# Modeling Firm Search and Innovation Trajectory Using Swarm Intelligence
## 4 Results  
In this section, we discuss how to fit our model to actual data in order to calibrate the parameters. We start by illustrating how to fit a partial model that captures specific behavior--in this case, the firm following its own performance feedback and following the leader. We then fit the entire model. Firm's personal best (path dependency)  
We begin by only allowing for the firm's own feedback to guide its innovation trajectory. In Equation (1) we set \(w_{P}=1\) and all other weights to 0. Equation (1) then converges to Equation (5) only. We should note, that in theory there is no limitation on the value of velocity, yet in our empirical work, the velocity is confined to be the difference between two consecutive landscapes. We define data-based (empirical) positions as \(\overset{\rightarrow}{\xi}_{t}^{(i)}\) and calculate velocity \(\overset{\rightarrow}{v}_{t}^{(i)}\) as follows:  
\[\overset{\rightarrow}{v}_{t}^{(i)}=\overset{\rightarrow}{\xi}_{t}^{(i)}- \overset{\rightarrow}{\xi}_{t-1}^{(i)} \tag{8}\]  
Figure 1: Obtaining a firm’s position in topic space. Our objective function is to minimize the sum of squared errors between \(\overset{\rightarrow}{v}_{t}^{(i)}\) and \(\overset{\rightarrow}{v}_{t}^{(i)}\):  
\[\underset{a_{p,t}^{(i)}}{\min}(\overset{\rightarrow}{v}_{t}^{(i)}-\overset{ \rightarrow}{v}_{t}^{(i)})^{\prime}(\overset{\rightarrow}{v}_{t}^{(i)}- \overset{\rightarrow}{v}_{t}^{(i)})=\underset{a_{p,t}^{(i)}}{\min}\sum_{j=1}^ {n}\left(v_{j,t}^{(i)}-v_{j,t}^{(i)}\right)^{2} \tag{9}\]  
We then solve the coefficient analytically as:  
\[\alpha_{p,t}^{(i)}=\frac{\sum_{j=1}^{n}(v_{j,t}^{(i)}-u_{j,t}^{(i)})(p_{j,t}^{ (i)}-x_{j,t}^{(i)}-u_{j,t}^{(i)})}{\sum_{j=1}^{n}\left(p_{j,t}^{(i)}-x_{j,t}^{(i) }-u_{j,t}^{(i)}\right)^{2}} \tag{10}\]  
where  
\[\overset{\rightarrow}{p}_{t}^{(i)}=\left\{\overset{\rightarrow}{x}_{\tau \leq t}^{(i)}|\underset{\overline{\mathrm{r}}}{\max}\phi(\overset{\rightarrow }{x}_{\tau}^{(i)})\right\} \tag{11}\]  
is the personal best and \(\phi(\cdot)\) is the fitness (performance) function. To calculate fitness or performance for firm \(i\), we need some measure of firm performance that can be directly tied to its position in the topic landscape. We use a patent-based measure of performance--forward citations. Forward citations--citations from other patents the focal patent receives--have been shown to be a good proxy for financial value of an innovation and are commonly used as a measure in the economics and strategic management literature [74]. To calculate performance, we take the average number of forward citations across all the successfully granted patents that the firm applied for in the technology class in year \(t\). We provide descriptive statistics to summarize results (across firms and across time) for each technology class in Table 2.  
Results in Table 2 indicate that firms closely follow their own feedback, with \(\alpha_{p,t}^{(i)}\) very close to 1. Note that this result is similar to a partial derivative with respect to the personal best. That is, holding other behavioral patterns constant, firms always \"remember\" their best performing locations in topic space. Following The Leader  
We demonstrate two simple ways to model leader-following behavior. \[ \begin{array}{l}\overset{\rightarrow}{v}_{L,t}^{(i)}=\alpha_{L,t}^{(i)} \overset{\rightarrow}{\bigotimes}{\delta}_{t-1}-\overset{\rightarrow}{x}_{t -1}^{(i)})\\ \overset{\rightarrow}{v}_{L,t}^{(i)}=\alpha_{L,t}^{(i)}-\overset{\rightarrow}{ x}_{t-1}^{(i)})\end{array}  \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Patent Class** & **Mean** & **Median** & **Standard Deviation** \\ \hline
340 & 0.985 & 0.988 & 0.022 \\ \hline
341 & 0.978 & 0.987 & 0.025 \\ \hline
342 & 0.976 & 0.983 & 0.025 \\ \hline
343 & 0.984 & 0.991 & 0.021 \\ \hline
367 & 0.966 & 0.980 & 0.049 \\ \hline
370 & 0.990 & 0.995 & 0.013 \\ \hline
375 & 0.99 where in the former each firm has its own leaders (\(\alpha_{L,t}^{(i)}\)) and in the latter all firms share the same leader (\(\alpha_{L,t}\)).