Bayesian Principal Component Analysis

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Aim

The aim is to extend principal component analysis of interest rates to use a Bayesian framework. We aim to show how incorporating expert-informed priors can improve forecasts when changes in the economic environment are not yet reflected in historical data. In particular, we consider how:

  • A classical model would have overestimated interest rates during the prolonged low-rate period following the 2008 financial crisis
  • A Bayesian model could have adjusted expectations more appropriately
  • Similar adjustments can be applied in response to global tariff policies imposed by the US administration
why chosen spot rates over forward rates
no arbitrage considerations

Process

PCA no adjustments

  • Choose dataset
  • Load into dataframe
  • Determine raw calibration dataset
    • Truncate
    • Judgements on blanks and problem values
  • Transform into calibration dataset
    • logarithms
    • differencing
    • de-mean
  • Calculate Co-variance Matrix

PCA in years following financial crisis

Determining Priors from prediction errors

  • form of prior

Determining Future Looking Priors

  • choice of data
supporting narative: the impace of quantitative easing (money supply) on interest rates.
supporting narative: the impact of wholesale funding activity on interest rates.

Notation

We begin with an element-wise formulation to ground intuition...

  • Number of observations is: \(N\)

  • each \(t \in \left\lbrace 1, 2, \ldots, N \right\rbrace \)is an index representing each observation where each t maps onto a specific observation date \(t' \in \left\lbrace 31/1/1970 , 28/2/1970, \ldots, 31/12/2024 \right\rbrace \)

  • \(R_i\) is a random variable for the spot rate for term i

  • \(r_i(t)\) is a dataset for the spot rate for term i over a series of observations \(t \in \left\lbrace 1, 2, \ldots, N \right\rbrace \)

  • \(S'\) is a random variable for the de-meaned logged spot rate differences over whole term structure

  • \(S'_i\) is a random variable for the de-meaned logged spot rate differences for term i

  • \(s_i(t)\) is the dataset of observed log difference in interest rates over the period \((t-1,t)\) i.e.

  • \(\Sigma_{{i_1},{i_2}} = \mathrm{Cov}(S'_{i_1}, S'_{i_2})\) is an element within the covariance matrix \(\Sigma\)

use of random variable
ensure lowercase for observatinos (data)

Creating One Combined DataFrame

We have 2 spreadsheets of spot yields from the Bank of England website that we will load into dataframes

Download GLC Nominal month end data_1970 to 2015.xlsx
Download GLC Nominal Month End Data (2016 to Present)

A v>

Basic Reasonableness Tests

We perform a couple of reasonableness checks to ensure the spreadsheet data has loaded correctly into the combined dataframe

A Check on the Number of Rows

Dataframe 1 - 1970 to 2015

the first date is 1970-01-31 and the last is 2015-12-31
one would therefore expect 12 x 46yrs = 552 entries
and indeed we see the number of rows in df is 552

Dataframe 2 - 2015 to present

the first dates is 2016-01-31 and the last is 2024-12-31
one would therefore expect 12 x 9yrs = 108 entries
and indeed we see the number of rows in df is 108

Combined DataFrame

The length of combined dataframe is 660 rows"
whereas the two separate dataframes come to 552 + 108

A Check on Sum of Values

Dataframe 1 - 1970 to 2015

manual inspection of the sum of all values in first spreadsheet is 191503.172322029
the sum of 1st dataframe is also 191503.17232202887

Dataframe 2 - 2015 to present

manual inspection of the sum of all values in second spreadsheet is 17844.9993308767
the sum of 1st dataframe is also 17844.999330876683

Combined DataFrame

the sum of combined dataframe is 209348.17165290558
and the sum of the manually observed 191503.172322029 + 17844.9993308767 = 209348.1716529057

Truncation & Interpolation of the Dataset

Principal component analysis requires same number of datapoints for each term so as to produce a rectangular matrix from which covariances can be calculated.

The dataset of spot yields contains gaps insofar that the whole set of observation dates is not consistently available for all terms. We want to choose a range of observation dates and terms that reduces the need to fill in gaps in the dataset.

We have spot yield data for terms 0.5 up to 40. The first step to identify a calibration dataset is to identify the first and last data point for each term. This gives us an initial idea of the size of the dataset available.

We make a judgement call about which terms to retain (and observation dates) to retain. If there are gaps in the data we use linear interpolation to fill them.

Matplotlib figures, subplots, axes

  • Figure The whole canvas or image
  • Axes One chart (with x/y axes, labels, data)
  • Subplot One chart within a grid layout (i.e., an Axes)
  • Grid of subplots Arrangement of multiple Axes in a Figure

Data Boundaries by Term

visual

The maiximum range of observation dates for each term is found by the earliest and latest non NaN entry. We see that for beyond term 25 data is only available from 31st January 2016 and that for earlier terms available from 31st January 1970 (with an exception for term 0.5).

My chart
tabular
start_term end_term earliest_date last_date
0.5 0.5 1970-07-31 2024-12-31
1.0 25.0 1970-01-31 2024-12-31
25.5 40.0 2016-01-31 2024-12-31

Interpolation

Summary statistics on interpolated/truncated dataset

  • term
  • actual data points

Rows untouched by interpolation should have same total as before. totals for those with interpolation should could be checked for reasonableness.

Decisions

  • Data for terms greater than 25 isn't available before 2016. We will therefore not model beyond term 25 in order to facilitate sufficient history of data-points.
  • we ignore term 0.5 and start at term 1 due to missing datapoints for term 0.5
  • we ignore terms beyond 20 since the proportion of missing datapoints is too great.
  • we replace -ve values with NaN and then interpolate

Null Counts

Histogram

An initial inspection of the data shows signficantly more nulls for greater terms. Beyond term 25 we see the number levels off and we later discover this is because data for term 25 onwards doesn't begin until 2016 meaning there is a significant block of NaN values from 1970 to 2016 for these terms.

My chart

Tabulated

We identify non contiguous blocks of data by determining the expected number of data points, based on first and last data point, and comparing with actual number of data points.

These are the columns which will be interpolated.

term actual no. data-points expected no. data-points missing data points
0.5 538 654 116
1 658 660 2
16.5 658 660 2
17 652 660 8
17.5 644 660 16
18 632 660 28
18.5 625 660 35
19 619 660 41
19.5 605 660 55
20 581 660 79
20.5 558 660 102
21 542 660 118
21.5 527 660 133
22 510 660 150
22.5 489 660 171
23 465 660 195
23.5 443 660 217
24 419 660 241
24.5 400 660 260
25 378 660 282

Removing Negatives

Logarithms are only defined for positive arguments. We therefore need to consider the small number of -ve values observable in the dataset:

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
563 -0.029262 -0.001047 0.043756 0.109305 0.189006 0.276475 0.367344 0.458905 0.549509 0.638135 0.724153 0.807198 0.887065 0.963633 1.036835 1.106630 1.173004 1.235965 1.295547 1.351811 1.404839 1.454735 1.501617 1.545612 1.586846 1.625441 1.661513 1.695176 1.726532 1.755679 1.782703 1.807684 1.830696 1.851804 1.871069 1.888547 1.904289 1.918343 1.930750
604 0.013280 -0.008485 -0.026463 -0.038360 -0.045366 -0.048285 -0.047483 -0.043166 -0.035471 -0.024514 -0.010422 0.006623 0.026389 0.048598 0.072939 0.099068 0.126623 0.155230 0.184513 0.214115 0.243703 0.272976 0.301683 0.329611 0.356588 0.382482 0.407196 0.430662 0.452836 0.473698 0.493245 0.511489 0.528452 0.544158 0.558634 0.571912 0.584025 0.595003 0.604871
605 -0.009801 -0.045821 -0.069679 -0.082474 -0.087813 -0.087702 -0.083129 -0.074636 -0.062575 -0.047221 -0.028843 -0.007721 0.015849 0.041565 0.069117 0.098202 0.128522 0.159784 0.191701 0.223987 0.256358 0.288540 0.320278 0.351335 0.381499 0.410597 0.438483 0.465038 0.490166 0.513801 0.535908 0.556471 0.575490 0.592980 0.608966 0.623480 0.636566 0.648269 0.658634
606 -0.021850 -0.060837 -0.093727 -0.117590 -0.132942 -0.140738 -0.141903 -0.137297 -0.127686 -0.113755 -0.096113 -0.075301 -0.051791 -0.026006 0.001680 0.030934 0.061458 0.092975 0.125219 0.157930 0.190848 0.223717 0.256297 0.288360 0.319703 0.350154 0.379560 0.407790 0.434736 0.460316 0.484474 0.507173 0.528394 0.548129 0.566381 0.583166 0.598503 0.612420 0.624943
607 -0.044205 -0.060842 -0.072683 -0.077088 -0.074019 -0.064180 -0.048489 -0.027846 -0.003051 0.025198 0.056286 0.089672 0.124869 0.161443 0.199012 0.237246 0.275859 0.314604 0.353255 0.391597 0.429425 0.466545 0.502782 0.537975 0.571987 0.604711 0.636056 0.665950 0.694343 0.721202 0.746517 0.770289 0.792526 0.813245 0.832466 0.850214 0.866519 0.881407 0.894903
608 -0.013335 -0.031917 -0.053845 -0.071520 -0.082226 -0.085494 -0.081745 -0.071701 -0.056157 -0.035908 -0.011696 0.015798 0.045965 0.078254 0.112178 0.147299 0.183232 0.219629 0.256177 0.292589 0.328609 0.364007 0.398583 0.432165 0.464610 0.495807 0.525665 0.554117 0.581113 0.606625 0.630642 0.653169 0.674213 0.693791 0.711921 0.728628 0.743936 0.757870 0.770452
609 -0.026419 -0.038957 -0.053194 -0.063327 -0.067327 -0.064971 -0.056681 -0.043049 -0.024727 -0.002351 0.023489 0.052251 0.083436 0.116576 0.151235 0.187014 0.223545 0.260494 0.297556 0.334453 0.370930 0.406762 0.441750 0.475723 0.508542 0.540096 0.570296 0.599073 0.626380 0.652191 0.676501 0.699315 0.720645 0.740513 0.758942 0.775962 0.791605 0.805897 0.818866
610 -0.021041 -0.018181 -0.020028 -0.020669 -0.017662 -0.010179 0.001872 0.018243 0.038508 0.062170 0.088734 0.117723 0.148696 0.181239 0.214973 0.249557 0.284683 0.320072 0.355470 0.390640 0.425362 0.459443 0.492706 0.525004 0.556215 0.586239 0.614997 0.642423 0.668475 0.693129 0.716376 0.738220 0.758670 0.777739 0.795446 0.811814 0.826865 0.840619 0.853095
611 -0.150365 -0.143101 -0.135252 -0.128029 -0.120277 -0.110666 -0.098319 -0.082835 -0.064186 -0.042572 -0.018300 0.008284 0.036825 0.066975 0.098400 0.130781 0.163819 0.197237 0.230777 0.264203 0.297302 0.329889 0.361801 0.392904 0.423088 0.452263 0.480356 0.507309 0.533080 0.557645 0.580993 0.603122 0.624032 0.643726 0.662211 0.679494 0.695579 0.710465 0.724151
612 -0.113058 -0.102450 -0.095639 -0.087447 -0.075699 -0.059686 -0.039392 -0.015160 0.012502 0.043034 0.075896 0.110587 0.146651 0.183678 0.221307 0.259217 0.297133 0.334812 0.372041 0.408633 0.444425 0.479279 0.513077 0.545726 0.577154 0.607305 0.636140 0.663630 0.689764 0.714540 0.737972 0.760073 0.780862 0.800356 0.818575 0.835541 0.851268 0.865767 0.879051
For ease of analysis we set these values to NaN.
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
563 NaN NaN 0.043756 0.109305 0.189006 0.276475 0.367344 0.458905 0.549509 0.638135 0.724153 0.807198 0.887065 0.963633 1.036835 1.106630 1.173004 1.235965 1.295547 1.351811 1.404839 1.454735 1.501617 1.545612 1.586846 1.625441 1.661513 1.695176 1.726532 1.755679 1.782703 1.807684 1.830696 1.851804 1.871069 1.888547 1.904289 1.918343 1.930750
604 0.01328 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.006623 0.026389 0.048598 0.072939 0.099068 0.126623 0.155230 0.184513 0.214115 0.243703 0.272976 0.301683 0.329611 0.356588 0.382482 0.407196 0.430662 0.452836 0.473698 0.493245 0.511489 0.528452 0.544158 0.558634 0.571912 0.584025 0.595003 0.604871
605 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.015849 0.041565 0.069117 0.098202 0.128522 0.159784 0.191701 0.223987 0.256358 0.288540 0.320278 0.351335 0.381499 0.410597 0.438483 0.465038 0.490166 0.513801 0.535908 0.556471 0.575490 0.592980 0.608966 0.623480 0.636566 0.648269 0.658634
606 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.001680 0.030934 0.061458 0.092975 0.125219 0.157930 0.190848 0.223717 0.256297 0.288360 0.319703 0.350154 0.379560 0.407790 0.434736 0.460316 0.484474 0.507173 0.528394 0.548129 0.566381 0.583166 0.598503 0.612420 0.624943
607 NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.025198 0.056286 0.089672 0.124869 0.161443 0.199012 0.237246 0.275859 0.314604 0.353255 0.391597 0.429425 0.466545 0.502782 0.537975 0.571987 0.604711 0.636056 0.665950 0.694343 0.721202 0.746517 0.770289 0.792526 0.813245 0.832466 0.850214 0.866519 0.881407 0.894903
608 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.015798 0.045965 0.078254 0.112178 0.147299 0.183232 0.219629 0.256177 0.292589 0.328609 0.364007 0.398583 0.432165 0.464610 0.495807 0.525665 0.554117 0.581113 0.606625 0.630642 0.653169 0.674213 0.693791 0.711921 0.728628 0.743936 0.757870 0.770452
609 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.023489 0.052251 0.083436 0.116576 0.151235 0.187014 0.223545 0.260494 0.297556 0.334453 0.370930 0.406762 0.441750 0.475723 0.508542 0.540096 0.570296 0.599073 0.626380 0.652191 0.676501 0.699315 0.720645 0.740513 0.758942 0.775962 0.791605 0.805897 0.818866
610 NaN NaN NaN NaN NaN NaN 0.001872 0.018243 0.038508 0.062170 0.088734 0.117723 0.148696 0.181239 0.214973 0.249557 0.284683 0.320072 0.355470 0.390640 0.425362 0.459443 0.492706 0.525004 0.556215 0.586239 0.614997 0.642423 0.668475 0.693129 0.716376 0.738220 0.758670 0.777739 0.795446 0.811814 0.826865 0.840619 0.853095
611 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.008284 0.036825 0.066975 0.098400 0.130781 0.163819 0.197237 0.230777 0.264203 0.297302 0.329889 0.361801 0.392904 0.423088 0.452263 0.480356 0.507309 0.533080 0.557645 0.580993 0.603122 0.624032 0.643726 0.662211 0.679494 0.695579 0.710465 0.724151
612 NaN NaN NaN NaN NaN NaN NaN NaN 0.012502 0.043034 0.075896 0.110587 0.146651 0.183678 0.221307 0.259217 0.297133 0.334812 0.372041 0.408633 0.444425 0.479279 0.513077 0.545726 0.577154 0.607305 0.636140 0.663630 0.689764 0.714540 0.737972 0.760073 0.780862 0.800356 0.818575 0.835541 0.851268 0.865767 0.879051
We now populate this values with interpolated values moving down the columns (terms)
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
563 0.050886 0.090516 0.043756 0.109305 0.189006 0.276475 0.367344 0.458905 0.549509 0.638135 0.724153 0.807198 0.887065 0.963633 1.036835 1.106630 1.173004 1.235965 1.295547 1.351811 1.404839 1.454735 1.501617 1.545612 1.586846 1.625441 1.661513 1.695176 1.726532 1.755679 1.782703 1.807684 1.830696 1.851804 1.871069 1.888547 1.904289 1.918343 1.930750
604 0.013280 0.041635 0.040681 0.044429 0.051206 0.059899 0.039947 0.048182 0.058168 0.059801 0.076369 0.006623 0.026389 0.048598 0.072939 0.099068 0.126623 0.155230 0.184513 0.214115 0.243703 0.272976 0.301683 0.329611 0.356588 0.382482 0.407196 0.430662 0.452836 0.473698 0.493245 0.511489 0.528452 0.544158 0.558634 0.571912 0.584025 0.595003 0.604871
605 0.012128 0.042606 0.046637 0.054873 0.066015 0.079125 0.033601 0.043192 0.054891 0.048267 0.069675 0.034306 0.015849 0.041565 0.069117 0.098202 0.128522 0.159784 0.191701 0.223987 0.256358 0.288540 0.320278 0.351335 0.381499 0.410597 0.438483 0.465038 0.490166 0.513801 0.535908 0.556471 0.575490 0.592980 0.608966 0.623480 0.636566 0.648269 0.658634
606 0.010976 0.043578 0.052592 0.065317 0.080824 0.098351 0.027255 0.038202 0.051615 0.036732 0.062981 0.061989 0.070359 0.101504 0.001680 0.030934 0.061458 0.092975 0.125219 0.157930 0.190848 0.223717 0.256297 0.288360 0.319703 0.350154 0.379560 0.407790 0.434736 0.460316 0.484474 0.507173 0.528394 0.548129 0.566381 0.583166 0.598503 0.612420 0.624943
607 0.009824 0.044549 0.058548 0.075760 0.095634 0.117576 0.020909 0.033212 0.048338 0.025198 0.056286 0.089672 0.124869 0.161443 0.199012 0.237246 0.275859 0.314604 0.353255 0.391597 0.429425 0.466545 0.502782 0.537975 0.571987 0.604711 0.636056 0.665950 0.694343 0.721202 0.746517 0.770289 0.792526 0.813245 0.832466 0.850214 0.866519 0.881407 0.894903
608 0.008672 0.045521 0.064504 0.086204 0.110443 0.136802 0.014564 0.028222 0.045061 0.037522 0.039888 0.015798 0.045965 0.078254 0.112178 0.147299 0.183232 0.219629 0.256177 0.292589 0.328609 0.364007 0.398583 0.432165 0.464610 0.495807 0.525665 0.554117 0.581113 0.606625 0.630642 0.653169 0.674213 0.693791 0.711921 0.728628 0.743936 0.757870 0.770452
609 0.007521 0.046493 0.070459 0.096648 0.125253 0.156028 0.008218 0.023233 0.041784 0.049846 0.023489 0.052251 0.083436 0.116576 0.151235 0.187014 0.223545 0.260494 0.297556 0.334453 0.370930 0.406762 0.441750 0.475723 0.508542 0.540096 0.570296 0.599073 0.626380 0.652191 0.676501 0.699315 0.720645 0.740513 0.758942 0.775962 0.791605 0.805897 0.818866
610 0.006369 0.047464 0.076415 0.107092 0.140062 0.175253 0.001872 0.018243 0.038508 0.062170 0.088734 0.117723 0.148696 0.181239 0.214973 0.249557 0.284683 0.320072 0.355470 0.390640 0.425362 0.459443 0.492706 0.525004 0.556215 0.586239 0.614997 0.642423 0.668475 0.693129 0.716376 0.738220 0.758670 0.777739 0.795446 0.811814 0.826865 0.840619 0.853095
611 0.005217 0.048436 0.082371 0.117535 0.154871 0.194479 0.095777 0.124159 0.025505 0.052602 0.082315 0.008284 0.036825 0.066975 0.098400 0.130781 0.163819 0.197237 0.230777 0.264203 0.297302 0.329889 0.361801 0.392904 0.423088 0.452263 0.480356 0.507309 0.533080 0.557645 0.580993 0.603122 0.624032 0.643726 0.662211 0.679494 0.695579 0.710465 0.724151
612 0.004065 0.049407 0.088326 0.127979 0.169681 0.213705 0.189681 0.230075 0.012502 0.043034 0.075896 0.110587 0.146651 0.183678 0.221307 0.259217 0.297133 0.334812 0.372041 0.408633 0.444425 0.479279 0.513077 0.545726 0.577154 0.607305 0.636140 0.663630 0.689764 0.714540 0.737972 0.760073 0.780862 0.800356 0.818575 0.835541 0.851268 0.865767 0.879051
checks we can perform on the interpolated values .....

Taking Logarithmns¶

Purpose

We want to calculate the natural log of spot yield returs. 41508.92158837641

.apply() function

This is the second column. Same flexibility as the first.

further complications with dtype:object

sometimes pandas is treating values as generic python objects not efficient numeric types even if they look like floats

it seems to happen when slicing rows.

a fix is to use .astype(float) before applying functions like np.log

Checking the Log Calculation

given that:

\[ \sum_i \log(x_i) = \log\left( \prod_i x_i \right) \]

we can perform a check on the log calculation. however the product approach doesn't work since there are so many values we get overflow for the product side of the equation we can instead chunk up the calculation to make it more manageable we therefore calculate the product for each row then take the log the sum the log of products for each row

The Product of All Entries

the product of each row the log of each row product the sum of log of row products
1.500157e+36 83.298633 41508.921588
4.498612e+35 82.094247 41508.921588
1.520994e+35 81.009842 41508.921588
7.778928e+35 82.641897 41508.921588
2.583242e+35 81.539523 41508.921588

Comparing Calculations

The sum of individual 'logged values is: 41508.92158837642. The sum of the log of row product generates: 41508.92158837641. The difference between the two is 7.275957614183426e-12.

Differencing Data

Logged Values

  1 1.500000 2 2.500000 3 3.500000 4 4.500000 5 5.500000
0 2.155865 2.164177 2.163407 2.159182 2.153934 2.148557 2.143398 2.138608 2.134239 2.130306
1 2.129794 2.127907 2.124743 2.120779 2.116447 2.112078 2.107884 2.103977 2.100422 2.097250
2 2.046943 2.051911 2.053485 2.053239 2.052194 2.050950 2.049814 2.048926 2.048347 2.048100
3 2.029005 2.062340 2.076126 2.079747 2.078543 2.075374 2.071704 2.068308 2.065603 2.063789
4 2.000277 2.045864 2.062064 2.064012 2.059325 2.051845 2.043562 2.035601 2.028615 2.022938

Differenced Values

  1 1.500000 2 2.500000 3 3.500000 4 4.500000 5 5.500000
0 nan nan nan nan nan nan nan nan nan nan
1 -0.026071 -0.036270 -0.038663 -0.038403 -0.037487 -0.036478 -0.035515 -0.034631 -0.033817 -0.033056
2 -0.082851 -0.075995 -0.071259 -0.067540 -0.064253 -0.061128 -0.058069 -0.055051 -0.052075 -0.049150
3 -0.017938 0.010429 0.022642 0.026507 0.026349 0.024423 0.021889 0.019381 0.017256 0.015688
4 -0.028727 -0.016476 -0.014062 -0.015735 -0.019218 -0.023528 -0.028141 -0.032706 -0.036988 -0.040851

Spot Checks

2.079746736818641 minus 2.06401192442566 equals 0.015734812392981024

Aggregate Checks

the sum of the differences is:-23.727449287990904

the sum of the first row minus the last row is:22.52306353296322

narrow down the difference

Differencing Data

We calculate differences since we are modelling changes in the yield curve:

Checks that can be made to ensure data has been differenced correctly:

  • spot check a small sample of values
  • total of differences = sum of first row - sum of last row

Logged Values

Differenced Values

De-meaning Data

Principal Component Analysis (PCA) aims to identify directions of maxium variance. De-meaning data removes bias introduced by non zero means and ensures that dimensions of the original vector are ranked on a standardised basis.

We now have a dataset of log differences \(s_i(t)\) where \( s_i(t)=\log{{ r_i(t)}}-\log {{ r_{{i-1}}(t)}} \)

Each dimension \(i \in \left\lbrace 1, 2, \ldots, m \right\rbrace \) represents to a specific maturity term. In this example the \(i\)maps onto the set \(j \in \left\lbrace 1, 1.5, 2, \ldots, 19.5, 20 \right\rbrace \)

Since we are evaluating the variance at each term \(t\) a mean \(\bar{{s}}(t)\) is calculated for each term using: \(\bar{{s}}(t)=\frac{{\sum_{{t=1}}^{{N}} s(t)}}{{N}}\)

A de-meaned dataset is calculated such that \(s'_i(t)=s_i(t)-\bar{{s}}(t)\)

Logged Differences

  1 1.500000 2 2.500000 3 3.500000 4 4.500000 5 5.500000
0 nan nan nan nan nan nan nan nan nan nan
1 -0.026071 -0.036270 -0.038663 -0.038403 -0.037487 -0.036478 -0.035515 -0.034631 -0.033817 -0.033056
2 -0.082851 -0.075995 -0.071259 -0.067540 -0.064253 -0.061128 -0.058069 -0.055051 -0.052075 -0.049150
3 -0.017938 0.010429 0.022642 0.026507 0.026349 0.024423 0.021889 0.019381 0.017256 0.015688
4 -0.028727 -0.016476 -0.014062 -0.015735 -0.019218 -0.023528 -0.028141 -0.032706 -0.036988 -0.040851

The Means

  0
1 -0.001055
1.500000 -0.001096
2 -0.001106
2.500000 -0.001104
3 -0.001099

Checks that can be made to ensure data has been differenced correctly:

  • spot check a small sample of values
  • sum or original table - sum of demeaned table =

De-meaned Values

  1 1.500000 2 2.500000 3 3.500000 4 4.500000 5 5.500000
0 nan nan nan nan nan nan nan nan nan nan
1 -0.025016 -0.035174 -0.037557 -0.037298 -0.036389 -0.035388 -0.034435 -0.033564 -0.032763 -0.032017
2 -0.081796 -0.074899 -0.070153 -0.066436 -0.063154 -0.060038 -0.056990 -0.053984 -0.051022 -0.048111
3 -0.016883 0.011525 0.023747 0.027612 0.027447 0.025513 0.022969 0.020448 0.018310 0.016727
4 -0.027672 -0.015380 -0.012956 -0.014630 -0.018120 -0.022438 -0.027062 -0.031639 -0.035935 -0.039812

Co-variance Matrix

The covariance matrix is calculated as:

\( \Sigma = \text{Cov}(S') = \begin{bmatrix} \text{Var}(s'_1) & \text{Cov}(s'_1, s'_2) & \cdots & \text{Cov}(s'_1, s'_n) \\ \text{Cov}(s'_2, s'_1) & \text{Var}(s'_2) & \cdots & \text{Cov}(s'_2, s'_n) \\ \vdots & \vdots & \ddots & \vdots \\ \text{Cov}(s'_n, s'_1) & \text{Cov}(s'_n, s'_2) & \cdots & \text{Var}(s'_n) \end{bmatrix} \)

where each element \(\Sigma_{{i_1},{i_2}} = \mathrm{Cov}(S'_{i_1}, S'_{i_2})\) is calculated as:


\(\text{Cov}(S'_{i_1}, S'_{i_2}) = \frac{1}{N-1} \sum_{t=1}^{N} (s'_{i_1}(t) - \bar{s'_{i_1}}) (s'_{i_2}(t) - \bar{s'_{i_2}}) \)

Checks that can be made to ensure that co-variance matrix calculated correctly:

  • spot check variance on some terms
  • spot check co-variance on some pairs of terms
singular value decomposition as an alternative method

Calculated Covariance Matrix

3 3.5 4 4.5 5 ... 18 18.5 19 19.5 20
3 0.029422 0.026623 0.023759 0.021909 0.020242 ... 0.006954 0.006828 0.006717 0.006611 0.006476
3.5 0.026623 0.024391 0.021868 0.020335 0.018906 ... 0.006748 0.006629 0.006522 0.00642 0.006287
4 0.023759 0.021868 0.048979 0.032092 0.016822 ... 0.005586 0.005496 0.005414 0.005334 0.005222
4.5 0.021909 0.020335 0.032092 0.024238 0.016722 ... 0.006061 0.005956 0.005858 0.005765 0.005639
5 0.020242 0.018906 0.016822 0.016722 0.034605 ... 0.008418 0.00826 0.008113 0.007975 0.007808
... ... ... ... ... ... ... ... ... ... ... ...
18 0.006954 0.006748 0.005586 0.006061 0.008418 ... 0.004862 0.004819 0.004783 0.004697 0.004617
18.5 0.006828 0.006629 0.005496 0.005956 0.00826 ... 0.004819 0.004799 0.004767 0.00468 0.004599
19 0.006717 0.006522 0.005414 0.005858 0.008113 ... 0.004783 0.004767 0.004755 0.004667 0.004586
19.5 0.006611 0.00642 0.005334 0.005765 0.007975 ... 0.004697 0.00468 0.004667 0.004655 0.004576
20 0.006476 0.006287 0.005222 0.005639 0.007808 ... 0.004617 0.004599 0.004586 0.004576 0.004565

Spot Checks

Variance at term 1

The variance of term 1 observations should equal the first entry in the co-variance matrix.

\(\Sigma_{1,1} = \text{Var}(s'_1) = \frac{1}{N-1} \sum_{t=1}^{N} (s_1'(t) - \bar{s_1'}) (s_1'(t) - \bar{s_1'}) \)

which comes out at: 0.04721808653443263

Eigenvector & Eigenvalue Decomposition

Eigenvalues are expressed in the formula:

\(\Sigma v = \lambda v\)

\(v\) denote eigenvectors
\(\lambda\) denote eigenvalues

Checks that can be made to ensure eigenvalues and eigenvectors have been calculated correctly:

  • recreate covariance matrix columns via matrix multiplication

Eigenvalues

eigenvalue % explained cummulative explained
34 0.359411 63.19 63.19
33 0.083377 14.66 77.85
32 0.057684 10.14 87.99
31 0.024939 4.38 92.37
30 0.018676 3.28 95.65
... ... ... ...
4 0.0 0.0 99.98
3 0.0 0.0 99.98
2 0.0 0.0 99.98
1 0.0 0.0 99.98
0 0.0 0.0 99.98

Eigenvectors

34 33 32 31 30 ... 4 3 2 1 0
0 0.212471 0.285834 0.036498 -0.132427 0.233877 ... -0.000011 -0.000006 -0.000001 0.0 -0.0
1 0.200404 0.256169 0.031873 -0.100597 0.166598 ... 0.000021 0.000012 0.000002 -0.0 0.000001
2 0.171044 0.589391 -0.205448 -0.387422 -0.124932 ... 0.000018 0.000011 0.000002 -0.0 0.000001
3 0.180292 0.358521 -0.06521 -0.218701 -0.015158 ... -0.000036 -0.000023 -0.000003 0.000001 -0.000001
4 0.244793 0.142936 0.120947 0.512247 0.344771 ... -0.000011 -0.000007 -0.000001 0.0 -0.0
... ... ... ... ... ... ... ... ... ... ... ...
30 0.102949 -0.008186 -0.007536 0.067154 -0.202482 ... 0.000001 -0.000001 0.0 -0.0 0.0
31 0.101158 -0.007643 -0.007285 0.067833 -0.206179 ... -0.000001 0.0 -0.0 -0.0 -0.0
32 0.099509 -0.007089 -0.007131 0.068473 -0.20994 ... 0.0 -0.0 -0.0 0.0 -0.0
33 0.09778 -0.006549 -0.006886 0.067668 -0.207863 ... -0.000001 0.0 0.0 0.0 0.0
34 0.095788 -0.006365 -0.006825 0.067205 -0.207058 ... -0.0 0.0 -0.0 -0.0 -0.0

Dimensionality Reduction

The first 3 eigenvalues 0.359411 , 0.083377 , 0.057684 , explain 63.19% , 14.66% , 10.14% of variance respectively.    variance in what specifically

In total the first 3 principal components explain 87.99% how to get from eigenvalue to principal component

Projecting Co-ordinates

Plotting Principal Components

Calculating Stress from 1st Principal Component

add back in mean or not