnace
1827	Pharma fabrication
1831	Calciumchlorate
1833	Telegraphy
1833	Urban gas
1835	Rolled rails
1837	Electric motor
1838	Photography
1839	Bicycle
1840	Vulcanized rubber
1841	Arc lamp
1844	Jacquard loom
1845	Lathe
1846	Inductor
1846	Electrodynamic measuring
1846	Rotary press
1846	Anesthetics
1849	Steel (puddling process)
1851	Sewing machine
1852	Plaster of paris
1854	Aluminum
1855	Safety match
1855	Bunsen burner
1856	Refined steel/Bessemer steel
1856	Steel pen / Fountain pen
1856	Tare colors industry
1856	Baking powder
1857	Elevator
1859	Lead battery
1859	Drilling for oil
1860	Internal combustion engine
1861	Soda works
1863	Aniline dyes
1864	Siemens-Martin steel
1865	Paper from wood
1866	Deep sea cable
1867	Dynamite
1867	Dynamo
1869	Commutator
1870	Typewriter
1870	Celluloid
1870	Combine harvester
1871	Margarine
1872	Thomas steel
1872	Reinforced concrete
1872	Drum rotor
1873	Preservatives
1875	Sulphuric acid
1876	Four-stroke engine
1877	Telephone
1878	Nickel
1879	Electric Railway
1880	Incandescent lamp
1880	Water turbine
1880	Jodoforme
1880	Half-tone process
1881	Electric power station
1882	Veronal
1882	Cable
1883	Antipyrin
1883	Coals whisks
1884	Steam turbine
1884	Chloroform
1884	Punched card
1884	Cash register
1885	Synthetic fertilizers
1885	Transformer
1885	Synthetic alkaloids
1886	Magnesium
1886	Electric welding
1886	Linotype
1887	Phonograph
1887	Electrolyze
1888	Motor car
1888	Pneumatic tire
1888	Electric counter
1888	Portable camera
1888	Alternating-current generator
1890	Man-made fibers
1890	Chemical fibers
1891	Melting by induction
1892	Acetylene welding
1892	Accounting machine
1894	Cinematography
1894	Antitoxins
1894	Motor cycle
1894	Monotype
1895	Diesel engine
1895	Drilling machine for mining
1895	Electric automobile
1896	X-rays
1898	Aspirin
1898	Arc welding
1900	Air ship
1900	Synthesis of indigo
1900	Submarine
1901	Holing machine
1902	Electric steel making
1903	Safety razor
1905	Viscose rayon
1905	Vacuum cleaner
1906	Acetylene
1906	Chemical accelerator for rubber vulcanization
1907	Electric washing machine
1909	Gyro compass
1910	Airplane
1910	Bakelite (Phenol plastics)
1910	High voltage isolation
1913	Vacuum tube
1913	Assembly line
1913	Thermal cracking
1913	Domestic refrigerator
1914	Ammonia synthesis
1914	Tractor
1914	Stainless steel
1915	Tank
1916	Synthetic rubber
1917	Cellophane
1918	Zip fastener
1920	AM Radio
1920	Acetate rayon
1920	Continuous thermal cracking
1922	Synthetic detergents
1922	Insulin
1922	Synthesis of methanol
1923	Continuous rolling
1924	Dynamic loudspeaker
1924	Leica camera
1925	Deep frozen food
1925	Electric record player
1927	Coal hydrogenation
1930	Power steering
1930	Polystyrene
1930	Rapid freezing
1931	Freon refrigerants
1932	Crease-resistant fabrics
1932	Gas turbine
1932	Polyvinylchloride
1932	Antimalaria drugs
1932	Sulfa drugs
1934	Fluor lamp
1934	Diesel locomotive
1934	Fischer-Tropsch procedure
1935	Radar
1935	Ballpoint pen
1935	Rockets/guided missiles
1935	Plexiglas
1935	Magnetophone
1935	Catalytic cracking
1935	Color photo
1935	Gasoline
1936	Television
1936	Photoelectric cell
1936	FM radio
1937	Vitamins
1937	Electron microscope
1938	Helicopter
1938	Nylon
1939	Polyethylene
1939	Automatic gears
1939	Hydraulic gear
1940	Antibiotics (penicillin)
1941	Cotton picker
1942	Jet engine/plane
1942	DDT
1942	Heavy water
1942	Continuous catalytic cracking
1943	Silicones
1943	Aerosol spray
1943	High-energy accelerators
1944	Streptomycin
1944	Titan reduction
1945	Sulzer loom
1946	Oxygen steelmaking
1946	Phototype
1948	Numerically controlled machine tools
1948	Continuous steel making
1948	Orlon
1948	Cortisone
1948	Long-playing record
1948	Polaroid land camera
1949	Thonet furniture
1949	Polyester
1950	Computer
1950	Transistor
1950	Xerography
1950	Terylene
1950	Radial tire
1951	Double-floor railway
1953	Cinerama
1953	Color television
1954	Nuclear energy
1954	Gas chromatograph
1954	Remote control
1954	Silicon transistor
1956	Air compressed building
1957	Atomic ice breaker
1957	Space travel
1958	Stitching bond
1958	Holography
1958	Transistor radio
1958	Diffusion process
1958	Fuel cell
1959	Quartz clocks
1959	Polyacetates
1959	Float glass
1960	Maser
1960	Micro modules
1960	Polycarbonates
1960	Contraceptive pill
1960	Hovercraft
1961	Integrated circuit
1961	Planar process
1962	Laser
1962	Communication satellite
1963	Implementation of ions
1963	Epitaxy
1964	Synthetic leather
1964	Transistor laser
1966	Optoelectronic diodes
1967	Wankel-motor
1968	Video
1968	Light emitting fluor display
1968	Minicomputers
1970	Quartz watches
1971	Microprocessor
1971	Electronic calculator
1972	Light-tunnel technology
1975	16-bit microprocessor
1976	16,384 bit RAM
1976	Microcomputer