Below is the R code and comments of data analysis project-1
Some theoretical concepts before actual work:
Linear regression assumpes below:
Step 1: explore relationships with the help of scatterplots
Step 2: explore relationships with the help of correlations
Correlation coeficient measures the stregth of relationship between two variables.
Relationship could be postivie or negative
Correlation value varies from -1 to +1.
Types of correlation coefficients:
a. Pearson's correlation coefficient (X & Y are continuous)
b. Point bi-serial correlation - when one variable is continous and the other is di-chotomous (0,1)
c.Phi coefficient - Both are dichotomous (0,1)
d. Spear's rank - (For ranked data)
Pearson's correlation coefficent = Degree to which X &Y vary together Cov(X,Y)
----------------------------------- = ------------
Degree to which X, Y vary independently Var of X & Y
Step 3 : Fit a linear regression equation between X & Y
Step 4; Explore residuals: a.Compute residuals (Y- Y') Y' is computed value from regression eq b. Normalize residuals c.Make residual plot d.Check normality
Step 5 : Quantify model fit using adjusted-R squared value;
R-sqaured value is square of correlation coefficient (Explained above)
R-sqaured value measures how well the model containing (x) values explains the varaibility in (Y)
R-squared value will be between 0 to 1. Any model with R-squared value above 0.64 is good (i.e correlation coefficient value - 0.8 or +0.8)
In case of multiple explanatory variables (X), all variables may not be there in linear regression model i.e. all explantatory variables may not contribute to explain variablity in Y.
Regression coefficient = Sum of squares of regression values/ Total sum of sqaures
Total sum of squares = Sum of squares of regression + Sum of squares of Residuals SSTO (Total sum of squares) = SST(Sum of squares of treatments) +SSE(Sum of squares of Errors)
The use of an adjusted R-squared (R2) is an attempt to take account of the phenomenon of the R-squared automatically and spuriously increasing when extra explanatory variables are added to the model. It is a modification due to Theil of R2 that adjusts for the number of explanatory terms in a model relative to the number of data points. The adjusted R2 can be negative, and its value will always be less than or equal to that of R2. Unlike R2, the adjusted R2 increases when a new explanator is included only if the new explanator improves the R2 more than would be expected in the absence of any explanatory value being added by the new explanator. If a set of explanatory variables with a predetermined hierarchy of importance are introduced into a regression one at a time, with the adjusted R2 computed each time, the level at which adjusted R2 reaches a maximum, and decreases afterward, would be the regression with the ideal combination of having the best fit without excess/unnecessary terms. The adjusted R2 is defined as
= R2 - (1-R2)(p/(n-p-1))
where p is the total number of regressors in the linear model (not counting the constant term), and n is the sample size.
Adjusted R2 can also be written as
= (Sum of sqaures of residuals/degrees of freedom of residuals) 1 - ————————————————————— (Sum of sqaures of total/degrees of freedom of total)
Going by above, R2 can also be written as
= (Sum of sqaures of residuals/n) 1 - ————————————————————— (Sum of sqaures of total/n)
Here degrees of freedom = n, number of data points
where dft is the degrees of freedom n– 1 of the estimate of the population variance of the dependent variable, and dfe is the degrees of freedom n – p – 1 of the estimate of the underlying population error variance.
Adjusted R2 does not have the same interpretation as R2—while R2 is a measure of fit, adjusted R2 is instead a comparative measure of suitability of alternative nested sets of explanators.As such, care must be taken in interpreting and reporting this statistic. Adjusted R2 is particularly useful in the feature selection stage of model building.
library(gclus)
## Loading required package: cluster
dt <- read.csv("/media/APOLLO M250/4_New_Coursera/coursera-master/dataanalysis-002/loansData.csv")
# dt<-read.csv(file.choose())
a <- strsplit(as.character(dt$FICO.Range), "-") #Splitting FICO RANGE STRING
j <- do.call("rbind", a) # CONVERTS list created above to a dataframe
colnames(j) <- c("F1", "F2") # Renaming columns of dataframe j
p <- cbind(dt, j) # Making one data frame with all data
p[!complete.cases(p), ] # This will give list of rows where missing data is present
## Amount.Requested Amount.Funded.By.Investors Interest.Rate
## 101596 5000 4525 7.43%
## 101515 3500 225 10.28%
## Loan.Length Loan.Purpose Debt.To.Income.Ratio State Home.Ownership
## 101596 36 months other 1% NY NONE
## 101515 36 months other 10% NY RENT
## Monthly.Income FICO.Range Open.CREDIT.Lines
## 101596 NA 800-804 NA
## 101515 15000 685-689 NA
## Revolving.CREDIT.Balance Inquiries.in.the.Last.6.Months
## 101596 NA NA
## 101515 NA NA
## Employment.Length F1 F2
## 101596 < 1 year 800 804
## 101515 < 1 year 685 689
# There are two rows in data framw with NAs
p <- p[complete.cases(p), ] # Data frame with complete set of values
Below is to find numerical values of Interest Rate
m <- sub("%", "", p$Interest.Rate) #repalces % in Interest Rate and adds to dataframe 'm'
m <- as.numeric(m) #converts string to numeric
q <- cbind(p, m) # makes one data frame with interest rate added to original
head(q)
## Amount.Requested Amount.Funded.By.Investors Interest.Rate
## 81174 20000 20000 8.90%
## 99592 19200 19200 12.12%
## 80059 35000 35000 21.98%
## 15825 10000 9975 9.99%
## 33182 12000 12000 11.71%
## 62403 6000 6000 15.31%
## Loan.Length Loan.Purpose Debt.To.Income.Ratio State
## 81174 36 months debt_consolidation 14.90% SC
## 99592 36 months debt_consolidation 28.36% TX
## 80059 60 months debt_consolidation 23.81% CA
## 15825 36 months debt_consolidation 14.30% KS
## 33182 36 months credit_card 18.78% NJ
## 62403 36 months other 20.05% CT
## Home.Ownership Monthly.Income FICO.Range Open.CREDIT.Lines
## 81174 MORTGAGE 6542 735-739 14
## 99592 MORTGAGE 4583 715-719 12
## 80059 MORTGAGE 11500 690-694 14
## 15825 MORTGAGE 3833 695-699 10
## 33182 RENT 3195 695-699 11
## 62403 OWN 4892 670-674 17
## Revolving.CREDIT.Balance Inquiries.in.the.Last.6.Months
## 81174 14272 2
## 99592 11140 1
## 80059 21977 1
## 15825 9346 0
## 33182 14469 0
## 62403 10391 2
## Employment.Length F1 F2 m
## 81174 < 1 year 735 739 8.90
## 99592 2 years 715 719 12.12
## 80059 2 years 690 694 21.98
## 15825 5 years 695 699 9.99
## 33182 9 years 695 699 11.71
## 62403 3 years 670 674 15.31
names(q)[names(q) == "m"] <- "Interest" # Colname is updated with Interest
Below is to categorize Debt to Income Ratio
q$n <- sub("%", "", q$Debt.To.Income.Ratio) #Replaces %in debt to income ratio
names(q)[names(q) == "n"] <- "DI_Ratio"
q$DI_Ratio <- as.numeric(q$DI_Ratio)
q$Category[q$DI_Ratio < 10] <- "A" # categorizing DI ratio as A
q$Category[q$DI_Ratio >= 10 & q$DI_Ratio < 15] <- "B" # categorizing DI ratio as B
q$Category[q$DI_Ratio >= 15 & q$DI_Ratio < 20] <- "C" # categorizing DI ratio as C
q$Category[q$DI_Ratio >= 20] <- "D" # categorizing DI ratio as D
Below is to find numerical vlaues of loadn length
q$LL <- sub(" months", "", q$Loan.Length)
names(q)[names(q) == "LL"] <- "LLen"
q$LLen <- as.numeric(q$LLen)
lapply(q, class) # To understand type of each column
## $Amount.Requested
## [1] "integer"
##
## $Amount.Funded.By.Investors
## [1] "numeric"
##
## $Interest.Rate
## [1] "factor"
##
## $Loan.Length
## [1] "factor"
##
## $Loan.Purpose
## [1] "factor"
##
## $Debt.To.Income.Ratio
## [1] "factor"
##
## $State
## [1] "factor"
##
## $Home.Ownership
## [1] "factor"
##
## $Monthly.Income
## [1] "numeric"
##
## $FICO.Range
## [1] "factor"
##
## $Open.CREDIT.Lines
## [1] "integer"
##
## $Revolving.CREDIT.Balance
## [1] "integer"
##
## $Inquiries.in.the.Last.6.Months
## [1] "integer"
##
## $Employment.Length
## [1] "factor"
##
## $F1
## [1] "factor"
##
## $F2
## [1] "factor"
##
## $Interest
## [1] "numeric"
##
## $DI_Ratio
## [1] "numeric"
##
## $Category
## [1] "character"
##
## $LLen
## [1] "numeric"
Graphs to understand data
attach(q)
boxplot(Interest ~ F1, data = q)
plot(q$Amount.Requested, q$Interes)
plot(q$F2, q$Interest)
plot(q$Home.Ownership, q$Interest)
plot(q$Inquiries.in.the.Last.6.Months, q$Interest)
plot(q$Loan.Length, q$Interest) # Higher the loan length more the interest
plot(q$Open.CREDIT.Lines, q$Interest)
plot(q$Employment.Length, q$Interest)
a <- subset(q, q$Revolving.CREDIT.Balance < 50000)
plot(a$Revolving.CREDIT.Balance, a$Interest) # lower the credit balance, lower the interest
plot(q$Loan.Purpose, q$Interest)
Graphs to understand impact of Debt_to_income_ratio
A <- subset(q, q$Category == "A")
B <- subset(q, q$Category == "B")
C <- subset(q, q$Category == "C")
D <- subset(q, q$Category == "D")
plot(D$Interest) # plot of DI ratio >=20
plot(C$Interest) # plot of DI ratio <20 & >=15
plot(B$Interest) # plot of DI ratio <15 & >=10
Correlation analysis to find out the relation between interest and other variables
cdf <- q[, c(1, 9, 11, 12, 13, 16, 17, 18, 20)] # a new dataframe with variables of interest
cdf$F2 <- as.numeric(cdf$F2) # Conversting FICO range upper value to a numeric to conduct cor
cor(cdf) # Generates coorelation matrix
## Amount.Requested Monthly.Income
## Amount.Requested 1.00000 0.39118
## Monthly.Income 0.39118 1.00000
## Open.CREDIT.Lines 0.19594 0.17140
## Revolving.CREDIT.Balance 0.29337 0.35968
## Inquiries.in.the.Last.6.Months -0.02956 0.03395
## F2 0.08334 0.12277
## Interest 0.33183 0.01292
## DI_Ratio 0.08129 -0.16234
## LLen 0.41230 0.07454
## Open.CREDIT.Lines Revolving.CREDIT.Balance
## Amount.Requested 0.19594 0.293365
## Monthly.Income 0.17140 0.359684
## Open.CREDIT.Lines 1.00000 0.290085
## Revolving.CREDIT.Balance 0.29009 1.000000
## Inquiries.in.the.Last.6.Months 0.11074 0.012186
## F2 -0.08950 0.002735
## Interest 0.09031 0.061109
## DI_Ratio 0.37085 0.189221
## LLen 0.04089 0.055436
## Inquiries.in.the.Last.6.Months F2
## Amount.Requested -0.02956 0.083338
## Monthly.Income 0.03395 0.122766
## Open.CREDIT.Lines 0.11074 -0.089497
## Revolving.CREDIT.Balance 0.01219 0.002735
## Inquiries.in.the.Last.6.Months 1.00000 -0.092142
## F2 -0.09214 1.000000
## Interest 0.16465 -0.709283
## DI_Ratio 0.01198 -0.216921
## LLen 0.02384 0.012736
## Interest DI_Ratio LLen
## Amount.Requested 0.33183 0.08129 0.41230
## Monthly.Income 0.01292 -0.16234 0.07454
## Open.CREDIT.Lines 0.09031 0.37085 0.04089
## Revolving.CREDIT.Balance 0.06111 0.18922 0.05544
## Inquiries.in.the.Last.6.Months 0.16465 0.01198 0.02384
## F2 -0.70928 -0.21692 0.01274
## Interest 1.00000 0.17220 0.42351
## DI_Ratio 0.17220 1.00000 0.02499
## LLen 0.42351 0.02499 1.00000
ABove results suggest that there is high corection between Interest Rate and a. Amount.requested (0.3318) b. Inquiries.in.the.Last.6.Months (0.1646) c. DI_Ratio (0.172) d. LLen (0.423) e FICO vlaue (-0.7092) f.Open.CREDIT.Lines ( 0.09030695)
Below plot will graphically illustrate the impact of correlation
cdr <- abs(cor(cdf))
cd.col <- dmat.color(cdr)
cd.or <- order.single(cdr)
cpairs(cdf, cd.or, panel.colors = cd.col, gap = 0.5, main = "Correlation")
Fitting regression equation with above varaibles
Linear <- lm(cdf$Interest ~ cdf$F2)
par(mfrow = c(2, 2))
plot(Linear)
residual_values <- residuals(Linear)
hist(residual_values)
plot(cdf$Interest, residual_values, ylab = "Residuals", xlab = "Interest Rate",
main = "Graph of Interest Rate vs Residuals")
Below will stanadardize residuals. Should not expect to see any residual beyond -3 to +3. If there is a residual beyond this range, then further investigation need to be done.
standard_residual <- rstandard(Linear)
hist(standard_residual)
As all values are within range of -3 to +3, further investigation is not required. To determine coefficient of linerar model:
coefs <- coefficients(Linear)
coefs
## (Intercept) cdf$F2
## 19.0719 -0.4235
To find a interest rate for a new value of F2, let new value of F2 be 100, then
newdata <- data.frame(F2 = 100)
predict(Linear, newdata) # - need to further check on why predict is not working
## Warning: 'newdata' had 1 rows but variable(s) found have 2498 rows
## 1 2 3 4 5 6 7 8 9 10
## 10.602 12.296 14.413 13.990 13.990 16.107 11.872 13.143 14.837 12.296
## 11 12 13 14 15 16 17 18 19 20
## 16.107 16.531 16.107 10.602 11.449 11.025 13.990 10.178 11.025 8.484
## 21 22 23 24 25 26 27 28 29 30
## 16.531 13.990 16.531 13.990 16.107 13.143 15.684 15.684 8.061 8.484
## 31 32 33 34 35 36 37 38 39 40
## 14.837 14.837 11.872 14.837 15.684 6.790 11.872 2.979 12.296 16.954
## 41 42 43 44 45 46 47 48 49 50
## 16.107 11.872 16.954 16.954 15.684 12.296 12.719 16.107 6.367 13.143
## 51 52 53 54 55 56 57 58 59 60
## 9.331 16.954 13.566 16.531 15.260 11.449 16.107 12.296 14.413 8.908
## 61 62 63 64 65 66 67 68 69 70
## 13.143 12.296 15.260 16.531 11.025 11.449 14.837 14.837 13.143 13.990
## 71 72 73 74 75 76 77 78 79 80
## 13.990 12.296 10.602 16.531 16.107 16.107 5.943 13.566 16.531 11.449
## 81 82 83 84 85 86 87 88 89 90
## 12.719 8.484 15.260 14.413 13.990 11.449 4.249 15.684 9.331 14.837
## 91 92 93 94 95 96 97 98 99 100
## 16.531 8.061 16.107 15.684 15.684 9.331 8.061 10.602 16.531 16.107
## 101 102 103 104 105 106 107 108 109 110
## 13.566 13.143 10.178 14.413 11.872 8.061 12.296 12.719 14.413 12.719
## 111 112 113 114 115 116 117 118 119 120
## 7.214 3.826 9.755 16.954 14.413 12.719 11.449 15.684 8.061 10.602
## 121 122 123 124 125 126 127 128 129 130
## 13.566 10.602 14.413 8.484 15.260 13.990 15.684 13.566 13.990 15.684
## 131 132 133 134 135 136 137 138 139 140
## 11.025 16.531 15.684 15.260 11.449 16.954 4.673 9.755 10.178 13.566
## 141 142 143 144 145 146 147 148 149 150
## 11.872 16.954 10.602 13.566 13.143 11.449 16.954 10.602 11.449 13.143
## 151 152 153 154 155 156 157 158 159 160
## 16.107 11.872 12.296 13.143 12.719 14.413 13.143 14.837 16.954 16.531
## 161 162 163 164 165 166 167 168 169 170
## 9.755 10.178 16.531 13.990 15.260 13.566 12.296 12.719 11.872 16.531
## 171 172 173 174 175 176 177 178 179 180
## 13.566 14.837 11.449 14.837 15.260 9.331 5.096 10.178 11.025 11.872
## 181 182 183 184 185 186 187 188 189 190
## 13.566 13.990 13.990 13.566 16.107 9.755 15.684 10.178 16.954 11.025
## 191 192 193 194 195 196 197 198 199 200
## 9.755 13.566 15.684 16.531 11.025 9.755 8.908 5.943 11.449 16.107
## 201 202 203 204 205 206 207 208 209 210
## 16.531 14.837 9.331 12.296 13.566 9.331 11.025 8.908 14.413 14.413
## 211 212 213 214 215 216 217 218 219 220
## 10.602 13.566 14.413 15.684 16.531 11.025 13.143 15.260 8.484 11.449
## 221 222 223 224 225 226 227 228 229 230
## 14.837 12.719 16.531 11.872 16.954 9.331 12.719 11.025 16.531 14.413
## 231 232 233 234 235 236 237 238 239 240
## 16.107 16.954 16.107 16.531 16.531 9.331 11.872 15.260 14.837 8.484
## 241 242 243 244 245 246 247 248 249 250
## 13.566 13.990 9.331 15.260 9.331 5.943 10.178 12.296 12.719 11.449
## 251 252 253 254 255 256 257 258 259 260
## 8.484 11.025 14.837 13.143 16.107 15.684 16.107 13.566 16.531 16.531
## 261 262 263 264 265 266 267 268 269 270
## 9.331 11.872 11.025 5.096 4.673 15.260 11.025 11.449 13.990 14.413
## 271 272 273 274 275 276 277 278 279 280
## 13.990 12.296 16.531 16.107 16.531 14.837 16.107 12.296 12.296 11.872
## 281 282 283 284 285 286 287 288 289 290
## 16.531 6.790 13.143 14.413 16.107 13.143 15.684 14.837 15.684 11.025
## 291 292 293 294 295 296 297 298 299 300
## 15.260 16.107 9.331 9.755 17.378 14.413 14.413 7.214 9.331 16.531
## 301 302 303 304 305 306 307 308 309 310
## 16.954 11.025 5.096 8.908 5.943 14.837 9.331 13.990 14.837 7.637
## 311 312 313 314 315 316 317 318 319 320
## 13.143 5.520 12.296 14.837 16.531 13.143 16.954 10.178 15.260 11.025
## 321 322 323 324 325 326 327 328 329 330
## 6.367 8.908 16.531 16.531 16.954 14.837 15.260 13.566 16.107 11.025
## 331 332 333 334 335 336 337 338 339 340
## 16.107 12.719 12.719 8.908 6.790 12.296 12.719 14.837 14.413 16.954
## 341 342 343 344 345 346 347 348 349 350
## 6.367 11.872 11.872 12.719 15.684 14.413 14.837 15.684 15.684 14.413
## 351 352 353 354 355 356 357 358 359 360
## 14.413 5.943 12.719 15.684 8.484 16.107 12.296 10.178 15.684 13.990
## 361 362 363 364 365 366 367 368 369 370
## 12.719 12.719 14.837 14.413 10.602 16.954 12.719 10.178 10.602 14.413
## 371 372 373 374 375 376 377 378 379 380
## 16.954 7.637 8.908 12.719 16.531 13.566 13.143 5.943 14.837 14.413
## 381 382 383 384 385 386 387 388 389 390
## 12.296 13.143 13.566 11.449 14.413 11.872 14.413 11.872 14.837 16.531
## 391 392 393 394 395 396 397 398 399 400
## 13.566 15.260 16.107 15.260 16.107 14.413 13.566 6.790 7.214 10.602
## 401 402 403 404 405 406 407 408 409 410
## 14.837 8.908 12.296 11.025 15.260 16.107 14.837 9.331 13.143 15.260
## 411 412 413 414 415 416 417 418 419 420
## 15.684 16.531 13.990 14.413 13.566 13.566 13.566 11.025 12.296 14.413
## 421 422 423 424 425 426 427 428 429 430
## 15.684 11.449 11.025 10.602 14.837 16.107 14.837 12.296 13.143 14.413
## 431 432 433 434 435 436 437 438 439 440
## 13.990 12.296 12.719 10.602 8.484 12.296 13.143 11.449 11.025 15.684
## 441 442 443 444 445 446 447 448 449 450
## 13.990 16.954 8.484 15.260 11.025 11.025 13.143 11.872 13.143 14.837
## 451 452 453 454 455 456 457 458 459 460
## 16.107 11.025 11.025 14.837 13.566 15.684 16.531 11.872 11.449 13.566
## 461 462 463 464 465 466 467 468 469 470
## 16.107 6.790 11.449 11.872 16.954 13.566 14.837 11.025 15.260 15.260
## 471 472 473 474 475 476 477 478 479 480
## 10.178 16.107 6.790 11.025 14.413 13.990 16.954 16.531 13.990 15.260
## 481 482 483 484 485 486 487 488 489 490
## 14.837 14.413 15.684 16.531 11.872 16.531 12.296 14.413 13.143 13.143
## 491 492 493 494 495 496 497 498 499 500
## 15.684 11.449 15.684 14.413 16.107 11.449 13.566 16.107 15.260 16.107
## 501 502 503 504 505 506 507 508 509 510
## 13.566 15.260 14.413 14.837 15.684 9.755 15.260 14.837 16.107 10.602
## 511 512 513 514 515 516 517 518 519 520
## 11.025 14.413 11.449 8.484 9.755 7.214 12.719 8.484 12.296 12.296
## 521 522 523 524 525 526 527 528 529 530
## 9.331 13.566 8.908 16.107 16.531 9.331 16.107 15.684 15.260 16.107
## 531 532 533 534 535 536 537 538 539 540
## 7.214 16.531 16.954 16.531 15.684 15.260 15.260 16.107 11.025 13.143
## 541 542 543 544 545 546 547 548 549 550
## 10.178 14.837 16.107 11.872 14.837 15.684 11.449 10.178 12.296 16.107
## 551 552 553 554 555 556 557 558 559 560
## 13.566 14.837 14.837 16.954 11.872 16.107 12.719 10.178 15.260 10.178
## 561 562 563 564 565 566 567 568 569 570
## 16.531 12.719 14.837 16.107 13.990 11.872 7.637 13.143 16.107 14.413
## 571 572 573 574 575 576 577 578 579 580
## 11.025 14.837 14.413 13.990 11.449 11.025 15.684 15.260 11.872 14.413
## 581 582 583 584 585 586 587 588 589 590
## 16.531 14.413 16.954 15.260 14.837 13.990 14.413 10.602 8.484 9.331
## 591 592 593 594 595 596 597 598 599 600
## 5.943 10.178 11.872 16.954 13.566 13.566 15.260 14.413 13.566 16.107
## 601 602 603 604 605 606 607 608 609 610
## 6.790 16.531 13.566 13.143 15.684 14.837 14.837 14.413 5.096 11.449
## 611 612 613 614 615 616 617 618 619 620
## 14.413 9.331 13.566 5.943 14.837 11.872 15.684 13.143 15.260 10.602
## 621 622 623 624 625 626 627 628 629 630
## 13.143 15.684 10.178 8.908 13.143 12.719 13.566 6.367 14.413 16.107
## 631 632 633 634 635 636 637 638 639 640
## 12.719 11.872 13.566 13.990 11.872 12.719 13.990 12.296 11.449 10.602
## 641 642 643 644 645 646 647 648 649 650
## 12.719 15.684 15.684 12.719 11.025 11.025 11.872 18.648 13.566 15.260
## 651 652 653 654 655 656 657 658 659 660
## 10.602 11.872 15.684 14.413 11.449 11.872 11.025 12.719 8.484 5.520
## 661 662 663 664 665 666 667 668 669 670
## 16.954 15.260 13.990 10.602 16.107 14.837 13.143 14.413 13.990 16.531
## 671 672 673 674 675 676 677 678 679 680
## 15.260 9.755 14.837 16.531 9.755 15.684 11.449 11.449 12.719 11.449
## 681 682 683 684 685 686 687 688 689 690
## 14.413 16.107 13.990 11.872 6.367 15.684 13.990 14.837 16.531 16.954
## 691 692 693 694 695 696 697 698 699 700
## 13.566 14.413 16.531 13.566 11.872 16.107 11.449 13.143 14.413 9.755
## 701 702 703 704 705 706 707 708 709 710
## 18.648 13.143 16.531 16.531 15.684 6.790 14.413 16.531 16.531 15.684
## 711 712 713 714 715 716 717 718 719 720
## 15.260 16.954 12.296 12.719 16.107 7.214 15.260 16.531 16.531 15.684
## 721 722 723 724 725 726 727 728 729 730
## 12.719 13.990 15.260 12.719 12.296 16.107 12.296 13.990 16.107 14.837
## 731 732 733 734 735 736 737 738 739 740
## 13.990 15.684 13.990 15.260 7.637 10.602 12.296 11.872 16.531 14.837
## 741 742 743 744 745 746 747 748 749 750
## 15.260 16.107 15.684 16.107 11.449 16.954 13.566 15.684 13.143 14.413
## 751 752 753 754 755 756 757 758 759 760
## 11.872 16.531 15.260 15.684 15.260 16.107 15.260 14.837 15.260 14.413
## 761 762 763 764 765 766 767 768 769 770
## 11.872 16.531 11.025 16.107 15.260 16.531 16.954 13.143 14.413 11.872
## 771 772 773 774 775 776 777 778 779 780
## 13.566 13.143 11.872 12.719 16.954 12.719 16.531 9.755 12.719 9.755
## 781 782 783 784 785 786 787 788 789 790
## 16.954 11.025 11.025 16.107 11.025 13.143 11.025 16.531 13.566 16.107
## 791 792 793 794 795 796 797 798 799 800
## 9.755 16.107 13.566 11.449 13.566 16.107 16.531 9.755 12.719 16.107
## 801 802 803 804 805 806 807 808 809 810
## 11.872 13.143 15.260 12.296 10.602 5.943 8.908 9.331 15.260 16.107
## 811 812 813 814 815 816 817 818 819 820
## 16.954 14.837 16.107 11.449 6.790 14.413 13.143 14.837 11.872 16.954
## 821 822 823 824 825 826 827 828 829 830
## 16.531 15.260 15.684 8.061 13.566 13.566 11.025 11.449 16.954 12.719
## 831 832 833 834 835 836 837 838 839 840
## 13.566 5.943 7.214 14.413 9.755 11.025 15.684 14.413 15.260 10.178
## 841 842 843 844 845 846 847 848 849 850
## 16.107 6.790 4.249 13.566 13.990 16.107 11.872 8.061 8.484 15.260
## 851 852 853 854 855 856 857 858 859 860
## 15.684 14.837 11.872 15.684 8.061 16.531 14.413 15.684 15.260 13.566
## 861 862 863 864 865 866 867 868 869 870
## 11.872 10.602 13.143 9.331 13.566 16.954 7.637 16.954 12.296 15.684
## 871 872 873 874 875 876 877 878 879 880
## 10.178 13.990 11.449 6.790 13.143 9.755 10.602 16.531 16.107 13.566
## 881 882 883 884 885 886 887 888 889 890
## 16.107 6.367 13.143 10.178 9.755 8.061 15.260 13.990 13.990 13.143
## 891 892 893 894 895 896 897 898 899 900
## 11.025 14.413 7.637 11.449 11.449 12.296 16.107 16.107 14.413 16.531
## 901 902 903 904 905 906 907 908 909 910
## 6.790 14.837 6.790 15.684 16.107 15.684 10.602 11.025 11.872 15.684
## 911 912 913 914 915 916 917 918 919 920
## 15.684 16.531 16.107 15.684 14.837 13.566 10.602 9.331 7.637 16.107
## 921 922 923 924 925 926 927 928 929 930
## 8.484 15.684 11.025 13.143 11.025 16.954 14.837 12.719 13.990 10.178
## 931 932 933 934 935 936 937 938 939 940
## 13.990 8.484 15.260 11.872 16.107 12.719 11.872 14.837 15.684 15.684
## 941 942 943 944 945 946 947 948 949 950
## 14.413 11.025 12.719 13.990 15.260 11.025 13.566 13.990 13.566 10.178
## 951 952 953 954 955 956 957 958 959 960
## 13.143 8.061 11.449 8.908 13.566 14.413 16.954 12.719 11.872 13.143
## 961 962 963 964 965 966 967 968 969 970
## 16.531 12.719 9.331 13.990 16.107 12.719 13.566 8.061 11.449 13.566
## 971 972 973 974 975 976 977 978 979 980
## 8.061 16.531 13.990 12.719 13.566 9.755 13.143 15.684 8.484 14.837
## 981 982 983 984 985 986 987 988 989 990
## 10.178 5.943 15.684 9.331 8.484 14.413 16.107 12.719 16.954 16.954
## 991 992 993 994 995 996 997 998 999 1000
## 16.531 11.449 13.990 14.837 8.908 15.260 16.107 13.990 11.025 8.484
## 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010
## 13.143 14.837 11.872 15.684 5.943 11.872 10.602 5.943 14.837 15.260
## 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020
## 9.755 14.413 14.837 13.990 13.990 10.602 12.296 16.954 15.260 14.413
## 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030
## 15.684 13.990 13.143 13.990 11.449 15.684 10.602 15.684 13.143 13.990
## 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040
## 15.684 13.990 11.025 14.837 16.107 16.954 14.413 6.367 6.367 14.413
## 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050
## 13.990 12.719 16.531 8.061 16.107 11.872 12.719 9.755 15.684 15.260
## 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060
## 11.872 15.260 14.413 14.837 16.107 15.260 11.025 13.990 13.566 8.484
## 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070
## 11.872 16.531 15.260 13.566 10.602 13.143 6.790 14.413 16.954 12.296
## 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080
## 16.954 15.260 15.260 12.296 15.260 16.107 12.296 15.260 11.025 8.484
## 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090
## 9.755 11.872 13.143 15.260 15.684 12.719 16.107 12.296 16.107 11.872
## 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100
## 13.143 8.484 15.684 13.990 15.684 11.872 15.260 12.719 9.331 13.990
## 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110
## 15.684 4.673 11.449 11.872 14.837 16.954 15.684 13.990 11.025 5.520
## 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120
## 16.107 14.837 16.531 15.260 16.107 13.566 11.449 11.872 13.990 8.061
## 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130
## 8.061 9.331 16.954 8.061 10.178 16.107 15.684 10.178 15.684 9.755
## 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140
## 11.872 8.908 11.872 16.107 8.061 16.954 16.531 16.954 15.684 8.908
## 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150
## 12.296 7.637 12.296 13.143 8.484 15.260 12.296 14.837 14.837 16.954
## 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160
## 11.872 8.908 15.684 13.990 9.331 13.566 14.413 14.413 13.566 5.943
## 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170
## 10.178 12.719 17.378 9.331 12.719 8.061 13.143 12.296 16.107 9.755
## 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180
## 11.449 8.908 9.755 12.296 8.484 12.296 16.954 12.719 16.107 13.990
## 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190
## 16.107 14.413 11.872 13.143 12.719 15.684 13.990 11.449 8.908 11.872
## 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200
## 13.990 9.331 13.143 12.719 16.107 15.684 11.025 13.143 16.531 15.684
## 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210
## 7.637 11.025 13.566 10.178 13.990 12.719 9.755 16.531 10.602 13.990
## 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220
## 13.990 13.143 13.990 12.719 8.908 13.990 15.260 15.684 14.413 14.837
## 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230
## 13.990 16.107 7.214 14.837 13.566 13.143 9.755 11.449 16.107 15.684
## 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240
## 12.719 11.025 9.331 12.296 16.107 12.719 12.296 13.566 12.719 7.637
## 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250
## 10.602 11.449 13.990 11.449 15.684 8.908 16.954 14.837 18.225 8.484
## 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260
## 11.025 14.837 15.260 15.684 13.990 13.143 15.684 11.025 8.061 15.260
## 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270
## 15.684 13.143 16.531 14.413 12.296 13.990 16.531 11.872 14.413 13.566
## 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280
## 16.107 16.531 13.143 13.566 13.990 16.954 8.908 12.296 14.413 13.143
## 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290
## 16.954 14.413 11.449 13.566 15.684 7.214 15.260 9.331 14.837 8.484
## 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300
## 8.061 13.143 8.484 11.449 4.249 14.837 12.296 9.331 14.837 13.566
## 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310
## 13.566 16.954 16.531 18.225 15.260 13.990 16.107 5.096 10.602 14.837
## 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320
## 9.755 16.531 12.296 5.096 13.990 14.413 12.719 8.061 12.719 16.107
## 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330
## 6.367 13.990 13.990 13.143 5.096 11.025 13.566 11.025 9.331 13.143
## 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340
## 15.684 12.719 7.214 11.025 16.954 11.872 11.025 11.025 6.367 13.990
## 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350
## 7.637 13.143 6.790 16.107 14.837 8.908 14.413 8.484 16.107 13.566
## 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360
## 13.143 16.954 6.790 16.531 8.484 13.990 16.531 16.954 11.449 15.684
## 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370
## 13.990 8.908 14.837 14.837 9.331 13.143 8.908 10.178 11.872 9.331
## 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380
## 16.107 13.566 14.413 11.872 16.954 15.260 11.025 11.872 10.602 15.260
## 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390
## 10.602 6.367 13.566 18.225 11.872 15.260 9.331 15.684 15.684 11.449
## 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400
## 6.790 10.602 16.954 11.872 12.296 10.602 8.484 11.449 10.602 14.837
## 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410
## 16.954 11.872 14.413 15.684 16.107 6.367 16.107 12.296 8.061 11.025
## 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420
## 14.413 16.954 9.755 8.061 12.296 8.061 13.566 11.449 13.990 6.367
## 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430
## 16.107 5.943 11.449 5.520 16.107 11.025 12.719 16.954 13.990 11.872
## 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440
## 15.684 13.143 11.872 8.484 14.837 16.531 11.449 16.954 12.719 16.954
## 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450
## 9.331 16.107 12.296 11.872 13.143 5.096 16.531 9.331 12.296 13.143
## 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460
## 14.837 16.107 16.531 10.602 15.260 15.260 13.990 16.531 16.107 15.684
## 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470
## 13.566 15.260 8.484 14.837 14.837 15.684 5.943 15.684 16.954 16.954
## 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480
## 10.178 9.755 13.990 16.531 11.872 14.413 15.684 9.755 11.025 15.260
## 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490
## 16.107 11.872 15.260 15.260 14.413 13.990 16.954 13.566 14.413 16.531
## 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500
## 16.954 11.025 15.260 16.107 13.143 8.484 13.566 15.684 13.990 13.990
## 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510
## 15.684 15.260 13.990 16.531 16.107 13.143 11.449 8.908 13.143 13.990
## 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520
## 11.449 13.566 11.449 6.790 15.260 15.260 15.684 12.719 14.837 7.214
## 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530
## 15.260 13.143 13.566 13.143 13.566 16.107 16.531 10.602 12.296 11.449
## 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540
## 13.990 11.025 10.602 15.260 9.755 9.331 14.837 15.260 11.449 16.107
## 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550
## 14.413 13.566 16.531 14.413 15.684 13.143 12.296 12.296 10.602 9.755
## 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560
## 16.954 16.531 16.954 12.296 14.837 11.025 16.954 7.637 15.260 13.990
## 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570
## 10.602 14.413 13.566 12.296 13.566 13.566 14.837 13.143 12.719 8.061
## 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580
## 15.260 14.837 13.990 12.719 11.872 14.837 8.908 16.107 9.755 16.107
## 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590
## 9.331 16.954 11.025 13.143 7.214 11.449 10.602 5.520 14.837 13.143
## 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600
## 16.954 10.178 12.719 14.413 15.684 16.954 11.025 13.566 11.872 16.531
## 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610
## 14.413 13.143 9.331 13.143 11.872 4.673 14.837 16.107 10.178 15.260
## 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620
## 11.449 13.566 16.531 8.908 16.531 14.837 9.331 10.602 10.178 3.826
## 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630
## 13.990 6.790 13.990 15.260 16.107 10.602 8.061 16.531 14.413 12.719
## 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640
## 18.648 11.025 10.178 12.719 11.449 16.531 11.449 13.990 13.990 16.107
## 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650
## 8.484 5.943 15.260 13.990 11.025 11.872 16.531 8.908 16.954 9.755
## 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660
## 16.954 16.954 8.484 11.872 16.531 8.908 4.249 13.566 8.061 13.143
## 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670
## 14.413 3.402 15.684 14.837 14.837 13.143 16.107 16.107 13.566 14.413
## 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680
## 15.684 16.531 15.260 12.296 12.719 14.837 13.143 11.025 11.025 15.684
## 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690
## 16.531 13.990 13.566 4.249 18.648 15.260 7.214 15.260 15.260 16.107
## 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700
## 15.684 11.449 16.954 13.990 16.107 3.826 11.025 14.837 13.143 13.143
## 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710
## 12.296 11.025 13.990 16.954 12.719 15.260 13.143 15.260 7.214 14.837
## 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720
## 8.908 15.260 15.260 11.872 9.755 10.602 14.837 8.484 14.837 14.837
## 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730
## 11.449 13.990 13.990 13.143 7.214 13.566 14.413 12.296 16.531 14.837
## 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740
## 14.413 11.872 10.178 16.107 14.837 10.178 13.566 13.990 15.684 11.872
## 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750
## 16.954 11.025 16.531 15.684 13.566 15.260 11.025 12.719 14.413 13.566
## 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760
## 9.331 16.531 16.954 14.413 16.107 11.872 13.990 15.684 12.296 12.719
## 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770
## 16.954 17.378 13.566 15.684 8.908 11.449 16.107 13.566 7.637 11.872
## 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780
## 9.331 4.673 15.260 13.566 12.719 13.566 13.990 15.260 9.755 12.296
## 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790
## 12.719 5.096 14.837 13.990 15.684 11.449 16.954 12.296 15.260 14.837
## 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800
## 14.837 12.719 16.954 13.566 12.296 8.908 13.143 13.566 13.990 14.413
## 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810
## 11.449 10.178 16.531 13.990 9.331 12.296 11.872 12.719 12.296 8.908
## 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820
## 10.178 13.143 6.790 13.566 13.143 16.531 8.908 13.990 15.684 14.837
## 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830
## 15.260 16.954 16.107 12.719 13.143 11.872 11.449 11.449 13.566 13.990
## 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840
## 13.566 16.107 13.566 15.260 11.449 16.107 15.260 11.872 13.566 13.566
## 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850
## 14.413 14.837 14.413 15.260 16.107 12.296 9.755 14.837 15.260 4.673
## 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860
## 4.249 11.449 16.107 14.837 10.178 16.531 4.673 15.684 15.260 11.025
## 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870
## 15.260 16.954 15.260 7.214 5.943 15.684 10.178 15.260 12.296 16.107
## 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880
## 8.484 12.719 13.143 13.143 16.107 13.143 13.990 14.413 9.331 14.413
## 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890
## 11.449 16.107 15.260 9.755 11.025 16.107 13.143 11.449 8.908 6.790
## 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900
## 11.025 13.143 16.531 16.107 10.602 10.178 14.413 14.837 13.143 12.296
## 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910
## 8.484 15.684 12.296 14.413 13.990 9.755 9.755 12.296 16.954 6.367
## 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920
## 13.566 15.260 13.990 14.413 11.872 13.143 13.566 16.531 13.990 14.837
## 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930
## 12.719 16.954 16.531 6.790 12.296 13.143 15.684 13.566 11.872 15.260
## 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940
## 15.260 13.566 13.990 8.484 15.684 11.872 10.602 14.837 7.214 15.684
## 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950
## 9.331 13.990 8.908 16.954 12.719 13.566 10.602 16.107 3.826 11.025
## 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960
## 12.296 13.566 14.413 12.296 14.837 11.872 5.520 13.143 16.531 16.531
## 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970
## 16.531 12.719 14.413 15.684 11.449 14.413 13.990 11.872 14.413 12.296
## 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980
## 13.143 9.331 16.531 12.719 9.755 14.837 15.684 16.954 17.378 8.908
## 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990
## 12.719 12.719 9.755 8.908 15.260 15.260 16.107 6.790 12.296 13.143
## 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
## 12.719 12.719 12.719 12.719 9.331 15.260 13.143 15.684 10.178 13.566
## 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
## 15.260 16.954 13.143 16.107 15.260 14.413 12.719 16.531 12.719 15.260
## 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
## 12.296 13.990 16.107 16.531 14.413 9.331 13.990 13.143 15.684 15.684
## 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030
## 10.178 15.684 13.990 14.413 11.872 16.531 4.673 5.943 14.413 12.296
## 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040
## 15.260 16.954 7.637 10.602 14.837 10.602 16.531 15.684 10.602 11.872
## 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050
## 6.790 10.602 14.413 16.531 15.684 8.908 16.107 16.107 9.331 9.755
## 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060
## 12.296 14.413 5.520 16.531 15.684 16.107 9.755 4.673 10.602 11.872
## 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070
## 7.637 16.107 9.331 13.990 11.449 15.260 11.025 15.260 7.214 16.531
## 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080
## 15.684 5.520 15.260 16.531 10.178 6.367 16.531 5.096 8.908 16.531
## 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090
## 15.684 16.107 16.531 14.837 9.755 13.990 15.260 14.837 16.107 15.684
## 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100
## 15.684 16.954 16.107 16.531 8.061 16.954 13.143 14.837 14.413 7.214
## 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110
## 16.107 13.143 15.260 14.837 15.260 13.143 11.449 13.990 14.413 12.719
## 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120
## 11.872 16.531 16.954 13.566 16.107 12.296 13.566 16.531 13.990 15.260
## 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130
## 3.826 8.908 13.566 12.296 15.684 14.837 15.260 14.413 16.954 9.331
## 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140
## 5.520 14.837 5.520 16.531 11.025 10.178 15.684 11.872 12.719 13.143
## 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150
## 13.566 14.413 13.143 14.413 16.954 13.990 15.684 13.990 16.954 14.413
## 2151 2152 2153 2154 2155 2156 2157 2158 2159 2160
## 16.954 10.178 11.872 16.107 13.566 10.602 13.990 13.143 16.531 14.837
## 2161 2162 2163 2164 2165 2166 2167 2168 2169 2170
## 18.648 13.566 8.061 9.755 16.954 14.413 11.872 13.143 11.025 15.684
## 2171 2172 2173 2174 2175 2176 2177 2178 2179 2180
## 15.684 6.367 8.908 14.837 10.178 12.296 5.520 15.684 10.178 8.484
## 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190
## 11.449 9.331 5.096 8.484 13.990 13.566 14.837 13.990 15.684 6.367
## 2191 2192 2193 2194 2195 2196 2197 2198 2199 2200
## 16.954 15.684 16.954 8.061 15.260 13.143 16.107 16.107 13.143 15.684
## 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210
## 11.449 11.025 16.107 11.872 15.684 11.449 13.143 16.954 13.990 8.061
## 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220
## 15.260 14.837 8.061 12.719 13.990 13.990 16.107 9.755 13.143 10.602
## 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230
## 8.908 14.413 16.954 6.790 10.602 15.260 13.566 16.531 13.143 15.684
## 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240
## 14.837 13.990 16.954 16.954 14.413 13.990 15.260 16.954 5.520 11.872
## 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250
## 9.331 11.025 13.566 13.990 16.531 14.413 15.684 14.837 14.413 16.531
## 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260
## 10.602 16.107 11.872 16.107 12.719 12.719 15.260 15.684 15.684 15.684
## 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270
## 15.260 8.484 11.025 10.602 9.755 7.214 9.331 16.107 13.990 14.837
## 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280
## 16.107 8.484 13.566 10.602 13.143 11.449 10.602 15.260 15.684 13.990
## 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290
## 10.178 16.107 13.990 13.990 14.837 13.566 14.413 15.684 15.684 16.531
## 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300
## 15.260 16.954 16.107 16.954 16.954 11.872 13.143 16.107 11.872 14.413
## 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310
## 16.531 16.531 15.260 5.520 11.449 13.990 8.061 16.107 10.178 13.990
## 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320
## 13.143 14.413 12.296 13.990 15.684 7.637 12.296 9.755 16.107 16.107
## 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330
## 16.107 13.990 11.872 12.719 16.954 9.331 6.367 13.990 11.872 13.990
## 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340
## 11.025 11.872 16.531 16.954 11.449 15.260 9.755 15.260 13.566 4.249
## 2341 2342 2343 2344 2345 2346 2347 2348 2349 2350
## 15.260 6.790 15.684 16.531 14.413 9.331 6.790 14.413 14.413 11.449
## 2351 2352 2353 2354 2355 2356 2357 2358 2359 2360
## 13.143 14.413 16.954 14.413 13.566 13.143 4.673 15.260 11.449 4.673
## 2361 2362 2363 2364 2365 2366 2367 2368 2369 2370
## 11.025 13.990 16.531 6.367 13.990 16.107 11.449 10.602 12.296 16.954
## 2371 2372 2373 2374 2375 2376 2377 2378 2379 2380
## 15.260 12.719 8.061 11.449 13.143 11.449 16.107 12.719 12.719 12.719
## 2381 2382 2383 2384 2385 2386 2387 2388 2389 2390
## 15.260 16.531 10.178 8.908 12.719 13.990 7.214 13.143 13.990 15.684
## 2391 2392 2393 2394 2395 2396 2397 2398 2399 2400
## 13.990 15.684 16.954 5.096 9.331 15.260 3.826 9.755 13.143 14.413
## 2401 2402 2403 2404 2405 2406 2407 2408 2409 2410
## 16.954 14.413 13.143 11.449 15.684 12.719 15.684 7.214 5.943 12.719
## 2411 2412 2413 2414 2415 2416 2417 2418 2419 2420
## 10.178 10.602 16.954 16.954 11.025 16.954 15.260 16.107 13.990 8.484
## 2421 2422 2423 2424 2425 2426 2427 2428 2429 2430
## 16.531 11.872 16.107 16.531 16.954 14.413 9.755 16.107 16.954 15.684
## 2431 2432 2433 2434 2435 2436 2437 2438 2439 2440
## 13.566 13.990 13.566 15.684 12.719 9.331 14.837 16.531 14.413 13.143
## 2441 2442 2443 2444 2445 2446 2447 2448 2449 2450
## 15.684 16.954 9.331 11.449 10.178 12.296 15.260 16.531 12.719 16.531
## 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460
## 8.061 6.790 15.260 14.413 12.719 13.143 14.837 11.872 12.719 11.872
## 2461 2462 2463 2464 2465 2466 2467 2468 2469 2470
## 14.413 10.602 14.837 8.908 16.107 11.025 12.296 8.908 16.531 7.637
## 2471 2472 2473 2474 2475 2476 2477 2478 2479 2480
## 14.837 17.801 16.954 15.684 15.684 11.025 11.449 15.260 8.484 4.249
## 2481 2482 2483 2484 2485 2486 2487 2488 2489 2490
## 11.872 15.684 14.413 8.061 16.531 16.954 14.837 16.107 12.719 11.872
## 2491 2492 2493 2494 2495 2496 2497 2498
## 12.719 15.684 14.837 13.143 10.178 15.260 15.684 16.107
Null Hypothesis of linear regression, says that slope of the line between response variable and predicted variable is zero. However, from below summary, p-value is less than zero. So, null hypothesis will be rejected and concluded that slope of regression line is not zero. Further, adjusted R2 value is ~ 0.5, above model explains 50% of variation.
summary(Linear)
##
## Call:
## lm(formula = cdf$Interest ~ cdf$F2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.990 -2.136 -0.456 1.835 10.194
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.07189 0.13314 143.2 <2e-16 ***
## cdf$F2 -0.42350 0.00842 -50.3 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.95 on 2496 degrees of freedom
## Multiple R-squared: 0.503, Adjusted R-squared: 0.503
## F-statistic: 2.53e+03 on 1 and 2496 DF, p-value: <2e-16
confint(Linear)
## 2.5 % 97.5 %
## (Intercept) 18.81 19.333
## cdf$F2 -0.44 -0.407
anova(Linear)
## Analysis of Variance Table
##
## Response: cdf$Interest
## Df Sum Sq Mean Sq F value Pr(>F)
## cdf$F2 1 21928 21928 2527 <2e-16 ***
## Residuals 2496 21659 9
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Below will produce a graph of Interest vs FICO values, with fitted line and residual segments
plot(cdf$Interest ~ cdf$F2)
abline(lm(cdf$Interest ~ cdf$F2))
segments(cdf$F2, Fitted_values, cdf$F2, cdf$Interest)
## Error: object 'Fitted_values' not found
Below will produce a scatter plot of fitted values vs residual values
Fitted_values <- fitted(Linear)
plot(Fitted_values, residual_values)
qqnorm(residual_values)
Pre?dic?tion inter?vals and con?fi?dence inter?vals are not the same thing. Unfor?tu?nately the terms are often con?fused, and I am often fre?quently cor?rect?ing the error in stu?dents? papers and arti?cles I am review?ing or editing.
A pre?dic?tion inter?val is an inter?val asso?ci?ated with a ran?dom vari?able yet to be observed, with a spec?i?fied prob?a?bil?ity of the ran?dom vari?able lying within the inter?val. For exam?ple, I might give an 80% inter?val for the fore?cast of GDP in 2014. The actual GDP in 2014 should lie within the inter?val with prob?a?bil?ity 0.8. Pre?dic?tion inter?vals can arise in Bayesian or fre?quen?tist statistics.
Confidence interval - an interval estimate of population parameter based on a random sample
Degree of confidence - Probability that the confidence interval captures the true populaiton parameter
A con?fi?dence inter?val is an inter?val asso?ci?ated with a para?me?ter and is a fre?quen?tist con?cept. The para?me?ter is assumed to be non-??random but unknown, and the con?fi?dence inter?val is com?puted from data. Because the data are ran?dom, the inter?val is ran?dom. A 95% con?fi?dence inter?val will con?tain the true para?me?ter with prob?a?bil?ity 0.95. That is, with a large num?ber of repeated sam?ples, 95% of the inter?vals would con?tain the true parameter.
Summary shows that FICO values explaines 50% of varaiation for Interest. Because R square value si 0.50
To explain the remainaing variation, multiple regression with variables identified after correlation analysis is done
Rg1 <- lm(cdf$Interest ~ cdf$Inquiries.in.the.Last.6.Months + cdf$DI_Ratio +
cdf$LLen + cdf$F2 + cdf$Open.CREDIT.Lines + cdf$Amount.Requested)
summary(Rg1)
##
## Call:
## lm(formula = cdf$Interest ~ cdf$Inquiries.in.the.Last.6.Months +
## cdf$DI_Ratio + cdf$LLen + cdf$F2 + cdf$Open.CREDIT.Lines +
## cdf$Amount.Requested)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.570 -1.381 -0.168 1.205 9.982
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.21e+01 2.31e-01 52.24 < 2e-16
## cdf$Inquiries.in.the.Last.6.Months 3.52e-01 3.39e-02 10.38 < 2e-16
## cdf$DI_Ratio 1.23e-03 6.05e-03 0.20 0.84
## cdf$LLen 1.34e-01 4.57e-03 29.41 < 2e-16
## cdf$F2 -4.36e-01 6.10e-03 -71.61 < 2e-16
## cdf$Open.CREDIT.Lines -5.03e-02 1.01e-02 -4.98 6.7e-07
## cdf$Amount.Requested 1.47e-04 5.96e-06 24.67 < 2e-16
##
## (Intercept) ***
## cdf$Inquiries.in.the.Last.6.Months ***
## cdf$DI_Ratio
## cdf$LLen ***
## cdf$F2 ***
## cdf$Open.CREDIT.Lines ***
## cdf$Amount.Requested ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.06 on 2491 degrees of freedom
## Multiple R-squared: 0.757, Adjusted R-squared: 0.757
## F-statistic: 1.3e+03 on 6 and 2491 DF, p-value: <2e-16
confint(Rg1)
## 2.5 % 97.5 %
## (Intercept) 11.6054002 12.5106546
## cdf$Inquiries.in.the.Last.6.Months 0.2857053 0.4187286
## cdf$DI_Ratio -0.0106278 0.0130914
## cdf$LLen 0.1253267 0.1432347
## cdf$F2 -0.4484503 -0.4245450
## cdf$Open.CREDIT.Lines -0.0701221 -0.0305200
## cdf$Amount.Requested 0.0001354 0.0001588
Summary shows that R-squared value is 0.757 Other variables that affect interest rate are a. Amount.requested (0.3318) b. Inquiries.in.the.Last.6.Months (0.1646) c. DI_Ratio (0.172) d. LLen (0.423) e.#$FICO vlaue (-0.7092) f.Open.CREDIT.Lines ( 0.09030695)
Howeve, to check whether no.of open credit lines really affect interest rate, one more regressiona analysis without Open.credit.lines is done.
Rg2 <- lm(cdf$Interest ~ cdf$Inquiries.in.the.Last.6.Months + cdf$DI_Ratio +
cdf$LLen + cdf$F2 + cdf$Amount.Requested)
summary(Rg2)
##
## Call:
## lm(formula = cdf$Interest ~ cdf$Inquiries.in.the.Last.6.Months +
## cdf$DI_Ratio + cdf$LLen + cdf$F2 + cdf$Amount.Requested)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.518 -1.363 -0.146 1.229 9.946
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.17e+01 2.23e-01 52.61 <2e-16
## cdf$Inquiries.in.the.Last.6.Months 3.32e-01 3.38e-02 9.80 <2e-16
## cdf$DI_Ratio -9.39e-03 5.69e-03 -1.65 0.099
## cdf$LLen 1.35e-01 4.58e-03 29.55 <2e-16
## cdf$F2 -4.36e-01 6.12e-03 -71.19 <2e-16
## cdf$Amount.Requested 1.41e-04 5.88e-06 24.05 <2e-16
##
## (Intercept) ***
## cdf$Inquiries.in.the.Last.6.Months ***
## cdf$DI_Ratio .
## cdf$LLen ***
## cdf$F2 ***
## cdf$Amount.Requested ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.07 on 2492 degrees of freedom
## Multiple R-squared: 0.755, Adjusted R-squared: 0.755
## F-statistic: 1.54e+03 on 5 and 2492 DF, p-value: <2e-16
confint(Rg2)
## 2.5 % 97.5 %
## (Intercept) 11.3093363 12.185073
## cdf$Inquiries.in.the.Last.6.Months 0.2652968 0.397959
## cdf$DI_Ratio -0.0205383 0.001766
## cdf$LLen 0.1264324 0.144403
## cdf$F2 -0.4479058 -0.423891
## cdf$Amount.Requested 0.0001299 0.000153
R-square value of 0.755 shows that Open.credit.lines does not signifcantly explain or account for interest rate, Hence variables that affect interest rate are a. Amount.requested (0.3318) b. Inquiries.in.the.Last.6.Months (0.1646) c. DI_Ratio (0.172) d. LLen (0.423) e.$FICO vlaue (-0.7092)