HeatSpace for NetLogo

I have put up a beta version of heatspace. This program generates visualizations for Netlogo's BehaviorSpace. Find the program and manual here:

NetLogo is an agent-based modeling platform whose ease of use and functionality are hard to beat. The goal of this manual is to succinctly show how to generate visualizations that present significant portions of a model's parameter space. Those who are not accustomed to agent-based modeling may expect visualizations to come in the form of line plots. Line plots are an easy way to convey a limited amount of information to a user, but it is difficult to convey more than a few lines without confusing the observer. Instead of representing the value of an output spatially, heatmaps represent value using color.

When we visualize the parameter space, it is not enough to represent this space only using a single run for each combination of parameters. Each combination must be mapped using a substantial number of runs. Usually 20 runs may produce a large enough sample to faithfully represent the parameter space. More is better. If generating data from model of interest is not a high cost endeavor, use more runs.

HeatSpace averages the values generated at each time period during a run. These values can then mapped onto two axes whose values are fixed throughout a run . A single map can be generated for each time period of a run, creating series of frames analogous to a movie. Alternately, the x-axis of a heatmap can represent time so that the change in the value of an output in light of a change in either the x or y parameter value at period t can be represented.

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I am a graduate lecturer and candidate for a Ph.D. in economics at George Mason University. I have been recipient of the F. A. Hayek Fellowship from Mercatus, the I.H.S. Humane Studies Fellowship, and their Summer Research Fellowship. I hold an M.A. in Economics from San Jose State University where I was a participant in the Student Faculty Partnership and received the Award for Excellence in Economics. I have co-edited Macroeconomics, a 2 volume set which is a collection of essays and primary sources that represent the core of macroeconomic thought. I have also published articles in the Review of Austrian Economics and Advances in Austrian Economics and published book reviews for EH.net, The Journal of Markets and Morality and History: Review of New Books.

Ex Ante, Ex Post: Making Sense of Rational Expectations and the Efficient Market Hypothesis

The move from a predominantly Keynesian paradigm in macroeconomics to the success of monetarist and the new classical macroeconomics that followed represents a shift from a general skepticism of markets within economics to a belief in market efficiency. The most extreme version of this is represented by real business cycle theorists who model the macroeconomy without using an upward sloping short-run aggregate supply curve. (This is actually how I prefer to teach disequilibrium effects associated with changes in aggregate demand.) The emphasis of these models is equilibrium which is, on average, reached in the economic system these models are built on. The results obtained from these models have been successful. In the long-run, the quantity theory holds true. In the long-run, the economy tends to grow at a steady rate, affected mostly by impediments to production and exchange driven by policy. What could be wrong about a field that has generated a tremendous amount of explanatory power?

John Maynard Keynes objected to long-run analysis of the classical economists, meaning most economists who preceded him. This included many of his contemporaries. He claimed that the agents of economic theory were assumed to have higher quality knowledge and decision-making abilities than they actually had. In a sense, Keynes was correct, but he overstated his case. "In the long-run, we're all dead" is a catchy slogan. It is also an abuse of economic ontology. The efficacy of the assumption of rational expectations and of the efficient market hypothesis depend on this distinction between long-run and short-run. Given time for adjustment and a lack of external perturbations, we expect markets will reach equilibrium prices and outputs for different goods. In the long-run, markets select for agents whose knowledge, as reflected by their action, is superior in light of the outcomes these actions generate. In the long-run, agent action is tightly constrained by one's budget constraint. In the short-run, an agent can act with little regard for one's budget constraint. This may be unwise, especially if taken to the extreme case, but eventually, bills must be paid or else that agent loses his power. Second, we must consider the rate of feedback that economic agents receive concerning their investments. Information cascades dominate human decision-making (Bikchandani, Hirshleifer, and Welch, 1998; Earl, Peng, and Potts, 2007). That is, agents learn from one another, copying those who appear to be successful and what appears to be common wisdom. The intelligence of one or a few are shared among many. Think about responses of investors, lay and professional, to perceived opportunities in the last two booms that dominated U.S. financial markets. Common perception went something like:

"The housing market is looking good and remember that the average price of real estate almost never falls!" 

"Have you heard about investments in tech? Better profit from the rise of the internet while we still can!"

Why didn't investors see these "bubbles" coming if markets are truly rational?

Markets are rational, but we sometimes forget to ask what it is that makes them rational. Failure makes them rational. Rational expectations and the efficient market hypothesis reflect the equilibrium arrived at by the market process. If there are above market rates of profit to be gained by pursuing one investment over another, those who invest in those markets will grow their total wealth and, in the process, push the rate of profit for that investment back down toward the market average. Those who miss out on these opportunities will grow at a lower rate relative to those who gained from them. Likewise, those whose investments return a below average rate of profit will be encouraged to pursue other avenues in light of opportunity cost. Those from this category whose profits are actually negative receive a strong signal to scale down their efforts or leave the market. In this process, many agents may guess wrong.

Ex post, after the fact, the market tends to select out those whose efforts fail. This tends to empower agents whose expectations ex ante, before hand, are more correct. Market selection make rational expectations and efficient market hypotheses hold in the long-run. The existence of short-run fluctuations are well observed. In the short-run, changes in stock prices follow a Cauchy distribution (Fama 1965). Whatever their form, we do not experience these distributions themselves. Rather, we experience states that, over time, comprise these distributions. Rational expectations and the efficient market hypothesis don't dispute this. The long-run models derived from them acknowledge that markets are, on average, right, even if they are unstable at times.

How do we describe the short-run? Criteria for efficiency hold constant agent belief, but it is this belief that can experience tremendous flux in the short-run.Without a theory of agent knowledge and the market process driven by this knowledge, a robust theory of the short-run in economics lies out of reach. History matters. Changes in nominal aggregates may not affect real aggregates in equilibrium, but they do affect the structure of society and they matter outside of equilibrium. We must take care in elaborating the implications of macroeconomic models. Short-run deviations can have long-run affects on capital structure. This is true in terms of physical capital in markets as well as social capital. Thus, the Great Depression radically changed economic, political, and academic landscape. In our own field, it allowed for the "Keynesian Diversion" that lasted for several decades! From a given point of departure, it may lead to a realization of an inferior adjacent possible, which economic theory should consider.

The Magic of a One Percent Increase in Growth

Knowledge, Lanugage and Coordination

Check out my latest post at Political Entrepreneurs

Visualize the Behavior Space in Netlogo: 1.1

Last month, I posted code for visualizing the behavior-space using Python to process data from NetLogo. See that post for the netlogo template you will want to use. The py file I posted before was not the easiest to interpret or to work with. It required that a programmer plan ahead, noting the position of runs with particular values for exogenously determined parameters. I rewrote the code so that the data processing is more efficient and easier to work with. The instance I am posting works with a behavior space where only two variables have been set exogenously across runs.  However, if it was necessary, a third or fourth variable could be altered in the same set of runs so long as that third and/or fourth variable was assigned a particular value as runs are collected in lines 196-209. Also note that I have reduced the variety of heatmaps presented since only one is required to convey the process. I loop the image generator code to create a series of frames.

I may eventually post such an extension of the code if I generate output that requires this in the future. For now, I wanted to leave you with something easier to work with. Remember, you should end up with something that looks like this (from earlier post):

See the updated code below:

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from __future__ import print_function import numpy as np import matplotlib.pyplot as plt import matplotlib.cm as cm import pandas as pd from matplotlib.backends.backend_pdf import PdfPages # of csv files generated by netlogo = # of runs files = 1280 # length of each run of the model ticks = 50 # these are the values of interest # in this case we adjusted water metabolism and sugar metabolism axis_values = np.array([.45, .5, .55, .6, .65, .7, .75, .8]) # length of x-axis and y-axis for heatmap num_x = len(axis_values) num_y = num_x runs_per_setting = 20 # min and max x and y parameters for heatmap min_sugar_metabolism = .45 max_sugar_metabolism = .8 min_water_metabolism = .45 max_water_metabolism = .8 # increments tested (here we assume constant increment, could be logged, quadratic, etc...) inc_sugar = .05 inc_water = .05 # set up names of each column of data from csv names = pd.Series([ 'sugar_metabolism_rate', 'water_metabolism_rate', 'sugar', 'water', 'mean_price', 'price_variance', 'population', 'basic_only', 'basic_herder', 'basic_arbitrageur', 'basic_herder_arbitrageur', 'switcher_only', 'switcher_herder', 'switcher_arbitrageur', 'switcher_herder_arbitrageur', 'percent_basic', 'percent_arbitrageur', 'percent_herder', 'percent_switcher', 'basic_only_wealth', 'basic_herder_wealth', 'basic_arbitrageur_wealth', 'basic_herder_arbitrageur_wealth', 'switcher_only_wealth', 'switcher_herder_wealth', 'switcher_arbitrageur_wealth', 'switcher_herder_arbitrageur_wealth', 'sugar_flow', 'water_flow', 'distance_from_equilibrium_price', 'fifty_period_RAP', 'mean_rate_of_price_change']) # names dict used to instantiate an m X n array that will be used # to keep track of the average value of each variable at each tick for each category # will be inserted into mean_values dataframe names_dict = { 'sugar_metabolism_rate': [np.arange(ticks)], 'water_metabolism_rate': [np.arange(ticks)], 'sugar': [np.arange(ticks)], 'water': [np.arange(ticks)], 'mean_price': [np.arange(ticks)], 'price_variance': [np.arange(ticks)], 'population': [np.arange(ticks)], 'basic_only': [np.arange(ticks)], 'basic_herder': [np.arange(ticks)], 'basic_arbitrageur': [np.arange(ticks)], 'basic_herder_arbitrageur': [np.arange(ticks)], 'switcher_only': [np.arange(ticks)], 'switcher_herder': [np.arange(ticks)], 'switcher_arbitrageur': [np.arange(ticks)], 'switcher_herder_arbitrageur': [np.arange(ticks)], 'percent_basic': [np.arange(ticks)], 'percent_arbitrageur': [np.arange(ticks)], 'percent_herder': [np.arange(ticks)], 'percent_switcher': [np.arange(ticks)], 'basic_only_wealth': [np.arange(ticks)], 'basic_herder_wealth': [np.arange(ticks)], 'basic_arbitrageur_wealth': [np.arange(ticks)], 'basic_herder_arbitrageur_wealth': [np.arange(ticks)], 'switcher_only_wealth': [np.arange(ticks)], 'switcher_herder_wealth': [np.arange(ticks)], 'switcher_arbitrageur_wealth': [np.arange(ticks)], 'switcher_herder_arbitrageur_wealth': [np.arange(ticks)], 'sugar_flow': [np.arange(ticks)], 'water_flow': [np.arange(ticks)], 'distance_from_equilibrium_price': [np.arange(ticks)], 'fifty_period_RAP': [np.arange(ticks)], 'basic_only_wealth_per_capita': [np.arange(ticks)], 'basic_herder_wealth_per_capita': [np.arange(ticks)], 'basic_arbitrageur_wealth_per_capita': [np.arange(ticks)], 'basic_herder_arbitrageur_wealth_per_capita': [np.arange(ticks)], 'switcher_only_wealth_per_capita': [np.arange(ticks)], 'switcher_herder_wealth_per_capita': [np.arange(ticks)], 'switcher_arbitrageur_wealth_per_capita': [np.arange(ticks)], 'switcher_herder_arbitrageur_wealth_per_capita': [np.arange(ticks)], 'percent_basic_only': [np.arange(ticks)], 'percent_basic_herder': [np.arange(ticks)], 'percent_basic_arbitraguer': [np.arange(ticks)], 'percent_basic_herder_arbitraguer': [np.arange(ticks)], 'percent_switcher_only': [np.arange(ticks)], 'percent_switcher_herder': [np.arange(ticks)], 'percent_switcher_arbitraguer': [np.arange(ticks)], 'percent_switcher_herder_arbitraguer': [np.arange(ticks)], } # Lists for used to generate new categories of data new_category = pd.Series([ 'basic_only_wealth_per_capita', 'basic_herder_wealth_per_capita', 'basic_arbitrageur_wealth_per_capita', 'basic_herder_arbitrageur_wealth_per_capita', 'switcher_only_wealth_per_capita', 'switcher_herder_wealth_per_capita', 'switcher_arbitrageur_wealth_per_capita', 'switcher_herder_arbitrageur_wealth_per_capita', 'percent_basic_only', 'percent_basic_herder', 'percent_basic_arbitrageur', 'percent_basic_herder_arbitrageur', 'percent_switcher_only', 'percent_switcher_herder', 'percent_switcher_arbitrageur', 'percent_switcher_herder_arbitrageur']) numerator_category = pd.Series([ 'basic_only_wealth', 'basic_herder_wealth', 'basic_arbitrageur_wealth', 'basic_herder_arbitrageur_wealth', 'switcher_only_wealth', 'switcher_herder_wealth', 'switcher_arbitrageur_wealth', 'switcher_herder_arbitrageur_wealth', 'basic_only', 'basic_herder', 'basic_arbitrageur', 'basic_herder_arbitrageur', 'switcher_only', 'switcher_herder', 'switcher_arbitrageur', 'switcher_herder_arbitrageur']) denominator_category = pd.Series([ 'basic_only', 'basic_herder', 'basic_arbitrageur', 'basic_herder_arbitrageur', 'switcher_only_wealth', 'switcher_herder_wealth', 'switcher_arbitrageur_wealth', 'switcher_herder_arbitrageur', 'population', 'population', 'population', 'population', 'population', 'population', 'population', 'population']) def process_data(): # Dataframe will house the average values of each variable across runs for some given # pair of parameters mean_values = pd.DataFrame(columns = [np.arange(num_x)], index = [np.arange(num_y)]) for x in range(0,num_x): for y in range(0,num_y): mean_values[x][y] = pd.DataFrame(names_dict) # import csvs using names array as headers for each column # order of elements in names array should match order of objects recorded # during runs in NetLogo for i in range(1, files + 1): filename = str(i) + 'sugarscapeGlobalTradeBasics.csv' runs = pd.read_csv(filename, names = names) for x in range(len(new_category)): add_per_capita_categories(runs, new_category[x], numerator_category[x], denominator_category[x]) c = 0 # Find runs whose fixed parameter values match the values for the target # coordinates in mean values, will eventually transfer to heatmap for a in axis_values: if runs.iloc[0]['sugar_metabolism_rate'] == a: # round(a,2): for b in axis_values: if runs.iloc[0]['water_metabolism_rate'] == b: # round(b,2): x = int(round(((b - min_water_metabolism) / inc_water),0)) y = int(round(((a - min_sugar_metabolism) / inc_sugar),0)) if c < 1: # avoid error in 0th element of each category mean_values[x][y] = runs else: # once the first run data has been placed mean_values[x][y] = mean_values[x][y].add(runs, fill_value = 0) c += 1 mean_values = mean_values / runs_per_setting print_behavior_space_representation(mean_values) ############################################################################################## ############################################################################################## def add_per_capita_categories(target_array, per_capita_name, per_capita_numerator, population_denominator): target_array[per_capita_name] = target_array[per_capita_numerator] / target_array[population_denominator] def initialize_image(num_x, num_y): image = [] for i in range(num_y): x_colors = [] for j in range(num_x): x_colors.append(0) image.append(x_colors) return image def color_points(title, filename, category, mean_values,pp, tick, min_val, max_val): # x_p = num_x # y_p = num_y image = initialize_image(num_x, num_y) for i in range(0, num_x): for j in range(0, num_y): image[i][j] = mean_values[i][j].iloc[tick][category] print(image) plt.imshow(image, origin='lower', extent=(min_sugar_metabolism, max_sugar_metabolism, min_water_metabolism, max_water_metabolism), cmap=cm.Greys_r, interpolation = 'nearest') plt.colorbar() plt.clim(min_val, max_val) plt.xlabel('Water Consumption Rate') plt.ylabel('Sugar Consumption Rate') plt.title(title + " " + str(tick + 1) + " Ticks") fig = plt.gcf() plt.show() plt.draw() pp.savefig(fig) # fig.savefig(filename + ".pdf") def print_behavior_space_representation(data): interval = 1 pp = PdfPages('population_local_basic_tick_50.pdf') for q in range(1, ticks, interval): color_points("Population\nGlobal Trade Basic","Population Global Trade Basic", "population", data, pp, q, 0, 300) plt.close('all') pp.close() if __name__ == '__main__': np.save('data', process_data()) np.load('data.npy')

Cleaner Code

In light of Gene Callahan's comment yesterday, I have posted some cleaner code below. Those who are not programmers may not understand the process of passing objects through multiple methods (i.e., "plotLines (. . .)). If you don't get it, don't worry about it. Those who prefer tighter organization should use this code.

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from __future__ import print_function import numpy as np import matplotlib.pyplot as plt from matplotlib.backends.backend_pdf import PdfPages pp = PdfPages('SupplyAndDemandFloor.pdf') def supplyAndDemandFloor(supply, demand, floor, equilibrium): fig = plt.figure(dpi=128, figsize=(10,6)) plotLines(supply, demand, equilibrium, floor) frame = plt.gca() plt.title('Pizza Market', fontsize = 32) placeText() setupAxes(frame) pp.savefig(fig) pp.close() def placeText(): #plt.text(x,y,text,fontsize) plt.text(-500, 10000, "$p$", fontsize=24) plt.text(-550, 7900, "$p_s$", fontsize=24) plt.text(-550, 5000, "$p_e$", fontsize=24) plt.text(8200, 8800,"$S$", fontsize = 24) plt.text(8200, 2000,"$D$", fontsize = 24) plt.text(1800, -650, "$Q_d$", fontsize=24) plt.text(7800, -650, "$Q_s$", fontsize=24) plt.text(4800, -650, "$Q_e$", fontsize=24) plt.text(10000, -650, "$Q$", fontsize=24) def plotLines(supply, demand, equilibrium, floor): # plt.plot((x1,x2), (y1,y2), linestyle/color, linewidth) plt.plot((2000, 2000), (0, 8000), 'r--', linewidth=1.5) plt.plot((8000, 8000), (0, 8000), 'r--', linewidth=1.5) plt.plot((5000, 5000), (0, 5000), 'k--', linewidth=1.5) plt.plot(supply, 'k-', linewidth=3) plt.plot(demand, 'k-', linewidth=3) plt.plot(equilibrium, 'k--', linewidth=1.5) plt.plot(floor, 'r--',label= "Price Floor", linewidth=1.5) def setupAxes(frame): frame.axes.get_xaxis().set_visible(False) frame.axes.get_yaxis().set_visible(False) plt.xlabel("Labor", fontsize=20) plt.ylabel("Wage", fontsize = 20) plt.tick_params(axis='both', which='major', labelsize=16) Supply = np.arange(10000) Demand = np.arange(10000,0, -1) priceFloor = np.arange(1, 10000) priceFloor[priceFloor > 0] = 8000 priceEquilibrium = np.arange(1, 5000) priceEquilibrium[priceEquilibrium > 0] = 5000 supplyAndDemandFloor(Supply, Demand, priceFloor, priceEquilibrium)