# In Situ Technological Innovation Diffusion Rate Accuracy Assessment

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Micro vs. Macro

#### 2.2. Diffusion Model

#### 2.3. Logistic Model

#### 2.4. Bass Model

## 3. Research Methodology

#### 3.1. Step 1: Data Collection

#### 3.2. Step 2: Data Diffusion Step Bounds

#### 3.3. Step 3: Diffusion Model Fit and Diffusion Rate Extraction

#### 3.4. Step 4: Percent-Error and Its Statistical Characteristics

## 4. Results

#### 4.1. Diffusion Model Fit and Diffusion Rate Extraction Results

#### 4.2. Percent-Error Results and Statistical Characteristics

## 5. Discussion of Results

#### 5.1. Percent-Error Trends and Patterns

#### 5.2. Diffusion Rate Assessment Model Comparision

#### 5.3. Assumptions and Limitations

#### 5.4. Practitioner Implications and Significance

#### 5.5. Model Errors

#### 5.6. Micro-Effects

## 6. Conclusions

- The Bass and logistic models are more likely to overestimate a technological innovation’s diffusion rate when assessed between the 50% point and the 70% point of its diffusion lifecycle.
- Diffusion rate percent-errors have a positive bias as a technological innovation’s diffusion rate increases.
- The data analysis resultant trend indicates that the Bass and logistic models are more disposed to extreme outliers when diffusion rate assessment is made early in a technological innovation’s lifecycle.
- A normative pattern is observable in diffusion rate percent-error as lifecycle percentage increase, indicating a lack of randomness, signifying and supporting that there is likely an underlying predictable pattern. Decision makers can leverage this pattern to simplify decisions and be used to make informed predictions on diffusion rate outcomes.
- The result trends suggest that, if over-assessing diffusion rate is more desirable than under-assessing diffusion rate, a decision maker should favor the logistic model over the Bass model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphical representation of the model derivatives adapted from Parvin and Beruvides [49].

**Figure 2.**Example of a technological innovation’s fitted raw diffusion data and fitted Bass and logistic models, at full maturation.

**Figure 3.**Example of a technological innovation’s fitted raw diffusion data and fitted Bass and logistic models, at 50% of its lifecycle (respective to full lifecycle maturation).

**Figure 4.**Bass model percent-error at each lifecycle percentage bound (defined in Section 3.2), shown in relation to maximum diffusion rate at full maturation for each technological innovation.

**Figure 5.**Logistic model percent-error at each lifecycle percentage bound (defined in Section 3.2), shown in relation to maximum diffusion rate at full maturation for each technological innovation.

**Figure 6.**Bass model percent-error inclusive quartile box plot, shown with outliers represented as •. Mean represented by x.

**Figure 7.**Logistic model percent-error inclusive quartile box plot, shown with outliers represented as •. Mean represented by x.

Logistic model equation | $Y\left(t\right)=\frac{C}{1+A{e}^{-rt}}$ |

1st derivative | $\frac{d}{dt}\left(Y\left(t\right)\right)=\frac{ACr{e}^{rt}}{{({e}^{rt}+A)}^{2}}$ |

2nd derivative | $\frac{{d}^{2}}{d{t}^{2}}\left(Y\left(t\right)\right)=-\frac{AC{r}^{2}{e}^{rt}\left({e}^{rt}-A\right)}{{({e}^{rt}+A)}^{3}}$ |

Bass model equation | $Y\left(t\right)=\frac{1-{e}^{-\left(p+q\right)t}}{1-\frac{q}{p}{e}^{-\left(p+q\right)t}}C$ |

1st derivative | $\frac{d}{dt}\left(Y\left(t\right)\right)=\frac{Cp\left(q-p\right)\left(q+p\right){e}^{\left(q+p\right)t}}{{\left(p{e}^{\left(q+p\right)t}+q\right)}^{2}}$ |

2nd derivative | $\frac{{d}^{2}}{d{t}^{2}}\left(Y\left(t\right)\right)=\frac{Cq\left(q-p\right){\left(q+p\right)}^{2}{e}^{\left(q+p\right)t}\left(p{e}^{\left(q+p\right)t}-q\right)}{{\left(p{e}^{\left(q+p\right)t}+q\right)}^{3}}$ |

**Table 3.**Resultant logistic model RMSE descriptive statistics, established and adopted from Parvin and Beruvides [4].

Count | Mean RMSE | Standard Deviation | Median RMSE | Min RMSE | Max RMSE | RMSE Skewness | RMSE Kurtosis |
---|---|---|---|---|---|---|---|

42 | 0.786 | 0.145 | 0.802 | 0.450 | 1.086 | –0.250 | –0.574 |

Count | Mean RMSE | Standard Deviation | Median RMSE | Min RMSE | Max RMSE | RMSE Skewness | RMSE Kurtosis |
---|---|---|---|---|---|---|---|

42 | 0.778 | 0.148 | 0.768 | 0.425 | 1.011 | –0.375 | –0.774 |

**Table 5.**Resultant maximum diffusion rates of model fit, fully matured dataset, expanded and adapted from Parvin and Beruvides [4].

# | Technological Innovation | Logistic Model Max Diffusion Rate | Bass Model Max Diffusion Rate | Percent Difference |
---|---|---|---|---|

1 | Air Conditioning | 2.22 | 2.10 | 5.56 |

2 | Automatic Transmission | 4.53 | 4.63 | 2.18 |

3 | Automobile | 1.44 | 1.49 | 3.41 |

4 | Automobile Air Conditioning | 7.94 | 7.90 | 0.51 |

5 | Automobile Disk Brakes | 16.14 | 15.64 | 3.15 |

6 | Automobile Electronic Ignition | 34.67 | 34.74 | 0.20 |

7 | Automobile Fuel Injection | 13.31 | 13.30 | 0.08 |

8 | Blast Oxygen Furnace | 11.39 | 11.09 | 2.67 |

9 | Broadband Internet | 8.37 | 8.05 | 3.90 |

10 | Cellular Phone | 6.16 | 6.49 | 5.22 |

11 | Chlorine-Free Paper Production | 10.70 | 10.52 | 1.70 |

12 | Color Television | 6.03 | 5.68 | 5.98 |

13 | Diesel Locomotive | 9.22 | 9.24 | 0.22 |

14 | Digital Camera | 10.45 | 10.26 | 1.83 |

15 | Digital Computer | 4.82 | 4.77 | 1.04 |

16 | DVD | 15.17 | 14.49 | 4.59 |

17 | DVR | 13.89 | 13.80 | 0.65 |

18 | Electric Clothes Dryer | 2.21 | 2.18 | 1.37 |

19 | Electric Clothes Washer | 1.60 | 1.63 | 1.86 |

20 | Electric Dishwasher | 1.40 | 1.36 | 2.90 |

21 | Front Wheel Drive | 8.62 | 8.33 | 3.42 |

22 | Gas Range/Stove | 2.09 | 2.16 | 3.29 |

23 | HDTV | 16.85 | 17.00 | 0.89 |

24 | Internet | 4.96 | 5.23 | 5.30 |

25 | Lockup Automatic Transmission | 5.52 | 8.35 | 40.81 |

26 | Medical MRI Units | 3.23 | 3.10 | 4.11 |

27 | Microwave Oven | 6.29 | 5.89 | 6.57 |

28 | Mobile PC | 4.64 | 4.64 | 0.00 |

29 | MP3 Player | 12.29 | 11.74 | 4.58 |

30 | Multi-Valve Engine (% of cars equipped) | 4.11 | 5.55 | 29.81 |

31 | Power Steering | 5.49 | 5.48 | 0.18 |

32 | Radial Tire | 21.55 | 20.43 | 5.34 |

33 | Refrigerator | 4.49 | 4.37 | 2.71 |

34 | Residential Electric power | 2.74 | 2.98 | 8.39 |

35 | Smart Meter | 9.01 | 9.09 | 0.88 |

36 | Smartphone | 10.59 | 10.46 | 1.24 |

37 | Tablet | 11.57 | 11.21 | 3.16 |

38 | Telephone (Landline) | 1.32 | 1.51 | 13.43 |

39 | TV | 9.20 | 11.83 | 25.01 |

40 | Vacuum Cleaner | 2.39 | 2.44 | 2.07 |

41 | Variable Valve Timing Automobile | 6.69 | 6.64 | 0.75 |

42 | VCR | 8.14 | 9.01 | 10.15 |

50% Lifecycle | 60% Lifecycle | 70% Lifecycle | 80% Lifecycle | 90% Lifecycle | |
---|---|---|---|---|---|

Mean | 13.74 | 12.30 | 1.44 | −0.11 | −2.15 |

Standard Error | 8.03 | 7.54 | 2.67 | 2.29 | 0.76 |

Median | 0.73 | 2.93 | −0.55 | −2.15 | −1.97 |

Standard Deviation | 52.04 | 48.86 | 17.31 | 14.83 | 4.92 |

Sample Variance | 2708.58 | 2387.19 | 299.70 | 219.81 | 24.24 |

Kurtosis | 12.95 | 14.50 | 1.51 | 9.64 | 1.87 |

Skewness | 3.01 | 3.28 | 0.81 | 2.40 | −0.66 |

Range | 310.14 | 295.81 | 88.55 | 91.05 | 25.01 |

Minimum | −44.17 | −41.93 | −32.70 | −25.16 | −16.09 |

Maximum | 265.97 | 253.88 | 55.85 | 65.90 | 8.92 |

Sum | 577.27 | 516.73 | 60.42 | −4.49 | −90.17 |

Count | 42 | 42 | 42 | 42 | 42 |

First Quartile | −11.13 | −13.32 | −9.32 | −7.40 | −4.40 |

Second Quartile | 0.73 | 2.93 | −0.55 | −2.15 | −1.97 |

Third Quartile | 23.86 | 19.61 | 9.48 | 3.59 | 0.94 |

50% Lifecycle | 60% Lifecycle | 70% Lifecycle | 80% Lifecycle | 90% Lifecycle | |
---|---|---|---|---|---|

Mean | 28.17 | 23.05 | 6.65 | 2.31 | −1.36 |

Standard Error | 9.82 | 9.31 | 3.03 | 2.30 | 0.71 |

Median | 14.21 | 7.99 | 4.48 | 0.68 | −1.08 |

Standard Deviation | 63.61 | 60.32 | 19.66 | 14.92 | 4.57 |

Sample Variance | 4046.19 | 3638.17 | 386.53 | 222.52 | 20.92 |

Kurtosis | 7.88 | 10.03 | 0.73 | 9.88 | 1.77 |

Skewness | 2.54 | 2.96 | 0.62 | 2.35 | −0.85 |

Range | 339.76 | 320.26 | 99.04 | 94.74 | 22.08 |

Minimum | −53.43 | −47.48 | −37.84 | −24.76 | −14.64 |

Maximum | 286.33 | 272.79 | 61.20 | 69.98 | 7.45 |

Sum | 1183.23 | 967.93 | 279.47 | 96.83 | −57.21 |

Count | 42 | 42 | 42 | 42 | 42 |

First Quartile | −3.54 | −8.69 | −6.41 | −5.33 | −3.11 |

Second Quartile | 14.21 | 7.99 | 4.48 | 0.68 | −1.08 |

Third Quartile | 38.32 | 29.78 | 17.19 | 5.42 | 1.23 |

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**MDPI and ACS Style**

Parvin, A.J., Jr.; Beruvides, M.G.; Tercero-Gómez, V.G.
In Situ Technological Innovation Diffusion Rate Accuracy Assessment. *Systems* **2022**, *10*, 25.
https://doi.org/10.3390/systems10020025

**AMA Style**

Parvin AJ Jr., Beruvides MG, Tercero-Gómez VG.
In Situ Technological Innovation Diffusion Rate Accuracy Assessment. *Systems*. 2022; 10(2):25.
https://doi.org/10.3390/systems10020025

**Chicago/Turabian Style**

Parvin, Albert Joseph, Jr., Mario G. Beruvides, and Víctor Gustavo Tercero-Gómez.
2022. "In Situ Technological Innovation Diffusion Rate Accuracy Assessment" *Systems* 10, no. 2: 25.
https://doi.org/10.3390/systems10020025