TOP500 HPC Race by Linear Regression: Power
Alexander Levakov, Senior Research Fellow, Ph.D
November, 2020
IBM-Summit
1 Introduction
We had discussed HPC race previously. See: https://rpubs.com/alex-lev/696179, https://rpubs.com/alex-lev/694840, https://rpubs.com/alex-lev/693131, https://rpubs.com/alex-lev/553777
2 TOP500 data
Data for November 2020 can be downloaded here: https://www.top500.org/lists/top500/2020/11/
3 Filtering Data
Note: the original names of variables were changed (truncated, concatenated and simplified) for the purpose of data exploration and visualization as much as possible.
library(readxl)
library(tidyverse)
library(tidyquant)
library(broom)
library(DT)
library(knitr)
TOP500_202011 <- read_excel("top500/TOP500_202011.xlsx")
names(TOP500_202011)## [1] "Rank" "PreviousRank"
## [3] "FirstAppearance" "FirstRank"
## [5] "Name" "Computer"
## [7] "Site" "Manufacturer"
## [9] "Country" "Year"
## [11] "Segment" "TotalCores"
## [13] "AcceleratorCoProcessorCores" "Rmax"
## [15] "Rpeak" "Nmax"
## [17] "Nhalf" "HPCG"
## [19] "Power" "PowerSource"
## [21] "PowerEfficiency" "Architecture"
## [23] "Processor" "ProcessorTechnology"
## [25] "ProcessorSpeed" "OperatingSystem"
## [27] "OSFamily" "AcceleratorCoProcessor"
## [29] "CoresperSocket" "ProcessorGeneration"
## [31] "SystemModel" "SystemFamily"
## [33] "InterconnectFamily" "Interconnect"
## [35] "Continent" "SiteID"
## [37] "SystemID"4 Linear Regression
Now we can compare five countries (USA,China,Japan,France,Germany), leading TOP500 race by total HPC mainframes (see https://rpubs.com/alex-lev/694840). Here we apply linear regression to observe pace of race in terms of coefficients \[ln(Y_i)=B_0+B_1ln(X_i) +E_i \]that is \[ln(Rpeak_i) = B_0+B_1ln(Power_i) +E_i\]
4.1 USA
TOP500.US <- TOP500_202011 %>% filter(Country=="United States") %>% select(Country,Name,Rpeak,TotalCores,Power) %>% na.omit()
TOP500.US %>% datatable()fit.us <- lm(log(Rpeak)~log(Power),TOP500.US)
summary(fit.us)##
## Call:
## lm(formula = log(Rpeak) ~ log(Power), data = TOP500.US)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.7135 -0.6632 -0.1560 0.4305 2.5762
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.9225 1.0829 4.546 6.99e-05 ***
## log(Power) 0.5110 0.1469 3.479 0.00144 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.052 on 33 degrees of freedom
## Multiple R-squared: 0.2683, Adjusted R-squared: 0.2461
## F-statistic: 12.1 on 1 and 33 DF, p-value: 0.001437 ggplot(TOP500.US,aes(x=log(Power),y=log(Rpeak)))+
geom_smooth(method="lm") + geom_point(col="darkblue") +
theme_bw() + ggtitle("Rpeak ~ Power linear regression USA")+
theme(plot.title = element_text(hjust = .5))
4.2 CHINA
TOP500.CH <- TOP500_202011 %>% filter(Country=="China") %>% select(Country,Name,Rpeak,TotalCores,Power) %>% na.omit()
TOP500.CH %>% datatable()fit.china <- lm(log(Rpeak)~log(Power),TOP500.CH)
summary(fit.china)##
## Call:
## lm(formula = log(Rpeak) ~ log(Power), data = TOP500.CH)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.10168 -0.39250 0.00672 0.26678 1.20704
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.49628 0.57882 6.040 9.66e-07 ***
## log(Power) 0.72988 0.08571 8.515 9.90e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5 on 32 degrees of freedom
## Multiple R-squared: 0.6938, Adjusted R-squared: 0.6842
## F-statistic: 72.51 on 1 and 32 DF, p-value: 9.899e-10 ggplot(TOP500.CH,aes(x=log(Power),y=log(Rpeak)))+
geom_smooth(method="lm") + geom_point(col="red") +
theme_bw() + ggtitle("Rpeak ~ Power linear regression CHINA")+
theme(plot.title = element_text(hjust = .5))
4.3 JAPAN
TOP500.JP <- TOP500_202011 %>% filter(Country=="Japan") %>% select(Country,Name,Rpeak,TotalCores,Power) %>% na.omit()
TOP500.JP %>% datatable()fit.jap <- lm(log(Rpeak)~log(Power),TOP500.JP)
summary(fit.jap)##
## Call:
## lm(formula = log(Rpeak) ~ log(Power), data = TOP500.JP)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4362 -0.4584 -0.2440 0.4569 1.2598
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.1674 1.0379 3.052 0.0101 *
## log(Power) 0.8669 0.1500 5.780 8.75e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7693 on 12 degrees of freedom
## Multiple R-squared: 0.7357, Adjusted R-squared: 0.7137
## F-statistic: 33.4 on 1 and 12 DF, p-value: 8.747e-05 ggplot(TOP500.JP,aes(x=log(TotalCores),y=log(Rpeak)))+
geom_smooth(method="lm") + geom_point(col="yellow") +
theme_bw() + ggtitle("Rpeak ~ Power linear regression JAPAN")+
theme(plot.title = element_text(hjust = .5))
4.4 GERMANY
TOP500.GR <- TOP500_202011 %>% filter(Country=="Germany") %>% select(Country,Name,Rpeak,TotalCores,Power) %>% na.omit()
TOP500.GR %>% datatable()fit.gr <- lm(log(Rpeak)~log(Power),TOP500.GR)
summary(fit.gr)##
## Call:
## lm(formula = log(Rpeak) ~ log(Power), data = TOP500.GR)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.50218 -0.40176 -0.01438 0.21393 2.10039
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.4451 3.4255 1.006 0.338
## log(Power) 0.7524 0.4726 1.592 0.142
##
## Residual standard error: 0.8922 on 10 degrees of freedom
## Multiple R-squared: 0.2022, Adjusted R-squared: 0.1224
## F-statistic: 2.534 on 1 and 10 DF, p-value: 0.1425 ggplot(TOP500.GR,aes(x=log(TotalCores),y=log(Rpeak)))+
geom_smooth(method="lm") + geom_point(col="green") +
theme_bw() + ggtitle("Rpeak ~ Power linear regression GERMANY")+
theme(plot.title = element_text(hjust = .5))
4.5 FRANCE
TOP500.FR <- TOP500_202011 %>% filter(Country=="France") %>% select(Country,Name,Rpeak,TotalCores,Power) %>% na.omit()
TOP500.FR %>% datatable()fit.fr <- lm(log(Rpeak)~log(Power),TOP500.FR)
summary(fit.fr)##
## Call:
## lm(formula = log(Rpeak) ~ log(Power), data = TOP500.FR)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4505 -0.6125 0.0034 0.5399 1.4488
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.5838 1.7582 3.176 0.00674 **
## log(Power) 0.4287 0.2491 1.721 0.10732
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8122 on 14 degrees of freedom
## Multiple R-squared: 0.1746, Adjusted R-squared: 0.1156
## F-statistic: 2.961 on 1 and 14 DF, p-value: 0.1073 ggplot(TOP500.FR,aes(x=log(TotalCores),y=log(Rpeak)))+
geom_smooth(method="lm") + geom_point(col="blue") +
theme_bw() + ggtitle("Rpeak ~ Power linear regression FRANCE")+
theme(plot.title = element_text(hjust = .5))
4.6 Comparison
TOP500.5 <- rbind(TOP500.US,TOP500.CH,TOP500.JP,TOP500.FR,TOP500.GR)
TOP500.5%>%group_by(Country)%>%
do(tidy(lm(log(Rpeak)~log(Power),data = .))) %>%
kable(caption = "Linear model parameters by country",digits = 4)| Country | term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|---|
| China | (Intercept) | 3.4963 | 0.5788 | 6.0403 | 0.0000 |
| China | log(Power) | 0.7299 | 0.0857 | 8.5154 | 0.0000 |
| France | (Intercept) | 5.5838 | 1.7582 | 3.1758 | 0.0067 |
| France | log(Power) | 0.4287 | 0.2491 | 1.7207 | 0.1073 |
| Germany | (Intercept) | 3.4451 | 3.4255 | 1.0057 | 0.3383 |
| Germany | log(Power) | 0.7524 | 0.4726 | 1.5919 | 0.1425 |
| Japan | (Intercept) | 3.1674 | 1.0379 | 3.0516 | 0.0101 |
| Japan | log(Power) | 0.8669 | 0.1500 | 5.7796 | 0.0001 |
| United States | (Intercept) | 4.9225 | 1.0829 | 4.5458 | 0.0001 |
| United States | log(Power) | 0.5110 | 0.1469 | 3.4786 | 0.0014 |
ggplot(TOP500.5,aes(x=log(Power),y=log(Rpeak),col=Country))+
geom_smooth(method="lm",se=F) + geom_point() +
theme_bw() + ggtitle("Rpeak ~ Power linear regression ")+
theme(plot.title = element_text(hjust = .5))
TOP500.5%>%ggplot(.,aes(x=log(Power),y=log(Rpeak),col=Country))+
geom_density2d(binwidth = 0.01, na.rm = T)+
theme_bw() + ggtitle("Density contour plot for Rpeak~Power")+
theme(plot.title = element_text(hjust = .5))
5 Conclusion
Applying linear regression lm(log(Rpeak)~log(Power)) that is \(ln(Rpeak_i) = B_0+B_1ln(Power_i)\) we can arrange three leaders in the HPC race by the following order according to the regression slope \(B_1\): 1. Japan - 0.867, 2. China - 0.730, 3. USA - 0.511. We excluded Germany and France due to the P-value as not significant. The slope coefficient \(B_1\) displays how much performance \(Rpeak\) could be produced by \(Power\) i.e. more efficient or less power consuming. However China has got about 43% of the total HPC mainframes in top500 list while Japan has produced Fugaku - HPC number one in the world today. The race is not over!